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Introduction to Economic and Strategic BehaviorIntroduction

Levent Kockesen

Koc University

Levent Kockesen (Koc University) Introduction 1 / 19

Description and Objectives

Description

Analysis of strategic interactions among individuals using game theory

Applications to economics, business, politics, law, biology, history

Evolution of social norms and other social institutions

Learning Objectives

Recognize real strategic situations to which game theory can beapplied

Model strategic situations as games

Apply tools of game theory to gain insights about real life situations

View social and economic issues from a completely new perspective


Game Theory: Definition and Assumptions

Game theory studies strategic interactions within a group ofindividuals

◮ Actions of each individual have an effect on the outcome◮ Individuals are aware of that fact

Individuals are rational◮ have well-defined objectives over the set of possible outcomes◮ implement the best available strategy to pursue them

Rules of the game and rationality are common knowledge


Example

10 people go to a restaurant for dinner

Order expensive or inexpensive fish?◮ Expensive fish: value = 18, price = 20◮ Inexpensive fish: value = 12, price = 10

Everbody pays own bill◮ What do you do?◮ Single person decision problem

Total bill is shared equally◮ What do you do?◮ It is a GAME


Split or Steal


Split or Steal


Split or Steal

Van den Assem, Van Dolder, and Thaler, “Split or Steal? CooperativeBehavior When the Stakes Are Large” Management Science, 2012.

Individual players on average choose “split” 53 percent of the time

Propensity to cooperate is surprisingly high for consequential amounts

Less likely to cooperate if opponent has tried to vote them offpreviously

◮ Evidence for reciprocity

Little evidence that contestants propensity to cooperate dependspositively on the likelihood that their opponent will cooperate

◮ Little evidence for conditional cooperation

Young males are less cooperative than young females

Old males are more cooperative than old females


Guess the Average

We will play a game

I will distribute slips of paper that looks like this

Round 1

Name:

Your guess:

Write your name and a number between 0 and 100

We will collect them and enter into Excel

The number that is closest to half the average wins

Winner gets 5TL (in case of a tie we choose randomly)Click here for the EXCEL file


Another Example: A Single Person Decision Problem

Ali is an investor with $100

StateGood Bad

Bonds 10% 10%Stocks 20% 0%

Which one is better?

Probability of the good state p

Assume that Ali wants to maximize the amount of money he has atthe end of the year.

Bonds: $110

Stocks: average (or expected) money holdings:

p× 120 + (1− p)× 100 = 100 + 20× p

If p > 1/2 invest in stocks

If p < 1/2 invest in bonds


An Investment Game

Ali again has two options for investing his $100:◮ invest in bonds

⋆ certain return of 10%◮ invest it in a risky venture

⋆ successful: return is 20%⋆ failure: return is 0%

◮ venture is successful if and only if total investment is at least $200

There is one other potential investor in the venture (Beril) who is inthe same situation as Ali

They cannot communicate and have to make the investment decisionwithout knowing the decisions of each other

Ali

BerilBonds Venture

Bonds 110, 110 110, 100Venture 100, 110 120, 120


Games We Play

Group projects: How much effort should you put?◮ Coordination

Who is going to clean the kitchen?◮ War of attrition

Bargaining: How low the seller is willing to go?◮ Information, commitment, signaling

Poker: Bluff or not◮ Randomizing

Grading on the curve: Work hard or not?◮ Prisoners’ Dilemma


Why Should We Learn Game Theory?

Media says so:

Managers have much to learn from game theory provided they use itto clarify their thinking, not as a substitute for business experience

◮ The Economist, 15 June 1996

Game Theory, long an intellectual pastime, came into its own as abusiness tool

◮ Forbes, 3 July 1995

Game theory is hot◮ The Wall Street Journal, 13 February 1995



Managers too...

Game theory forces you to see a business situation over many periodsfrom two perspectives: yours and your competitors

◮ Judy Lewent - CFO, Merck

To help predict competitor behavior and determine optimal strategy,our consulting teams use techniques such as pay-off matrices andcompetitive games

◮ John Stuckey & David White - McKinsey

At Bell Atlantic, we’ve found that the lessons of game theory give usa wider view of our business situation and provide us a more nimbleapproach to corporate planning

◮ Raymond Smith - CEO, Bell Atlantic



More Importantly

Life is sequence of strategic situations

Game theory does not provide recipes for them

But provides a method and toolbox to understand them better

You can even change the game for the better if you understand it well



It also helps us understand how and why certain norms andinstitutions arise throughout history and why some disappear

Used in◮ Economics◮ Political science◮ Law◮ Diplomacy◮ Business◮ Philosophy◮ Biology◮ History


Strategic Thinking: Change the Game

I have 26 black cards while each one of you has one red card.

If one of us has one black and one red card, we will get paid 100 TL

How much are you willing to accept to give me your card?

Suppose now I throw away 3 black cards

How much are you willing to accept now?


Correct Strategic Thinking

You have to understand the game correctly

You should try to predict what the other players are going to do◮ Who are the other players?◮ What are their possible moves?◮ What do they want from the game?◮ What do they know about the game?

Modeling the strategic interaction correctly is very important


Types of Games

Games differ along two main dimensions:

1. Order of Moves◮ Simultaneous: sealed-bid auction◮ Sequential: ascending-bid auction

2. Information◮ Complete Information: valuations known◮ Incomplete Information: valuations unknown


Game Forms

Information

Moves

Complete Incomplete

Simultaneous

Sequential

Strategic FormGames withCompleteInformation

Strategic FormGames withIncompleteInformation

Extensive FormGames withCompleteInformation

Extensive FormGames withIncompleteInformation


Method

Look at examples and play games to get intuition

Understand general principles by studying theory

Study real life cases


Introduction to Economic and Strategic BehaviorStrategic Form Games

Levent Kockesen

Koc University

Levent Kockesen (Koc University) Strategic Form Games 1 / 1

Contribution Game

Everybody starts with 10 TL

You decide how much of 10 TL to contribute to joint fund

Amount you contribute will be doubled and then divided equallyamong everyone

I will distribute slips of paper that looks like this

Name:

Your Contribution:

Write your name and an integer between 0 and 10

We will collect them and enter into Excel

We will choose one player randomly and pay herClick here for the EXCEL file


Contribution Game

Who are the other players?

What are their possible moves?

What do they want from the game?

What do they know about the game?


Strategic Form Games

It is used to model situations in which players choose strategieswithout knowing the strategy choices of other players

Also known as normal form games

A strategic form game is composed of

1. Set of players

2. A set of strategies for each player

3. A payoff function for each player

An outcome is a collection of strategies, one for each player.◮ Also known as a strategy profile


Price Competition

You will be matched randomly with another student

Each will independently choose High or Low price

If you both choose High: Each gets 10

If both Low: Each gets 5

If one High the other Low: High gets 2, Low gets 15

Make you choice and write on a piece of paper


Price Competition

Player 1

Player 2High Low

High 10, 10 2, 15Low 15, 2 5, 5

Bimatrix

This way of writing the game is known as a bimatrix


Price Competition

Player 1

Player 2High Low

High 10, 10 2, 15Low 15, 2 5, 5

What should Player 1 play?

Does that depend on what she thinks Player 2 will do?

Low is an example of a dominant strategy

It is optimal independent of what other players do

How about Player 2?

(Low, Low) is a dominant strategy equilibrium


Dominant Strategies

A strategy A strictly dominates another strategy B if it yields a strictlyhigher payoff irrespective of how the other players play.

A weakly dominates B if it never does worse than B and sometimes doesstrictly better.

A strategy A is strictly dominant if it strictly dominates every otherstrategy of that player. It is called weakly dominant if it weakly dominatesevery other strategy.


Dominant Strategy Equilibrium

If every player has a (strictly or weakly) dominant strategy, then thecorresponding outcome is a (strictly or weakly) dominant strategyequilibrium.



T

WH L

H 10, 10 2, 15L 15, 2 5, 5

L strictly dominates H

(L,L) is a strictly dominantstrategy equilibrium

T

WH L

H 10, 10 5, 15L 15, 5 5, 5

L weakly dominates H

(L,L) is a weakly dominantstrategy equilibrium



A reasonable solution concept

It only demands the players to be rational

It does not require them to know that the others are rational too

But it does not exist in many interesting games


Prisoners’ Dilemma


Player 1

Player 2Confess Deny

Confess −5,−5 0,−10Deny −10, 0 −1,−1

How would you play?


Case Study: Advertising Ban in Tobacco Industry

1960s: tobacco companies advertise heavily on TV and radio◮ TV ads accounted for more than 80% of total advertising budget

1964: US Surgeon General’s 1964 report finds cigarette smokinghazardous to health

1970: Public Health Cigarette Smoking Act banned all cigarette TVand radio ads

Question

What is the effect on advertising expenditures? On profits?



Model Assumptions

Symmetric oligopolistic industry

Firms choose how much to spend on advertising

Total demand is perfectly inelastic

Advertising Game

PM

RJRAd No

Ad 30, 30 60, 20No 20, 60 50, 50



What Happened:

Advertising expenditures fell sharply in 1971

Profits rose as stock returns for the major tobacco companies reachedabnormal heights

Aggregate sales remained unchanged

After 5 years: Advertising spending began to recover and even exceedpre-ban levels

See Shi Qi, “The Impact of Advertising Regulation on Industry TheCigarette Advertising Ban of 1971,” RAND J. (2013)


Rationality

Players have consistent preferences over outcomes◮ Can rank outcomes◮ Payoffs represent rankings

They have beliefs regarding future events◮ Other players’ moves◮ Chance events

On the basis of these beliefs they choose the best strategy◮ The strategy that maximizes their expected payoffs


Rationality

Suppose there are 4 possible outcomes: a, b, c, d

Corresponding payoffs: u(a), u(b), u(c), u(d)

and probabilities p(a), p(b), p(c), p(d)

Expected payoff is

p(a)u(a) + p(b)u(b) + p(c)u(c) + p(d)u(d)


Risk Aversion

Choose between◮ S: I will give you $100◮ R: I will flip a fair coin. If heads I will give you $200; If tails I will give

you nothing

If you prefer S over R you are risk averse

If you don’t care you are risk neutral

If you prefer R over S you are risk lover

An individual is risk averse is he prefers to receive the expected value of alottery to playing the lottery


Risk Aversion

Payoff function (of certain outcome) is

Linear if risk neutral◮ u(x) = x

Concave if risk averse◮ u(x) =

√x

Convex if risk lover◮ u(x) = x2


Risk Preferences

b

b

bb

x

u(x)

x1 x2E[x]

Risk averse: u(E[x]) > E[u(x)]

b

b

b

x

u(x)

x1 x2E[x]

Risk neutral: u(E[x]) = E[u(x)]

b

b

b

b

x

u(x)

x1 x2E[x]

Risk lover: u(E[x]) < E[u(x)]


Rationality

Rationality does not mean selfishness◮ Payoff functions must include all relevant considerations such as

altruism

It is a consistency requirement

Should be thought as an approximation to reality

In evolutionary games we do not assume rationality◮ Genes govern behavior◮ Those that are successful reproduce faster◮ An evolutionary stable state is the eventual outcome of this dynamics


Common Knowledge

We also assume that the game as well as the rationality is commonknowledge

A fact X is common knowledge if everybody knows it

If everybody knows that everybody knows it

If everybody knows that everybody knows that everybody knows it

... ad infinitum


A Puzzle

Three girls sitting in a circle

Each is wearing a hat: Red or White

Each can see the others’ hat but not her own

Teacher asks each to guess her hat color◮ This is done sequentially◮ Assume a wrong guess is very costly

In fact all have red

Could any one of them guess correctly?

Suppose now that the teacher says

At least one of you is wearing a red hat

How about now?


Price Matching

Wal-Mart: Our Ad Match Guarantee

We’re committed to providing low prices every day. On everything. So ifyou find a lower advertised price on an identical product, tell us and we’llmatch it. Right at the register.


Price Matching

Sounds like a good deal for customers

How does this change the game?

Toys“R”us

Wal-MartHigh Low Match

High 10, 10 2, 15 10, 10Low 15, 2 5, 5 5, 5

Match 10, 10 5, 5 10, 10

Is there a dominant strategy for any of the players?

There is no dominant strategy equilibrium for this game

So, what can we say about this game?


Price Matching

Toys“R”us

Wal-MartHigh Low Match

High 10, 10 2, 15 10, 10Low 15, 2 5, 5 5, 5

Match 10, 10 5, 5 10, 10

High is weakly dominated and Toys“R”us is rational◮ Toys“R”us should not use High

High is weakly dominated and Wal-Mart is rational◮ Wal-Mart should not use High

Each knows the other is rational◮ Toys“R”us knows that Wal-Mart will not use High◮ Wal-Mart knows that Toys“R”us will not use High◮ This is where we use common knowledge of rationality


Price Matching

Therefore we have the following “effective” game

Toys“R”us

Wal-MartLow Match

Low 5, 5 5, 5Match 5, 5 10, 10

Low becomes a weakly dominated strategy for both

Both companies will play Match and the prices will be high

The above procedure is known as Iterated Elimination of DominatedStrategies (IEDS)

To be a good strategist try to see the world from the perspective of yourrivals and understand that they will most likely do the same


Dominated Strategies

A “rational” player should never play an strategy when there isanother strategy that gives her a higher payoff irrespective of how theothers play

We call such a strategy a dominated strategy

A strategy A is strictly dominated by another strategy B if B does strictlybetter than A irrespective of how other players play.

A is weakly dominated by B if B never does worse than A and sometimesdoes strictly better.


Iterated Elimination of Dominated Strategies

Common knowledge of rationality justifies eliminating dominatedstrategies iteratively

This procedure is known as Iterated Elimination of DominatedStrategies

If every strategy eliminated is a strictly dominated strategy◮ Iterated Elimination of Strictly Dominated Strategies

If at least one strategy eliminated is a weakly dominated strategy◮ Iterated Elimination of Weakly Dominated Strategies

Remember Guessing Game?

Pick a number between 0 and 100

You win if your number is closest to half the average

What do you pick?


Iterated Elimination of Dominated Strategies

Half the average can be at most 50

If everybody is rational◮ Nobody should choose higher than 50

If everybody knows that everybody is rational◮ Nobody should choose higher than 25

If everybody knows that everybody knows that everybody is rational◮ Nobody should choose higher than 12.5

Can eliminate all down to zero

Keynes’ Beauty Contest

Number closest to the average wins.

Shiller’s NYT article


Envelope Paradox

Ali and Berna has each an envelope with 5, 10, 20, 40, 80, or 160 TL.

One of the envelopes has twice the amount in the other

Each looks into their envelope and independently decide whether theywant to exchange envelopes

If both want to do so, they exchange envelopes

Should they exchange?

Suppose Berna has 20 TL. Should she exchange?


Chairman’s Paradox

A committee of 3 members, 1, 2, 3 must choose among 3 options,A,B,C

The alternative that gets two votes is chosen

In case of a tie the chairman, player 3, chooses

Preferences1 2 3

A B CB C AC A B



1 2 3

A B CB C AC A B

1

2A B C

A 2, 0, 1 2, 0, 1 2, 0, 1B 2, 0, 1 1, 2, 0 0, 1, 2C 2, 0, 1 0, 1, 2 0, 1, 2

A

2A B C

A 2, 0, 1 1, 2, 0 0, 1, 2B 1, 2, 0 1, 2, 0 1, 2, 0C 0, 1, 2 1, 2, 0 0, 1, 2

B

2A B C

A 2, 0, 1 0, 1, 2 0, 1, 2B 0, 1, 2 1, 2, 0 0, 1, 2C 0, 1, 2 0, 1, 2 0, 1, 2

C

What happens if everyone votes sincerely?◮ There is a tie◮ Chairman (3) chooses C

Is this a reasonable outcome?



1

2A B C

A 2, 0, 1 2, 0, 1 2, 0, 1B 2, 0, 1 1, 2, 0 0, 1, 2C 2, 0, 1 0, 1, 2 0, 1, 2

A

2A B C

A 2, 0, 1 1, 2, 0 0, 1, 2B 1, 2, 0 1, 2, 0 1, 2, 0C 0, 1, 2 1, 2, 0 0, 1, 2

B

2A B C

A 2, 0, 1 0, 1, 2 0, 1, 2B 0, 1, 2 1, 2, 0 0, 1, 2C 0, 1, 2 0, 1, 2 0, 1, 2

C

Are there dominated strategies?◮ Player 1: C is weakly dominated by A◮ Player 2: A and C are weakly dominated by B◮ Player 3: A and B are weakly dominated by C

The reduced game

2B

A 0, 1, 2B 1, 2, 0

C

Final outcome: (B,B,C) and B wins◮ Worst outcome for the chairman!


