game theory lecture notes

5/20/2018 Game Theory lecture notes

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GAME THEORY:

INSIDE OLIGOPOLY

Dr. Gong Jie

National University of Singapore


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Course Road Map

Managerial Economics

Determination ofPrices

Introduction

Demand and Supply

Consumer Theory

Production and Cost Theory

Market Structure &Profit-MaximizingPricing Decisions

Competitive Markets

Market Power & Monopoly

Pricing with Market Power

Game Theory&

Oligopoly Markets

Game Theory Fundamentals

Simultaneous-Move Game

Sequential-Move Game


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Oligopoly and Strategic Thinking

Two extremes of market structure competitive market

monopoly

Oligopoly: the market structure between the extremes A small number of sellers

Each of them secures a considerably large marketshare.

The behavior of each seller has a strong influence onothers' behavior.


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Oligopoly and Strategic Thinking

How to manage an oligopolistic firm: StrategicThinking! Strategic thinking:put yourself in others' shoes

You must figure out the action and the intention of others

when you take your action, while others think in the sameway.

The outcome depends on how you and your opponentinteract with each other.

Strategic Decision Making If I believe that my competitors are rational and act to

maximize their own profits, how should I take theirbehavior into account when making my own profit-maximizing decisions?


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Game and Game Theory

Game is any situation in whichplayers (the

participants) make strategic decisions.

Firms compete with each other by setting prices, spending

on advertising, R&D, merger & acquisition, etc. Group of consumers bid against each other in an auction.

Many continentals believe life is a game; the English think

cricket is a game. (George Mikes, Hungarian humorist)

Game Theory Game theory systemizes the strategic thinking.


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Agenda

Oligopoly and strategic thinking

Game theory as the tool for strategic thinking

Setup of a game

Nash Equilibrium

Solve for the equilibrium

Games with multiple equilibria

Mixed strategy


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Grade Game

Without showing your neighbor what you are doing, write

down on a form either the letter or the letter . Think of this

as a grade bid. We will randomly pair your form with one

other form. Neither you nor your pair will ever know with

whom you were paired.

Grades may be assigned by the following rule:

If you put and your pair puts , then you will get A and your

pair grade C.

If both put , then both will get B-.

If you put and your pair puts , then you will get C and your

pair A.

If both put , then you will both get B+.


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How to describe the Grade Game?

My grades Pairs grades

me

pair

B- A

B+C

me

pair

B- C

B+A


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What strategy should a rational person

choose in the Grade Game?

Outcome Matrix Payoff Matrix

me

pair

B-, B- A, C

B+, B+C, A

me

pair

0, 0 3, -1

1, 1-1, 3


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Lessons

To figure out what actions you should choose, a good

first step is to figure out what are your payoffs (what

do you care about) and what the other players

payoffs.You should never play a strictly dominated strategy.

Rational play by rational players can lead to bad

outcomes.

NUS students are evil.


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How to specify a game?

Who are the players?

What are the possible actionsthese players can take?

What are the payoffs associated

with each possible outcome?

Who are the participants in the market?

What is the set of potential entrants?

What is the set of bidders?

Enter; launch; merge (discrete) R&D expenditure; capacity/price

setting; production level; advertising

spending (continuous)

What are everyones profits given theprices charged, capacity installed,

products launched, advertising spent,

number of entrants, etc.?


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Key Concepts in Games

Actions and Strategies

Strategy is the decision rule that describes the actions a

player will take at each decision point.

Strategy is not a single action, but a plan of actions.

Best responses The strategy of one player that results in the best payoffs

to him/her, given the combination of other players

strategies.


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What is the likely outcome of a game?

Game theory predicts optimal strategy for each player. The optimal strategy maximizes a players payoffs given others

strategic plays.

Game theory predicts how the game is going to be played in

obviously reasonable ways.

Solution concept: Nash Equilibrium Nash Equilibrium is a strategy profile (a combination of

strategy), where no player in the game has the incentive todeviate.

In a Nash Equilibrium, a player is unable to do strictly better byunilaterally switching his/her strategy.

In a Nash Equilibrium, each player's strategy is the best responseto other players' strategies.


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How to classify a game?

The sequence of moves: sequential or simultaneous?

Are the players' interests in total conflicts or is there

any commonality?

Is the game played once or repeatedly?

Do the players have full or equal information?


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One-Shot, Simultaneous-Move Games

We use normal-form (strategic form) to represent thegame.

Players simultaneously decide their strategies.

A representation of a game indicating the players, theirpossible strategies, and the payoffs resulting fromalternative strategies.

One player chooses strategy from the row, while theother player chooses strategy from the column.

In each cell, the first entry indicates the payoff of the rowplayer, while the second entry indicates the payoff of thecolumn player.


