Investment vs. reward in a competitive knapsack problem
This addresses the evolutionary problem of brain size optimization for species, though it is incremental as it applies existing methods to a specific biological-inspired scenario.
The study investigated the trade-off between brain size (metabolic cost) and competitive advantage in solving combinatorial problems using a two-player knapsack game with neural networks of varying sizes. It found that relative win rate follows a simple relation, with success increasing linearly when network sizes differ greatly but diminishing when they are comparable.
Natural selection drives species to develop brains, with sizes that increase with the complexity of the tasks to be tackled. Our goal is to investigate the balance between the metabolic costs of larger brains compared to the advantage they provide in solving general and combinatorial problems. Defining advantage as the performance relative to competitors, a two-player game based on the knapsack problem is used. Within this framework, two opponents compete over shared resources, with the goal of collecting more resources than the opponent. Neural nets of varying sizes 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 of a net with $N_A$ neurons against one with $N_B$. Success increases linearly with investments in additional resources when the networks sizes are very different, i.e. when $N_A \ll N_B$, with returns diminishing when both networks become comparable in size.