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 co- operate, 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 CO- OPERATION 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 co- operation increases. – Incomplete information about players’ opinions, motivations or behaviours can explain the observed co- operation. 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 co- operative one.

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