The replicator dynamics of zero-sum games arise from a novel poisson algebra

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 geome-try for zero-sum games that extends the Symplectic geometry of the Poisson bracket and potentially an alternate approach to quantizing evolutionary games. (C) 2021 Elsevier Ltd. All rights reserved.

Automated summary

The replicator dynamics for zero-sum games emerge from a non-canonical bracket that combines elements of a Poisson Bracket and a Nambu Bracket. This bracket is parameterized by both the skew-symmetric payoff matrix and a mediating function, which plays a critical role in the dynamics. The mediating function also leads to the definition of a natural metric for phase flow conservation, showing potential implications for quantizing evolutionary games.