GTAP Sessions 16-21

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gtapProf. J. Ajith Kumar, TAPMI, Manipal GAME THEORY NOTES

GAME THEORY & APPLICATIONSJ. Ajith Kumar, TAPMI, Manipal

Extensive Games with Perfect Information

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Two players A and B have to share Rs. 100 between themselves. The game is played as follows. Player A offers a share of the Rs. 100 to Player B – say Rs. X. Player B can choose to accept or decline the offer. If Player B accepts the offer, then she gets the Rs. X, while Player A gets Rs. 100–X. If Player B declines the offer, both players get nothing.

For example, assume that Player A offers Rs. 27 to Player B. If Player B accepts it, then Player A gets Rs. 73 and Player B, Rs. 27. If Player B declines, then both players get Rs. 0.

2

THE ULTIMATUM GAME

Let’s play this game between ourselves!

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gtapProf. J. Ajith Kumar, TAPMI, Manipal GAME THEORY NOTES 3

ENTRY GAMERead Problem 1 of Exercise Set D.

Set of Players

In Out

0, 0

1, 2

Challenger

Incumbent

2, 1

Acquiesce Fight

(Challenger, Incumbent)

Terminal history: a sequence of actions that leads to a terminal node.(In, Acquiesce), (In, Fight), (Out)

Decision node: A node where a player chooses an action

Terminal node: A terminal point at the end of a sequence of actions.

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ENTRY GAMEPlayer function: a function that assigns a player to every node in the game.

A node can be represented by the sub-history of the game that precedes it.P(Φ) = Challenger, P(In) = Incumbent.

Action sets availableThis is specified node-wise, not player-wise

A(Φ) = {In, Out}, A(In) = {Acquiesce, Fight}

Preferences of the playersRepresents for each player, his payoff for each terminal history in the game.

If ‘A’ denotes the Challenger and ‘B, the Incumbent, then,

uA(In, Acquiesce) = 2; uA(In, Fight) = 0; uA(Out) = 1

uB(In, Acquiesce) = 1; uB(In, Fight) = 0; uB(Out) = 2

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EXERCISE – Problem 2 of Exercise Set D.C D

3, 4

2

2, 1

E F

1

1, 24, 3

HG

2

EXERCISE – Problem 3 of Exercise Set D.

E F

G H 3, 1

0, 0

2, 0

1, 2

1

2

1

C DPlayers: 1, 2.

Terminal histories: (C, E, G), (C, E, H), (C, F), (D).

Player functions: P(Φ) = 1, P(C) = 2, P(C, E) = 1

Player preferences:u1(CG, E) = 1, u1(C, E, H) = 0, u1(C, F) = 3, u1(D) = 2.u2(C, E, G) = 2, u1(C, E, H) = 0, u1(C, F) = 1, u1(D) = 0.

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EXERCISE Problem 4 of Exercise Set D.

R E

B H HB

Ernesto

Karl

Rosa

2, 3, 2 1, 1, 1

HB

Rosa

1, 1, 1 3, 2, 3

HB

Rosa

2, 3, 2 1, 1, 1

HB1, 1, 1 3, 2, 3

HB

ErnestoErnesto

Players: Karl, Rosa, Ernesto (the utilities are listed in this order).Terminal histories: (R, B, B), (R, B, H), (R, H, B), (R, H, H), (E, B, B), (E, B, H), (E, H, B), (E, H, H).

Player functions: P(Φ) = Karl, P(R) = Rosa, P(E) = Ernesto, P(R, B) = Ernesto, P(R, H) = Ernesto….Player preferences: Write them….

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Strategies and Outcomes

C D

3, 4

2

2, 1

E F

1

1, 24, 3

HG

2

A player’s strategy specifies the action the player chooses for every history after which it is her turn to move.

in extensive games with perfect information

Strategies of Player 1{C, D}

Strategies of Player 2{EG, EH, FG, FH}

Strategy profiles in the game(C, EG), (C, EH), (C, FG), (C, FH), (D, EG), (D, EH), (D, FG), (D, FH)

In other words, a player’s strategy reveals the action she takes at each of the nodes at which it is her turn to move.

Here, ‘strategy’ does not mean ‘plan of action’.

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Strategies and Outcomesin extensive games with perfect information

A strategy profile is as if each player has decided before hand the action that she

would take if the game reaches each node where she has to make a move.

