A Bayesian graphical model for matching law behavior

In environments that have multiple sources of reward available simultaneously, organisms generally invest effort proportionately to exploit them. This general and robust behavioral finding is known as “the matching law” and has been documented in humans and several non-human species, under a variety of experimental variations as well as in observational settings. In a typical matching preparation, two alternatives of reward would pay at different rates. For example, alternative A would deliver, on average, two rewards per minute, while alternative B would deliver only one. Under these constraints organisms tend to invest, on average, twice as many resources exploiting alternative A than B. In other words, the matching law states that the relative rate of investment is a linear function of the relative rate of reward. The two parameters of the linear matching model represent, respectively, sensitivity to relative reinforcement rates and bias to some alternative regardless of its relative richness. The perfect matching relation is the special case of this linear model in which the organism shows equal sensitivity to all alternatives (slope 1) and no bias towards any (intercept 0). Deviations from that equilibrium constitute an active field of research in that they may account for suboptimal behavior. Crucial to this endeavor, especially when sample sizes are small, is the proper accounting of statistical uncertainty over inferred parameter values. In this work we present a novel Bayesian graphical model to quantify matching behavior and show its potential by analyzing previously published datasets. The key contribution of the model lies in its generative nature: while most published analyses under the matching law framework summarize and collapse data across sessions, subjects, or both, our model is able to generate raw counts of responses directly for each individual under every experimental condition. Furthermore, hierarchical extensions of the model allow the inclusion of differences and effects both at individual and session level, paving the way for explanatory extensions to account for potential sources of optimal or suboptimal behavior. The Bayesian implementation we propose naturally quantifies the evidence in favor or against the matching equilibrium for each unit and for the hyperparameters that control their hierarchical distribution without loss of uncertainty. These novel tools may shed new light on a behavioral finding that has been central in animal decision-making over the last half century.