# Schedule - Parallel Session 5 - Beliefs and Utility

WMG IMC Room 246 - 14:00 - 15:30#### Belief Formation and Updating in a Signalling Game Without Common Prior: an Experiment

Alex Possajennikov

#### Abstract

The paper investigates the formation and the evolution of beliefs in strategic interaction between two players, modelled as a signalling game. In the game one player (Sender) has private information, represented by type. Sender sends a message to the other player (Receiver). Receiver observes the message but not the type of Sender and chooses an action. The payoffs of the players depend on the type of Sender, the message and the action in such a way that there are two pure strategy equilibria of the game where different types of Sender send different messages and Receiver takes appropriate actions. Depending on the probabilities of different Sender’s types, either one or the other equilibrium is more efficient. In the experiment, the game is played for a number of rounds with random partners. In some sessions the underlying probability distribution of two Sender’s types is not revealed to the experiment participants while in others it is. Along with playing the game, subjects are asked to report their beliefs about the prior probability of Sender’s type (in sessions where this probability is not revealed), about the posterior probability of Sender’s type (after receiving a message) and about the action of Receiver (after sending a message). Belief elicitation is incentivized using a quadratic scoring rule (as in e.g. Nyarko and Schotter, 2002; see also the discussion in Offerman et al., 2009). The result of the experiment show that experimental subjects often start with diffuse beliefs about types and about choices of the other player, centred on 50% probabilities of each of the two possible outcomes (“principle of insufficient reason”, see e.g. Sinn, 1980). Subjects then update the beliefs in view of experienced outcomes (learning-from-experience, e.g. Hertwig et al., 2004; Yechiam and Busemeyer, 2005). For some parameter values, initial beliefs centred on 50% have a medium-run effect compared with the case where the probability of Sender types is revealed: the frequency of the efficient equilibrium is lower if an asymmetric probability distribution is not revealed although subjects report beliefs closer to the underlying distribution as play progresses. The results also indicate that the speed of updating beliefs is different for events involving objective uncertainty (Sender’s types) and events involving strategic uncertainty (player’s choices). Fitting a model of belief updating that incorporates a strength of initial belief and a forgetting parameter, the estimated initial belief strength is higher for beliefs about Sender’s types than for beliefs about players’ choices. It thus appears that beliefs about strategic uncertainty are updated faster than beliefs about objective underlying distribution. Comparing the results with those from a previous experiment on the same game but without belief elicitation, we also show that eliciting beliefs does not significantly affect the play itself.

#### Heterogeneity in Preferences for Skewness

Peter Moffatt; Morten Lau; Daniel Maller; Lionel Page

#### Abstract

We analyse data from a risky choice experiment with 80 subjects each solving 60 binary choice problems. The lotteries were presented to subjects graphically, in the form of the probability distribution of outcomes. The lotteries vary in expected outcome, variance of outcome, and skewness of outcome. The choice data is modelled within the framework of the mean variance utility function, extended to include a skewness term. Such an approach allows estimation of risk aversion and also preference for skewness (a priori we expect agents to be skewness-loving). Our model assumes between-subject heterogeneity in both risk aversion and skewness preference, and moreover allows non-zero correlation between these two preference parameters. Estimation is performed using the Method of Maximum Simulated Likelihood (MSL), with Halton draws used for simulation. Non-nested testing procedures are used to compare the performance of the model with skewness to more mainstream models that assume probability weighting. Preliminary estimation of a homogeneous model indicates that allowance for skewness preference does not add much statistically over models that allow for probability weighting. However, of greater interest is which type of model performs better when between-subject heterogeneity is assumed.

#### Measuring the Utility for Money in a Riskless Context: Evidence on Separable Representations

Michele Bernasconi; Christine Choirat; Raffaello Seri

#### Abstract

The utility of money, which assign measures of subjective value to physical amounts of money, is a fundamental concept in decision theories. Most of the modern empirical literature (experimental and from the fields) focus on preferences over money gambles, eliciting jointly the utility for money and the decision rules used to combine probabilities of money-utilities. Very few recent experiments have investigated the possibility to measure the utility of money directly. In this paper we investigate the possibility to measure the utility of money in a riskless context. We conduct an experiment in which participants are asked to suppose to receive a monetary gift d1 and are asked to indicate a monetary response d2 which they would want in order to feel in some proportion p as happy as the amount d1 would make them to feel. The experimental framework is similar to one used in a classical investigation by Galanter (1962), based on what in psychophysics Stevens (1946, 1951, 1975) called ratio magnitude production. It is well known in behavioral sciences that Stevens’ approach can be criticized because of lack of mathematical and psychological foundations justifying the proposition that, when assessing a ratio judgment, a “subject is, in a scientific sense, ‘computing ratios’ ” (Narens, 1996, p. 109). However, in recent years, an important stream of research has clarified the conditions and given the precise set of axioms that can justify ratio estimations and ratio productions (Luce 2002, 2004, 2008, Narens 1996, 2002, 2006). In particular, it has been shown that two axioms are essential to obtain meaningful representation measure from subjects’ ratio scaling: multiplicatively and commutativity. Several experiments conducted in recent years have tested the two axioms in various contexts (Ellermeier and Faulhammer, 2000; Zimmer, 2005; Steingrimsson and Luce, 2005b,a, 2006, 2007; Augustin and Maier, 2008; Bernasconi, Choirat, Seri, 2008, 2010; Steingrimssonl Luce, Narens 2012). The evidence in the various domain is in general in favor of commutativity, but against multiplicativity. This is consistent with models of so called separable representations. Separable representation incorporate the notion that various and independent distortions may occur both in the assessment of subjective intensities and in the determination of subjective ratios. In the context of binary monetary gambles, when one of the outcome is zero, an example of a model separable representations is offered by Prospect Theory. Both axioms of multiplicativity and commutativity are tested in our experiments. We find substantial evidence in favour of commutativity, but against multiplicativity. We provide estimate of the psychophysical functions of interest, including the utility function for money.

#### Pareto Optima and Competitive Equilibria in Markets with Expected and Dual Utility

Tim Boonen

#### Abstract

This paper analyzes optimal risk sharing between agents that are endowed with either expected utility preferences or with dual utility preferences. We find that Pareto optimal risk redistributions and the competitive equilibria are obtained via bargaining with an hypothetical representative agent of expected utility maximizers and an hypothetical representative agent of dual utility maximizers. The representative agent of expected utility maximizers resembles an average risk-averse agent, whereas representative agent of dual utility maximizers resembles an agent that has lowest aversion to mean-preserving spreads. This leads to an allocation of the aggregate risk to both groups of agents. The optimal contract for the expected utility maximizers is proportional to their allocation, and the optimal contract for the dual utility maximizing agents is given by ‘tranching’ of their allocation. We show a method to derive the equilibrium prices. We identify a condition under which prices are locally independent of the expected utility functions, and given in closed form. Moreover, we characterize uniqueness of the competitive equilibrium.