Investment vs. Reward in a Competitive Knapsack Problem

Abstract

Natural selection drives species to develop larger brains to tackle harder and harder general and combinatorial tasks. We seek to determine the metabolic cost of a larger brain compared to the advantage it gives in solving these problems, where advantage is defined as performance in comparison to competitors. For this purpose a two-player game based on the knapsack problem is used. In effect, two players compete over shared resources, with the goal to collect more resources than the opponent. Neural nets with respectively $N_A$ and $N_B$ hidden neurons are trained using a variant of the AlphaGo Zero algorithm. A surprisingly simple relation, $N_A/(N_A+N_B)$, is found for the relative win rate. Success increases linearly with investments in additional resources when the networks sizes are very different, i.e. when $N_A \ll N_B$, with diminishing returns when both networks become comparable in size.