Capital Games and Growth Equilibria
Ben Abramowitz
TL;DR
Capital games reinterpret payoffs as capital observables governed by dynamic rules, and define growth equilibria as fixed points of best responses when players maximize the time-average growth rate of their capital. The core result is a precise correspondence: for positive capital games with linearlizable dynamics, growth equilibria are exactly the Nash equilibria of a derived standard game where utilities are obtained from the dynamics (additive: $u_i(a)=x_i(a)-w_i$, multiplicative: $u_i(a)=\ln x_i(a)-\ln w_i$). Existence is guaranteed under these conditions, and the computation of growth equilibria is PPAD-complete; the framework is reversible, allowing a standard Nash analysis to carry over to capital-game growth analysis. The approach clarifies when observable capital determines underlying preferences and offers a principled method to derive VNM utilities from growth dynamics, with implications for mechanism design and economic modeling. Overall, the paper connects ergodic growth, dynamic capital processes, and Nash theory to provide a rigorous pathway from time-evolving payoffs to equilibrium analysis.
Abstract
We examine formal games that we call "capital games" in which player payoffs are known, but their payoffs are not guaranteed to be von Neumann-Morgenstern utilities. In capital games, the dynamics of player payoffs determine their utility functions. Different players can have different payoff dynamics. We make no assumptions about where these dynamics come from, but implicitly assume that they come from the players' actions and interactions over time. We define an equilibrium concept called "growth equilibrium" and show a correspondence between the growth equilibria of capital games and the Nash equilibria of standard games.