Repeated games. (Lecture 6)

Сентябрь 14, 2022

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Repeated games. (Lecture 6)

Содержание

2. Introduction Lectures 1-5: One-shot games The game is played just once, then the interaction ends. Players
3. Introduction Players may behave differently when a game is repeated. They are less opportunistic and prioritize
4. A model of price competition Two firms compete in prices. The NE is to set low
5. A model of price competition The equilibrium that arises from using dominant strategies is worse for
6. Finite repetition Games where players play the same game for a certain finite number of times.
7. Finite repetition When a one-shot game with a unique PSNE is repeated a finite number of
8. Infinite Repetition What if the interaction never ends? No final period, so no rollback. Players may
9. Trigger Strategies Tit-for-tat (TFT): choose the action chosen by the other player last period Defect Defect
10. Trigger Strategies Grim strategy: cooperate until the other player defects, then if he defects punish him
11. Trigger Strategies Tit-for-Tat is most forgiving shortest memory proportional credible but lacks deterrence Grim trigger is
12. Firm 1 Firm 2
13. Infinite repetition and defection Is it worth defecting? Consider Firm1. Cooperation: Firm 1 defects: gain 36
14. Infinite repetition and defection If Firm 1 keeps defecting: If Firm 1 reverts back to cooperation:
15. Discounting future payoffs Recall from the analysis of bargaining that players discount future payoffs. The discount
16. Discounting future payoffs Considering an infinitely repeated game, suppose that an outcome of this game is
17. Defection? Defecting once vs. always cooperate against a TFT player. Gain 36 in period 1; Lose
18. Defection? When r is high (r>minimum{1,2}, i.e. r>1 in this example), cooperation cannot be sustained. When
19. Games of unknown length Interactions don’t last forever: Suppose there is a probability p present value
20. Games of unknown length The effective rate of return R is the rate of return used
21. Games of unknown length We found that the condition for defecting against a TFT player is:
22. Example with asymmetric payoffs Firm 1 Firm 2
23. Example with asymmetric payoffs Firm 1: no change Defect once better than cooperate if: Defect forever
24. Example with asymmetric payoffs Firm 2: Defect once better than cooperate if: Defect forever better than
25. Experimental evidence from a prisoner’s dilemma game From Duffy and Ochs (2009), Games and Economic Behavior.
26. The Axelrod Experiment: Assessing trigger strategies Axelrod (1980s) invited selected specialists to enter strategies for cooperation
27. The Axelrod Experiment: Assessing trigger strategies In another experiment, some “players” were programmed to defect, some
28. The Axelrod Experiment: Assessing trigger strategies According to Axelrod, TFT follow the following rules: “Don’t be
29. q1 q2 NE=(240,240) Cournot in repeated games (180,180)
30. Cournot in repeated games In a one-shot Cournot game, the unique NE is that producers defect
31. Cournot in repeated games How to reconcile the Cournot model with the fact that many cartels
32. Cournot in repeated games High p also helps. Cartels are more likely to be stable in
33. Other factors affecting the possibilities of collusion I The more complex the negotiations, the greater the
34. Other factors affecting the possibilities of collusion II The incentive to defect from the cartel are
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Introduction
Lectures 1-5: One-shot games
The game is played just once, then the

interaction ends.
Players have a short term horizon, they are opportunistic, and are unlikely to cooperate (e.g. prisoner’s dilemma).
Firms, individuals, governments often interact over long periods of time
Oligopoly
Trade partners

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Introduction
Players may behave differently when a game is repeated. They are

less opportunistic and prioritize the long-run payoffs, sometimes at the expense of short-term payoffs.
Types of repeated games:
Finitely repeated: the game is played for a finite and known number of rounds, e.g. 2 rounds/repetitions.
Infinitely: the game is repeated infinitely.
Indefinitely repeated: the game is repeated for an unknown number of times. The interaction will eventually end, but players don’t know when.

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A model of price competition
Two firms compete in prices. The NE

is to set low prices to gain market shares.
They could obtain a higher payoff by cooperating (Prisoner’s dilemma situation)

Firm 1

Firm 2

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A model of price competition
The equilibrium that arises from using dominant

strategies is worse for every player than cooperation.
Why does defection occur?
No fear of punishment
Short term or myopic play
What if the game is played “repeatedly” for several periods?
The incentive to cooperate may outweigh the incentive to defect.

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Finite repetition
Games where players play the same game for a certain

finite number of times. The game is played n times, and n is known in advance.
Nash Equilibrium:
Each player will defect in the very last period
Since both know that both will defect in the last period, they also defect in the before last period.
etc…until they defect in the first period

Defect

Player 1

Player 2

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Finite repetition
When a one-shot game with a unique PSNE is repeated

a finite number of times, repetition does not affect the equilibrium outcome. The dominant strategy of defecting will still prevail.
BUT…finitely repeated games are relatively rare; how often do we really know for certain when a game will end? We routinely play many games that are indefinitely repeated (no known end), or infinitely repeated games.

