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Unformatted text preview: INTRODUCTION TO REPEATED GAMES …--PRISONER’S DILEMMA and OTHER EXAMPLES A REPEATED GAME CONSISTS OF REPETITION OF THE SAME STAGE GAME. THE STAGE GAME ITSELF COULD BE ANY FINITE GAME… …. … ROUND 1 ROUND 2 ROUND N IN A REPEATED GAME, A PLAYER’S STRATEGY CAN DEPEND ON THE HISTORY OF THE GAME… WHAT OTHERS HAVE DONE IN PREVIOUS ROUNDS… Examples of repeated games… 1. Buyer-seller repeated purchases : (In each round , the buyer B decides on whether to buy from seller S; if B decides to buy, S either offers “good quality” or “bad quality”. The next round they play the same game…) 2. Employer and worker interact repeatedly , playing the same “stage game”: (Each month , Worker decides on whether to work hard or shirk, and the employer decides on whether to fire or keep the worker…) 3. Rival firms compete in the same market, they repeatedly play the same price setting stage game. (Each “period” Pınar and SEK decide simultaneously determine their prices). 4. Interaction between countries can be thought of as a repeated play of a stage game. (Each period, each OPEC member country decides on whether to respect its production quota or produce more than its quota) 5. Family members play the same stage game over and over again (the parent each day decides on whether to enforce rules (punishing or not), the child decides each day on whether to obey the rule // - // husband and wife decide each day on how much time to allocate to the family). 1 STAGE GAME STAGE GAME STAGE GAME RULES IN REPEATED GAMES (which we analyze in this course) • All the actions played in a stage game are observed at the end of that stage game (before the new stage game is played). • The payoff of a player in the repeated game is the (discounted) sum of his payoffs from each stage game (round). Fix a strategy profile. When this strategy profile is played, suppose that player 1 obtains the payoffs U 1 in round 0, U 1 1 in round 1, …., U N 1 in round N. Player 1’s discounted payoff in the repeated game is U 1 = U 1 + δU 1 1 + …. δ N U N 1 where δ is his discount factor. • The overall repeated game is itself a dynamic game. (therefore the appropriate solution concept is backward induction (if the stage game is one of perfect information) or SPE (if the stage game involves simultaneous moves). • Each time a stage game is played can be called “A ROUND” or “A PERIOD.” If the game changes from one round to the other, we have a different dynamic game, NOT a repeated game… EXAMPLE OF STAGE GAME… THE PRISONER’S DILEMMA Story.. The police has enough evidence to convict two suspects, A and B, but wants to obtain further evidence to strengthen their hands. To this end, they put the two suspects in separate cells. Each suspect is asked the following question, “is the other suspect guilty?” to which the suspects answer “yes” or “no.” If they both say “no” A and B will each get a three-month imprisonment term (suppose the payoffs they get is “0”). imprisonment term (suppose the payoffs they get is “0”)....
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- Fall '12
- Game Theory, SPE, stage game