Replicator dynamics of the Hawk-Dove game with different stochastic noises in infinite populations

Abstract

This paper investigates the dynamic system of the Hawk-Dove game with deterministic and stochastic interference and considers the two-player game and N-player game, respectively. First, we introduce the replicator dynamics equations of the Hawk-Dove game for the two-player and N-player cases and prove the stability of the unique equilibrium point. When the system reaches a stable state, we find that the frequency of individuals adopting D-strategy depends upon the number of players. As the number of players increases, the frequency of D-strategy increases. Second, we ponder two forms of stochastic noise, additive noise and multiplicative noise, and use the potential function method to prove the stochastic stability of equilibria of the two-player and N-player game systems respectively. The additive noise causes the frequency of players who use D-strategy to oscillate at the stochastically stable equilibrium. Then, we use Itô’s formula and the strong law of large numbers to prove that the frequency of D-strategy is almost surely exponentially stable (ASES) and persistent. We obtain a threshold value for multiplicative noise, when the multiplicative noise value is greater than the threshold value, the frequency of D-strategy x=0 is ASES. Conversely, when the multiplicative noise value is less than the threshold value, the frequency of D-strategy is persistent. The results show that noise interference has an important effect on the replicator dynamics of Hawk-Dove game, this improves the study of game dynamics without noise. Finally, numerical simulations are used to show our theoretical results.