Expected utility theory without the completeness axiom

We study the problem of obtaining an expected utility representation for a potentially incomplete preference relation over lotteries by means of a set of von Neumann-Morgenstern utility functions. It is shown that, when the prize space is a compact metric space, a preference relation admits such a multi-utility representation provided that it satisfies the standard axioms of expected utility theory. Moreover, the representing set of utilities is unique in a well-defined sense.     

Ambiguity Without a State Space

Many decisions involve both imprecise probabilities and intractable states of the world. Objective expected utility assumes unambiguous probabilities; subjective expected utility assumes a completely specified state space. This paper analyses a third domain of preference: sets of consequential lotteries. Using this domain, we develop a theory of objective ambiguity without explicit reference to any state space. We characterize a representation that integrates a non-linear transformation of first-order expected utility with respect to a second-order measure. The concavity of the transformation and the weighting of the measure capture ambiguity aversion. We propose a definition for comparative ambiguity aversion.     

Reducing ambiguity in lotteries: evidence from a field experiment

Ambiguity about the chances of winning represents a key aspect in lotteries. By means of a controlled field experiment, we exogenously vary the degree of ambiguity about the winning chances of lotteries organized to incentivize the contribution for a public good. In one treatment, people have been simply informed about the maximum number of potential participants (i.e. the number of lottery tickets released). In a second treatment, this information has been omitted as in all traditional lotteries. Our general finding shows that simply reducing the degree of ambiguity of the lottery leads to a sizable and significant increase (67%) in the participation rate. This result is robust to alternative prize configurations.     

Growth in economies with non convexities: sunspots and lottery equilibria

Summary. We investigate the relation between lotteries and sunspot allocations in a dynamic economy where the utility functions are not concave. In an intertemporal competitive economy, the household consumption set is identified with the set of lotteries, while in the intertemporal sunspot economy it is the set of measurable allocations in the given probability space of sunspots. Sunspot intertemporal equilibria whenever they exist are efficient, independently of the sunspot space specification. If feasibility is, at each point in time, a restriction over the average value of the lotteries, competitive equilibrium prices are linear in basic commodities and intertemporal sunspot and competitive equilibria are equivalent. Two models have this feature: Large economies and economies with semi-linear technologies. We provide examples showing that in general, intertemporal competitive equilibrium prices are non-linear in basic commodities and, hence, intertemporal sunspot equilibria do not exist. The competitive static equilibrium allocations are stationary, intertemporal equilibrium allocations, but the static sunspot equilibria need not to be stationary, intertemporal sunspot equilibria. We construct examples of non-convex economies with indeterminate and Pareto ranked static sunspot equilibrium allocations associated to distinct specifications of the sunspot probability space.Received: 25 August 2003, Revised: 16 March 2004, JEL Classification Numbers: D84, D90.Correspondence to: Paolo SiconolfiWe thank Herakles Polemarchakis for helpful conversations on the topic. The research of Aldo Rustichini was supported by the NSF grant NSF/SES-0136556.     

Comparing risk aversion in a probability triangle

An agent's acceptance set consists of the probability distributions preferred to the status quo. One agent is more risk averse than another if the more risk averse agent's acceptance set is a proper subset of the less risk averse agent's acceptance set. An agent's odds premium expresses the odds in favor of winning the largest cash prize in a lottery over the best and worst alternatives that is indifferent to the the agent's initial wealth. Comparisons of two agents odds premia completely characterizes the risk aversion relations between them when facing lotteries in a probability triangle. The result applies to expected utility and some non-expected utility theories. Received: December 30, 1998; revised version: February 10, 1999     

Financing Public Goods by Means of Lotteries

When viewed as taxes, lotteries are routinely criticized as being both inequitable and inefficient. But is this an entirely fair comparison? Frequently lotteries are used in lieu of voluntary contributions by private charities and governments when taxes are not feasible. When heterogeneous individuals with quasi-linear preferences participate in lotteries whose proceeds will be used to fund a public good, we find that, relative to voluntary contributions, wagers in the unique lottery equilibrium (a) increase the provision of the public good, (b) are welfare improving, and (c) provide levels of the public good close to first-best as the lottery prize increases.     

