ECON20005
COMPETITION AND STRATEGY
Lecture 10: Repeated Games
Recommended reading: Dixit, Skeath and McAdams, ch. 10
1
Recap:
• In many strategic situations, a player’s success depends upon his/her
actions being unpredictable.
• In such games, players may want to use mixed strategies.
• A mixed strategy is a probability distribution that a player uses to
randomly choose among available actions.
• Many humans are bad at randomising properly, so exploit the patterns of
your opponents!
• To avoid being exploited, use the mixed strategy that keeps your opponent
guessing what is best for him/her!
• If Nadal simply alternates between serving L and R, Federer can exploit this.
2
Recap:
• We can find the mixed strategy NE by using the following simple recipe:
1. Calculate player A’s expected payoff from choosing each pure strategy
taking player B’s mixed strategy as given.
2. Calculate player B’s mixed strategy that keeps player A indifferent
between her pure strategies. This gives player B’s equilibrium mixed
strategy.
3. Repeat steps 1 and 2 with reversed players to get player A’s equilibrium
mixed strategy.
3
Mixed Strategy NE – Empirical Evidence
• Do we see players (students subjects in laboratory experiments or
professionals) mixing between multiple strategies?
• If yes, are the mixing proportions in line with the equilibrium predictions?
• Is there any correlation between the actions used in different periods or are
players able to randomize?
• In laboratory experiments, the behavior of student subjects is largely
inconsistent with mixed strategy NE.
• Students do not choose actions according to the equilibrium proportions and they
exhibit serial correlation in their actions, rather than the serial independence
predicted by theory.
4
Mixed Strategy NE – Empirical Evidence
• Evidence from professional sports contests suggests that the on-the-
field behavior of professionals in situations requiring unpredictability is
more likely conform to the theory of mixed strategy NE.
• Walker and Wooders (2001) study first serves in tennis.
• They test whether the probability of winning is the same whether the player
who serves the ball serves to the right or to the left.
• Palacios-Huerta (2003) study penalty kicks in soccer and show that:
i. winning probabilities are statistically identical across strategies, and
ii. players’ choices are serially independent.
5
Mixed Strategy NE – Empirical Evidence
• This evidence suggests that behavior is consistent with game theory in
settings where the financial stakes are large and where the players have
devoted their lives to becoming experts.
• However, randomness poses a problem.
• Walker and Wooders (2001) find that the tennis players switch their
serves from left to right and vice versa too often to be consistent with
random play.
• This behaviour is consistent with the overwhelming experimental
evidence in psychology that when people try to generate “random”
sequences they generally “switch too often” to be consistent with
randomly generated choices (Wagenaar, 1972).
6
NEXT TOPIC: REPEATED GAMES
(Let’s play - MobLab)
7
Overview:
1. Introduction to the prisoners’ dilemma and repeated games
2. Repeated prisoners’ dilemma games with
• finite repetitions
• infinite repetitions
• unknown number of repetitions
3. Other solutions to the prisoners’ dilemma
4. Evidence
Prisoners’ Dilemma:
Prisoners’ dilemma games have three essential features:
1. Each player has two (or more) strategies.
2. Each player has a dominant strategy.
3. The dominance solution is worse for all players than one of the non-
equilibrium outcomes.
Remember the original prisoner’s dilemma game:
Repeated Games:
• Repeated games consist of the same (simultaneous-move) game
played repeatedly in successive periods.
• A repeated prisoners’ dilemma game is a one-shot prisoners’ dilemma
game played repeatedly.
Repeated Games:
• Repeated interactions give players the possibility to:
• learn about their opponents.
• punish their opponents in later rounds if they cheat/defect.
• build reputation.
• Hence, in repeated games, players can use the past to determine what
action they are going to take in the future.
• Players compare future benefits from continued cooperation (collusive
behaviour) to the immediate benefit of cheating.
Repeated Prisoners’ Dilemma – An Example:
• Recall the pizza-pricing game between Papa Gino’s and Corretto.
• The Nash Equilibrium was ($11, $11) where each restaurant was making
profits of $98.
• As shown in this week’s tutorial:
• By colluding and each choosing a price of $14.50, they could obtain
maximum profits of $110.25 each.
• In a one-shot game, they could however not commit to the profit
maximizing prices of $14.50.
• But what if the two restaurants choose prices each week?
Repeated Prisoners’ Dilemma – An Example:
• Assume the restaurants have only two actions:
• they can collude and charge $14.50, or
• they can “cheat” and charge $11.
• If Papa Gino’s charges Pp = 11 and Corretto Pc = 14.5, their profits are:
• We have the following Prisoners’ Dilemma game:
Finitely Repeated Prisoners’ Dilemma
• Assume Papa Gino’s and Corretto only have four weeks left in their
leases.
• We know how to solve this game: each restaurant uses backward
induction to determine what price to charge each week.
• Start with the final week: What strategies will the restaurants choose in
the fourth week?
Finitely Repeated Prisoners’ Dilemma
• Given what they do in the fourth week, what will they do in the third
week?
• We add 98 to both players’ payoffs when doing rollback.
• What about the second and first week?
• With finite number of repetitions, the same logic applies period by
period.
→ Use backward induction to solve for the rollback equilibrium.