Three-strategy N-person snowdrift game incorporating loners

The N-person snowdrift game is generalized to incorporate a third strategy. In addition to the cooperative C and non-cooperative D strategies, a strategy L representing a loner behavior is introduced. Agents taking on the L strategy (L-agents) do not contribute to the game as the C-agents do but they do not take advantage of the C-agents. Instead, they would rather settle with a fixed payoff L. Dynamical equations governing the time evolution of the frequencies of the strategies in a well-mixed population are derived. The dynamics and the frequencies of the steady state reveal the rich behavior resulting from the interplay between the payoff r, which promotes the non-cooperative behavior, and L. Detailed studies on how a system evolves indicated that the steady state could be an AllL, AllC, or C+D state, depending on the parameters r, L, and group size N. In contrast, only a C+D state results for r>0 and an AllC state is possible only at r=0 without the strategy L. With the strategy L, the AllC phase occupies a finite, though tiny, region of the r-L parameter space. The L-agents play an important role in the dynamics leading to the AllC phase. They help eliminate the D strategy in the transient and later only to be replaced by the C strategy. Phase diagrams in the r-L space are presented for different values of N. The strategy L plays two roles. It leads to an AllL phase and helps give an AllC phase. An algorithm for simulating the model numerically is described and validated. The algorithm will be useful in studying our model in various structured populations.