Effort Game

You choose how much effort to expend for a joint project◮ An integer between 1 and 7

The quality of the project depends on the smallest effort: e◮ Weakest link

Effort is costly

If you choose e your payoff is

6 + 2e− e

We will play this twice

We will randomly choose one round and one student and pay her

Enter your name and effort choice for Round 1Click here for the EXCEL file


Effort Game

Two effort level - two player version

Low HighLow 7, 7 7, 1High 1, 7 13, 13

Is there a dominated strategy?

If player 1 expects player 2 to choose Low, what is her best strategy(best response)?

If player 2 expects player 1 to choose Low what is its best response?

(Low, Low) is an outcome such that◮ Each player best responds, given what she believes the other will do◮ Their beliefs are correct

It is a Nash equilibrium

Is (Low,Low) the only Nash equilibrium?


Nash Equilibrium

Nash Equilibrium

Nash equilibrium is a strategy profile (a collection of strategies, one foreach player) such that each strategy is a best response (maximizes payoff)to all the other strategies

Nash equilibrium is self-enforcing: no player has an incentive todeviate unilaterally

One way to find Nash equilibrium is to first find the best responsecorrespondence for each player

◮ Best response correspondence gives the set of payoff maximizingstrategies for each strategy profile of the other players

... and then find where they “intersect”


Nash Equilibrium

Low HighLow 7, 7 7, 1High 1, 7 13, 13

1’s best response to Low is Low

Her best response to High is High

Similarly for player 2

Best response correspondences intersect at (Low, Low) and (High,High)

These two strategy profiles are the two Nash equilibria of this game

We would expect in the long-run one of these outcomes to prevail

Risk dominance

Payoff dominance


Stag Hunt

Jean-Jacques Rousseau in A Discourse on Inequality

If it was a matter of hunting a deer, everyone well realized thathe must remain faithful to his post; but if a hare happened topass within reach of one of them, we cannot doubt that he wouldhave gone off in pursuit of it without scruple...

Stag HareStag 2, 2 0, 1Hare 1, 0 1, 1


The Bar Scene


The Bar Scene

Blonde BrunetteBlonde 0, 0 2, 1

Brunette 1, 2 1, 1

See S. Anderson and M. Engers: Participation Games: Market Entry,Coordination, and the Beautiful Blonde, Journal of EconomicBehavior and Organization, 2007


John Nash


Nash Demand Game

Each of you will be randomly matched with another student

You are trying to divide 10 TL

Each writes independently how much she wants (in multiples of 1 TL)

If two numbers add up to greater than 10 TL each gets nothing

Otherwise each gets how much she wrote

Write your name and demand on the slips

I will match two randomly

Choose one pair randomly and pay themClick here for the EXCEL file


Some Other Commonly Used Games

Hawk-Dove

Player 1

Player 2D H

D 3, 3 1, 5H 5, 1 0, 0

Coordination

Player 1

Player 2L R

L 1, 1 0, 0R 0, 0 1, 1

Player 1

Player 2L R

L 1, 1 0, 0R 0, 0 2, 2



Battle of the Sexes

Player 1

Player 2B S

B 2, 1 0, 0S 0, 0 1, 2

Chicken

Jim

BuzzQuit Stay

Quit 0, 0 −1, 1Stay 1,−1 −2,−2



Matching Pennies

Player 1

Player 2H T

H 1,−1 −1, 1T −1, 1 1,−1


Nash Equilibrium: Discussion

Nash equilibrium as steady-state

Nash equilibrium as self-enforcing agreement

Nash equilibrium as stable norm or institution


Evidence from Experiments

Davis and Holt, Experimental Economics◮ In simple games with unique Nash equilibrium, that outcome has

considerable drawing power

Multiple equilibria: Importance of focal points

Interdependent preferences


Difficulties with Nash Equilibrium

Why should players play Nash equilibrium?◮ Roger Myerson: Why not?

How would you play the following?

Player 1

Player 2A B C

A 2, 2 3, 1 0, 2B 1, 3 2, 2 3, 2C 2, 0 2, 3 2, 2

What would Myerson say?◮ If you really think the other will play C, you should play B◮ But if the other goes through this reasoning, she should play A◮ in which case, you might as well play a best response: A


Introduction to Economic and Strategic BehaviorStrategic Form Games: Applications

Levent Kockesen

Koc University

Levent Kockesen (Koc University) Applications 1 / 1

Auctions

Many economic transactions are conducted through auctions

treasury bills

foreign exchange

publicly owned companies

mineral rights

airwave spectrum rights

art work

antiques

cars

houses

government contracts

Also can be thought of as auctions

takeover battles

queues

wars of attrition

lobbying contests


Auction Formats

1. Open bid auctions1.1 ascending-bid auction

⋆ aka English auction⋆ price is raised until only one bidder remains, who wins and pays the

final price

1.2 descending-bid auction⋆ aka Dutch auction⋆ price is lowered until someone accepts, who wins the object at the

current price

2. Sealed bid auctions2.1 first price auction

⋆ highest bidder wins; pays her bid

2.2 second price auction⋆ aka Vickrey auction⋆ highest bidder wins; pays the second highest bid


Auction Formats

Auctions also differ with respect to the valuation of the bidders

1. Private value auctions◮ each bidder knows only her own value◮ artwork, antiques, memorabilia

2. Common value auctions◮ actual value of the object is the same for everyone◮ bidders have different private information about that value◮ oil field auctions, company takeovers


Strategically Equivalent Formats

!

!

Open Bid Sealed Bid

English Auction

Dutch Auction

Second Price

First Price

We will study sealed bid auctions

For now we will assume that values are common knowledge◮ value of the object to player i is vi dollars

For simplicity we analyze the case with only two bidders

Assume v1 > v2 > 0


Second Price Auctions

Highest bidder wins and pays the second highest bid

In case of a tie, the object is awarded to player 1

Suppose player 1’s value is 5 and player 2’s is 3

Assume they can bid 1, 2, 3, 4, 5, 6

1 2 3 4 5 61 4, 0 0, 2 0, 2 0, 2 0, 2 0, 22 4, 0 3, 0 0, 1 0, 1 0, 1 0, 13 4, 0 3, 0 2, 0 0, 0 0, 0 0, 04 4, 0 3, 0 2, 0 1, 0 0,−1 0,−15 4, 0 3, 0 2, 0 1, 0 0, 0 0,−26 4, 0 3, 0 2, 0 1, 0 0, 0 −1, 0



1 2 3 4 5 61 4, 0 0, 2 0, 2 0, 2 0, 2 0, 22 4, 0 3, 0 0, 1 0, 1 0, 1 0, 13 4, 0 3, 0 2, 0 0, 0 0, 0 0, 04 4, 0 3, 0 2, 0 1, 0 0,−1 0,−15 4, 0 3, 0 2, 0 1, 0 0, 0 0,−26 4, 0 3, 0 2, 0 1, 0 0, 0 −1, 0

For player 1: bidding 4 and 5 weakly dominates all others◮ What if player 2 could also bid 4.5?◮ 5 would weakly dominate 4

For player 2: bidding 3 and 4 weakly dominates all others◮ If player 1 could also bid between 3 and 4?◮ 3 would weakly dominate 4

If players could bid any number, everybody bidding own value is a weaklydominant strategy equilibrium.



I. Bidding your value weakly dominates bidding higher

Suppose your value is $10 but you bid $15. Three cases:

1. The other bid is higher than $15 (e.g. $20)◮ You loose either way: no difference

2. The other bid is lower than $10 (e.g. $5)◮ You win either way and pay $5: no difference

3. The other bid is between $10 and $15 (e.g. $12)◮ You loose with $10: zero payoff◮ You win with $15: loose $2

5

10 value

12

15 bid

20



II. Bidding your value weakly dominates bidding lower


1. The other bid is higher than $10 (e.g. $12)◮ You loose either way: no difference

2. The other bid is lower than $5 (e.g. $2)◮ You win either way and pay $2: no difference

3. The other bid is between $5 and $10 (e.g. $8)◮ You loose with $5: zero payoff◮ You win with $10: earn $2

Weakly dominant strategy equilibrium = (v1, v2)

There are many Nash equilibria. For example (v1, 0) 2

10 value

8

5 bid

12


First Price Auctions

Highest bidder wins and pays her own bid

In case of a tie, the object is awarded to player 1

Suppose player 1’s value is 5 and player 2’s is 3

Assume they can bid 1, 2, 3, 4, 5, 6

1 2 3 4 5 61 4, 0 0, 1 0, 0 0,−1 0,−2 0,−32 3, 0 3, 0 0, 0 0,−1 0,−2 0,−33 2, 0 2, 0 2, 0 0,−1 0,−2 0,−34 1, 0 1, 0 1, 0 1, 0 0,−2 0,−35 0, 0 0, 0 0, 0 0, 0 0, 0 0,−36 −1, 0 −1, 0 −1, 0 −1, 0 −1, 0 −1, 0



1 2 3 4 5 61 4, 0 0, 1 0, 0 0,−1 0,−2 0,−32 3, 0 3, 0 0, 0 0,−1 0,−2 0,−33 2, 0 2, 0 2, 0 0,−1 0,−2 0,−34 1, 0 1, 0 1, 0 1, 0 0,−2 0,−35 0, 0 0, 0 0, 0 0, 0 0, 0 0,−36 −1, 0 −1, 0 −1, 0 −1, 0 −1, 0 −1, 0

Player 2: bidding 2 is weakly dominant

Player 1’s best response to 2 is 2

(2, 2) is a Nash equilibrium◮ What if player 2 could bid between 2 and 3?

Other Nash equilibria? (3, 3), (4, 4), (5, 5)


Nash Equilibria of First Price Auctions

Assume players can bid any number

What are the Nash equilibria?

We can compute the best response correspondences

or we can adopt a different approach◮ First find the necessary conditions for a Nash equilibrium

⋆ If a strategy profile is a Nash equilibrium then it must satisfy theseconditions

◮ Then find the sufficient conditions⋆ If a strategy profile satisfies these conditions, then it is a Nash

equilibrium


Necessary ConditionsLet (b∗1, b

∗2) be a Nash equilibrium. Then,

1. Player 1 wins: b∗1 ≥ b∗2

Proof

Suppose not: b∗1 < b∗2. Two possibilities:

◮ b∗2 ≤ v2: Player 1 could bid v2 and obtain a strictly higher payoff

◮ b∗2 > v2: Player 2 has a profitable deviation: bid zero

Contradicting the hypothesis that (b∗1, b∗2) is a Nash equilibrium.

2. b∗1 = b∗2

Proof

Suppose not: b∗1 > b∗2. Player 1 has a profitable deviation: bid b∗2

3. v2 ≤ b∗1 ≤ v1

Proof

ExerciseLevent Kockesen (Koc University) Applications 13 / 1

Sufficient Conditions

So, any Nash equilibrium (b∗1, b∗2) must satisfy

v2 ≤ b∗1 = b∗2 ≤ v1.

Is any pair (b∗1, b∗2) that satisfies these inequalities an equilibrium?

Set of Nash equilibria is given by

{(b1, b2) : v2 ≤ b1 = b2 ≤ v1}


Price Competition Models

Quantity (or capacity) competition: Cournot Model◮ Augustine Cournot (1838)

Price Competition: Bertrand Model◮ Joseph Bertrand (1883)

Two main models:

1. Bertrand Oligopoly with Homogeneous Products

2. Bertrand Oligopoly with Differentiated Products


Bertrand Duopoly with Homogeneous Products

Two firms, each with unit cost c ≥ 0

They choose prices◮ The one with the lower price captures the entire market◮ In case of a tie they share the market equally

Total market demand is equal to one (not price sensitive)

Both firms are profit maximizers


Nash Equilibrium

Suppose P ∗1 , P

∗2 is a Nash equilibrium. Then

1. P ∗1 , P

∗2 ≥ c. Why?

2. At least one charges c◮ P ∗

1 > P ∗2 > c?

◮ P ∗2 > P ∗

1 > c?◮ P ∗

1 = P ∗2 > c?

3. P ∗2 > P ∗

1 = c?

4. P ∗1 > P ∗

2 = c?

The only candidate for equilibrium is P ∗1 = P ∗

2 = c, and it is indeed anequilibrium.

The unique Nash equilibrium of the Bertrand game is (P ∗1 , P

∗2 ) = (c, c)


Models of Spatial Competition

There is a population of individuals (consumers, voters, etc.)characterized by some trait (type)

◮ income◮ taste◮ political inclination

Individuals have preferences (which may depend on their types) overoutcomes

◮ tax rates◮ product characteristics◮ environmental regulations

Players (firms, political parties, etc.) choose outcomes

Given players’ choices, each individual chooses a player

Players have preferences over outcomes and the choices of theindividuals

◮ Firms choose product characteristic and care about how much they sell◮ Political parties choose policy positions and care about whether they

win the election and which position gets implemented


Electoral Competition: Downs Model

Originally due to Hotelling (1929)◮ Applied to electoral competition by Downs (1957)

There is a large number of voters each with an ideal policy

Assume equal number of voters for each ideal policy◮ Uniform distribution

Each voter has single peaked preferences over policies◮ They prefer policies that are closer to their ideal policy


Electoral Competition: Downs Model

Candidates announce a policy◮ 10% tax rate vs. 25% tax rate◮ pro-EU vs anti-EU

Voters vote sincerely◮ each voter votes for the candidate whose policy is closest to her ideal

policy◮ if indifferent they vote for each with equal probabilities

Candidate with the highest amount of votes wins

Candidates care only about the outcome of the election◮ preferences: win ≻ tie ≻ lose

Say the two candidates choose 0 < p1 < p2 < 1

0 1p1 p2p1+p22

Vote for 1 Vote for 2


Nash Equilibrium

Suppose p∗1, p∗2 is a Nash equilibrium. Then

1. Outcome must be a tie◮ Whatever your opponent chooses you can always guarantee a tie

2. p∗1 6= p∗2?

0 11/2p1 p2

3. p∗1 = p∗2 6= 1/2?

0 11/2p1

p2

The only candidate for equilibrium is p∗1 = p∗2 = 1/2, which is indeed anequilibrium.

The unique Nash equilibrium of the game is (p∗1, p∗2) = (1/2, 1/2)


Extensions

Median Voter Theorem

Consider the above model but assume that the voters are distributedaccording to any distribution function. Then the unique Nash equilibriumis to choose the median voter’s ideal policy for both candidates.

Other Models

Models with participation costs

Models with more than two players

Models with multidimensional policy space

Models with ideological candidates: Wittman (1973) Model


Game Theory and Traffic

4 cars to go from A to D

If n cars travel, it takes n hours from A to B and C to D

Takes 6 hours from A to C and C to D

A

B

C

D

n

6

6

n

What is the Nash equilibrium?



A

B

C

D

nB

6

6

nC

nB + 6 > nC + 6 ⇒ switch to C

nC + 6 > nB + 6 ⇒ switch to B

This implies nB = nC = 2 in any equilibrium

nB = nC = 2 is indeed a Nash equilibrium

Travel time is 8 hours



A fast highway is built from B to C

A

B

C

D

n

6

6

n

1

How does the travel time change?



A

B

C

D

nB

6

6

nC

1

What is the Nash equilibrium nB and nC?No car will go to C directly because going through B takes at most 5hrsTherefore, all the cars must go through BOnce at B no car will go directly to D because going through C takesat most 5 hrsnB = nC = 4Total travel time is 9 hours!


Braess’s Paradox

German mathematician Dietrich Braess in a 1969 paper

In South Korea, traffic sped up after a motorway was removed

42nd Street in NYC was closed to traffic on Earth Day (1990) ⇒ lesscongestion


Game Theory of Legal Regimes

A driver and a pedestrian choose care level◮ Due care: Socially optimal◮ No care: Less than socially optimal

Due care costs 10

Accident inflicts injury on pedestrian with cost 100

An accident happens for certain unless both choose due care

It happens with one-in-ten chance if both choose due care


No Liability

If driver is never liable for accident

Pedestrian

DriverNo Care Due Care

No Care −100, 0 −100,−10Due Care −110, 0 −20,−10

What do you expect to happen?

