AGTB

Lecture 12: Rollback reasoning

Election Game

• Seat currently occupied by Mr. A

• Likely challenger Ms. B

• Should Mr. A launch preemptive ad campaign

• Should Ms. B enter

Game Tree - Partial

A

BB

B

Ads

No Ads

In

In

Out

Out

Outcomes

Outcome Payoff

Mr. A Ms. B Mr. A Ms. B

Not advertize Not enter best - 4 Third best - 2

advertize Not enter second best - 3 second best - 3

Not advertize enter third best - 2 best - 4

Advertize enter fourth best - 1 fourth best - 1

Game Tree - Full

BB

B

Ads

No Ads

In

In

Out

Out

1, 1

4,2

2,4

3,3

A

The Game Tree

• Consists of nodes and branches ( connecting nodes)• Two types of nodes

– Decision nodes : represent specific points in the game at which decisions are made. Each decision node is associated with the player who chooses an action at that node

• Initial node: decision node at which the game begins

– Terminal node : end point of the game. Each terminal node has associated with it a set of outcomes for the players involved in the game. These outcomes are mapped to payoffs

• Branches represent the possible actions from each decision node

• Each branch leads from a decision node to another, generally for another player, or to the terminal node

What is strategy?

BB2

B1

Ads

No Ads

In

Out

Out

In

Complete plan of actionRecommendation of action at each decision node

B’s strategies

_I/O_ _ I/O _Possible ActionB1 B2 Decision node

_I_ _ I _ Possible ActionB1 B2 Decision node

_O_ _ I _ Possible ActionB1 B2 Decision node

_I_ _ O _ Possible ActionB1 B2 Decision node

_O_ _ O _ Possible ActionB1 B2 Decision node

• B– In In (II)– In Out ( IO)– Out In (OI)– Out Out (OO)

The first letter represents B’s action at her first decision node B1, and the second at her second decision node B2.

A simple way of denoting strategies

Total number of strategies

• Suppose there are k decision nodes

• Let N1 be number of actions from decision node 1, N2 from decision node 2, etc.

• Total number of strategies = N1 X N2X N3X…X Nk

How many strategies does B have?

BB

B

U

D In

Out

Out

In

MIn

Out

BA

BB3

B1

U

D In

Out

Out

In

MIn

Out

B2A

B’s strategiesIIIIIOIOIIOOOIIOIOOOIOOO

What is equilibrium?

• Nash Equilibrium

• How to find?

Rollback Equilibrium: Pruning the tree

BB

B

Ads

No Ads

In

In

Out

Out

1, 1

4,2

2,4

3,3

Fully pruned tree

BB

B

Ads

No Ads

In

In

Out

Out

1, 1

4,2

2,4

3,3

A

Outcome & Equilibrium

• Mr. A gives ads and Ms. B stays out

• In equilibrium Mr. A’ strategy is ‘A” and Ms. B’s strategy is ‘OI’

• E = (A, OI)

Practicing rollback reasoning

A

B

N

S

t

b

0,2

2,1

1, 0

A

B

A

B

A

B

0,12, 3

5,4

3,2

4,5

1,0

2,2

Practicing rollback reasoning

How many strategies does A have?How many strategies does B have?What is the equilibrium?What is the equilibrium payoff?

Word Problem

• Consider the rivalry between Airbus and Boeing to develop a new commercial jet aircraft. Suppose Boeing is ahead in he development process and Airbus is considering whether to enter the competition. If Airbus stays out, it earns zero profit while Boeing enjoys a monopoly and earns a profit of $ 1 billion. If Airbus decides to enter and develop the rival plane, then Boeing has to decide whether to accommodate Airbus peacefully or to wage a price war. In the event of peaceful competition each firm will make a profit of $ 300 million. If there is a price war, each will lose $ 100 million because prices of airplanes will fall so low that neither will be able to recoup development costs.

