We combine choice data in the ultimatum game with the expectations of proposers elicited by subjective probability questions to estimate a structural model of decision making under uncertainty. The model, estimated using a large representative sample of subjects from the Dutch population, allows both nonlinear preferences for equity and expectations to vary across socioeconomic groups. Our results indicate that inequity aversion to one's own disadvantage is an increasing and concave function of the payoff difference. We also find considerable heterogeneity in the population. Young and highly educated subjects have lower aversion for inequity than other groups. Moreover, the model that uses subjective data on expectations generates much better in- and out-of-sample predictions than a model which assumes that players have rational expectations.

We derive bounds on the causal effect of belief-dependent preferences (reciprocity and guilt aversion) on choices in sequential two-player games without data on the (higher-order) beliefs of players. We show how informative bounds can be derived by exploiting a specific invariance property common to those preferences. We illustrate our approach by analysing data from an experiment conducted in Denmark. Our approach produces tight bounds on the causal effect of reciprocity in the games we consider. These bounds suggest there exists significant reciprocity in our population—a result also substantiated by the participants’ answers to a post-experimental questionnaire. On the other hand, our approach yields high implausible estimates of guilt aversion—participants would be willing, in some games, to pay at least three Danish crowns (DKK) to avoid letting others down by one DKK. We contrast our estimated bounds with point estimates obtained using data on stated higher-order beliefs, keeping all other aspects of the model unchanged. We find that point estimates fall within our estimated bounds, suggesting that elicited higher-order belief data in our experiment is weakly (if at all) affected by various reporting biases.