Lecture 7 - Repeated Games

Repeated GamesEconomics 302 - Microeconomic Theory II: Strategic Behavior

Instructor: Songzi Du

compiled by Shih En LuSimon Fraser University

February 26, 2015

ECON 302 (SFU) Lecture 7 February 26, 2015 1 / 18

Introduction to Repeated Games

One class of much studied sequential games is repeated games.

These are games formed by playing a stage game over and overagain.

Applications: competition in an oligopoly, customer relations,cleaning a shared apartment, etc.

We will look at cases where this stage game has simultaneous moves,and where actions are perfectly observed after each stage.

We will focus on SPEs.


Discount Factors

Recall that if a player has discount factor and a payoff streamu0, u1, u2, ... in periods 0, 1, 2, ..., then the value of that payoff streamfrom period 0s point of view is the time-discounted sum:

t=0

tut = u0 + u1 + 2u2 + ...

We usually assume [0, 1].People are impatient.

You have a small chance of dying each day.

Firms (whose payoffs are usually assumed to be their profits) caninvest $1 today to get more than $1 (on average) tomorrow.

Aside: there is evidence that on top of this exponential discounting,people place a premium on present payoffs. Search for hyperbolicdiscounting and behavioral economics for more information.


If Stage Game Has One NE

Recall the prisoners dilemma:

C DC -2, -2 -5, -1D -1, -5 -3, -3

Unique NE is (D, D).

Suppose the game is played twice, and players have discount factor = 1. Lets draw the game tree.

What are the subgames? What are the strategies? Which one is thetit-for-tat strategy?

In SPE, what must happen in the last stage?

So what will players do in the first stage?


Twice-repeated prisoners dilemma ( = 1)



Both Player 1 and Player 2 play (D2, D3, D4, D5) in stage 2:



Both Player 1 and Player 2 play (D2, D3, D4, D5) in stage 2:

(C1, D2, D3, D4, D5) (D1, D2, D3, D4, D5)(C1, D2, D3, D4, D5) (5,5) (8,4)(D1, D2, D3, D4, D5) (4,8) (6,6)

Both players playing (D1, D2, D3, D4, D5) is the only subgameperfect equilibrium.


If Stage Game Has One NE (II)

This reasoning applies whenever a stage game has just one NE, andwhenever the number of stages is finite.

In SPE, in the last stage, the unique stage-game NE must be playedno matter what happened in earlier stages.

But given that, what players do in the second-to-last stage does notimpact what happens in the last stage.

Therefore, the stage games NE must be played in the second-to-laststage: no player has a reason not to play a best response.

Can keep going with this reasoning.

Conclusion: When a stage game with a unique NE is repeatedfinitely many times, the unique SPE of the repeated game is toplay the stage games NE at each stage.


If Stage Game Has Multiple NEs

Things get more interesting!

Its still true that an NE must be played in the last stage.

But now, which NE is played can depend on what has happenedbefore.

Sometimes, this can sustain a non-NE outcome in stages before thelast one!

(Note that this is allowed in SPE: any stage game but the last one,taken by itself, is NOT a subgame.)


Twice-repeated Polite Rude Game

Suppose that the following stage game is played twice, and thediscount factor is 1:

Nice MeanPolite 5,0 0,0Rude 6,-10 1,-10

In the stage game, Rude is player 1s dominant strategy.


Twice-repeated Polite Rude Game


Easy SPEs in Twice-repeated Polite Rude Game

Player 1 plays (R2, R3, R4, R5) in stage 2, Player 2 plays (N2, N3, N4,N5) in stage 2. ( = 1)



Player 1 plays (R2, R3, R4, R5) in stage 2, Player 2 plays (M2, M3, M4,M5) in stage 2. ( = 1)

It is a SPE to play an NE in the first stage, followed by a fixedNE in the second stage.



Player 1 plays (R2, R3, R4, R5) in stage 2, Player 2 plays (M2, M3, M4,M5) in stage 2. ( = 1)

It is a SPE to play an NE in the first stage, followed by a fixedNE in the second stage.ECON 302 (SFU) Lecture 7 February 26, 2015 13 / 18

A More Efficient SPE in Twice-repeated Polite Rude Game

A more efficient SPE:

Player 1: In stage 1, Polite; in stage 2, RudePlayer 2: In stage 1, Nice; in stage 2, Nice if player 1 was polite in stage 1, Mean

if player 1 was rude in stage 1

What if the discount factor is 0.1?



Player 1 plays (R2, R3, R4, R5) in stage 2, Player 2 plays (N2, N3, M4,M5) in stage 2. ( = 1)



Player 1 plays (R2, R3, R4, R5) in stage 2, Player 2 plays (N2, N3, M4,M5) in stage 2.

(N1, N2, N3, M4, M5) (M1, N2, N3, M4, M5)(P1, R2, R3, R4, R5) (5, 0) + (6,10) (0, 0) + (6,10)(R1, R2, R3, R4, R5) (6,10) + (1,10) (1,10) + (1,10)

(P1, R2, R3, R4, R5) and (N1, N2, N3, M4, M5) is a subgameperfect equilibrium when = 1.

(P1, R2, R3, R4, R5) and (N1, N2, N3, M4, M5) is not a subgameperfect equilibrium when = 0.1.


If Stage Game Has Multiple NEs

Playing an NE in each stage, without conditioning on previous play, isstill subgame-perfect.

There are also SPE where in some stages, the stage games NEis not played.

The stage-game best response may not be the overall best response,because it may lead to lower payoffs in the future.

But this can only happen when players care sufficiently about futurepayoffs.


Recap of Finitely Repeated Games

In a SPE of a repeated game, does a stage game NE have to beplayed every period?

No: sometimes, it is possible for players to incentivize each other toplay something else (usually an efficient outcome) by promisinghigher payoffs in later periods.

Example: if a NE of the stage game Pareto dominates another one,then players might play the good NE in a later period if they didwhat they were supposed to do, and the bad NE otherwise.

This is more likely to work when players are more patient.

However, a stage game NE must be played in the last period.

Takeaway message: repetition promotes efficiency/cooperation.


Lecture 7 - Repeated Games

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