PRISONERS DILEMMA By Ajul Shah Hiten Morar Pooja

PRISONER’S DILEMMA By Ajul Shah, Hiten Morar, Pooja Hindocha, Amish Parekh & Daniel Castellino

PRISONER’S DILEMMA EXPLAINED • The Prisoner’s Dilemma constitutes a problem in game theory. In its classical model Prisoner’s Dilemma is presented as follows: • Hannah and Sam are interviewed separately. • They have the option to either cooperate or defect.

PRISONER’S DILEMMA EXPLAINED • The Payoff Matrix:

NASH EQUILIBRIUM • The dominant strategy for Rachel is to defect. • This is also true for Sam. • In the one shot game the equilibrium which will be set up is both convicts defecting. • Nash Equilibrium: - “A combination of strategies, one for each player, such that neither player can do better by picking a different strategy, given that the other player adheres to their own strategy. ”

ARTICLE 1 – “ HOW REPUTATION EFFECTS DUE TO INFORMATIONAL ASYLUM CAN GENERATE SOME SORT OF CO-OPERATIONAL BEHAVIOUR” BY KREPS, MILGROM, ROBERTS & WILSON • Incomplete information about another player can lead to some measure of co-operation in finitely repeated Prisoner’s Dilemma. • Consider the following game: • Assumptions: • a>1 • b<0 • a+b<2

ARTICLE 1 – “ HOW REPUTATION EFFECTS DUE TO INFORMATIONAL ASYLUM CAN GENERATE SOME SORT OF CO-OPERATIONAL BEHAVIOUR” BY KREPS, MILGROM, ROBERTS & WILSON Each player is informed simultaneously about the action of their opponent and recalls their own previous action. • The game above has a unique Nash Equilibrium which involves each player choosing to ‘fink’ at each stage. • For sequential equilibrium in a game where there is incomplete information: - Player has to think about evolution of the game. - Come up with an optimal strategy.

ARTICLE 1 – “ HOW REPUTATION EFFECTS DUE TO INFORMATIONAL ASYLUM CAN GENERATE SOME SORT OF CO-OPERATIONAL BEHAVIOUR” BY KREPS, MILGROM, ROBERTS & WILSON • Model 1 – Player 1 plays ‘tit for tat’ as he is not sure Player 2 will act rationally. • Model 2 – There is 2 sided uncertainty about stage payoffs. • Conclusion: – Although it is more beneficial for both players to cooperate, it cannot be guaranteed that it will occur in every single sequential equilibrium.

ARTICLE 2 – “END BEHAVIOUR IN SEQUENCE OF FINITE PRISONER’S DILEMMA SUPER GAME” BY SELTON & STOECKER • Assumptions: – Game is repeated for a fixed number of times known to both players in advance. – Players act in a rational manner. • Conclusions: – Tacit co-operation > non co-operation. – Explained by player’s monetary incentive – incentive to gain utility from co-operation. – Results that don’t follow suit maybe justified by Krep’s idea of incomplete information of the other player’s payoff.

ARTICLE 3 – “BOUNDED COMPLEXITY JUSTIFIES COOPERATION IN THE FINITELY REPEATED PRISONER’S DILEMMA’’ BY ABRAHAM NEYMAN • The paper suggested that repeated prisoner’s dilemma is different from other Nash Equilibria. • Suggestions from the paper: – As the number of repetitions increases, the chance of cooperation increases. – Incomplete information about players’ opinions, motivations or behaviours can explain the observed cooperation. • Conclusions : If the players are restricted to using finite automata of a fixed size, then for a sufficiently large number of repetitions, there is an equilibrium that yields a payoff close to the cooperative one.

t p k • Probability that in round t a randomly chosen subject has intention to deviate periods 1 – k. t k • S k = ∑ m=1 t p m • Probability that in t the subject has intention to deviate.

THE QUESTION Can co-operation ever be achieved in the finitely repeated Prisoner’s Dilemma?