Iterated elimination of dominated strategies → (No Care, No Care)

Driver has too little incentive to take care


Pure Strict Liability

Driver pays for all injuries

Pedestrian


No Care 0,−100 0,−110Due Care −10,−100 −10,−20


Iterated elimination of dominated strategies → (No Care, No Care)

Pedestrian has too little incentive to take care


Negligence with Contributory Negligence

Driver pays only if he is negligent and pedestrian is not

Pedestrian


No Care −100, 0 −100,−10Due Care −10,−100 −20,−10

Prevailing principle of Anglo-American tort law


Iterated elimination of dominated strategies → (Due Care, Due Care)


Strict Liability with Contributory Negligence

If both exercise care driver pays

Pedestrian


No Care −100, 0 −100,−10Due Care −10,−100 −10,−20

Same prediction: (Due Care, Due Care)

Only the costs of accident are shared differently


Political Reform

Victorious parties rewards their followers with government jobs◮ Patronage

Leads to corruption and inefficiencies

Meritocracy is better◮ A reforms is needed to achieve it◮ Has proved to be difficult◮ Why?

Model by Geddes (1991)


Political Reform

Both patronage and reform have advantages in elections

Majority party and minority party

With no patronage: Chances of winning (p, 1− p)

Patronage increases chances by v1 and v2

NP PNP p, 1− p p− v2, 1− p+ v2P p+ v1, 1− p− v1 p+ v1 − v2, 1− p− v1 + v2

P: Patronage, NP: No Patronage

Row player: Majority party, Column player: Minority Party

Patronage is dominant strategy equilibrium


Political Reform

Supporting reform may help during campaign

Suppose reform passes if and only if majority party supports it

If majority party supports reform, reform laws pass and neither partycan use patronage

If majority party opposes reform, then both parties use patronage◮ This is from the first model: Patronage is dominant strategy without

reform

However, minority may still support reform during the campaign

There is a benefit of supporting reform when the other does not: e


Political Reform

R NRR p, 1− p p+ e, 1− p− e

NR p+ v1 − v2 − e, 1− p− v1 + v2 + e p+ v1 − v2, 1− p− v1 + v2

Supporting reform is dominant for minority party◮ Reform is a common theme with challengers

Reform happens if

v1 − v2 < e

Reform is more likely if v1 − v2 is small◮ If parties are relatively equal in electoral strength◮ If majority party is much stronger, it may expect greater benefits from

patronage◮ If majority party has had disproportionate access to patronage in the

past it will have more entrenched workers and more to benefit frompatronage

Reform is also more likely if there is a strong support for reform◮ That is e is high◮ Likely to be case after scandals


Financial Crises

Consumers demand liquidity

Banks provide them: Demand deposits

But they make loans that are less liquid

This makes them vulnerable to bank runs

Diamond-Dybvig Model (1983)◮ See also: Diamond (2007)

September 2007


Bank Runs

Consumers demand liquidity because they are uncertain about whenthey need liquid funds

Assume three dates: t = 0, 1, 2

Each consumer need to consume either at date 1 or date 2

At date 0, they don’t know when◮ probability of date 1: p◮ probability of date 2: 1− p

At date 0 each has 1 TL that they can invest◮ If they liquidate on date 1: r1◮ If they liquidate on date 2: r2◮ r1 < r2 ⇒ the asset is illiquid

Expected utility

pu(r1) + (1− p)u(r2)


Bank Runs

Banks can invest 1 TL in an asset that returns◮ 1 if liquidated on date 1◮ R > 1 if liquidated on date 2

They offer a contract to consumers that promise◮ r1 > 1 if withdrawn on date 1◮ r2 < R if withdrawn on date 2◮ This is more liquid than banks’ asset

Banks create liquidity◮ Without them there may be no investment in illiquid assets

Why do the consumers prefer deposit in the bank?◮ Because they are risk averse. For example

u(c) = 1− 1

c


Bank Runs

Expected utility of less liquid asset

p(1− 1

1) + (1− p)(1 − 1

R)

Expected utility of bank deposit

p(1− 1

r1) + (1− p)(1− 1

r2)

If R = 2, p = 0.25, r1 = 1.28, r2 = 1.813◮ Expected utility of illiquid asset is 0.375◮ Expected utility of bank deposit is 0.391

Would a bank be willing to offer these rates?


Bank Runs

Suppose 100 consumers deposit 1 TL each on date 0

There is no aggregate uncertainty◮ A fraction p of all consumers is definitely date 1 type

Fraction p withdraws on date 1 gets r1◮ Date 1 payout is 100pr1

Remaining 100(1 − pr1) brings 100(1 − pr1)R on date 2

Out of that 100(1 − p)r2 is paid to the depositors

Banks’ profit is

100(1 − pr1)R− 100(1 − p)r2


Bank Runs

Break-even r2 is

r2 =(1− pr1)R

1− p

1 < r1 < 1/p ⇒ r2 < R

R = 2, p = 0.25, r1 = 1.28, r2 = 1.813◮ Gives zero profits to banks and consumers prefer bank deposits

At date 1 bank has 100 TL

If 25 of them withdraws, it pays out 25 × 1.28 = 32

Remaining 68 TL brings 68× 2 = 136 on date 2

Each of 75 depositor receives

136

75= 1.813

So everything works out fine if only 25 withdraw at date 1


Bank Runs

At date 1 all type 1 consumers choose to withdraw

Remaining fraction 1− p has to decide whether to withdraw or not◮ They can withdraw at date 1 and consume at date 2

This defines a game with 100 players

What are the Nash equilibria?

Let the equilibrium fraction who withdraw at date 1 be f

We must have f ≥ p

If fr1 > 1, the bank becomes insolvent and does not survive to date 2

If fr1 ≤ 1, at date 2 the bank pays out

r2(f) =(1− fr1)R

1− f


Bank Runs

Type 2 players have two options:◮ Withdraw at date 1 and get

r1(f) =

{r1, fr1 ≤ 11f , fr1 > 1

◮ or withdraw at date 2 and get

r2(f) =

{(1−fr1)R

1−f , fr1 ≤ 1

0, fr1 > 1

In any Nash equilibrium◮ They choose date 1 if r1(f) > r2(f)◮ and choose date 2 otherwise


Bank Runs

Assume 1 < r1 < 1/p, R > 1, and

r2 =(1− pr1)R

1− p

Then r2 > r1

We can show that

f = p is a Nash equilibrium.

f = p and pr1 < 1 imply

r2(f) = r2 > r1 = r1(f)

So, all type 2 consumers prefer to withdraw at date 2

This is an equilibrium without a bank run


Bank Runs

Claim

f = 1 is a Nash equilibirum.

r1 > 1 and f = 1 imply fr1 > 1

Therefore

r1(f) =1

f= 1 > 0 = r2(f)

So, all type 2 consumers prefer to withdraw at date 1

This is a bank-run equilibrium

f = 1 is a self-fulfilling prophecy


Bank Runs

f > f ⇒ r1(f) > r2(f) ⇒ Everybody withdraws at date 1

f < f ⇒ r1(f) < r2(f) ⇒ Only fraction p withdraws at date 1

f = 1 and f = p are the only Nash equilibria

f11/r1fp

r2(f)

r1(f)

Rr2

r1

1


Bank RunsA simpler model:

4 depositors, each with 1 TL, half need to consume on date 1

R = 4, r1 = 1.5, r2 = 2

Suppose on date 1, 2 depositors withdraw

Bank is left with 1 TL

This brings 4 TL on date 2, which the bank returns to the other twodepositorsWhat happens if 3 depositors withdraw on date 1?

◮ The bank goes bankrupt◮ They each get 4/3, the other gets nothing

What happens if 4 depositors withdraw on date 1?◮ The bank goes bankrupt◮ They each get 1

The game between two date 2 depositors

W NW 1, 1 4/3, 0N 0, 4/3 2, 2


Bank Runs

Starting from good equilibrium a rumor that the bank is not doingwell could cause a run

This is so even if most know it not to be true

You only need people believe that the others will withdraw


Bank Runs

How do you prevent bank runs?

Suspend convertibility of deposits to cash: force f ≤ p◮ before deposit insurance, banks regularly suspended convertibility to

stop runs

Deposit insurance: Government promises (r1, r2)◮ Does it with taxpayers’ money◮ May create incentive problems for banks


page.1

Introduction to Economic and Strategic BehaviorMixed Strategies

Levent Kockesen

Koc University

Levent Kockesen (Koc University) Mixed Strategies 1 / 20

page.2

Matching Pennies

Player 1

Player 2H T

H −1, 1 1,−1T 1,−1 −1, 1

How would you play?

No solution?

You should try to be unpredictable

Choose randomly


page.3

Audit

Tax authority wants to prevent tax evasion with minimal cost.

Decides whether to audit or not◮ Audit always catches a cheater◮ But costs 1

The tax payer decides whether to cheat or not◮ Cheating saves 20◮ If you get caught you pay a fine of 10◮ There is also non-monetary cost equivalent to 30

Tax Payer

TreasuryAudit No Audit

Cheat −20, 9 20, 0Not Cheat 0, 19 0, 20


page.4

Audit

What is the set of Nash equilibria?

Tax Payer


Cheat −20, 9 20, 0Not Cheat 0, 19 0, 20

Both parties want to be unpredictable


page.5

Mixed Strategy Equilibrium

Tax Payer


Cheat −20, 9 20, 0Not Cheat 0, 19 0, 20

A mixed strategy is a probability distribution over the set of actions.

Suppose the treasury audits half the cases◮ Probability of audit is 1/2

What should the tax payer do?◮ He is indifferent between cheating and not cheating◮ He is also indifferent over cheating with any probability

Suppose in general 10% of tax payers cheat◮ Probability of cheating is 1/10

What should the treasury do?◮ It is indifferent between audit or not◮ Also indifferent over auditing with any probability


page.6


Auditing with probability 1/2 and cheating with probability 1/10 is amixed strategy equilibrium

Both players are best responding to each other


page.7


In a mixed strategy equilibrium every action played with positiveprobability must be a best response to other players’ mixed strategies

In particular players must be indifferent between actions played withpositive probability

Tax payer’s probability of cheating p

Treasury’s probability of auditing q

Tax payer’s expected payoff to◮ Cheating is q × (−20) + (1− q)× 20 = 20− 40q◮ Not cheating is 0

Indifference condition0 = 20− 40q

or q = 1/2


page.8


The treasury’s expected payoff to◮ Audit is p× 9 + (1− p)× 19 = 19− 10p◮ Not is p× 0 + (1− p)× 20 = 20− 20p

Indifference condition

19 − 10p = 20− 20p

or p = 1/10

(p = 1/10, q = 1/2) is a mixed strategy equilibrium

Since there is no pure strategy equilibrium, this is also the unique Nashequilibrium


page.9

Audit

What determines cheating rate?

Let auditing cost be c

Fine if caught cheating f

Other costs of cheating r

Tax Payer


Cheat 20− f − r, f − c 20, 0Not Cheat 0, 20 − c 0, 20


page.10

Audit

Tax Payer

TreasuryAudit (q) No Audit (1-q)

Cheat (p) 20− f − r, f − c 20, 0Not Cheat (1-p) 0, 20 − c 0, 20

f + r > 20◮ Otherwise always cheats

c < f◮ Otherwise never audits

Equilibrium condition I

p(f − c) + (1− p)(20 − c) = (1− p)20 ⇒ p =c

f

Equilibrium condition II

q(20 − f − r) + (1− q)20 = 0 ⇒ q =20

f + r


page.11

Audit

Cheating rate is c/f◮ Increase the fine◮ Decrease the cost of audit◮ Other costs of getting caught has no effect!

Audit rate is 20/(f + r)◮ Increase the total penalty of cheating◮ Increase audit rate as the amount of cheating increases


page.12

Mixed and Pure Strategy Equilibria

How do you find the set of all (pure and mixed) Nash equilibria?

In 2× 2 games we can use the best response correspondences interms of the mixed strategies and plot them

Consider the Battle of the Sexes game

Player 1

Player 2B S

B 2, 1 0, 0S 0, 0 1, 2

Denote Player 1’s strategy as p and that of Player 2 as q (probabilityof choosing B)


page.13

B SB 2, 1 0, 0S 0, 0 1, 2

What is Player 1’s best response?

Expected payoff to◮ B is 2q◮ S is 1− q

If 2q > 1− q or q > 1/3◮ best response is B (or equivalently p = 1)

If 2q < 1− q or q < 1/3◮ best response is S (or equivalently p = 0)

If 2q = 1− q or q = 1/3◮ he is indifferent◮ best response is any p ∈ [0, 1]

Player 1’s best response correspondence:

B1(q) =

{1} , if q > 1/3

[0, 1], if q = 1/3

{0} , if q < 1/3


page.14

B SB 2, 1 0, 0S 0, 0 1, 2

What is Player 2’s best response?

Expected payoff to◮ B is p◮ S is 2(1− p)

If p > 2(1− p) or p > 2/3◮ best response is B (or equivalently q = 1)

If p < 2(1− p) or p < 2/3◮ best response is S (or equivalently q = 0)

If p = 2(1− p) or p = 2/3◮ she is indifferent◮ best response is any q ∈ [0, 1]

Player 2’s best response correspondence:

B2(p) =

{1} , if p > 2/3

[0, 1], if p = 2/3

{0} , if p < 2/3


page.15

B1(q) =

{1} , if q > 1/3

[0, 1], if q = 1/3

{0} , if q < 1/3

B2(p) =

{1} , if p > 2/3

[0, 1], if p = 2/3

{0} , if p < 2/3

q

p1

1

0 2/3

1/3

b

b

b

B1(q)

B2(p)

Set of Nash equilibria

{(0, 0), (1, 1), (2/3, 1/3)}


page.16

Dominated Actions and Mixed Strategies

Up to now we tested actions only against other actions

An action may be undominated by any other action, yet bedominated by a mixed strategy

Consider the following game

L RT 1, 1 1, 0M 3, 0 0, 3B 0, 1 4, 0

No action dominates T

But playing M and B with 1/2 probability each strictly dominates T

A strictly dominated action is never used with positive probability in amixed strategy equilibrium


page.17

Dominated Actions and Mixed Strategies

An easy way to figure out dominated actions is to compare expectedpayoffsLet player 2’s mixed strategy given by q = prob(L)

L RT 1, 1 1, 0M 3, 0 0, 3B 0, 1 4, 0

u1(T, q) = 1

u1(M, q) = 3q

u1(B, q) = 4(1 − q)

0

1

2

3

4

0 14/7

12/7

q

u1(., q)

u1(T, q)

u1(M, q)

u1(B, q)An action is a never bestresponse if there is no beliefthat makes that action a bestresponseT is a never best responseAn action is a NBR iff it isstrictly dominated


page.18

What if there are no strictly dominated actions?

L RT 2, 0 2, 1M 3, 3 0, 0B 0, 1 3, 0

Denote player 2’s mixed strategy by q = prob(L)u1(T, q) = 2, u1(M, q) = 3q, u1(B, q) = 3(1 − q)

q

u1(., q)

1/2

3/2

2/31/3 1

3

2

0

u1(T, q)

u1(M, q)u1(B, q)

Pure strategy Nash eq. (M,L)

Mixed strategy equilibria?◮ Only one player mixes? Not possible◮ Player 1 mixes over {T,M,B}? Not possible◮ Player 1 mixes over {M,B}? Not possible◮ Player 1 mixes over {T,B}? Let p = prob(T )

q = 1/3, 1− p = p → p = 1/2◮ Player 1 mixes over {T,M}? Let p = prob(T )

q = 2/3, 3(1− p) = p → p = 3/4


page.19

Real Life Examples?

Ignacio Palacios-Huerta (2003): 5 years’ worth of penalty kicks

Empirical scoring probabilities

L RL 58, 42 95, 5R 93, 7 70, 30

R is the natural side of the kicker

What are the equilibrium strategies?


page.20

Penalty Kick

L RL 58, 42 95, 5R 93, 7 70, 30

Kicker must be indifferent

58p + 95(1 − p) = 93p + 70(1 − p) ⇒ p = 0.42

Goal keeper must be indifferent

42q + 7(1− q) = 5q + 30(1 − q) ⇒ q = 0.39

Theory Data

Kicker 39% 40%Goallie 42% 42%

Also see Walker and Wooders (2001): Wimbledon


Introduction to Economic and Strategic BehaviorStrategic Form Games with Incomplete Information

Levent Kockesen

Koc University

Levent Kockesen (Koc University) Bayesian Games 1 / 1

Games with Incomplete Information

Some players have incomplete information about some components ofthe game

◮ Firm does not know rival’s cost◮ Bidder does not know valuations of other bidders in an auction

We could also say some players have private information


Games with Incomplete Information

Does incomplete information matter?

Suppose you make an offer to buy out a company

If the value of the company is V it is worth 1.5V to you

The seller accepts only if the offer is at least V

If you know V what do you offer?

