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 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.
