Investment vs. reward in a competitive knapsack problem

TL;DR

This paper explores the trade-off between brain size and problem-solving advantage using a competitive knapsack game, revealing a simple relation for win rates based on neural network sizes.

Contribution

It introduces a novel game-based framework to quantify the advantage of larger neural networks in competitive resource allocation tasks.

Findings

Win rate ratio approximates N_A/(N_A+N_B)

Success increases linearly when one network is much smaller

Diminishing returns observed as network sizes become similar

Abstract

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.