You know only that V is uniformly distributed over [0, 100]. Whatshould you offer?

Enter your name and your bid

Click here for the EXCEL file


Bayesian Games

We will first look at incomplete information games where playersmove simultaneously

◮ Bayesian games

Later on we will study dynamic games of incomplete information


Bayesian Games

What is new in a Bayesian game?

Each player has a type: summarizes a player’s private information

Players’ payoffs depend on types

Incomplete information can be anything about the game◮ Payoff functions◮ Actions available to others◮ Beliefs of others; beliefs of others’ beliefs of others’...

Harsanyi (1967-1968) showed that introducing types in payoffs isadequate

Each player has beliefs about others’ types


Bayesian Equilibrium

Since payoffs depend on types different types of the same player mayplay different strategies

We assume that players maximize their expected payoffs.

Bayesian Equilibrium

A strategy profile is a Bayesian equilibrium if each type of each playermaximizes her expected payoff given other players’ strategies.


An Example: Hawk-Dove

Two animals are trying to share a prey

They can be hawkish (H) or dovish (D)

Value of the prey = 2

There is a fight if both of them choose to be hawkish

If they get into a fight they both suffer injuries◮ Cost to player 1= 2◮ Cost to player 2= c2

If only one is hawkish, he gets the entire prey

Otherwise they share

H DH −1, 1 − c2 2, 0D 0, 2 1, 1

c2 can be 0 (tough type) or 2 (weak type)◮ Pl. 2 knows◮ Pl. 1 believes it is 2 with probability q


Hawk-Dove

H DH −1,−1 2, 0D 0, 2 1, 1

Weak (q)

H DH −1, 1 2, 0D 0, 2 1, 1

Tough (1− q)

Player 1 chooses H or D

Player 2’s strategy may depend on type◮ If both do the same: pooling◮ If different types play differently: separating

Four possible types of pure strategy Bayesian equilibria◮ Separating: (W : H,T : D) or (W : D,T : H)◮ Pooling: (W : H,T : H) or (W : D,T : D)

H is strictly dominant for tough type → only two possibilities


Hawk-Dove: Separating EquilibriumH D

H −1,−1 2, 0D 0, 2 1, 1

Weak (q)

H DH −1, 1 2, 0D 0, 2 1, 1

Tough (1− q)

(W : D,T : H)

For Weak to play D, Pl. 1 must be playing HExpected payoff to H

q × 2 + (1− q)× (−1) = 3q − 1

Expected payoff to D

q × 1 + (1− q)× 0 = q

H is optimal if

3q − 1 ≥ q or q ≥ 1

2

1 plays H and 2 plays (W : D,T : H) is a Bayesian equilibrium iff q ≥ 1/2.


Hawk-Dove: Pooling Equilibrium

H DH −1,−1 2, 0D 0, 2 1, 1

Weak (q)

H DH −1, 1 2, 0D 0, 2 1, 1

Tough (1− q)

(W : H,T : H)

For Weak to play H, Pl. 1 must be playing D

Expected payoff to H is −1

Expected payoff to D is 0

So D is optimal

1 plays D and 2 plays (W : H,T : H) is a Bayesian equilibrium.


Auctions

Many economic transactions are conducted through auctions

treasury bills

foreign exchange

publicly owned companies

mineral rights

airwave spectrum rights

art work

antiques

cars

houses

government contracts

Also can be thought of as auctions

takeover battles

queues

wars of attrition

lobbying contests


Auction Formats

1. Open bid auctions1.1 ascending-bid auction

⋆ aka English auction⋆ price is raised until only one bidder remains, who wins and pays the

final price

1.2 descending-bid auction⋆ aka Dutch auction⋆ price is lowered until someone accepts, who wins the object at the

current price

2. Sealed bid auctions2.1 first price auction

⋆ highest bidder wins; pays her bid

2.2 second price auction⋆ aka Vickrey auction⋆ highest bidder wins; pays the second highest bid


Auction Formats

Auctions also differ with respect to the valuation of the bidders

1. Private value auctions◮ each bidder knows only her own value◮ artwork, antiques, memorabilia

2. Common value auctions◮ actual value of the object is the same for everyone◮ bidders have different private information about that value◮ oil field auctions, company takeovers


Independent Private Values

Each bidder knows only her own valuation

Valuations are independent across bidders

Bidders have beliefs over other bidders’ values

Risk neutral bidders◮ If the winner’s value is v and pays p, her payoff is v − p



Only one 4G license will be sold

There are 10 groups

I generated 10 random values between 0 and 100

I will now distribute the values: Keep these and don’t show it toanyone until the end of the experiment

I will now distribute paper slips where you should enter your name,value, and bid

Highest bidder wins, pays the second highest bid

I will pay the winner her net payoff: value - priceClick here for the EXCEL file



I. Bidding your value weakly dominates bidding higher


1. There is a bid higher than $15 (e.g. $20)◮ You loose either way: no difference

2. 2nd highest bid is lower than $10 (e.g. $5)◮ You win either way and pay $5: no difference

3. 2nd highest bid is between $10 and $15 (e.g. $12)◮ You loose with $10: zero payoff◮ You win with $15: loose $2

5

10 value

12

15 bid

20



II. Bidding your value weakly dominates bidding lower


1. There is a bid higher than $10 (e.g. $12)◮ You loose either way: no difference

2. 2nd highest bid is lower than $5 (e.g. $2)◮ You win either way and pay $2: no difference

3. 2nd highest bid is between $5 and $10 (e.g. $8)◮ You loose with $5: zero payoff◮ You win with $10: earn $2

2

10 value

8

5 bid

12


English Auction

Suppose you value the item at 100 TL

What is your optimal strategy?

Stay in bidding until the price exceeds 100 TL

This is a dominant strategy

If everyone plays this strategy what happens?◮ The bidder with highest value wins◮ Pays something close to second highest value



Only one 4G license will be sold

There are 10 groups

I generated 10 random values between 0 and 100



Highest bidder wins, pays her bid




Would you bid your value?

What happens if you bid less than your value?◮ You get a positive payoff if you win◮ But your chances of winning are smaller◮ Optimal bid reflects this tradeoff

Bidding less than your value is known as bid shading

Choose your bid b to maximize

π = (v − b) prob(win)

Probability of winning depends on your bid and others’ bids

Given what you believe about the others, it is increasing in your bid


Example

Suppose your value is 6 and the highest possible value is 10

b v − b prob π

10 -4 1 -49 -3 0.9 -2.78 -2 0.8 -1.67 -1 0.7 -0.76 0 0.6 05 1 0.5 0.54 2 0.4 0.83 3 0.3 0.92 4 0.2 0.81 5 0.1 0.5


Example

0

1

−1

−2

−3

−4

−5

1 2 3 4 5 6 7 8 9 10

bb b b

b

b

b

b

b

b


Equivalent Formats

English auction has the same equilibrium as Second Price auction

This is true only if values are private

Stronger equivalence between Dutch and First Price auctions

Open Bid Sealed Bid

Dutch Auction First Price

EnglishAuction

Second Price

Strategically Equivalent

Same Equilibrium

in Private Values


Which One Brings More Revenue?

Second Price◮ Bidders bid their value◮ Revenue = second highest bid

First Price◮ Bidders bid less than their value◮ Revenue = highest bid

Which one is better?

Turns out it doesn’t matter


Revenue Equivalence Theorem

Any auction with independent private values with a common distributionin which

1. the number of the bidders are the same and the bidders arerisk-neutral,

2. the object always goes to the buyer with the highest value,

3. the bidder with the lowest value expects zero surplus,

yields the same expected revenue.


Common Value Auctions

Value of 4G license = v◮ Uniform between 0 and 100

You observe v + x◮ x ∼ N(0, 10)◮ x is between −20 and 20 with 95% prob.

I generated a signal for each of 10 groups



Highest bidder wins, pays the second highest bid



Common Value Auctions and Winner’s Curse

If everybody bids her estimate winning is “bad news”

This is known as Winner’s Curse

Optimal strategies are complicated

Bidders bid much less than their value to prevent winner’s curse

To prevent winner’s curse

Base your bid on expected value conditional on winning

Auction formats are not equivalent in common value auctions

Open bid auctions provide information and ameliorates winner’s curse◮ Bids are more aggressive

Sealed bid auctions do not provide information◮ Bids are more conservative


Auction Design: Failures

New Zeland Spectrum Auction (1990)◮ Used second price auction with no reserve price◮ Estimated revenue NZ$ 240 million◮ Actual revenue NZ$36 million

Some extreme cases

Winning Bid Second Highest BidNZ$100,000 NZ$6,000

NZ$7,000,000 NZ$5,000NZ$1 None

Source: John McMillan, “Selling Spectrum Rights,” Journal of Economic Perspectives, Summer 1994

Problems◮ Second price format politically problematic

⋆ Public sees outcome as selling for less than its worth

◮ No reserve price


Auction Design: FailuresAustralian TV Licence Auction (1993)

◮ Two satellite-TV licences◮ Used first price auction◮ Huge embarrasment

High bidders had no intention of paying

They bid high just to guarantee winning

They also bid lower amounts at A$5 million intervalsThey defaulted

◮ licences had to be re-awarded at the next highest bid◮ those bids were also theirs

Outcome after a series of defaults

Initial Bid Final PriceA$212 mil. A$117 mil.A$177 mil. A$77 mil.

Source: John McMillan, “Selling Spectrum Rights,” Journal of Economic Perspectives, Summer 1994

Problem: No penalty for default


Auction Design: Failures

Turkish GSM licence auction

April 2000: Two GSM 1800 licences to be auctioned

Auction method:

1. Round 1: First price sealed bid auction2. Round 2: First price sealed bid auction with reserve price

⋆ Reserve price is the winning bid of Round 1

Bids in the first roundBidder Bid AmountIs-Tim $2.525 bil.

Dogan+ $1.350 bil.Genpa+ $1.224 bil.Koc+ $1.207 bil.Fiba+ $1.017 bil.

Bids in the second round: NONE!

Problem: Facilitates entry deterrence


Auction Design

Good design depends on objective◮ Revenue◮ Efficiency◮ Other

One common objective is to maximize expected revenue

In the case of private independent values with the same number ofrisk neutral bidders format does not matter

Auction design is a challenge when◮ values are correlated◮ bidders are risk averse

Other design problems◮ collusion◮ entry deterrence◮ reserve price


Auction Design

Correlated values: Ascending bid auction is better

Risk averse bidders◮ Second price auction: risk aversion does not matter◮ First price auction: higher bids

Collusion: Sealed bid auctions are better to prevent collusion

Entry deterrence: Sealed bid auctions are better to promote entry

A hybrid format, such as Anglo-Dutch Auction, could be better.

Anglo-Dutch auction has two stages:

1. Ascending bid auction until only two bidders remain

2. Two remaining bidders make offers in a first price sealed bid auction


page.1

Introduction to Economic and Strategic BehaviorExtensive Form Games

Levent Kockesen

Koc University

Levent Kockesen (Koc University) Extensive Form Games 1 / 23

page.2

Extensive Form Games

Strategic form games are used to model situations in which playerschoose strategies without knowing the strategy choices of the otherplayers

In some situations players observe other players’ moves before theymove

Removing Coins:◮ There are 21 coins◮ Two players move sequentially and remove 1, 2, or 3 coins◮ Winner is who removes the last coin(s)◮ We will determine the first mover by a coin toss◮ Volunteers?


page.3

Extensive Form Games

Strategic form has three ingredients:◮ set of players◮ sets of actions◮ payoff functions

Extensive form games provide more information◮ order of moves◮ actions available at different points in the game◮ information available throughout the game

Easiest way to represent an extensive form game is to use a game tree


page.4

Entry Game

Kodak is contemplating entering the instant photography market andPolaroid can either fight the entry or accommodate

K

P

Out In

F A

0, 20

−5, 0 10, 10


page.5

Game Trees

What’s in a game tree?

nodes◮ decision nodes◮ initial node◮ terminal nodes

branches

player labels

action labels

payoffs

information sets◮ to be seen later

K

P

Out In

F A

0, 20

−5, 0 10, 10


page.6

Extensive Form Game Strategies

A pure strategy of a player specifies an action choice at each decision nodeof that player

K

P

Out In

F A

0, 20

−5, 0 10, 10

Kodak’s strategies◮ SK = {Out, In}

Polaroid’s strategies◮ SP = {F,A}


page.7

Backward Induction Equilibrium

What should Polaroid do ifKodak enters?

Given what it knows aboutPolaroid’s response to entry,what should Kodak do?

This is an example of abackward induction equilibrium

K

P

Out In

F A

0, 20

−5, 0 10, 10

At a backward induction equilibrium each player plays optimally atevery decision node in the game tree

◮ plays a sequentially rational strategy

(In,A) is the unique backward induction equilibrium of the entrygame


page.8

Grab the Money

Two piles of money on the table◮ One has 20 Kr. the other 0

Player 1 begins◮ Grab one pile (the other goes to Player 2)◮ Pass

Grab → game ends

Pass → I add 10 Kr. to BOTH piles

Player 2 moves: Grab or Pass

This continues 4 rounds

Two volunteers?


page.9

Grab the Money

1 2 1 2 1 2 1 210, 8

2, 0 1, 3 4, 2 3, 5 6, 4 5, 7 8, 6 7, 9

G

P

G

P

G

P

G

P

G

P

G

P

G

P

G

P

How many strategies does player 1 have?

What is the backward induction equilibrium outcome?◮ G

What is the backward induction equilibrium?◮ (GGGG,GGGG)


page.10

Backward Induction

Backward Induction Algorithm

1. Go to the last decision node(s)◮ Find the optimal strategies there

2. Go to penultimate decision node(s)◮ Find the optimal strategies there

3. Repeat until you reach the first decision node


page.11

Pirates

5 pirates are trying to split 100 gold coins

Oldest proposes an allocation

All vote for or against

If at least half vote for it, it is shared that way

Otherwise the proposing pirate is thrown overboard

Repeated with the pirates that remain

They like to live and like gold

If you are the oldest, what do you propose?


page.12

Pirates

1 2 3 4 5

100 099 0 1

99 0 1 098 0 1 0 1


page.13

Strategic Form of an Extensive Form Game

If you want to apply a strategic form solution concept◮ Nash equilibrium◮ Dominant strategy equilibrium◮ IEDS

Analyze the strategic form of the game

Strategic form of an extensive form game

1. Set of playersand for each player

2. Strategies

3. Payoffs


page.14

Strategic Form of an Extensive Form Game

K

P

Out In

F A

0, 20

−5, 0 10, 10

1. Players: K and P

2. Strategies:SK = {Out, In}, SP = {F,A}

3. Payoffs in the bimatrix

K

PF A

Out 0, 20 0, 20In −5, 0 10, 10


page.15

Nash Equilibria of an Extensive Form Game

K

PF A

Out 0, 20 0, 20In −5, 0 10, 10

K

P

Out In

F A

0, 20

−5, 0 10, 10

Set of Nash equilibria = {(In,A), (Out, F )}(Out, F ) is sustained by an incredible threat by Polaroid

Backward induction equilibrium eliminates equilibria based uponincredible threats

Nash equilibrium requires rationality

Backward induction requires sequential rationality◮ Players must play optimally at every point in the game


page.16

Extensive Form Games with Imperfect Information

We have seen extensive form games with perfect information◮ Every player observes the previous moves made by all the players

What happens if some of the previous moves are not observed?

Consider the following game between Kodak and Polaroid

Kodak doesn’t know whetherPolaroid will fight oraccommodate

The dotted line is aninformation set:

◮ a collection of decision nodesthat cannot be distinguishedby the player

We cannot determine theoptimal action for Kodak atthat information set

K

P

K K

Out In

F A

F A F A

0, 20

−5,−5 5, 15 15, 5 10, 10


page.17

Subgame Perfect EquilibriumWe will introduce another solution concept: Subgame Perfect Equilibrium

Definition

A subgame is a part of the game tree such that

1. it starts at a single decision node,

2. it contains every successor to this node,

3. if it contains a node in an information set, then it contains all thenodes in that information set.

This is a subgame

P

K K

F A

F A F A

−5,−5 5, 15 15, 5 10, 10

This is not a subgame

P

K

A

F A

15, 5 10, 10


page.18

Subgame Perfect Equilibrium

Extensive form game strategies

A pure strategy of a player specifies an action choice at each informationset of that player

Definition

A strategy profile in an extensive form game is a subgame perfectequilibrium (SPE) if it induces a Nash equilibrium in every subgame of thegame.

