# On Zero-sum Game Representation for Replicator Dynamics

## TL;DR

This paper demonstrates that for polynomial replicator dynamics, there exists a skew-symmetric, polynomial payoff matrix that can generate the same dynamics, highlighting limitations in inferring payoff matrices from observed data.

## Contribution

It establishes the existence of a canonical skew-symmetric polynomial payoff matrix for any polynomial replicator dynamics, advancing understanding of the relationship between payoffs and dynamics.

## Key findings

- Payoff matrices cannot be uniquely inferred from observed strategy frequencies.
- A canonical skew-symmetric polynomial payoff matrix always exists for polynomial replicator dynamics.
- The main result connects polynomial replicator dynamics with skew-symmetric payoff matrices.

## Abstract

Replicator dynamics have been widely used in evolutionary game theory to model how strategy frequencies evolve over time in large populations. The so-called payoff matrix encodes the pairwise fitness that each strategy obtains when interacting with every other strategy, and it solely determines the replicator dynamics. If the payoff matrix is unknown, we show in this paper that it cannot be inferred from observed strategy frequencies alone -- distinct payoff matrices can induce the same replicator dynamics. We thus look for a canonical representative of the payoff matrix in the equivalence class. The main result of the paper is to show that for every polynomial replicator dynamics (i.e., the vector field is a polynomial), there always exists a skew-symmetric, polynomial payoff matrix that can induce the given dynamics.