Game Theory 17.1 Tit-for-Tat Reputation: Consider the following Prisoner\'s Dile
ID: 1216806 • Letter: G
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Game Theory
17.1 Tit-for-Tat Reputation: Consider the following Prisoner's Dilemma: Ti Player 2 C 1,1 -1,2 Player 1 2, -1 0,0 D Player 2 has payoffs given by the matrix while player 1 has one of two types: he is either a strategic type or a tit-for-tat TFT type. The TFT type uses the following strategy: in the first period he always cooperates; in every period t 1 thereafter he plays what his opponent played in period t 1. Let p be the probability that Nature chooses player 1 to be the TFT type. Assume that the players do not discount future payoffs a. What is the perfect Bayesian equilibrium if the game is played twice? b. If the game is played three times, is there an equilibrium in which the strategic type of player defects in the first period?Explanation / Answer
In a Bayesian game, one has to specify strategy spaces, type spaces, payoff functions and beliefs for every player. A strategy for a player is a complete plan of action that covers every contingency that might arise for every type that player might be. A strategy must not only specify the actions of the player given the type that he is, but must specify the actions that he would take if he were of another type. Strategy spaces are defined as above. A type space for a player is just the set of all possible types of that player. The beliefs of a player describe the uncertainty of that player about the types of the other players.
The tit for tat game theory is an expression in the mathematical area of game theory, relevant to a problem called the iterated prisoner's dilemm. "Tit for tat with forgiveness" is sometimes superior. When the opponent defects, the player will occasionally cooperate on the next move anyway. This allows for recovery from getting trapped in a cycle of defections. The exact probability that a player will respond with cooperation depends on the line-up of opponents. The reason for these issues is that tit for tat is not a subgame perfect equilibrium, except under knife-edge conditions on the discount rate. If one agent defects and the opponent cooperates, then both agents will end up alternating cooperate and defect, yielding a lower payoff than if both agents were to continually cooperate. While this subgame is not directly reachable by two agents playing tit for tat strategies, a strategy must be a Nash equilibrium in all subgames to be subgame perfect.
Tit for two tats is similar to tit for tat in that it is nice, retaliating, forgiving and non-envious, the only difference between the two being how forgiving the strategy is. A simultaneous move game in which each player has a dominant strategy — and in which the resulting Nash equilibrium is inefficient. This type of game is often referred to as the “Prisoners’ Dilemma,” and it occupies a particularly important place in microeconomics because it so starkly illustrates how strategic behavior can lead to outcomes that can be improved. Upon through some type of non-market institution. The name “Prisoners’ Dilemma” has its origins in the 1950’s when Albert Tucker (1905-1995), a mathematician and dissertation advisor to the young John Nash. he n players in the game. The equilibrium strategies are the strategies players pick in trying to maximize their individual payoffs, as distinct from the many possible strategy profiles obtainable by arbitrarily choosing one strategy per player. Equilibrium is used differently in game theory than in other areas of economics. In a general equilibrium model, for example, an equilibrium is a set of prices resulting from optimal behavior by the individuals in the economy. In game theory, that set of prices would be the equilibrium outcome, but the equilibrium itself would be the strategy profile– the individuals’ rules for buying and selling– that generated the outcome. People often carelessly say “equilibrium” when they mean “equilibrium outcome,” and “strategy” when they mean “action.”
The strategy sd i is a dominated strategy if it is strictly inferior to some other strategy no matter what strategies the other players choose, in the sense that whatever strategies they pick, his payoff is lower with sd i . Mathematically, sd i is dominated if there exists a single s0 i such that i(sd i , si) < i(s0 i, si) s
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