To find SPE

1. Find the Nash equilibria of the “smallest” subgame(s)

2. Fix one for each subgame and attach payoffs to its initial node

3. Repeat with the reduced game


page.19


Consider the following game

K

P

K K

Out In

F A

F A F A

0, 20

−5,−5 5, 8 8, 5 10, 10


page.20


The “smallest” subgame

P

K K

F A

F A F A

−5,−5 5, 8 8, 5 10, 10

Its strategic form

K

PF A

F −5,−5 8, 5A 5, 8 10, 10

Nash equilibrium of the subgame is (A,A)

Reduced subgame isK

Out In

0, 20 10, 10

Its unique Nash equilibrium is (In)

Therefore the unique SPE of the game is ((In,A),A)


page.21

Role of Chance

In some games chance also plays a role◮ Games like poker, bridge, etc.◮ But also in many economic or social situations

We introduce chance as another “player”

Chooses her action randomly◮ Using exogenously given probabilities


page.22

Role of Chance: An Example

Elites choose income distribution: Equitable (e) or Unequitable (u)

If unequitable, People decide whether to revolt (r) or not (n)

Exogenous factors ⇒ revolution is successful (s) or a failure (f)◮ Probability of success is p

Revolution is costly◮ Destroys half the income available

See Acemoglu and Robinson (2006) book


page.23

Role of Chance: An Example

Backward Induction

Last real decision node is P

Expected payoff to r is 5p◮ r if p > 0.2◮ n otherwise

Move up to E◮ e if p > 0.2◮ u otherwise

E

5, 5

P

C

9, 1

0, 5 5, 0

u e

r n

s f

Backward Induction Equilibrium

1. If p > 0.2 (e, r)

2. If p ≤ 0.2 (u, n)


page.1

Introduction to Economic and Strategic BehaviorExtensive Form Games: Applications

Levent Kockesen

Koc University

Levent Kockesen (Koc University) Extensive Form Games: Applications 1 / 47

page.2

Entry Game

Consider the following entry game

K

P

0, 100

−50, 20 50, 50

n e

f a


page.3

Commitment

K

P

0, 100

−50, 20 50, 50

n e

f a

Backward induction equilibrium (e, a)◮ Polaroid’s payoff is 50

Suppose Polaroid commits to fight (f)if entry occurs.

What would Kodak do?

Kodak would not enter and Polaroid would be better off

Is this commitment credible?


page.4

A U.S. air force base commander ordersthirty four B-52’s to launch a nuclearattack on Soviet Union

He shuts off all communications withthe planes and with the base

U.S. president invites the Russianambassador to the war room andexplains the situation

They decide to call the Russianpresident Dimitri


page.5

Dr. Strangelove

What is the outcome if the U.S. doesn’tknow the existence of the doomsdaydevice?

What is the outcome if it does?

Commitment must be observable

What if USSR can un-trigger thedevice?

Commitment must be irreversible

US

USSR

0, 0

−4,−4 1,−2

Attack Don’t

Retaliate Don’t


page.6

Commitment

Thomas Schelling

The power to constrain an adversary depends upon the power to bindoneself.


page.7

Credible Commitments: Burning Bridges

In non-strategic environments having more options is never worse

Not so in strategic environments

You can change your opponent’s actions by removing some of youroptions

1066: William the Conqueror ordered his soldiers to burn their shipsafter landing to prevent his men from retreating

1519: Hernan Cortes sank his ships after landing in Mexico for thesame reason

Sun-tzu in The Art of War, 400 BC

At the critical moment, the leader of an army acts like one who hasclimbed up a height, and then kicks away the ladder behind him.


page.8

Commitment

How can you achieve credible commitment in the entry game?

Change the game◮ Change payoffs◮ Remove accommodate as an option

Change others’ beliefs about your payoffs or options◮ Build reputation for toughness◮ Delegate to a tough CEO


page.9

Change Payoffs

Suppose you can invest in a new production facility

Profitable only if you sell enough

P

K

P

0, 100

−50, 20 50, 50

n e

f a

K

P

0, 90

−50, 10 50, 5

n e

f a

N I

What is the backward induction equilibrium?


page.10

Delegation

Delegate the decision to a CEO who is known to be aggressive

Or tie his compensation to market share◮ Pay 10 if market share is above 50%◮ Otherwise pay 5

K

CEO

0, 90

−50, 10 50, 5

n e

f a

What is the BIE?

Contract must be observable, non-renegotiable


page.11

Reputation

You might be playing this game repeatedly in different markets

You may want to build reputation for being tough

Proper analysis requires a model with incomplete information

But intuition is clear

Suppose other firms believe you are tough with probability p

And if you are tough you always fight entry

Payoff to entry is

−50p+ 50(1 − p) = 50 − 100p

If they believe you are more likely to be tough they will not enter

You might fight initially to increase the belief that you are tough evenif you are not

Acting “irrational” could be rational


page.12

A Simple Game

You have 10 TL to share

A makes an offer◮ x for me and 10− x for you

If B accepts◮ A’s offer is implemented

If B rejects◮ Both get zero

Half the class will play A (proposer) and half B (responder)◮ Proposers should write how much they offer to give responders◮ I will distribute them randomly to responders

⋆ They should write Yes or No

Click here for the EXCEL file


page.13

Ultimatum Bargaining

Two players, A and B, bargainover a cake of size 1

Player A makes an offerx ∈ [0, 1] to player B

If player B accepts the offer(Y ), agreement is reached

◮ A receives x◮ B receives 1− x

If player B rejects the offer (N)both receive zero

A

B

x, 1− x 0, 0

x

Y N


page.14

Subgame Perfect Equilibrium of Ultimatum BargainingWe can use backward induction

B’s optimal action◮ x < 1 → accept◮ x = 1 → accept or reject

1. Suppose in equilibrium B accepts any offer x ∈ [0, 1]◮ What is the optimal offer by A? x = 1◮ The following is a SPE

x∗ = 1

s∗B(x) = Y for all x ∈ [0, 1]

2. Now suppose that B accepts if and only if x < 1◮ What is A’s optimal offer?

⋆ x = 1?⋆ x < 1?

Unique SPE

x∗ = 1, s∗B(x) = Y for all x ∈ [0, 1]


page.15

Bargaining

Bargaining outcomes depend on many factors◮ Social, historical, political, psychological, etc.

Early economists thought the outcome to be indeterminate

John Nash introduced a brilliant alternative approach◮ Axiomatic approach: A solution to a bargaining problem must satisfy

certain “reasonable” conditions⋆ These are the axioms

◮ How would such a solution look like?◮ This approach is also known as cooperative game theory

Later non-cooperative game theory helped us identify critical strategicconsiderations


page.16

BargainingTwo individuals, A and B, are trying to share a cake of size 1If A gets x and B gets y,utilities are uA(x) and uB(y)If they do not agree, A gets utility dA and B gets dBWhat is the most likely outcome?

uA

uB

1

1

dA

dB


page.17

BargainingLet’s simplify the problem

uA(x) = x, and uB(x) = xdA = dB = 0A and B are the same in every other respectWhat is the most likely outcome?

uA

uB

1

1

(dA, dB)

45◦

0.5

0.5


page.18

BargainingHow about now? dA = 0.3, dB = 0.4

uA

uB

1

1

0.3

0.4

45◦

0.45

0.55

Let x be A’s share. Then

Slope = 1 =1− x− 0.4

x− 0.3

or x = 0.45

So A gets 0.45 and B gets 0.55


page.19

Bargaining

In general A gets

dA +1

2(1− dA − dB)

B gets

dB +1

2(1− dA − dB)

But why is this reasonable?

Two answers:

1. Axiomatic: Nash Bargaining Solution2. Non-cooperative: Alternating offers bargaining game


page.20

Bargaining: Axiomatic Approach

John Nash (1950): The Bargaining Problem, Econometrica1. Efficiency

⋆ No waste

2. Symmetry⋆ If bargaining problem is symmetric, shares must be equal

3. Scale Invariance⋆ Outcome is invariant to linear changes in the payoff scale

4. Independence of Irrelevant Alternatives⋆ If you remove alternatives that would not have been chosen, the

solution does not change


page.21

Nash Bargaining Solution

What if parties have different bargaining powers?

Remove symmetry axiom

Then A gets

xA = dA + α(1− dA − dB)

B gets

xB = dB + β(1− dA − dB)

α, β > 0 and α+ β = 1 represent bargaining powers

If dA = dB = 0

xA = α and xB = β


page.22

Alternating Offers Bargaining

Two players, A and B, bargain over a cake of size 1

At time 0, A makes an offer xA ∈ [0, 1] to B◮ If B accepts, A receives xA and B receives 1− xA

◮ If B rejects, then

at time 1, B makes a counteroffer xB ∈ [0, 1]◮ If A accepts, B receives xB and A receives 1− xB

◮ If A rejects, he makes another offer at time 2

This process continues indefinitely until a player accepts an offer

If agreement is reached at time t on a partition that gives player i ashare xi

◮ player i’s payoff is δtixi

◮ δi ∈ (0, 1) is player i’s discount factor

If players never reach an agreement, then each player’s payoff is zero


page.23 A

B

xA, 1− xA

B

xA

Y N

A

δA(1− xB), δBxB

A

xB

Y N

B

δ2AxA, δ2B(1− xA)

A

xA

Y N


page.24


Stationary No-delay Equilibrium

1. No Delay: All equilibrium offers are accepted

2. Stationarity: Equilibrium offers do not depend on time

Let equilibrium offers be (x∗A, x∗B)

What does B expect to get if she rejects x∗A?◮ δBx

∗B

Therefore, we must have

1− x∗A = δBx∗B

Similarly1− x∗B = δAx

∗A

A

B

xA, 1− xA

B

xA

Y N

A

δA(1 − xB), δBxB

A

xB

Y N

B

δ2AxA, δ2B(1− xA)

A

xA

Y N


page.25


There is a unique solution

x∗A =1− δB

1− δAδB

x∗B =1− δA

1− δAδB

There is at most one stationary no-delay SPE

Still have to verify there exists such an equilibrium

The following strategy profile is a SPE

Player A: Always offer x∗A, accept any xB with 1− xB ≥ δAx∗A

Player B: Always offer x∗B, accept any xA with 1− xA ≥ δBx∗B


page.26

Properties of the Equilibrium

Bargaining Power

Player A’s share

πA = x∗A =1− δB

1− δAδB

Player B’s share

πB = 1− x∗A =δB(1− δA)

1− δAδB

Share of player i is increasing in δi and decreasing in δj

Bargaining power comes from patience

Example

δA = 0.9, δB = 0.95 ⇒ πA = 0.35, πB = 0.65


page.27

Properties of the Equilibrium

First mover advantage

If players are equally patient: δA = δB = δ

πA =1

1 + δ>

δ

1 + δ= πB

First mover advantage disappears as δ → 1

limδ→1

πi = limδ→1

πB =1

2


page.28

Voting

We looked at voting in the context of candidate competition inelections

We assumed voters vote sincerely

If the voting body is small there may be strategic issues


page.29

Voting Procedures

How should the preferences of voters be aggregated?

Two alternatives (candidates, options): Easy◮ Majority Rule

More than two alternative: Problematic◮ A popular procedure: pairwise voting◮ Condorcet method: each alternative against each of the others◮ The alternative that defeats every other in pairwise voting wins

⋆ Condorcet winner

◮ Many other methods:⋆ Borda count⋆ Majority runoff, etc.


page.30

Example

Three voters: 1, 2, 3

Three alternatives: a, b, c

Preferences1 2 3

a c bb a cc b a

Between a and b: a wins

Between a and c: c wins

Between b and c: b wins

a ≻ b ≻ c ≻ a

Aggregate preferences are intransitive

No Condorcet winner (Condorcet Paradox)


page.31

Pairwise Voting

Alternatives are paired according to some order: Agenda

Winner of pairwise voting proceeds to next round

Winner of the entire election is the winner of the last round

Consider again

1 2 3

a c bb a cc b a

Agenda:1. a vs. b2. winner vs. c

Who is the winner?1. a vs. b: a wins2. a vs. c: c wins3. c is the winner

Is this an equilibrium?


page.32

Strategic Voting

What is the subgame perfect equilibrium?

Two possible outcome of the first round: a and b

Suppose the winner of the first round is a

Second round game:

1

2a c

a 2, 1, 0 2, 1, 0c 2, 1, 0 0, 2, 1

a

2a c

a 2, 1, 0 0, 2, 1c 0, 2, 1 0, 2, 1

c

Three Nash equilibria: (a, a, a), (a, c, c, ), (c, c, c)

Weakly dominant strategy equilibrium: (a, c, c) → c wins


page.33

Strategic Voting

1 2 3

a c bb a cc b a

Suppose the winner of the first round is bWeakly dominant strategy equilibrium: (b, c, b) → b wins

a wins → c

b wins → b

Weakly dominant strategies in first round:◮ Pl. 1: b◮ Pl. 2: a◮ Pl. 3: b

b wins in both roundsDifferent from sincere votingSuppose pl. 2 is the agenda setterWhat is the best she can do?


page.34

Agenda Setting

1 2 3

a c bb a cc b a

Round 1 Winner Round 2 Winner

(a, b) b (b, c) b(a, c) a (a, b) a(b, c) c (a, c) c

Can make any alternative a winner!


page.35

Political InstitutionsAcemoglu & Robinson (2006): Economic Origins of Dictatorship andDemocracy

Democracy typically emerged as a result of voluntary extension offranchise by elites

After democratization: redistribution, pro-poor policies

Why do elites democratize?

In response to social unrest

The British 1832 Reform Act extended franchise◮ In response of revolutionary threat◮ Riots in early 1800’s

Still, why democratize instead of introducing reforms?

Acemoglu & Robinson’s answer:◮ Political institutions are more durable and therefore provide credible

commitment to future policies◮ Democracy as a commitment to pro-poor policies


page.36

Political InstitutionsA simplified version of Acemoglu & Robinson model:

δ fraction is rich with income yr

1− δ is poor with income yp < yr

Mean income

y = δyr + (1− δ)yp

Divide by y

1 =δyr

y+

(1− δ)yp

y

Measure of income inequality

θ =δyr

y

yp < yr ⇒ θ > δ


page.37

Political Institutions

Main political issue is taxation and redistribution

Tax rate τ

Government budget constraint

T = (τ − C(τ))y

T : transfer, C(τ) distortion cost of taxation

Assume C(τ) = 0

Net incomes:

ur(τ) = (1− τ)yr + τy = yr + τ(y − yr)

up(τ) = (1− τ)yp + τy = yp + τ(y − yp)

Ideal tax rates

τ r = 0, τp = 1


page.38

Non-Democratic Politics

Elites (rich) govern

Citizens (poor) get a chance to revolt

If it succeeds a fraction µ is destroyed, the rest divided among poor◮ Poor gets

(1− µ)y

(1− δ)

◮ Rich gets 0

Two possibilities:

1. Elites can commit to tax rate2. Elites cannot commit


page.39

Elites can commit E

C

(1−µ)y(1−δ)

, 0 yp + τ(y − yp), yr + τ(y − yr)

τ

R N

Assume no revolution if indifferent

R ⇔ (1− µ)y

(1− δ)> yp + τ(y − yp)

A little algebra

R ⇔ µ < θ − τ(θ − δ)Levent Kockesen (Koc University) Extensive Form Games: Applications 39 / 47

page.40

Elites can commit

R ⇔ µ < θ − τ(θ − δ)

µ < δ ⇒ even when τ = 1 revolt

µ ≥ θ ⇒ even when τ = 0 do not revolt

δ ≤ µ < θ ⇒ there is a tax rate τ∗ such that

µ = θ − τ∗(θ − δ)

Elites choose tax rate

τ∗ =θ − µ

θ − δ

and revolution is prevented

Payoffs

up =(1− µ)y

(1− δ), ur =

µy

δ


page.41

Elites can commit

θ1

τ∗

1

Inequality increases tax rate

Higher the inequality, higher the need for redistribution to dissuadepeople from revolting


page.42

Elites cannot commitC

(1−µ)y(1−δ)

, 0

E

yp + τ(y − yp), yr + τ(y − yr)

R N

τ

If no revolution elites set best tax rate τ = 0

R ⇔ (1− µ)y

(1− δ)> yp

Revolution iff µ < θ


page.43

Non-Democratic Politics

Assume that the cost of revolution is not too high or low

When elites are in power they have a commitment problem:◮ They would like to commit to some redistribution to poor◮ But their promise is not credible◮ Once the threat of revolution passes, they will renege

Acemoglu & Robinson Hypothesis: Democracy is a commitmentdevice

◮ It transfers power from elites to citizens and solves the commitmentproblem.


page.44

Democratization

Initially Elites hold power

They may democratize or not◮ Democratization: Citizens hold power

⋆ Choose tax rate⋆ Afterwards choose whether to revolt

◮ No Democratization: Elites hold power⋆ Promise tax rate⋆ Citizens choose whether to revolt⋆ If no revolution⋆ With probability q Elites keep promise, otherwise choose tax rate again


page.45

DemocratizationE

C

C

D

τ

(1−µ)y(1−δ)

, 0

R

y, y

N

N

E

C

τ

(1−µ)y(1−δ)

, 0

R

Chance

N

yp + τ(y − yp), yr + τ(y − yr)

q

yp, yr

1− q


page.46

Democratization

Let us assume Elites’ promises are not credible: q = 0

In the previous model elites had commitment problem whenδ ≤ µ < θ

Assume this is the case

µ < θ implies

(1− µ)y

(1− δ)> yp

So, if Elites do not democratize, Citizens revolt

µ ≥ δ implies

(1− µ)y

(1− δ)≤ y

Under democracy there is no revolution

It is best for Elites to democratize


page.47

Democratization

Democracy

If δ ≤ µ < θ, Elites democratize and there is no revolution.

