Investment vs. reward in a competitive knapsack problem

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 networks of varying sizes are trained using a variant of the AlphaGo Zero* algorithm, by generating training data out of self play games. 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 is much smaller than N_B , with returns diminishing when both networks become comparable in size.
References:
Silver, David, et al. "Mastering the game of go without human knowledge." nature 550.7676 (2017): 354-359.