Finite iterated prisoner's dilemma revisited: Belief change and end-game effect

We develop a novel Bayesian model for the finite Iterated Prisoner's Dilemma that takes into consideration belief change and end-game effect. According to this model, mutual defection is always the Nash equilibrium at any stage of the game, but it is not the only Nash equilibrium under some conditions. The conditions for mutual cooperation to be Nash equilibrium are deduced. It reveals that cooperation can be achieved if both players believe that their opponents are likely to cooperate not only at the current stage but also in future stages. End-game effect cannot be backward induced in repeated games with uncertainty. We illustrate this by analyzing the unexpected hanging paradox.