Discounting in the historic prisoner's dilemma

The standard spatial formulation of the Prisoner's Dilemma (PD) is ahistoric (memoryless): only results generated in the last round are taken into account to decide the next choice. In the standard historic model, all results coming from previous rounds are considered without discounting. The effect of geometric discounting in the historic spatial formulation of PD is assessed in this work. The fate of a single isolated defector surrounded by cooperators and that of some groups of cooperators surrounded by defectors are studied, varying both the discount factor (alpha) and the temptation (b, the parameter which characterizes the PD game). The effect of noise and stochastic rules are also assessed in the first scenario. The most important finding is that there are some range of parameter values (b, alpha) for which the evolution dynamics becomes unexpected.