An Information-Constrained Model for Ultimatum Bargaining

We argue for the use of the principle of maximum entropy to carry out inference in experimental eco- nomics. In particular we take the ultimatum game as a case study. We derive the Logit equilibrium by maximizing Shannon's informational entropy subject to behavioral constraints. This provides an effective way to translate behavioral hypotheses into theoretical distributions that are candidates to characterize em- pirical frequencies when performing experiments. Based on this approach we present two maximum entropy models applied to the ultimatum game. The first one assumes that the payoff functions of agents playing the game depend only on the portion of the money prize they obtain at the end of the game. The second one introduces an additional fairness constraint to represent the behavioral hypothesis that players also follow altruistic motivations. Each model suggests a particular distribution of offers that we can compare to empirical distributions from data gathered from experimental results. We build a database containing observed interactions of simple ultimatum game experiments conducted by Henrich et al. (2004), Ensminger & Henrich (2014), and Andreoni & Blanchard (2006).The data consists of 1,016 observations of demands made by proposers in the standard ultimatum game interaction. Out of these demands, a total of 636 report whether the demand was accepted or rejected, allowing us to derive the joint probability distribution of demands and acceptance/rejection. The experiments conducted by by Henrich et al., and Ensminger & Henrich consists on ultimatum experiments performed around the world on small scale societies. On the other hand, the experiments conducted by Andreoni & Blanchard were implemented to individuals from the University of Wisconsin-Milwaukee. The information distinguishability index shows that the fairness constrained model recovers 90% of the information in the marginal distribution of demands, in contrast with the 60% recovered by the non-fairness constrained model. We also estimate the fairness constrained model on the joint distribution of demands and quantal responses, recovering 87% of the information contained in the data, in contrast with the 52% recovered by the non-fairness constrained model.