Effect of the presence of empty sites on the evolution of cooperation by costly punishment in spatial games

Cooperation and spiteful behavior are still evolutionary puzzles. Costly punishment, for which the game payoff is the same as that of spiteful behavior, is one mechanism for promoting the evolution of cooperation. A spatially structured population facilitates the evolution of either cooperation or spite/punishment if cooperation is linked explicitly or implicitly with spite/punishment; a cooperator cooperates with another cooperator and punishes/spites the other type of player. Different updating rules in the evolutionary game produce different evolutionary outcomes: with one updating rule—the score-dependent viability model, in which a player dies with a probability inversely proportional to the game score and the resulting unoccupied site is colonized by one player chosen randomly—the evolution of spite/punishment is promoted more than with the other updating rule—the score-dependent fertility model, in which, after a player dies randomly, the site is colonized by a player with a higher game score. If the population has empty sites, spiteful players or punishers should have less chance to interact with others and then spite/punish others. Thus the presence of empty sites would affect the evolutionary dynamics of spite/punishment. Here, we investigated whether the presence of empty sites discourages the evolution of spite/punishment in both a lattice-structured population and a completely mixing population where players interact with others randomly, especially when the score-dependent viability model is adopted. In the lattice-structured population adopting this viability model, the presence of empty sites promoted the evolution of cooperation and did not reduce the effect of spite/punishment. In the completely mixing population, the presence of empty sites did not promote evolution of cooperation by punishment. The evolutionary dynamics of the score-dependent viability model with empty sites were close to those of the score-dependent fertility model.