If µ ≥ θ, Elites do not democratize and there is no revolution.

If µ < δ, there is always revolution.

Exercise

What happens if q > 0?


page.1

Introduction to Economic and Strategic BehaviorRepeated Games

Levent Kockesen

Koc University

Levent Kockesen (Koc University) Repeated Games 1 / 29

page.2

A Simple Repeated Game

Everybody is matched with somebody else

Choose C or N

1. N → You gain 1 TL2. C → Other gains 2 TL

Will repeat as long as Excel draws a number ≤ 0.6Click here for the EXCEL file


page.3


Cooperate DefectCooperate 2, 2 −1, 3

Defect 3,−1 0, 0

It is a parable for many social interactions◮ Contributing to a joint project◮ Common pool resources◮ Arms’ race

The prediction is quite bleak

But humanity seems to have solved at least some of these problems◮ Many social and economic interactions depend on trust and

cooperation

We could not have survived otherwise


page.4

Institutions

Institutions may establish cooperative outcomes◮ Formal institutions:

⋆ State⋆ Law⋆ Police, etc.

◮ Informal Institutions⋆ Social norms⋆ Religion⋆ Ethics, etc.

How do these institutions arise?

What kind of institutions survive?


page.5

Institutions

Institutions

Any institution must be a Nash equilibrium of the game of life.

Otherwise some actor is not acting optimally

In the long-run such non-Nash behavior will be eliminated◮ We will see later that such behavior is not evolutionary stable


page.6

Cooperation

Cooperation in PD type situations is a commonly observedphenomenon

But it is not a Nash equilibrium

How did it survive as an institution?

In early societies PD type games were used to be played in smallgroups in repeated manner

Could cooperation be sustained in such environments?


page.7

Repeated Interactions

In repeated games, reward-punishment mechanisms can be used

Tit-for-Tat◮ Start with cooperating◮ Then do what the other has done last period

Trigger Strategy◮ Start with cooperating◮ Cooperate as long as the other has cooperated◮ After a defection, play defect forever◮ Also known as Grim-Trigger Strategy

Both are forms of reciprocal altruism◮ It is one of the universal norms◮ Known as the Golden Rule

Observed in animal world too◮ Vampire bats, stickleback fish


page.8

Golden Rule


page.9

Reciprocal Altruism


Defect 3,−1 0, 0

Is Tit-for-Tat or Trigger strategy a subgame perfect equilibrium?

Suppose you play it once

Twice?

Need infinite repetition◮ But is this realistic?◮ Perhaps more realistic than finite repetition◮ Every period there is a positive probability that there will be another

period of interaction


page.10

Repeated Games

Players play a stage game repeatedly over time

If there is a final period: finitely repeated game

If there is no definite end period: infinitely repeated game◮ We could think of firms having infinite lives◮ Or players do not know when the game will end but assign some

probability to the event that this period could be the last one


page.11

Repeated Games

Today’s payoff of $1 is more valuable than tomorrow’s $1◮ This is known as discounting◮ Think of it as probability with which the game will be played next

period◮ ... or as the factor to calculate the present value of next period’s payoff◮ or some combination

Denote the discount factor by δ ∈ (0, 1)

In PV interpretation: if interest rate is r

δ =1

1 + r


page.12

Payoffs

If starting today a player receives an infinite sequence of payoffs

u1, u2, u3, . . .

The payoff consequence is

(1− δ)(u1 + δu2 + δ2u3 + δ3u4 · · · )

Example: Period payoffs are all equal to 2

(1− δ)(2 + δ2 + δ22 + δ32 + · · · ) = 2(1 − δ)(1 + δ + δ2 + δ3 + · · · )= 2(1 − δ)

1

1− δ= 2


page.13

Equilibria of Infinitely Repeated Games

There is no end period of the game

Cannot apply backward induction type algorithm

We use One-Shot Deviation Property to check whether a strategyprofile is a subgame perfect equilibrium

One-Shot Deviation Property

A strategy profile is a SPE of a repeated game if and only if no player cangain by changing her action after any history, keeping both the strategiesof the other players and the remainder of her own strategy constant

Take an history

For each player check if she has a profitable one-shot deviation (OSD)

Do that for each possible history

If no player has a profitable OSD after any history you have an SPE

If there is at least one history after which at least one player has aprofitable OSD, the strategy profile is NOT an SPE


page.14

Reciprocal Altruism


Defect 3,−1 0, 0

Consider the following trigger strategy◮ Start with Cooperate◮ Cooperate as long as everyone has always cooperated◮ Defect otherwise

Is it a subgame perfect equilibrium?

Suppose you think the other player will play according to that rule

Do you have an incentive to deviate from it?

There are two types of histories:1. Cooperative Phase

⋆ Nobody has ever defected ⇒ Cooperate

2. Punishment Phase⋆ Somebody has defected at some point ⇒ Defect


page.15

Reciprocal AltruismCooperate Defect

Cooperate 2, 2 −1, 3Defect 3,−1 0, 0

Cooperative Phase

If you cooperate you expect to receive 2 forever◮ Payoff is

(1− δ)(2 + δ2 + δ22 + δ32 · · · ) = 2

If you defect◮ You get 3 today and 0 from tomorrow onwards◮ Payoff is

(1− δ)3

Cooperate is optimal if 2 ≥ (1− δ)3 or

δ ≥ 1

3


page.16

Reciprocal Altruism


Defect 3,−1 0, 0

Punishment Phase

Defecting gets you 0

If you cooperate◮ You get −1 today◮ At most 0 starting tomorrows

Defect is optimal

Trigger is an equilibrium if and only if δ ≥ 1/3.

Cooperation can be sustained if players are patient enough and futureinteraction is likely enough.


page.17

Reciprocal Altruism

More general payoffs

Cost of Cooperation c > 0

Benefit to the other b > 0

Cooperate DefectCooperate b− c, b− c −c, b

Defect b,−c 0, 0

Condition for equilibrium

b− c ≥ (1− δ)b

or

δ ≥ c

b


page.18

Reciprocal Altruism

Reciprocal Altruism is an equilibrium norm if

1. Future interaction is likely enough

2. Cost of altruism is small enough

3. Benefit of altruism is large enough

This provides testable predictions◮ In times of instability we should see more defection◮ If short term gain is too large, cooperation is harder to sustain

Also important:◮ Must be able to identify cheaters◮ Trigger strategy is inefficient

⋆ Limited punishment may be enough


page.19

Reciprocal Altruism

David Hume (1740): A Treatise of Human Nature

I learn to do service to another, without bearing him any real kindness,because I foresee, that he will return my service in expectation of anotherof the same kind, and in order to maintain the same correspondence ofgood offices with me and others. And accordingly, after I have served himand he is in possession of the advantage arising from my action, he isinduced to perform his part, as foreseeing the consequences of his refusal.


page.20

Theory of Repeated Games

Other equilibria?

Many!

Folk Theorem

If the likelihood of future interaction is high enough, any individuallyrational payoff can be sustained as an equilibrium payoff.

Institutions have another role: Selecting equilibrium◮ Efficiency◮ Fairness

Societies whose institutions select efficient and fair allocations thrive,others perish.


page.21

Evidence: Stickleback Fish

When a potential predator appears, one or more sticklebacksapproach to check it out

This is dangerous but provides useful information◮ If hungry predator, escape◮ Otherwise stay

Milinski (1987) found that they use Tit-for-Tat like strategy◮ Two sticklebacks swim together in short spurts toward the predator

Cooperate: Move forward

Defect: Hang back


page.22

Evidence: Stickleback Fish

Milinski also run an ingenious experiment

Used a mirror to simulate a cooperating or defecting stickleback

When the mirror gave the impression of a cooperating stickleback◮ The subject stickleback move forward

When the mirror gave the impression of a defecting stickleback◮ The subject stickleback stayed back


page.23

Evidence: Vampire Bats

Vampire bats (Desmodus rotundus) starve after 60 hours

They feed each other by regurgitatingIs it kin selection or reciprocal altruism?

◮ Kin selection: Costly behavior that contribute to reproductive successof relatives

Wilkinson, G.S. (1984), Reciprocal food sharing in the vampire bat,Nature.

◮ Studied them in wild and in captivation


page.24

Evidence: Vampire Bats

If a bat has more than 24 hours to starvation it is usually not fed◮ Benefit of cooperation is high

Primary social unit is the female group◮ They have opportunities for reciprocity

Adult females feed their young, other young, and each other◮ Does not seem to be only kin selection

Unrelated bats often formed a buddy system, with two individualsfeeding mostly each other

◮ Reciprocity

Also those who received blood more likely to donate later on

If not in the same group, a bat is not fed◮ If not associated, reciprocation is not very likely

It is not only kin selection


page.25

Evidence: Medieval Trade Fairs

In 12th and 13th century Europe long distance trade took place infairs

Transactions took place through transfer of goods in exchange ofpromissory note to be paid at the next fair

Room for cheating

No established commercial law or state enforcement of contracts

Fairs were largely self-regulated through Lex mercatoria, the”merchant law”

◮ Functioned as the international law of commerce◮ Disputes adjudicated by a local official or a private merchant◮ But they had very limited power to enforce judgments

Has been very successful and under lex mercatoria, trade flourished

How did it work?


page.26


What prevents cheating by a merchant?

Could be sanctions by other merchants

But then why do you need a legal system?

What is the role of a third party with no authority to enforcejudgments?


page.27


If two merchants interact repeatedly honesty can be sustained bytrigger strategy

In the case of trade fairs, this is not necessarily the case

Can modify trigger strategy◮ Behave honestly iff neither party has ever cheated anybody in the past

Requires information on the other merchant’s past

There lies the role of the third party


page.28


Milgrom, North, and Weingast (1990) construct a model to show howthis can work

The stage game:

1. Traders may, at a cost, query the judge, who publicly reports whetherany trader has any unpaid judgments

2. Two traders play the prisoners’ dilemma game3. If queried before, either may appeal at a cost4. If appealed, judge awards damages to the plaintiff if he has been

honest and his partner cheated5. Defendant chooses to pay or not6. Unpaid judgments are recorded by the judge


page.29


If the cost of querying and appeal are not too high and players aresufficiently patient the following strategy is a subgame perfectequilibrium:

1. A trader querries if he has no unpaid judgments2. If either fails to query or if query establishes at least one has unpaid

judgement play Cheat, otherwise play Honest3. If both queried and exactly one cheated, victim appeals4. If a valid appeal is filed, judge awards damages to victim5. Defendant pays judgement iff he has no other unpaid judgements

This supports honest trade

An excellent illustration the role of institutions◮ An institution does not need to punish bad behavior, it just needs to

help people do so


page.1

Introduction to Economic and Strategic BehaviorExtensive Form Games with Incomplete Information

Levent Kockesen

Koc University

Levent Kockesen (Koc University) Ext. Form Inc. Info 1 / 38

page.2

Extensive Form Games with Incomplete Information

We have seen extensive form games with perfect information◮ Entry game

And strategic form games with incomplete information◮ Auctions

Many incomplete information games are dynamic

There is a player with private information


page.3

Used Car Market

Two types of used cars:◮ Bad (Lemon)◮ Good (Peach)

Seller knows

Potential buyer does not◮ Believes a fraction q is peach

Willingness to pay (or be paid)

Peach LemonBuyer 16,000 6,000Seller 12,500 3,000


page.4

Used Car Market


If there is complete information

Both types are brought to the market and sold◮ This is the efficient outcome

Price?

Depends on bargaining power◮ Assume seller has all the bargaining power

Lemon’s price = 6,000

Peach’s price = 16,000


page.5

Used Car Market


Back to incomplete information

Is there an efficient equilibrium?

If both are brought to the market expected value to the buyers

q × 16, 000 + (1− q)× 6, 000 = 6, 000 + 10, 000q

This is the most they are willing to pay for a car

Peach owner must be willing to sell at that price:

6, 000 + 10, 000q ≥ 12, 500

or q ≥ 0.65


page.6

Lemon’s Problem

There is an efficient equilibrium iff fraction of lemons is small

Otherwise only lemons are brought to the market

First analyzed by George Akerlof◮ Winner of 2001 Nobel in Economics◮ Akerlof, G. (1970), The Market for Lemons, Quarterly Journal of

Economics.


page.7

Adverse Selection

This is also known as adverse selection

Comes from the insurance industry

Suppose the average risk of accident is 10%

Cost of accident is 100,000

Insurance company breaks even if premium is 10,000

But then only those whose risk is greater than 10% will buy insurance

Insurance company will make losses

Increase premium?◮ Say 15,000

What happens?


page.8

Signaling and Screening

Two possible but usually costly solutions◮ Signal◮ Screen

Signaling Games: Informed player moves first◮ Warranties◮ Education

Screening Games: Uninformed player moves first◮ Insurance company offers different contracts◮ Price discrimination


page.9

Signaling Examples

Used-car dealer◮ How do you signal quality of your car?◮ Issue a warranty

An MBA degree◮ How do you signal your ability to prospective employers?◮ Get an MBA

Entrepreneur seeking finance◮ You have a high return project. How do you get financed?◮ Retain some equity

Stock repurchases◮ Often result in an increase in the price of the stock◮ Manager knows the financial health of the company◮ A repurchase announcement signals that the current price is low

Limit pricing to deter entry◮ Low price signals low cost

Peacock tails


page.10

An Example: Beer or Quiche

A stranger comes to the bar for breakfast

A native is also at the bar trying to decide whether to pick a fight

The stranger is either tough or weak◮ Native doesn’t know

Stranger’s payoff to◮ Fight is 2◮ No fight is 4

Native’s payoff to◮ Fight is 2 against weak, −1 against tough◮ No fight is 0

Stranger decides on breakfast◮ Quiche: no cost◮ Beer: costs

⋆ 1 to tough⋆ 3 to weak


page.11

Beer or Quiche

Nature

1

1

2, 2

4, 0

2,−1

4, 0

−1, 2

1, 0

1,−1

3, 0

W〈0.5〉

T〈0.5〉

Q

Q

2

f

r

f

r

B

B

2

f

r

f

r

How would you play?

Backward induction or subgame perfect equilibrium?

Beliefs are important

We need a new solution concept


page.12

Beer or Quiche

Nature

1

1

2, 2

4, 0

2,−1

4, 0

−1, 2

1, 0

1,−1

3, 0

W〈0.5〉

T〈0.5〉

Q

Q

2

f

r

f

r

B

B

2

f

r

f

r

Given beliefs we want players to play optimally at every informationset

◮ sequential rationality

We want beliefs to be consistent with chance moves and strategies◮ Bayes Law gives consistency


page.13

Beer or Quiche

Nature

1

1

2, 2

4, 0

2,−1

4, 0

−1, 2

1, 0

1,−1

3, 0

W〈0.5〉

T〈0.5〉

Q

Q

2

f

r

f

r

B

B

2

f

r

f

r

If you are tough can you avoid a fight?

Drink beer?

Is this an equilibrium?


page.14

Strategies and Beliefs

A solution in an extensive form game of incomplete information is acollection of

1. A behavioral strategy profile

2. A belief system

We call such a collection an assessment

A behavioral strategy specifies the play at each information set of theplayer

◮ This could be a pure strategy or a mixed strategy

A belief system is a probability distribution over the nodes in eachinformation set


page.15

Perfect Bayesian Equilibrium

Sequential Rationality

At each information set, strategies must be optimal, given the beliefs andsubsequent strategies

Weak Consistency

Beliefs are determined by Bayes Law and strategies whenever possible

The qualification “whenever possible” is there because if an informationset is reached with zero probability we cannot use Bayes Law to determinebeliefs at that information set.