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Strategies and Outcomesin extensive games with perfect information

C D

3, 4

2

2, 1

E F

1

1, 24, 3

HG

2The terminal history of a strategy profile is called the outcome of that strategy profile.

For strategy profile s, outcome O(s) is the terminal history of s.

O(C, EG) = (C, E).O(D, EH) = (D, H).O(C, FH) = (C, F).

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0

EXERCISEProblem 6, 7 of Exercise Set D.

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Problem 8 of Exercise Set D.

In Out

0, 0

1, 2

Challenger

Incumbent

2, 1

Acquiesce Fight

INCUMBENT

Acquiesce Fight

CHAL

LEN

GER

In 2, 1 0, 0

Out 1, 2 1, 2

How do we interpret (Out, Fight) as a NE?

o ‘fight’ is not a credible threat to the challenger, since the challenger knows that if it enters, the incumbent is better off choosing ‘acquiesce’.

o (out, fight) is not a robust steady state. Why?

Finding NE by converting to strategic form

o (In, Acquiesce) is a robust steady state and can be expected.

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Nash Equilibrium of an Extensive Game

A strategy profile s* in an extensive game with perfect information is a Nash Equilibrium iff for every player i, his payoff is at least good as those in another profile (ri, s-i*) where ri is any strategy of player i and s-i* is the set of strategies of other players in s*.

i.e. ui[ O(s*) ] ≥ ui[ O(ri, s-i*) ] for all i.

EXERCISE Problem 9 of Exercise Set D.

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Is it practical to always convert an extensive game to its strategic form and solve for Nash Equilibrium?

Question

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Voting By Alternating Veto Problem 10 in Exercise Set D.

YZ

1

2, 2 3, 1

YX1, 3 2, 2

ZY

1, 3 3, 1

ZX

X

22

2

1, 3 1, 3 1, 3 1, 3 2, 2 2, 2 2, 2 2, 2

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

2, 2 3, 1 2, 2 3, 1 2, 2 3, 1 2, 2 3, 1

X

Y

Z

YXX YXY YZX YZY ZXX ZXY ZZX ZZY

There are two Nash equilibria: (Z, YXX) and (Z, ZXX).

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Problem 11b in Exercise Set D.

YZ

1

2, 2 3, 1

YX1, 3 2, 2

ZY

1, 3 3, 1

ZX

X

22

2

There are two Nash equilibria of the full game: (Z, YXX) and (Z, ZXX).Which of these is subgame perfect?

This game has 4 subgames. The full game (with sub-history Φ ) is a sub-game. The other three subgames are:

1, 3 2, 2

ZY2

1, 3 3, 1

ZX

2

2, 2 3, 1

YX

2

Subgame Perfect Equilibrium

First ask: What are the NEs of each of the subgames?

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Each subgame perfect equilibrium is a Nash Equilibrium. However, each Nash Equilibrium need not be subgame perfect.

A subgame perfect equilibrium is a strategy profile s* of the full game such that for each player i, who plays ri* in s*, the payoff obtained from the corresponding portion of s* in any subgame of the full game, is at least as good as what he would obtain by choosing an alternative strategy ri’, given that all other players collectively play s -i*.

A subgame perfect equilibrium is a strategy profile that induces a Nash equilibrium in every subgame.

In the above game, (Z, YXX) is a subgame perfect equilibrium, whereas (Z, ZXX) is not.

Problem 11b in Exercise Set D.Subgame Perfect Equilibrium (SPE)

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E F

G H 3, 1

0, 0

2, 0

1, 2

1

2

1

C D

Which NE of this game are SPE? Why?

1, 2 3, 1

0, 0 3, 1

2, 0 2, 0

2, 0 2, 0

CG

E F

CH

DG

DH

There are three NE in the game: (CH, F), (DG, E), (DH, E).

Of these, only (DG, E) is Subgame perfect.

Problem 12 in Exercise Set D.Subgame Perfect Equilibrium (SPE)

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q As the outcome of the players’ rational calculations about each others’ strategies.

E F

G H 3, 1

0, 0

2, 0

1, 2

1

2

1

C D

E.g. (DG, E) is the SPE of the above game. If D is actually played by Player 1, then Player 2 will not play E, nor will Player 1 ever play G… However, the SPE can be thought of as reflecting Player 1’s belief that if he plays C, then Player 2 will play E after reasoning that Player 1 will play G if the history CE is played out.

