The iterated Prisoner’s Dilemma in societies of deterministic players

Two players engaged in the Prisoner’s Dilemma have to choose between cooperation and defection, the pay-off of the players is determined by a weight w=(T,R,P,S)w=(T,R,P,S). For deterministic strategies p1,…,pnp1,…,pn we consider a society  S=S(ui:pi∣i=1,…,n)S=S(ui:pi∣i=1,…,n) formed by u=∑i=1nui individuals playing at random the IPD with weight ww. We introduce the concept of a ww-successful society as one where all individuals have eventually a non-negative pay-off. We discuss success of individuals and societies by means of quadratic forms associated to the pay-off matrix of the given set of strategies.