Perfect Bayesian Equilibrium

An assessment is a PBE if it satisfies

1. Sequentially rationality

2. Weak Consistency


page.16

Bayes Law

Suppose you take a test for a disease

Test is 99% reliable◮ prob(+|D) = 0.99◮ prob(−|N) = 0.99

On average 0.5% of population have disease

You tested positive

What is the likelihood that you have the disease?


page.17

Bayes Law

Suppose you test 1000 people

5 have disease◮ 5× 0.99 = 4.95 positive◮ 5× 0.01 = 0.05 negative

995 don’t have it◮ 995× 0.99 = 9.95 positive◮ 995× 0.01 = 985.05 negative

positive negative

Disease 5 4.95 0.05

No Disease 995 9.95 985.05

Disease prob. 33.22% 0.005%


page.18

Nature

1

1

〈1〉

〈0〉

〈0〉

〈1〉

2, 2

4, 0

2,−1

4, 0

−1, 2

1, 0

1,−1

3, 0

W〈0.5〉

T〈0.5〉

Q

Q

2

f

r

f

r

B

B

2

f

r

f

r

Separating Equilibria

1. T : B,W : Q◮ Bayes rule (BR) ⇒ B → T,Q → W◮ Sequential rationality (SR) of 2 ⇒ Q → f,B → r◮ SR of 1 ⇒ Tough: OK, Weak: OK


page.19


Separating Equilibrium

The following is a PBE

Behavioral Strategy Profile:◮ Stranger: (T : B,W : Q)◮ Native: (B : r,Q : f)

Belief System◮ prob(T |B) = 1◮ prob(W |Q) = 1


page.20

Nature

1

1

〈0〉

〈1〉

〈1〉

〈0〉

2, 2

4, 0

2,−1

4, 0

−1, 2

1, 0

1,−1

3, 0

W〈0.5〉

T〈0.5〉

Q

Q

2

f

r

f

r

B x

B

2

f

r

f

r


2. T : Q,W : B◮ Bayes rule (BR) ⇒ B → W,Q → T◮ Sequential rationality (SR) of 2 ⇒ Q → r, B → f◮ SR of 1 ⇒ Tough: OK, Weak: NO◮ No such PBE


page.21

Nature

1

1

〈〉

〈〉

〈0.5〉

〈0.5〉

2, 2

4, 0

2,−1

4, 0

−1, 2

1, 0

1,−1

3, 0

W〈0.5〉

T〈0.5〉

Q

Q

2

f

r

f

r

B

B

2

f

r

f

r

Pooling Equilibria

1. T : B,W : B◮ Bayes rule (BR) ⇒ prob(T |B) = 0.5, prob(T |Q) = free◮ Sequential rationality (SR) of 2 ⇒ B → f,Q →?◮ SR of 1 ⇒ Tough: NO, Weak: NO◮ No such PBE


page.22

Nature

1

1

〈0.5〉

〈0.5〉

〈〉

〈〉

2, 2

4, 0

2,−1

4, 0

−1, 2

1, 0

1,−1

3, 0

W〈0.5〉

T〈0.5〉

Q

Q

2

f

r

f

r

B

B

2

f

r

f

r

Pooling Equilibria

2. T : Q,W : Q◮ Bayes rule (BR) ⇒ prob(T |Q) = 0.5, prob(T |B) = free◮ Sequential rationality (SR) of 2 ⇒ Q → f,B →?◮ But SR of player 1 type T ⇒ B → f◮ SR of 2 ⇒ prob(W |B) = 1


page.23

Pooling Equilibria

Pooling Equilibrium

The following is a PBE

Behavioral Strategy Profile:◮ Stranger: (T : Q,W : Q)◮ Native: (B : f,Q : f)

Belief System◮ prob(W |B) = 1◮ prob(W |Q) = 0.5


page.24

Beer or Quiche

What if the cost of beer is equal to 1 for weak stranger too?

Nature

1

1

2, 2

4, 0

2,−1

4, 0

1, 2

3, 0

1,−1

3, 0

W〈0.5〉

T〈0.5〉

Q

Q

2

f

r

f

r

B

B

2

f

r

f

r

Can Beer signal toughness?

Signal must be costlier for the weak

Known as the handicap principle in biology


page.25

Peacock’s Tail

Why?


page.26

Peacock’s Tail

Could peahen be the reason?


page.27

Handicap Principle

Why do we observe traits that reduce fitness?

Because they are signals

Handicap Principle

Reliable signals must be costly to the signaler, costing the signalersomething that could not be afforded by an individual with less of aparticular trait

First proposed by biologist Amotz Zahavi◮ Zahavi, A. (1975) Mate selection - a selection for a handicap. Journal

of Theoretical Biology.

Grafen, A. (1990) Biological signals as handicaps. Journal ofTheoretical Biology

◮ Game theoretical modeling


page.28

Peacock’s Tail

Peacocks have different genetic quality◮ High (H) or Low (L)

Peahens’ fitness higher if they mate with high quality

Peacocks’ fitness higher if they are chosen as mates

Peacocks may exhibit different sized tails◮ Short (S) or Long (L)

Long tail is costly◮ cH to high quality◮ cL to low quality

Is there an equilibrium in which peahens select long tailed peacocks asmates?

◮ Known as sexual selection


page.29

Peacock’s Tail

Nature

Pc

Pc

1, 1

0, 0

1, 0

0, 1

1− cH , 1

0, 0

1− cL, 0

0, 1

H〈0.5〉

L〈0.5〉

S

S

Ph

m

r

m

r

L

L

Ph

m

r

m

r


page.30

Peacock’s Tail

Chance

Pc

Pc

1, 1

0, 0

1, 0

0, 1

1− cH , 1

0, 0

1− cL, 0

0, 1

H〈0.5〉

L〈0.5〉

S

S

Ph

m

r

m

r

L

L

Ph

m

r

m

r

Equilibrium with only High quality “choosing” Long tail?

Long must be interpreted as High quality

Short must be interpreted as Low quality

cH < 1

cL > 1


page.31

Alternative Theories of Education

Human Capital Theory◮ Education enhances productivity and increases earnings◮ Individuals decide comparing marginal benefit and cost of additional

education

Signaling Theory◮ Higher education signals higher ability and increases earnings◮ It pays only more able to get higher education


page.32

Signaling Theory of Education

First analyzed by Michael Spence◮ Winner of Economics Nobel in 2001◮ Spence, A. M. (1973). Job Market Signaling. Quarterly Journal of

Economics.

There are two types of workers◮ high ability (H): proportion q◮ low ability (L): proportion 1− q

Output is equal to◮ H if high ability◮ L if low ability

Workers can choose to have High education (H) or Low education (L)

Low education costs nothing but High costs◮ cH if high ability◮ cL if low ability

There are many employers bidding for workers◮ Wage of a worker is equal to her expected output

High education is completely useless in terms of worker’s productivity!


page.33

A Theory of Education

If employers can tell worker’s ability wages will be given by

wH = H,wL = L

Nobody gets High education

Best outcome for high ability workers

If employers can only see worker’s education, wage can only depend oneducation

Employers need to form beliefs about ability in offering a wage

wH = pH ×H + (1− pH)× L

wL = pL ×H + (1− pL)× L

where pH (pL) is employers’ belief that worker is high ability if shehas High (Low) education


page.34

Separating EquilibriaOnly High ability gets High education

What does Bayes Law imply?

pH = 1, pL = 0

What are the wages?wH = H,wL = L

What does High ability worker’s sequential rationality imply?

H − cH ≥ L

What does Low ability worker’s sequential rationality imply?

L ≥ H − cL


page.35


Only High ability gets High education

CombiningcH ≤ H − L ≤ cL

Again the same principle as in Beer or Quiche and Peacock’s Tail

There is an equilibrium in which only High ability gets High education iff

cH ≤ H − L ≤ cL

High education is a waste of money but High ability does it just tosignal her ability

The result would be even stronger if education increases productivity◮ This is an alternative but complementary theory of education◮ Known as human capital theory


page.36

Evidence

Two papers

Bedard, K. (2001), “Human Capital Versus Signalling Models:University Access and High School Drop-outs,” Journal of PoliticalEconomy 109, 749-775.

Land, K. and D. Kropp (1986), “Human Capital Versus Sorting: TheEffects of Compulsory Attendance Laws,” Quarterly Journal ofEconomics, 101, 609-624.


page.37

Bedard (2001)Tests signaling model of education against the human capital modelUses ease of access to university (presence of a local university) as aninstrumentHuman capital model:

◮ Education enhances productivity and increases earnings◮ Individuals decide comparing marginal benefit and cost of additional

education◮ Easier access to university ⇒ More high-school graduates go to

universitySignaling model

◮ Easier access to university ⇒ High ability high-school graduates go touniversity

◮ Average ability of high-school graduates becomes lower◮ Incentives to pool with them become weaker◮ Low ability students drop out of high school

The paper finds that easier university access1. increases the probability to dropout from high school2. increases the average skill of dropouts

Rejects human capital model in favor of the signaling modelLevent Kockesen (Koc University) Ext. Form Inc. Info 37 / 38

page.38

Signaling: Lang and Kropp (1986)

Tests signaling model of education against human capital model

Uses compulsory school attendance law (CAL) as an instrumentSignaling model:

◮ Suppose CAL increases minimum education from s− 1 to s years◮ Lower ability workers who would have left school after s− 1 will now

remain in school through s◮ Average ability and therefore wage of workers with s years of education

decrease◮ For the more able workers who previously stopped at s years, obtaining

s+ 1 years of education becomes more attractive, and so on◮ Prediction: CAL increases educational attainment of high-ability

individuals who are not directly affected by the law

Human capital model:◮ Prediction: No change in educational attainment of those who are not

directly affected by the law

They find CAL indeed increases the education received by those whoare not directly affected by the law and hence reject human capitalmodel in favor of the signaling model


page.1

Introduction to Economic and Strategic BehaviorExtensive Form Games with Incomplete Information

Levent Kockesen

Koc University


page.2

Screening

A contracting problem with Hidden Information

Uninformed party (principal) offers contract to informed party (agent)

Examples◮ Insurance

⋆ Insuree knows her risk, insurer does not⋆ Insurer offers several packages with different premiums and deductibles

◮ Finance⋆ Borrower knows the risk of project, lender does not⋆ Lender offers several packages with different interest rates and

collateral requirements

◮ Hiring⋆ Applicants know their ability, employer does not⋆ Employer offers different packages varying wages, bonuses, etc.

◮ Pricing⋆ Buyer knows her valuation of the product, seller does not⋆ Seller offers different qualities at different prices, or quantity discounts


page.3

Screening by a Monopolist: A Very Simple Example

You are a wine producer

A fraction c of your customers are connoisseurs while the rest arenovice

You can produce high quality and/or low quality wine

The following table gives the cost of producing the two qualities ofwine as well as the willingness to pay for them by the two types ofcustomers

Willingness to payCost Connoisseur Novice

Low Quality 2 10 6High Quality 10 30 12

You choose which qualities to produce and their prices to maximizeyour expected profit


page.4

Screening by a Monopolist

Suppose you can tell your customer’s type

What should you do?

Produce both qualities

Sell high quality to connoisseur at $30 and low quality to novice at $6

Your profit is

(30− 10)× c+ (6− 2)× (1− c) = 20c + 4(1− c)

Maximizes the total surplus:◮ Extra benefit from high quality is bigger (smaller) than the extra cost

for connoisseur (novice)

This is first degree (or perfect) price discrimination

It is great if you can implement it

But usually you cannot observe types

You may try to ask them their types. What would they tell you?


page.5

You cannot observe types

What happens if you offer high quality at $30 and low quality at $6?

Here are the net payoffs of the two types from purchasing the twoqualities

Net PayoffsConnoisseur Novice

Low Quality 10− 6 = 4 6− 6 = 0High Quality 30− 30 = 0 12− 30 = −18

High type wants to buy low quality

Everybody will buy low quality

Your profit will be 6− 2 = 4 only

Less than if you could observe types and perfectly discriminate

What is the best you can do in this case?


page.6

You cannot observe types

There are several things that you can do

1. Offer only low quality wine

2. Offer only high quality wine

3. Offer both so that Connoisseur (C) chooses high and Novice (N)chooses low quality


page.7

Offer only low quality



What price should you set?

If you set price at $6 both types buy◮ Your profit is 6− 2 = 4

If you set price at $10 only type C buys◮ Your profit is c(10− 2) = 8c

Therefore your best pricing policy and resulting profit is

Price Profitc ≤ 1/2 6 4c > 1/2 10 8c


page.8

Offer only high quality



What price should you set?

If you set price at $12 both types buy◮ Your profit is 12− 10 = 2

If you set price at $30 only type C buys◮ Your profit is c(30− 10) = 20c

Therefore your best pricing policy and resulting profit is

Price Profitc ≤ 1/10 12 2c > 1/10 30 20c


page.9

You offer both qualities

You want C to buy high and N to buy low quality wine

You want to choose PH and PL to maximize your expected profit

c(PH − 10) + (1− c)(PL − 2)

C must prefer high to low quality

30− PH ≥ 10− PL (ICC)

N must prefer low to high quality

6− PL ≥ 12− PH (ICN )

Called Incentive Compatibility (IC) constraints

And you want them to prefer to buy

30− PH ≥ 0 (IRC)

6− PL ≥ 0 (IRN )

Called Individual Rationality (IR) (or participation) constraints


page.10

Solving the problem1. (IRN ) holds with equality: PL = 6

Suppose instead that 6− PL > 0. Then

30− PH ≥ 10 − PL > 6− PL > 0

◮ We could increase PH and PL by the same amount without violatingIC constraints and increase our profit

◮ This also shows that we can neglect (IRC)

2. (ICC) holds with equality: 30 − PH = 10− PL

Suppose instead that 30− PH > 10− PL. Then

30− PH > 10 − PL > 6− PL = 0

◮ We could increase PH without violating any constraints and increaseour profit

3. Therefore, we have

PL = 6, PH = PL + (30 − 10) = 30− (10− 6) = 26


page.11

You offer both quantities

So prices are PL = 6 and PH = 26

Notice that you extract the full surplus from the Novice but leave $4surplus to the Connoisseur

Why?Remember what happens if you set prices at perfect informationlevels: i.e. PL = 6 and PH = 30

◮ Novice will happily buy his low quality wine◮ Too bad: Connoisseurs want to buy low quality as well

The amount of surplus you leave to Connoisseur is the minimum youhave to give her to buy high quality

◮ This is called information rent

What is your profit?

(26− 10)× c+ (6− 2)× (1− c) = 16c + 4(1− c)

Compare this with your profit under observable types

(30− 10)× c+ (6− 2)× (1− c) = 20c + 4(1− c)


page.12

What is the best?

Low Only High Only BothPrice Profit Price Profit Price Profit

c ≤ 1/10 6 4 12 2 (26, 6) 16c+ 4(1− c)1/10 < c ≤ 1/2 6 4 30 20c (26, 6) 16c+ 4(1− c)

c > 1/2 10 8c 30 20c (26, 6) 16c+ 4(1− c)

Both is always better than Low only

If c ≤ 1/2 producing both is the best

If c > 1/2 producing High only is the best

Getting surplus from Novice customers is costly◮ You have to leave the Connoisseurs some surplus◮ As the fraction of Connoisseurs increases the benefit decreases and the

cost increases

Therefore, if there are enough Connoisseurs around you don’t caremuch about getting the surplus from the Novice


page.13

Insurance

Similarly in insurance markets

Different policies with different premiums and coverage

High Premium High Coverage → High risk chooses

Low Premium Low Coverage → Low risk chooses


page.14

Insurance

Why are there insurance firms?

They only shift risk from one party to another

Insurance could be beneficial for both the insurer and the insuree

Suppose cost of accident is 100

Probability is 10%

If individual is risk averse◮ She is willing to pay a little bit more than 10, say 12

If the insurance firm is risk neutral◮ It is willing to sell full insurance at little bit more than 10

At 11 they are both better off


page.15

Insurance

This is a very general principle

Efficient Risk Sharing

More of the risk should be borne by less risk averse party. In particular, allthe risk should be borne by the risk neutral party.