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Prisoners Dilemma

Two prisoners are interrogated separately

Confess Dont Confess

Confess

Dont

Confess

Prisoner 2

-5, -5 0, -10

-1, -1-10, 0


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What would be the outcome (equilibrium)

of such a game?

How to solve the game? How to find the Nash

equilibrium?

Step 1:

Find one players best response to each of the possiblestrategiesplayed by the other.

Circle the payoffs of this player that result from his/her best

response and the given play of the other.

Step 2: Repeat this procedure to the other player.The combination of strategies that result in two

circles in one cell is a Nash-equilibrium.


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Prisoners Dilemma: Solve the Game

Confess Dont Confess

Confess

Dont

Confess

Prisoner 2

-5, -5 0, -10

-1, -1-10, 0


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The Nash-equilibrium of the game is given by a

strategy profile (confess, confess). Both prisoner will choose to confess.

Does this strategy profile maximize their collective

payoffs?

It doesnt. If they both deny, they can end up with -2 in total.

(dont confess, dont confess) is not an equilibrium.

These two prisoners have conflicting interests!

When economic agents have conflicting interests,

individual decision making without enforcement

cannot reach the collective optimum.


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Airlines Dilemma: the blessing from

terrorists!

Baggage policy is a nagging problem for airlines.

Passengers who carry multiple bags onto a plane slow

down the boarding process.

Why didnt airlines enforce tighter limits and force theirextra baggage?

Lets try to analyze this in a normal-form game.


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Airlines Dilemma

Generous Policy Tighter Policy

Generous

Policy

Tighter

Policy

Air l ine B

0, 0 5, -5

2, 2-5, 5


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Given the other airline adopting lenient policy, no

single airline would like to put tighter limit on carry-on baggage.

If one airline adopts tighter policy, the other airline

will benefit from a lenient policy.

Individual behavior does not make the most desirableoutcome.

After 911, the U.S. government started to enforce

carry-on baggage screening regulation.


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Dominant Strategies and Dominated

Strategy

Dominant Strategy is one that is optimal no matter

what an opponent does.

Dominated strategy is one that results in worse

payoffs than other strategies regardless of otherplayers strategic plays.

An example

A and B sell competing products and they are deciding

whether to undertake advertising campaigns.


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Payoff Matrix for Advertising Game

Advertise

Dont

Advertise

Advertise

DontAdvertise

Firm B

10, 5 15, 0

10, 26, 8


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Observations

A: regardless of B,

advertising is the best.

B: regardless of A,

advertising is the best.

Firm A

Advertise

Dont

Advertise

Advertise

Dont

Advertise

Firm B

10, 5 15, 0

10, 26, 8


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Observations

Dominant strategy for A

and B is to advertise.

Do not worry about theother player.

Equilibrium in

dominant strategyFirm A

Advertise

Dont

Advertise

Advertise

Dont

Advertise

Firm B

10, 5 15, 0

10, 26, 8


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Dominant Strategies and Dominated

Strategies

If you have a dominant strategy, play it! If you have a

dominated strategy, forget about it!

Equilibrium in dominant strategies

Outcome of a game in which each firm is doing the best itcan regardless of what its competitors are doing

Optimal strategy is determined without worrying about the

actions of other players


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Simplify the game by successive

elimination of dominated strategies

Player II

Player I U

M

B

L N R

2,1 1,-2 -4,0

0,1 0,-1 0,2

-1,4 -3,5 -2,0


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In this game, no matter what player II does, the

strategy B results in strictly worse payoffs than M

for player I. Then delete it!

Player II

Player I U

M

B

L N R

2,1 1,-2 -4,0

0,1 0,-1 0,2

-1,4 -3,5 -2,0


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In the revised game, no matter what player I does,

the strategy N is strictly dominated by both L and R.

Then delete it!

Player II

Player I U

M

B

L N R

2,1 1,-2 -4,0

0,1 0,-1 0,2

-1,4 -3,5 -2,0


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Thus, we end up with a two-by-two game.

Not every game has a dominant strategy for each

player.

The existence of a dominated strategy doesnt imply

there is a single (pure) dominant strategy.

With just two strategies for each player, if one strategy isdominant then the other must be dominated.

With more than two strategies available to each player, a

player might have dominated strategies but no dominant

strategy.

A game may not involve dominated strategy.


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Systematic Procedure for Identifying The

Nash Equilibria

Step 0: identify dominant or dominated strategies and

simplify the game.

Step 1: identify player 1s best responses to each of

player 2s strategies.Step 2: identify player 2s best responses to each of

player 1s strategies.

Step 3: see where those best responses occur together.


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Multiple Equilibria: Two drivers on the

road (in opposite directions)

Left Right

Left

Right

dr iver B

1, 1 -1, -1

1, 1-1, -1

Coordination game: Both drivers want to settle on the same choice


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Why Is It Called Coordination Game?