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Infinite Repetition
What if the interaction never ends?
No final period, so no

rollback.
Players may be using history-dependent strategies, i.e. trigger/contingent strategies:
e.g. cooperate as long as the rivals do
Upon observing a defection: immediately revert to a period of punishment (i.e. defect) of specified length.

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Trigger Strategies
Tit-for-tat (TFT): choose the action chosen by the other player

last period

Defect

Cooperate

Defect

CONDITIONAL COOPERATION

RECIPROCITY

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Trigger Strategies
Grim strategy: cooperate until the other player defects, then if

he defects punish him by defecting until the end of the game

Defect

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Trigger Strategies
Tit-for-Tat is
most forgiving
shortest memory
proportional
credible but lacks deterrence
Grim trigger is
least forgiving
longest

memory
not proportional
adequate deterrence but lacks credibility

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Firm 1
Firm 2

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Infinite repetition and defection
Is it worth defecting? Consider Firm1.
Cooperation:
Firm 1 defects:

gain 36 (360-324)
If Firm 2 plays TFT, it will also defect next period:

324

360

216

defect

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Infinite repetition and defection
If Firm 1 keeps defecting:
If Firm 1 reverts

back to cooperation:
If defection, trade-off defection - return to cooperation

360

216

288

360

216

360

216

324

Gain: 36

Loss: 108

Gain: 36

Loss: 36

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Discounting future payoffs
Recall from the analysis of bargaining that players discount

future payoffs. The discount factor is δ= 1/(1+r), with δ < 1.
r is the interest rate
Invest $1 today ? get $(1+r) next year
Want $1 next year ? invest $1/(1+r) today
For example, if r=0.25, then δ =0.8, i.e. a player values $1 received one period in the future as being equivalent to $0.80 right now.

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Discounting future payoffs
Considering an infinitely repeated game, suppose that an outcome

of this game is that a player receives $1 in every future play (round) of the game, starting from next period.
Present value of $1 every period (starting from next period):

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Defection?
Defecting once vs. always cooperate against a TFT player. Gain 36

in period 1; Lose 108 in period 2.
Defect if:
Defecting forever vs. always cooperate against a TFT player. Gain 36 in period 1; Lose 36 every period ever after.
Defect if:

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Defection?
When r is high (r>minimum{1,2}, i.e. r>1 in this example), cooperation

cannot be sustained.
When future payoffs are heavily discounted, present gains outweigh future losses.
Cooperation is sustainable only if r<1, i.e. if future payoffs are not too heavily discounted.
Lesson: Infinite repetition increases the possibilities of cooperation, but r has to be low enough.

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Games of unknown length
Interactions don’t last forever: Suppose there is a

probability p<1 that the interaction will continue next period ? Indefinitely repeated games.
present value of 1 tomorrow is
Future losses are discounted more heavily than in infinitely repeated games, because they may not even materialize. Cooperation is more difficult to sustain when p<1 than when p=1.

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Games of unknown length
The effective rate of return R is the

rate of return used to discount future payoffs when p<1. R is such that:
i.e. the discount factor δ is lower when p<1.
R>r, and future payoffs are more heavily discounted, which decreases the possibilities of cooperation.

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Games of unknown length
We found that the condition for defecting against

a TFT player is:
e.g. suppose that r=0.05 ? no defection
Now assume that there is each period a 10% chance that the game stops: p=0.90.
? R=0.16 (still <1, hence no defection)
If instead p=0.5, then R=1.1, and there is defection (1.1>minimum{1,2}).

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Example with asymmetric payoffs
Firm 1
Firm 2

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Example with asymmetric payoffs
Firm 1: no change
Defect once better than cooperate

if:
Defect forever better than cooperate if:

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Example with asymmetric payoffs
Firm 2:
Defect once better than cooperate if:
Defect forever

better than cooperate if:
Cooperation may not be stable when r>0.66

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Experimental evidence from a prisoner’s dilemma game
From Duffy and Ochs (2009),

Games and Economic Behavior.
Initially 30% of players cooperate, and this increase to 80% with more repetitions. Trust between players increases over time and fewer of them defect.

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The Axelrod Experiment: Assessing trigger strategies
Axelrod (1980s) invited selected specialists to

enter strategies for cooperation games in a round-robin computer tournament.
Strategies specified for 200 rounds.
TFT obtained the highest overall score in the tournament.
Why did TFT win?
TFT's can adapt to opponents. It resists exploitation by defecting strategies but reciprocates cooperation.
Programs that defect suffer against TFT programs.
Programs that never defect lost against programs that defect.

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The Axelrod Experiment: Assessing trigger strategies
In another experiment, some “players” were

programmed to defect, some to cooperate, some to play trigger strategies such as TFT and grim.
The programs that do well “reproduce” themselves and gain in population. The losing programs lose population.
After 1000 rounds, TFT accounted for 70% of the population.
TFT does well against itself and other cooperative strategies.
Defecting strategies fare badly when their own kind spreads, and against TFT.