There may be parts of a player’s strategy that are inconsistent with other parts… however these can thought of as that player’s belief about what other players think the focal player will do, if the history preceding that action is played out.

InterpretationsSubgame Perfect Equilibrium (SPE)

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q A slightly perturbed steady state in which all players, on rare occasions, take non-equilibrium actions, so that after long experience each player forms correct beliefs about entire players’ other strategies. Thus, each player knows how the other players will behave in every subgame. Given these beliefs, no player wishes to deviate from her strategy either at the start of the game or after any history.

E F

G H 3, 1

0, 0

2, 0

1, 2

1

2

1

C DInterpretationsSubgame Perfect Equilibrium (SPE)

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Backward Induction

C D

0, 2

2

2, 1

E F

1, 33, 0

HG

2

1

C D

0, 2

2

2, 1

EF

1, 33, 0

HG

2

1

2, 1 1, 3

2, 1

Subgame perfect equilibrium : (C, EH)

A procedure to locate Subgame Perfect Equilibria

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Backward InductionA procedure to locate Subgame Perfect Equilibria

Length of a subgame: length of the longest history in the subgame

q Find optimal actions of players who move in subgames of length 1.

qMoving back: Take these actions as given and then find the optimal actions of players in subgames of length 2. In general, first find optimal actions in subgame of length k and then in that of length k+1.

q If there are multiple optimal actions for a given subgame, then retain all of them.

q Repeat the procedure until we complete the full game.

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C D

2, 0

2

2, 1

E F

1, 01, 1

HG

2

1

C D

2, 0

2

2, 1

E F1, 01, 1

HG2

1There are 4 combinations of player 2’s optimal actions: EG, EH, FG, FH

For each combination, what is Player 1’s optimal action?

For EG, it is both C and D. For EH, it is only C. For FG, it is only D. For FH, it is both C and D. The subgame perfect equilibria are:-(C, EG), (D, EG), (C, EH), (D, FG), (C, FH) and (D, FH).

Identify the subgame perfect equilibria in this game using backward induction

Backward Induction

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Backward induction identifies only the subgameperfect equilibria of the game. It cannot find Nash

Equilibria that are not subgame perfect.

Possible only in games of finite horizon. Games of infinite horizon cannot provide a definite starting

point that backward induction needs.

Two important realizationsBackward Induction

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How ‘Burning Bridges’ poses a credible threat

The enemy knows that you have no option but to fight and hence you will certainly fight. This means that attacking would lead to a worse outcome for the enemy than not attacking.

Model as a simple entry game in which the incumbent signals the new entrant that he will increase the industry capacity or decrease the selling price and tries to create a credible threat.

Problem 14 of Exercise Set D

Assignment

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More challenging exercises

q Entry game- multiple periods (Prob 15 of Ex Set D)

q Firm-union Bargaining (Prob 16 of Ex Set D)

q Synergistic Relationship (Prob 17 of Ex Set D)

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Three ways one can be cheated in a partnership

q Adverse Selection: When the partner makes false claims about his skills, abilities and resources, while entering into the partnership.

qMoral Hazard: When the partnership is underway, the partner offers less than what he promised at the beginning.

q Hold-up: When the partner misuses the power he has gained after one (but not the partner) has made a transaction-specific investment into the partnership.

STRATEGIC THINKING & GAME THEORY

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Two players A and B have to share Rs. 100 between themselves. The game is played as follows. Player A offers a share of the Rs. 100 to Player B – say Rs. X. Player B can choose to accept or decline the offer. If Player B accepts the offer, then she gets the Rs. X, while Player A gets Rs. 100–X. If Player B declines the offer, both players get nothing.

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THE ULTIMATUM GAME

What are the NE and SPE of this game?

Assume that player 1 can make only discrete offers, in multiples of p, with the highest offer possible being 100, and the lowest, 0.

Only 2 of these are SPEs:§ Player 1 plays the strategy “0” and Player 2, “accept whatever

player 1 offers”, i.e. “AAAAA… AA”.§ Player 1 plays the strategy “p” (lowest offer above 0) and

Player 2, “accept whatever player 1 offers, except 0”, i.e. “AAAAA… AR”.

