  Game Theory has been frequently applied in biological research since 1970s.
While the key idea of Game Theory is Nash Equilibrium, it is critical to
understand and figure out the payoff matrix in order to calculate Nash
Equilibrium. In this paper we present a dynamic programming implemented method
to compute 2x2 non-cooperative finite resource allocation game's payoff matrix.
We assume in one population there exists two types of individuals, aggressive
and non-aggressive and each individual has equal and finite resource. The
strength of individual could be described by a function of resource consumption
in discrete development stages. Each individual undergoes logistic growth hence
we divide the development into three stages: initialization, quasilinear growth
and termination. We first discuss the theoretical frame of how to dynamic
programming to calculate payoff matrix then give three numerical examples
representing three different types of aggressive individuals and calculate the
payoff matrix for each of them respectively. Based on the numerical payoff
matrix we further investigate the evolutionary stable strategies (ESS) of the
games.