Insurance companies are risk neutral because they hold many policieswith imperfectly correlated risks

Same as you holding a diversified portfolio


page.16

Moral Hazard

But if all the risk is borne by the insurer, the insuree may act careless

This may increase accident probability◮ Known as moral hazard

If preventive care is unobservable, leave some risk on insuree

Moral hazard is a very pervasive phenomenon◮ Compensation◮ Development aid◮ Financial bail-outs


page.17

Moral Hazard

There is moral hazard when

Agent takes an action that affects his payoff as well as the principal’s◮ How hard to work

Principal only observes an outcome, an imperfect indicator of theaction

◮ How many items the salesperson sold

Agent chooses an inefficient action◮ Salesperson slacks off


page.18

Moral Hazard

Principal’s problem is to find a contract that induces high effort◮ Principal-Agent Problem

If the agent is risk neutral (and there is no limited liability) efficientoutcome can still be obtained

If he is risk averse then there is a trade off between◮ risk sharing: agent’s wage should not depend too strongly on outcome◮ incentives: to get high effort wage must depend on outcome


page.19

Risk Aversion

An individual is risk averse is he prefers to receive the expected value of alottery to playing the lottery

Payoff function (of certain outcome) is

Linear if risk neutral◮ u(x) = x

Concave if risk averse◮ u(x) =

√x

Convex if risk lover◮ u(x) = x2


page.20

Risk Preferences

b

b

bb

x

u(x)

x1 x2E[x]

Risk averse: u(E[x]) > E[u(x)]

b

b

b

x

u(x)

x1 x2E[x]

Risk neutral: u(E[x]) = E[u(x)]

b

b

b

b

x

u(x)

x1 x2E[x]

Risk lover: u(E[x]) < E[u(x)]


page.21

An Example

The principal will hire a manager for a new project

Revenue is◮ 500K if successful◮ 0 if not

Probability of success depends on the manager’s effort◮ High effort: 0.9◮ Low effort: 0.1

Effort is costly for the manager◮ High effort: 250K◮ Low effort: 50K

Outside option of the manager is 0


page.22

An Example: Risk Neutral Manager

Suppose the manager is risk neutral

If you can observe his effort level◮ Pay 250K for high and 0 for low effort◮ Manager expends high effort◮ Expected profit

0.9× 500− 250 = 200

◮ This is the first best profit for the principal


page.23


What is the optimal pay schedule if effort cannot be observed?

Constant wage?

Manager will choose low effort

Then the best wage is 50K

Expected profit0.1× 500− 50 = 0

Can the principal do better?


page.24


25K salary plus 250K bonus if successful

Manager:◮ Low effort: 25 + 0.1× 250− 50 = 0◮ High effort: 25 + 0.9× 250− 250 = 0◮ Chooses high effort

Expected profit

0.9× (500 − 250) − 25 = 200

Same as observable effort!


page.25

An Example: Risk Averse Manager

If you offer 25K salary plus 250K bonus if successful

Expected wage of the manager: 0

But there is a downside: With positive probability negative payoff

Outside option is a sure 0 payoff

A risk averse manager would not accept it

Principal has to increase expected wage◮ How much depends on how risk averse the manager is

To induce high effort there has to be a bonus

Profit will be less than the first best

For an example see Dixit et al. p. 558-561


page.26

Incentives

Tying compensation to performance may itself create problems

Sometimes it is not easy to measure performance◮ How do you measure a professor’s performance◮ Student evaluations?◮ Professor may become student serving

Sometime success is multi-dimensional◮ If compensation is tied to only one, other dimensions may suffer


page.27

Incentives

In Soviet Union incentives were tied to production volume◮ Quality suffered

AT&T used to pay according to line of programs◮ Unnecessarily long programs

1980s American football player Ken O’Brian◮ Many of his passes were being intercepted◮ His club gave a new contract

⋆ Penalty for every intercepted pass

◮ It worked: His passing percentage increases⋆ But he was passing a lot less


page.28

Incentives

Individual commissions may lead employees sabotage each other◮ See David Mamet film Glengarry Glen Ross

If measuring performance is difficult but can be subjectively assessed◮ Efficiency wage may work

⋆ If performance is good pay high wage⋆ If bad, fire


page.29

Incentives

If task is multidimensional whether they are complements orsubstitutes matter

Farmer may be working on both apple orchard and corn field◮ Substitutes

Farmer works on both apple orchard and beehive◮ Complements


page.30

Incentives

If complements◮ High powered incentives

If substitues◮ Low powered incentives

It is better to collect complementary tasks together

Tasks at an airport are complements◮ It is best if different airports are managed by different companies◮ and each airport managed by a single company

London airports (Heathrow, Gatwick, Stanstead) did exactly theopposite

◮ All three belong to British Airports Authority◮ Different companies manage different “tasks” in each

Outcome is not very good


page.1

Introduction to Economic and Strategic BehaviorEvolutionary Game Theory

Levent Kockesen

Koc University

Levent Kockesen (Koc University) Evolutionary Game Theory 1 / 34

page.2

Evolution

Evolution: Change over time in one or more inherited traits in populationsof individuals

Four mechanisms of evolutionary change

1. Mutation

2. Gene Flow

3. Genetic drift

4. Natural selection


page.3

Genetic Variation

Drift and natural selection require genetic variation top operate

Sources

1. Mutations: Changes in DNA2. Gene flow: Movement of genes from one population to another3. Sex: Genetic shuffling


page.4

Natural Selection

Darwin’s biggest idea: Main mechanism behind evolution

Three conditions:

1. Variation2. Differential reproduction.3. Heredity

Fitter trait becomes more common in the population

Fitness refers to how good a particular genotype (set of genes) is atleaving offspring relative to other genotypes

◮ Survival, mate-finding, reproduction all matter


page.5

Natural SelectionMany examples:

Finches’ beaks on the Galapagos Islands◮ after droughts, the finch population has deeper, stronger beaks that let

them eat tougher seeds

Moths in 19th century Manchester

◮ Both light and dark peppered moths exists◮ Industrial revolution blackened trees◮ Being dark became advantageous in hiding from predators◮ In 50 years nearly all were black


page.6

Sexual Selection

Another important mechanism of evolution

Acts on ability to obtain or successfully copulate with a mate

Peacocks, male redback spider


page.7

Evolution

It is a fascinating subject but we don’t have the time to study itfurther

Check the web: Understanding Evolution1

One of the best books:◮ Richard Dawkins (1976): The Selfish Gene

1http://evolution.berkeley.eduLevent Kockesen (Koc University) Evolutionary Game Theory 7 / 34

page.8

Evolutionary Game Theory

Fitness of an individual sometimes depends on what others do◮ Success of defending the territory depends also on whether others

defend as well

This creates a game◮ Players: Members of a population◮ Strategies: Traits - behavior, color, etc.◮ Payoffs: Fitness

Can game theory help understand animal or plant behavior?

The answer has been an enthusiastic yes


page.9

Evolutionary Game Theory: History

C. Darwin (1871): The Descent of Man◮ Used game theory informally to explain equal sex ratio◮ Eliminated it from the 2nd ed.

R. A. Fisher (1930): The Genetic Theory of Natural Selection

Reformulated the theory:◮ Suppose less female births◮ Females have better mating prospects and leave more offspring◮ Parents who give birth to more females have more grandchildren◮ Genes for female birth spread

R. C. Lewontin (1961): Evolution and the Theory of Games◮ First explicit application

John Maynard Smith: Introduced evolutionarily stable strategy◮ 1972: Game Theory and the Evolution of Fighting◮ with Price 1973: The Logic of Animal Conflict◮ 1982: Evolution and the Theory of Games

Theory of evolution influenced social sciences too


page.10

The Model

A large population of organisms with different traits

Asexual reproduction◮ Each organism reproduces on its own◮ No gene reshuffling

Within species evolution◮ Strategies and payoffs are the same◮ Symmetric games

With high probability each member adopts its parent’s strategy

With small probability chooses another strategy (mutation)

Random matching in pairs

Payoffs represent (reproductive) fitness

Central idea: Strategies with higher fitness spread, those with lowerfitness diminish


page.11

The Model

We can look at two types of population composition

1. Monomorphic: Each individual plays the same strategy2. Polymorphic: Different individuals may play different strategies


page.12

Evolutionary Game Theory

Players: Fish defending their territory

Strategies:

1. Cooperate: Join in defending2. Defect: Do not defend

Payoffs:

C DC 2, 2 0, 3D 3, 0 1, 1

Can cooperation be evolutionarily stable?◮ A strategy is evolutionarily stable if it drives out any mutant


page.13


Suppose everybody plays C

A small fraction ε of mutants appear◮ They play D

Large population ⇒◮ Probability of meeting C type: 1− ε◮ Probability of meeting D type: ε

What is the average payoff of a cooperating type?

(1− ε)× 2 + ε× 0 = 2(1− ε)

How about for the defecting type?

(1− ε)× 3 + ε× 1 = 3− ε

Defecting type has higher fitness for any ε

They will invade the population of cooperators

Cooperation is not evolutionarily stable


page.14


Is D evolutionarily stable?

A small fraction ε of mutants play C

Probability of meeting D type: 1− ε

Probability of meeting C type: ε

What is the average payoff of a cooperating type?

ε× 2 + (1− ε)× 0 = 2ε

How about for the defecting type?

ε× 3 + (1− ε)× 1 = 1 + 2ε

Defecting type has higher fitness for any ε

Defecting is evolutionarily stable


page.15


Lesson 1

A dominated strategy cannot be evolutionarily stable

Lesson 2

Evolution can be inefficient


page.16

Another Example: Hawk-Dove

H DH −1,−1 4, 0D 0, 4 2, 2

Is H ESS?

H → (1− ε)× (−1) + ε× 4 = −1 + 5ε

D → (1− ε)× 0 + ε× 2 = 2ε

A small population of D invades

So, H is not an ESS

What is wrong?

(H,H) is not a Nash equilibrium◮ There is a better response to H than H itself◮ This can invade a population of H players

⋆ Most of the time they meet H⋆ Rarely they meet each other


page.17

Another Example: Hawk-Dove

H DH −1,−1 4, 0D 0, 4 2, 2

D is not an ESS either◮ We will get back to this later◮ There maybe polymorphic equilibria

Remarkable fact

ESS vs. Nash

If a strategy s is an ESS, then (s, s) must be a Nash equilibrium

Only candidates for monomorphic (pure strategy) ESS are symmetricNash equilibria


page.18

Another Example

Is every Nash equilibrium ESS?

H DH 1, 1 2, 0D 0, 2 2, 2

(D,D) is a NE

Is D an ESS?

D → ε× 0 + (1− ε)× 2 = 2(1 − ε)

H → ε× 1 + (1− ε)× 2 = ε+ 2(1− ε)

H does as well as against D AND does better against H

D is not an ESS

What is special about (D,D)?

Every strict NE is ESS


page.19

Evolutionary Stability

NE vs. ESS

1. ESS ⇒ ESS

2. Strict NE ⇒ ESS

Evolutionarily Stable Strategy

A strategy s is an ESS if s yields a strictly higher expected payoff thandoes any mutant as long as the population of mutants is small enough.


A strategy s is an ESS if there exists ε > 0 such that for any s′ 6= s

(1− ε)u(s, s) + εu(s, s′) > (1− ε)u(s′, s) + εu(s′, s′)

for all ε with 0 < ε < ε.

This doesn’t look easy to check.


page.20


ESS

A strategy s is an ESS if

1. (s, s) is a Nash equilibrium

2. u(s′, s) = u(s, s) ⇒ u(s, s′) > u(s′, s′)

Intuitive

Easy to check


page.21


H DH 1, 1 2, 0D 0, 2 2, 2

H is an ESS

1. It is a Nash equilibrium2. No other best response

D is not an ESS

1. It is a NE2. H is another best response and against H , D does not do strictly

better than H


page.22

Polymorphic Populations

Hawk-Dove again

H DH −1,−1 4, 0D 0, 4 2, 2

There is no (pure strategy) ESS

Could there be a mixed strategy monomorphic ESS?◮ Each organism plays randomly◮ Has to be a mixed strategy Nash equilibrium◮ Not sufficient

Another way of thinking about it: Polymorphic population◮ Is there a stable population composed of both Hawks and Doves?


page.23


H DH −1,−1 4, 0D 0, 4 2, 2

Suppose a fraction p is H

Average payoffs:

H → p× (−1) + (1− p)× 4 = 4− 5p

D → p× 0 + (1− p)× 2 = 2− 2p

4− 5p > 2− 2p ⇒ H ր4− 5p < 2− 2p ⇒ D ր

Stable only if

4− 5p = 2− 2p or p = 2/3

But this is nothing other than the mixed strategy equilibrium


page.24


How do we know it is ESS?

A phase diagram helps

p1

4

2

−1

Fitness

2/3

We are not going to do dynamics but intuitively◮ Any payoff monotonic dynamic leads to p = 2/3

This is the unique stable population composition

Polymorphic


page.25


H DH −1,−1 4, 0D 0, 4 2, 2

How can we check if p = 2/3 an ESS?

Definitions is as before

Is it a NE? Yes

Are there other best responses? Yes

Does p = 2/3 do strictly better against them than they do againstthemselves? Yes

You can calculate it. Intuitively:◮ If mutant plays p′ > 2/3, they are hurt when they face each other◮ If mutant plays p′ < 2/3, p = 2/3 strategy takes advantage


page.26

Hawk-Dove: General Payoffs

H DH (v − c)/2, (v − c)/2 v, 0D 0, v v/2, v/2

What does the incidence of fight depend on?

Is D an ESS?◮ No: (D,D) is not NE

How about H?

Must have v ≥ c

Is H a better response to D than D itself?

u(H,D) = v > v/2 = u(D,D)

So, a population of all Hawks is stable iff v ≥ c

Whenever the prize of the contest is at least as high as the cost ofresulting fight we should observe fights


page.27

Hawk-Dove: General Payoffs

H DH (v − c)/2, (v − c)/2 v, 0D 0, v v/2, v/2

What if v < c

Must have polymorphic population

pv − c

2+ (1− p)v = (1− p)

v

2⇒ p =

v

c

Incidence of fighting increases in prize, decreases in cost

Average payoff of the population

(1− p)v

2= (1− v

c)v

2

Payoff increases in cost of fighting


page.28

Rock, Paper, Scissors

R P SR a, a −1, 1 1,−1P 1,−1 a, a −1, 1S −1, 1 1,−1 a, a

In RPS a = 0 but we will allow a > 0

Unique Nash Equilibrium (1/3, 1/3, 1/3)


page.29

Rock, Scissors, Paper

R P SR a, a −1, 1 1,−1P 1,−1 a, a −1, 1S −1, 1 1,−1 a, a

Let a > 0

Is p = (1/3, 1/3, 1/3) an ESS?

Consider S

S against S yields a

p against S yields a/3

So, p is not an ESS

If the game is like RPS, we should not find a stable population


page.30

Side-Blotched Lizards

Common Side-blotched Lizard (Uta stansburiana) common in PacificCoast of North AmericaEach males follows a fixed heritable mating strategy1. Orange-throated: Strongest, do not form strong pair bonds2. Blue-throated: Middle-sized, form strong pair bonds3. Yellow-throated: Smallest, coloration mimics females

Orange beats Blue, Blue beats Yellow, and Yellow beats OrangeSinervo and Lively (1996):

◮ 1990-91: Blue dominated◮ 1992: Orange◮ 1993-94: Yellow

Evolutionary game theory seems to be more than a theoreticalcuriosity


page.31

Asymmetric Games

Games played between different species◮ Insects and plants◮ Predator-Prey

Games played by asymmetrically positioned members of the samespecies

◮ Current owner of a nesting ground, food, or mates versus intruder◮ Different sized animals, etc.

Many confrontations are resolved peacefully◮ Current owner of food source keeps it


page.32

Asymmetric Games

Owner

IntruderH D

H (V − c)/2, (v − c)/2 V, 0D 0, v V/2, v/2

An individual may come to play either role in different times

A strategy conditions action on the role◮ For example: Hawk if Owner, Dove if Intruder

What is ESS?

Result

A conditional strategy is an ESS iff it is a strict Nash equilibrium.

V < c and v < c ⇒ only two strict NE◮ (H,D): Known as bourgeois strategy◮ (D,H): Known as paradoxical strategy


page.33

Asymmetric Hawk-DoveFraction of Hawks in Owner: pFraction of Hawks in Intruder: q

q < V/c ⇒ p րp < v/c ⇒ q ր

Stable points: (1, 0) and (0, 1)

p1v/c

q

1

V/c

1


page.34

Evidence

A male Hamadryas baboon forms a long-lasting relationship withseveral females

Kummer et al. (1974) experiment

Male A is allowed to interact with the female

Male B is in a cage and watches

When released, B does not challenge A

The same males change roles over a different female

The same outcome

Escalated fights if each perceives himself as the owner


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