The best choice depends on what each player thinks

the other party is likely to do.

If the two players communicate with each other

before they take every action, they will follow whatthey agree with when they take their action,

because

They have common interest!

Their agreement is a Nash equilibrium.


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Player 1

ARTS BIZ

ARTS

BIZ

Player 2

10, 10 -5,-8

8,8-8,-5

Coordination Game2: Meeting at Canteen

You call your groupmate

when you walk out of YIH.

You learn that your

groupmate is at Central

Library.

You two decide to discuss

homework at Canteen.

However, your cell phone

battery runs out of power.

You two havent agreed at

which Canteen you will meet.


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Player 1

ARTS BIZ

ARTS

BIZ

Player 2

10, 10 -5,-8

8,8-8,-5

Pure (common interest) Coordination

Which canteen will you head for?

How many Nash-equilibria arethere in the game?

Which NE will be picked?

(ARTS, ARTS) is more likely tobe played than (BIZ, BIZ). Thisis called a focal point.

A focal point may stem from

custom, common sense,tradition, etc.


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Matching Pennies

RULES Player A and B each has a

coin and must secretly turn

the coin to heads or tails.

Then both players reveal thatchoice simultaneously.

If the coins match, player A

keeps both coins. If the coins

dont match, player B keeps

both coins.

Player A

Heads Tails

Heads

Tails

Player B

1, -1 -1, 1

1, -1-1, 1


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Matching Pennies

Pure strategy: No

Nash equilibrium

No combination of

head and tails leaves

both players better off.

Player A

Heads Tails

Heads

Tails

Player B

1, -1 -1, 1

1, -1-1, 1


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Matching Pennies: How about randomized

actions?

Player A might flip coin playing heads with probability and tails with probability.

If both players follow this strategy, there is a Nashequilibriumboth players will be doing the best theycan given what their opponent is doing.

Although the outcome is random, the expected payoffis 0 for each player.


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Lets Check Whether (0.5, 0.5) is Nash

Equilibrium

Given that player 1 randomizes between head and tailwith the probabilities 0.5 and 0.5, if player 2 plays headwith a probability p, and tail with a probability 1-p, then

player 2 ends up with

(head, head) with probability 0.5*p, payoff=-1 (head, tails) with probability 0.5*(1-p), payoff=1

(tails, head) with probability 0.5*p, payoff=1

(tails, tails) with probability 0.5*(1-p), payoff=-1

Play 2s expected payoff with this mixed strategy is:

0.5*p*(-1)+0.5*(1-p)*1+0.5*p*1+0.5*(1-p)*(-1)=0

Player 2 has no incentive to deviate fromrandomizing with prob 0.5 and 0.5, since there is noother strategy with str ictly higherpayoff.


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Pure Strategy and Mixed Strategy

If a strategy involves a single action at each single

contingency, such a strategy is a pure strategy.

Player makes a specific choice or takes a specific action.

If a Nash equilibrium involves all players playingpure strategies, the equilibrium is called Pure Strategy

Equilibrium.

Sometimes a pure-strategy equilibrium doesnt exist.

When allowing for mixed strategies, every game has

a Nash equilibrium.


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Mixed Strategy

Mixed strategies: unpredictability can have strategicvalue.

Sports: Soccer, tennis, baseball

Pricing Strategy: randomized pricing as mixed strategy

A strategy of constantly changing prices:

decreases consumers incentive to shop around as they cannot learn

from experience which firm charges the lowest price

reduces the ability of rival firms to undercut a firms prices


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Is O. Henry Wrong?

In O. Henrys novel The Gift of Magi, Della and Jim are

the young couples who are poor but love each other and

are ready to sacrifice anything for each other.

They both wish to give the other a surprise Christmas gift.Della considers selling her hair to buy a chain for Jims

watch.

Jim considers selling his watch to buy a comb for Della.

In the novel, they both do that after struggling.


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Della

Sell Watch Keep Watch

Keep Hair

Sell Hair

J im

1, 2 0,0

2,10,0

Two pure strategyequilibria: (keep hair, sell

watch) and (sell hair, keepwatch)

But there still exists amixed strategy equilibrium:Della (Jim) sells hair

(watch) with theprobability 2/3.

Surprise is costly! Theoutcome of the noveloccurs with the probability

of 2/3*2/3=4/9, more thaneither of the two bestoutcomes (keep hair, sellwatch), (sell hair, keepwatch), both with prob 2/9!


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Takeaways

Game theory is useful when your payoff depend on

choices of the other parties.

It is important to think about how other players will

play, not how you think they ought toplay.Nash-equilibrium does not necessarily correspond to

the outcome that maximizes the aggregate profit of

the players.

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