There are many NEs in the game given by the family – Player 1 plays ‘x’ and Player 2 plays “accept x, reject offers below x, either accept or reject offers above x” i.e. “.……..ARRRRR”. Yet another NE is: Player 1 plays ‘0’ and Player 2 plays “reject all offers”, i.e. (0, RRRRR…R).

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THE ULTIMATUM GAME 1

2

100-X, X

Y N

0, 0

X

Assume that Plr 1 can make continuous offers with the highest offer possible being 100, and the lowest, 0. Plr 2 can do ‘A’ or ‘R’.

But, there is only 1 SPE:§ Player 1 plays the strategy “0” and Player 2, “accept whatever

player 1 offers”, i.e. “AAAAA… AA”.

Again many NEs in the game given by:–• Player 1 plays “x” and Player 2 plays “accept x, reject everything

below x and accept or reject offers above x” and • Player 1 plays “0” and Player 2 plays “reject all offers”.

Plr 1’s actions are continuous, while Plr 2’s are discrete.

What are the NE and SPE of this game?

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What it demonstratesAims to model people’s thinking in situations where they need to work with each other in alliances and/or teams and sharing of efforts and payoffs is involved.

Models relationships where one partner has a stronger say in the how the pie is divided between the two, while the other only has a power to nullify the pie. Can be relationships between people, communities, social classes, organizations, countries.

q In a stable equilibrium, one partner completely dominates the other and exploits him to the extreme.

q The equilibrium will not be reached if both partners perceive each other to be equally powerful, and/or if either or both partners have a sense of fear of rejection, sense fairness or sense of equity.

THE ULTIMATUM GAME

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The Hold-up Game

1

2

X-L, CL-X

A R

-L, 0

X

1

2

X-H, CH-X

A R

-H, 0

X

2

H L

Person 2 exerts effort L resulting in the smaller sized pie of size CL. Person 1 offers 0 amount to Person 2 and takes the small pie all for himself.

Problem 5 of Exercise Set E

Subgame Perfect Equilibrium

(L –AAAA…A–AAAA… A, 0) leading to the payoffs (–L, CL)

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Lessons: When human beings act perfectly rational in the “Homo Economicus” sense, it can lead to an outcomes that are sub-optimal for all. Here, Person 2 knows that Person 1 will try to grab the entire pie produced for himself, hence Person 2 is not motivated to produce a large pie.

If Person 1 (manager) is an ‘exploiter’ and Person 2 (worker) knows so, then both become losers.

The Hold-up GameProblem 5 of Exercise Set E

1

2

X-L, CL-X

A R

-L, 0

X

1

2

X-H, CH-X

A R

-H, 0

X

2

H L

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q Workers in a factory, who know that if they work hard, their owners will make a large profit. From past experience, they also know that the owners will appropriate most of it and give very little to the workers. The workers ask – “why should we work so hard?” and compromise on their work, leading to lesser productivity and lower quality. As a result, owners’ will make lesser profit, and will share even less with workers. Everyone loses.

q Corrupt politicians, IAS officers and leaders vs. staff working under them in government offices. Smaller pie means that the whole country suffers.

q Two companies A and B entering into a strategic alliance. Suppose B has to invest irreversibly in a large fixed asset as part of the alliance, A can do a hold-up on B after the investment is made. Suspecting this, B will be very cautious in entering into an alliance with A. As a result, the likelihood of alliance formation decreases… and potential benefit is lost.

Who should take the first step?



The Hold-up GameProblem 5 of Exercise Set E

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The Centipede GameProblem 6 in Exercise Set E.

1 2 1 2 1 2 1 2

S

C C C C C C C C

S S S S S S S

2, 0 1, 3 4, 2 6, 43, 5 5, 7 8, 6 7, 9

10, 8

What is the SPE?If players are made to play this game, will they actually play the SPE? Explain.

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Demonstrates how people reason about others’ application of rationality in real-life situations. Each person can potentially think – “what are other people more likely to do? Stop or Continue? If they are likely to continue, then I need not stop here too. But if they are highly likely to continue in the next round, let me continue too…!”

Shows how people trade-off between short-term ‘rational’ outcomes, and larger long-term ‘not-rational’ outcomes. Individual greed and mistrust between people can hamper larger outcomes for everyone that can accrue if there is long-term cooperation and understanding.

The Centipede GameProblem 6 in Exercise Set E.

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GTAP Sessions 16-21

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