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Natural selection drives species to develop brains, with sizes that increasewith the complexity of the tasks to be tackled. Our goal is to investigate thebalance between the metabolic costs of larger brains compared to the advantagethey provide in solving general and combinatorial problems. Defining advantageas the performance relative to competitors, a two-player game based on theknapsack problem is used. Within this framework, two opponents compete overshared resources, with the goal of collecting more resources than the opponent.Neural nets of varying sizes are trained using a variant of the AlphaGo Zeroalgorithm. A surprisingly simple relation, $N_A/(N_A+N_B)$, is found for therelative win rate of a net with $N_A$ neurons against one with $N_B$. Successincreases linearly with investments in additional resources when the networkssizes are very different, i.e. when $N_A \ll N_B$, with returns diminishingwhen both networks become comparable in size.

Investment vs. reward in a competitive knapsack problem