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The Axelrod Experiment: Assessing trigger strategies
According to Axelrod, TFT follow the

following rules:
“Don’t be envious, don’t be the first to defect, reciprocate both cooperation and defection, don’t be too clever.”
Folk theorem: two TFT strategies are best replies for each other (i.e. it is a Nash Equilibrium).
However, other Nash equilibria also exist, and may involve defecting strategies.

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q1
q2
NE=(240,240)
Cournot in repeated games
(180,180)

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Cournot in repeated games
In a one-shot Cournot game, the unique NE

is that producers defect rather than cooperate. Cooperation yields higher payoff, but is not stable.
Cartels do form, and governments may have to intervene to prevent cartel formation. Some cartels are unstable, but some are stable.

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Cournot in repeated games
How to reconcile the Cournot model with the

fact that many cartels are formed?
Repetition increases the possibilities of cooperation, provided that producers attach sufficient weight on future payoffs (low r).
“Short-termism” makes cartels less stable.

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Cournot in repeated games
High p also helps.
Cartels are more likely to

be stable in “static” industries, where producers know that they will have a very long-term relationship.
e.g. OPEC. The list of oil exporting countries is unlikely to change much over the next decades.
In “dynamic” industries, where market shares quickly change, collusion is less stable.

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Other factors affecting the possibilities of collusion I
The more complex the

negotiations, the greater the costs of cooperation (and create a cartel)
It is easier to form a cartel when…
Few producers are involved.
77% of cartels have six or fewer firms (Connor, 2003)
The market is highly concentrated.
Cartel members usually control 90%+ of the industry sales (Connor, 2003)
Producers have a nearly identical product.
If the products are different it is difficult to spot cheating because different products naturally have different prices

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Other factors affecting the possibilities of collusion II
The incentive to defect

from the cartel are larger when there are many producers. Consider an industry with N producers. π is the monopoly profit.
Profit if all producers cooperate: π /N
Profit if one defects: become a monopolist and get π
Profit if is being punished: 0
As the number of producers rises, the gain from defection increases:
π - π /N increases with N. With a high number of producers, the incentives to defect are strong.

Repeated games. (Lecture 6)

Содержание

IntroductionLectures 1-5: One-shot gamesThe game is played just once, then the

IntroductionPlayers may behave differently when a game is repeated. They are

A model of price competitionTwo firms compete in prices. The NE

A model of price competitionThe equilibrium that arises from using dominant

Finite repetitionGames where players play the same game for a certain

Finite repetitionWhen a one-shot game with a unique PSNE is repeated

Infinite RepetitionWhat if the interaction never ends?No final period, so no

Trigger StrategiesTit-for-tat (TFT): choose the action chosen by the other player

Trigger StrategiesGrim strategy: cooperate until the other player defects, then if

Trigger StrategiesTit-for-Tat ismost forgivingshortest memoryproportionalcredible but lacks deterrenceGrim trigger isleast forgivinglongest

Firm 1Firm 2

Infinite repetition and defectionIs it worth defecting? Consider Firm1.Cooperation:Firm 1 defects:

Infinite repetition and defectionIf Firm 1 keeps defecting:If Firm 1 reverts

Discounting future payoffsRecall from the analysis of bargaining that players discount

Discounting future payoffsConsidering an infinitely repeated game, suppose that an outcome

Defection?Defecting once vs. always cooperate against a TFT player. Gain 36

Defection?When r is high (r>minimum{1,2}, i.e. r>1 in this example), cooperation

Games of unknown lengthInteractions don’t last forever: Suppose there is a

Games of unknown lengthThe effective rate of return R is the

Games of unknown lengthWe found that the condition for defecting against

Example with asymmetric payoffsFirm 1Firm 2

Example with asymmetric payoffsFirm 1: no changeDefect once better than cooperate

Example with asymmetric payoffsFirm 2:Defect once better than cooperate if:Defect forever

Experimental evidence from a prisoner’s dilemma gameFrom Duffy and Ochs (2009),

The Axelrod Experiment: Assessing trigger strategiesAxelrod (1980s) invited selected specialists to

The Axelrod Experiment: Assessing trigger strategiesIn another experiment, some “players” were

The Axelrod Experiment: Assessing trigger strategiesAccording to Axelrod, TFT follow the

q1q2NE=(240,240)Cournot in repeated games(180,180)

Cournot in repeated gamesIn a one-shot Cournot game, the unique NE

Cournot in repeated gamesHow to reconcile the Cournot model with the

Cournot in repeated gamesHigh p also helps.Cartels are more likely to

Other factors affecting the possibilities of collusion IThe more complex the

Other factors affecting the possibilities of collusion IIThe incentive to defect

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Infinite repetition and defection
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Discounting future payoffs
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In another experiment, some “players” were

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