October 3, 2025
Assume a finite number of stategies S =
\{s_1,\ldots,s_k\} and a fitness function
\phi\colon S\times S\to\boldsymbol{R}.
Assume n players at elements if \boldsymbol{Z}_n and an initial distribution
of strategies \Delta_0\boldsymbol{Z}_n\to
S.
At each round player i gets a payoff of \phi_i = \phi(s_{i-1},s_i) + \phi(s_i, s_{i+1}) from its nearest neighbors where s_i = \Delta_0(i) is the strategy of player i. I.e., everyone plays their nearest neighbor. The distribution at time 1 is determined by competition. In the deterministic case \Delta_1(i) the strategy having the greatest fitness in the [i-1,i,i+1] arena. In the Malthusian case the strategy is choosen at random from \{s_{i-1},s_i,s_{i+1}\} weighted by the corresponding fitness. The distribution \Delta_t takes positions to a probability density on S.
\delta_a\star\delta_b = \delta_{a + b}.
\delta_a\star\delta_b(E) = \int_x\int_y 1_E(x+y)\,\delta_a(x)\star\delta_b(y) = 1_E(a+b).
Distribution is \Delta\boldsymbol{Z}_n\to\P(S) where \Delta(i) is a probability measure on S.
The distribution (assuming independence) of \phi(s_{i-1},s_i) + \phi(s_i, s_{i+1}) is the convolution of measures.
\Delta_t \in B(S)^n where \mathcal{P}(S) is all probability measures on S.
Play P\colon\mathcal{P}(S)^n\to\mathcal{P}(S)^n by P\Delta_(i) = \phi(s_{i-1},s_i) + \phi(s_i, s_{i+1})