Question

A. Why might cooperation emerge from a repeated game even though the single shot game yields non-cooperation as the dominant strategy?

B. Why do strategic moves affect the outcome of a one-shot sequential game, but not a one-shot simultaneous game? Does this change if the games change from one-shot to repeated?

Answer 1

A. There are two kinds of repeated games- finite and infinite horizon games.

A rather famous example in Game Theory is the Prisoner's Dilemma Game where the dominant strategy Nash equilibrium is (Fink,Fink). This is the non-cooperative equilibrium for a one shot game. However, it is found that when this game is repeated, the dominant strategy changes to the co-operative equilibrium (silence,silence) which yields a higher payoff for both prisoners.

In repeated games, it becomes easier to force players to stick to the co-operative equilibrium by punishing players who deviate from this strategy in the next period. The threat has to be credible so that the player finds more utility in co-operating every period rather than short term gains by deviating. Many games make use of these strategies such as the Grim Trigger or the Stick and Carrot strategy. In the Trigger strategy player 1 punishes player 2 for deviating by playing a strategy that reduces ALL his future payoffs. In contrast, the stick and carrot strategy is less severe. Player 1 punishes player 2 for deviating in one period (stick) but promises to choose the co-operative strategy next period if player 2 reverts back to good behaviour or the co-operative strategy(carrot). If player 2 continues with the bad behaviour, player 1 repeatedly chooses the "Stick" strategy in order to punish player 2.

Note- When there is a known, finite end to a game with a unique equilibrium there will be no change in its equilibrium outcome no matter how many times it is repeated. This is also true for zero or constant sum games.

B. In a one shot sequential game, players move one after the other. Hence, they are well aware of each others' moves and play strategies according to their respective Best Response functions. However, in a one shot simultaneous form game, the players move at the same time without knowing what action the other player chose. Simultaneous games are also refereed to as games with imperfect information as you know nothing about your opponent's moves.

As mentioned above, if a one shot simultaneous game with a unique equilibrium is repeated a finite number of times, there will be no change in its equilibrium outcome. But these results might change in the case of two or more Nash equilibria in a repeated simultaneous game where incentives and punishments can be introduced.