Spike-based Decision Learning of Nash Equilibria in Two-Player Games

Humans and animals face decision tasks in an uncertain multi-agent environment where an agent's strategy may change in time due to the co-adaptation of others strategies. The neuronal substrate and the computational algorithms underlying such adaptive decision making, however, is largely unknown. We propose a population coding model of spiking neurons with a policy gradient procedure that successfully acquires optimal strategies for classical game-theoretical tasks. The suggested population reinforcement learning reproduces data from human behavioral experiments for the blackjack and the inspector game. It performs optimally according to a pure (deterministic) and mixed (stochastic) Nash equilibrium, respectively. In contrast, temporal-difference(TD)-learning, covariance-learning, and basic reinforcement learning fail to perform optimally for the stochastic strategy. Spike-based population reinforcement learning, shown to follow the stochastic reward gradient, is therefore a viable candidate to explain automated decision learning of a Nash equilibrium in two-player games. Author Summary: Socio-economic interactions are captured in a game theoretic framework by multiple agents acting on a pool of goods to maximize their own reward. Neuroeconomics tries to explain the agent's behavior in neuronal terms. Classical models in neuroeconomics use temporal-difference(TD)-learning. This algorithm incrementally updates values of state-action pairs, and actions are selected according to a value-based policy. In contrast, policy gradient methods do not introduce values as intermediate steps, but directly derive an action selection policy which maximizes the total expected reward. We consider a decision making network consisting of a population of neurons which, upon presentation of a spatio-temporal spike pattern, encodes binary actions by the population output spike trains and a subsequent majority vote. The action selection policy is parametrized by the strengths of synapses projecting to the population neurons. A gradient learning rule is derived which modifies these synaptic strengths and which depends on four factors, the pre- and postsynaptic activities, the action and the reward. We show that for classical game-theoretical tasks our decision making network endowed with the four-factor learning rule leads to Nash-optimal action selections. It also mimics human decision learning for these same tasks.