Fibonacci Numbers and Equilibria in Large “Neighborhood” Games

Abstract

We deal with a game-theoretic framework involving a finite number of infinite populations, members of which have a finite number of available strategies. The payoff of each individual depends on her own action and distributions of actions of individuals in all populations. A method to find all equilibria is discussed which requires the search through all nonempty subsets of the types’ strategy sets, assigning equilibria to each of them. The method is then used to find equilibria in two types of “neighborhood” games in which there is one type of player who has strategies in V = {1, . . . , k} and payoff functions Ф(j; p) = α · p _j−1 + p _j + α · p _j+1 for j = 2, . . . , k − 1 and: in the case of “chain” games Ф(1; p) = p ₁ + α · p ²; Ф(k; p) = α p _k−1 + p k; in the case of “circular” games Ф(1; p) = α · p k + p ₁ + α · p ₂; Ф(k; p) = α · p _k−1 + p k + α p ₂; (in both cases 0 ≤ α ≤ ½; p is a distribution on V). The Fibonacci numbers are used to determine the coordinates of equilibria in the case α = 1/3, for other values of ? we need to construct numerical Fibonacci-like sequences which determine, in an analogous manner, coordinates of equilibria. An alternative procedure makes use of some numerical Pascal-like triangles, specially constructed for this purpose.