The Replicator Dynamics of Zero-Sum Games Arise from a Novel Poisson Algebra

TL;DR

This paper introduces a novel non-canonical bracket combining Poisson and Nambu brackets to explain the replicator dynamics in zero-sum games, revealing new geometric and algebraic structures.

Contribution

It presents a new hybrid bracket framework that generalizes the Poisson structure for zero-sum games, linking dynamics, conserved quantities, and geometry.

Findings

The non-canonical bracket satisfies all Poisson identities except Jacobi in general.

The mediating function influences the conservation properties and the metric structure.

Special cases of the mediating function restore the Jacobi identity, enabling a symplectic-like geometry.

Abstract

We show that the replicator dynamics for zero-sum games arises as a result of a non-canonical bracket that is a hybrid between a Poisson Bracket and a Nambu Bracket. The resulting non-canonical bracket is parameterized both the by the skew-symmetric payoff matrix and a mediating function. The mediating function is only sometimes a conserved quantity, but plays a critical role in the determination of the dynamics. As a by-product, we show that for the replicator dynamics this function arises in the definition of a natural metric on which phase flow-volume is preserved. Additionally, we show that the non-canonical bracket satisfies all the same identities as the Poisson bracket except for the Jacobi identity (JI), which is satisfied for special cases of the mediating function. In particular, the mediating function that gives rise to the replicator dynamics yields a bracket that satisfies JI. This neatly explains why the mediating function allows us to derive a metric on which phase flow is conserved and suggests a natural geometry for zero-sum games that extends the Symplectic geometry of the Poisson bracket and potentially an alternate approach to quantizing evolutionary games.