Order of play

B

A

B

In

Out

Ads

Ads

No ads

No ads

1, 1

2,4

3,3

4,2

A

First mover advantage

Game 1

BB2

B1

U

D L

R

R

L

A

W

L

L

D

Game 1

BB2

B1

U

D L

R

R

L

A

W

L

L

D B can force a win

Game 2

BB2

B1

U

D L

R

R

L

A

W

W

L

D

Game 2

BB2

B1

U

D L

R

R

L

A

W

W

L

D A can force a win

Game 3

BB2

B1

U

D L

R

R

L

A

L

W

D

W

Game 3

BB2

B1

U

D L

R

R

L

A

L

W

D

Both can force a draw

W

Zermelo’s Theorem

• In any finite 2 player game where the possible outcomes are win, lose or draw, one can draw the game tree

• Find the rollback equilibrium • Play according to the equilibrium dictates a certain

outcome, i.e. W, L, or D• If a player can achieve a certain outcome with the

equilibrium strategy against the other player’s equilibrium strategy, then he can do at least as well against the other player’s non-equilibrium strategy

• Therefore either player 1 can force a win, or player 2 can force a win, or both can force a draw

Creating a game table from a game tree

BB

B

Ads

No Ads

In

In

Out

Out

1, 1

4,2

2,4

3,3

1,1 1, 1 3,3 3, 3

2,4 4,2 2,4 4,2

Ads

No ads

In – in In out Out in Out out

Mr. A’s strategies

Ms. B’s strategies

• Two Nash equilibria– One arrived at by rollback reasoning– The other equilibrium involves a threat

strategy by Ms. B, that allow her to wrest the first mover advantage, despite moving second

– The equilibrium which is not rollback is not ‘subgame perfect’

Ultimatum Game

• Two persons use the following procedure to split Rs. 100. Person 1 offers person 2 an amount of money up to Rs. 100. If 2 accepts, then 1 receives the remainder. If 2 rejects, neither receives any payoff. Each person cares only about the amount of money she receives.

Assume that player 1 can offer any number ( not just integer value) of cents

Game tree

x

Y

N

100-x, x

0,0

0

100

• What are the rollback equilibria?

• Find the values of x for which there is a Nash Equilibrium of the ultimatum game in which Person 1 offers x

• Find the rollback cequilibria in which the amount of money is available only in multiples of a rupee

Experiments on the ultimatum game

Duopoly

• Firm 1– P= 100 – q1-q2– MC = 40q1

• Firm 2– P= 100 – q1-q2– MC = 40q2

• Equilibrium is a choice of q1,q2 s.t firm 1 maximizes S1 given q2 and firm 2 maximizes S2 given q1

Best response functions

• S1 = (100-q1-q2)q1 – 40q1

• dS1/dq1 = 60-2q1-q2 =0

• q1* = (60- q2)/2 (1)

• S2 = (100-q1-q2)q2 – 40q2

• dS2/dq2 = 60-2q2-q1 =0

• q2* = (60- q1)/2 (2)

Stackelberg Model: Leader and follower

• Firm 1 moves first, chooses q1• Firm 2 observes Firm 1 and then sets its quantity q2

B

Firm 2

In

4,2

Firm 1q1

q2

q2

Firm 2

• We know q2* = (60-q1)/2• Therefore S1 = (100 – q1 – (60-q1)/2)q1 – 40q1• Ds1/dq1 = 30-q1 = 0• Q1* = 30• Q2* = 15• S1* = Rs. 1650• S2* = Rs. 825• Firm 1 has first mover advantage• Prove: Firm 1 will always get more than it gets under

duopoly• Suppose firm 2 could change its output after seeing what

Firm 2 does

• q1*(15) = (60- q2)/2 = 22.5• Suppose firm 1 could change its output after

seeing what Firm 2 does• If we add a third stage to the game in which firm

1 chooses an output then the first stage is irrelevant

• Firm 2 becomes the first mover• In rollback equilibrium Firm 1 is worse off• Firm 1 prefers to be committed not to change its

mind

Sequential games with nature as a player

N

p

1-p

• If A advertises, expected payoff = 1p + 3(1-p)

• If A does not advertise, expected payoff = 2p + 32(1-p) = 2

• Mr. A should advertize if

• 1p + 3(1-p) > 2

• P< 1/2

Imperfect information

BB

B

Ads

No Ads

In

In

Out

Out

1, 1

4,2

2,4

3,3

Can’t decide what B should play?

Enter Not enter

Ads 1,1 3,3.

No ads 2,4 4,2