2×2 Games/Well-mixed populations: Difference between revisions

[[Image:Well-mixed 2x2 Games.png|thumb|300px|Four basic evolutionary scenarios of 2×2 Games in the \(S,T\)-plane (with \(R=1, P=0\) ): (a) Dominance of type \(B\) (Prisoner's Dilemma); (b) Stable co-existence of \(A\) and \(B\) types (Snowdrift Game, Hawk-Dove Game, Chicken Game); (c) Bi-stability - the evolutionary outcome depends on the initial configuration (Staghunt Game); (d) Dominance of type \(A\) (Bi-product mutualism). The color code indicates the equilibrium frequency of type \(A\) ranging from low, red to intermediate, green and high, blue. In region (c) the color indicates the size of the basin of attraction of state \(B\).]]

In well-mixed populations the equilibrium fractions of cooperators and defectors are easily calculated using the replicator equation. If \(x\) denotes the fraction of cooperators (and \(1-x\) the fraction of defectors) then their evolutionary fate is given by

where \(P_A\) and \(P_B\) denote the average payoffs of type \(A\) and type \(B\) players, respectively. The replicator equation basically states that the more successful strategy, i.e. the one with the higher payoff will increase in abundance. The above equation has three equilibria: two trivial ones with \(x_1=0\) and \(x_2=1\) as well as a non-trivial equilibrium for \(P_A=P_B\) which leads to

The replicator equation allows to shift and normalize the payoffs without affecting the dynamics because the performance of cooperators and defectors only depends on the relative payoffs, i.e. on payoff differences. For this reason we can set \(R=1\) and \(P=0\) without loss of generality. Note that the equilibrium \(x_3\) does not necessarily exist, i.e. lie in the interval \([0,1]\). This gives rise to four basic evolutionary scenarios discussed below.