Role of delay-based reward in the spatial cooperation

• We introduce delay-based reward into spatial cooperation, in which individuals can select various reward amounts on the basis of delay.
• We find that the intermediate differences of reward among kinds of delays are most important contributions to cooperation.
• We show that the individuals who select the fastest and smallest reward in evolving have strongest competition.

Strategy selection in games, a typical decision making, usually brings noticeable reward for players which have discounted value if the delay appears. The discounted value is measure: earning sooner with a small reward or later with a delayed larger reward. Here, we investigate effects of delayed rewards on the cooperation in structured population. It is found that delayed reward supports the spreading of cooperation in square lattice, small-world and random networks. In particular, intermediate reward differences between delays impel the highest cooperation level. Interestingly, cooperative individuals with the same delay time steps form clusters to resist the invasion of defects, and cooperative individuals with lowest delay reward survive because they form the largest clusters in the lattice.