Information acquisition and full surplus extraction

It is well known that when agents' types are correlated, the mechanism designer can extract the entire surplus. This creates an incentive for agents to acquire information about other agents' types. Robust lotteries (are payment schemes that) support full extraction and partially robust lotteries support efficient implementation in the presence of information acquisition opportunities. Necessary and sufficient conditions for existence of robust and partially robust lotteries are derived. If an agent's information signal spans other agents' types then robust lotteries do not exist. However, if all agents report their signal realizations then robust lotteries exist in an extended type space.     

Funding Public Goods with Lotteries: Experimental Evidence

Why do individuals participate in charitable gambling activities? We conduct a laboratory investigation of a model that predicts risk-neutral expected utility maximizers will participate in lotteries when they recognize that lotteries are being used to finance public goods. As predicted by the model, we find that public goods provision is higher when financed by lottery proceeds than when financed by voluntary contributions. We also find support for other comparative static predictions of the model. In particular we find that ticket purchases vary with the size of the fixed prize and with the return from the public good: lotteries with large prizes are more effective, and ticket purchases drop dramatically when the public good is not valued by subjects.     

Recursive utility and preferences for information

Summary. This paper presents an axiomatic foundation for recursive utility that captures the role of the timing of resolution of uncertainty without relying on exogenously specified objective beliefs. Two main representation results are proved. In the first one, future utility enters the recursion through the type of general aggregators considered in Skiadas (1997a), and as a result the formulation is purely ordinal and free of any probabilities. In the second representation these aggregators are conditional expectations relative to subjective beliefs. A new recursive representation incorporating disappointment aversion is also suggested. The main methodological innovation of the paper derives from the fact that the basic objects of choice are taken to be pairs of state-contingent consumption plans and information filtrations, rather than the temporal (objective) lotteries of the existing literature. It is shown that this approach has the additional benefit of being directly applicable to the continuous-time version of recursive utility developed by Duffie and Epstein (1992). Received: February 18, 1997; revised version: July 18, 1997     

Irrational diversification in multiple decision problems

The paper deals with multiple decision problems, which are similar to the task of guessing the color outcomes of five independent spinnings of a roulette wheel, 60% of whose slots are red and 40% white. Each correct guess yields a prize of $1. The guess of 5 Reds clearly first order stochastic dominates any other strategy. In contrast, subjects diversify their choices when facing a multiple decision problem in which the choice is between lotteries with clear objective probabilities. The diversification is stronger when the subjects face uncertainty without objective probabilities and weaker when the choice problem involves real life actions.     

A note on linear utility

Summary We show that any complete, lower-semicontinuous, and translation-invariant preorder defined on a topological vector space admits a linear and continuous utility representation.Thanks are given to an anonymous referee for his/her valuable suggestions and comments.     

Charitable Lottery Structure and Fund Raising: Theory and Evidence

Do fixed-prize charitable lotteries generate more net revenue than do revenue-dependent lotteries? I present the results of an experiment designed to test a theoretical prediction of the relationship between the prize structure of a lottery funding a public good and individuals' participation in the lottery. I find that a fixed-prize lottery configuration induces significantly greater participation and a significantly higher level of public good funding than does a revenue-dependent lottery.     

Selective incentives and intragroup heterogeneity in collective contests

A group taking part in a contest has to confront the collective action problem among its members, and devices of selective incentives are possible means of resolution. We argue that heterogeneous prize‐valuations in a competing group normally prevent effective use of such selective incentives. To substantiate this claim, we adopt cost‐sharing as a means of incentivizing the individual group members. We confirm that homogeneous prize valuations within a group result in a cost‐sharing rule inducing the first‐best individual contributions. As long as the cost‐sharing rule is dependent only on the members’ contributions, however, such a first‐best rule does not exist for a group with intragroup heterogeneity. Our main result clarifies how unequal prize valuations affect the cost‐sharing rule and, in particular, the degree of cost‐sharing. If the relative rate of change of the marginal effort costs is decreasing, it is reduced by intragroup heterogeneity. If the rate is increasing, the cost is fully shared, but it cannot induce the first‐best contributions for the group.     

Difference-form contest success functions and effort levels in contests

We examine players' equilibrium effort levels in a contest with difference-form contest success functions in which two players compete with each other to win a prize. We show the following. At the pure-strategy Nash equilibrium of a simultaneous-move game, and in the subgame-perfect equilibrium of a sequential-move game, only one of the players expends effort or neither player expends effort. If one player's composite strength is far greater than the other player's, only the player with greater composite strength expends effort whether they move simultaneously or sequentially. If the players' valuations for the prize and their marginal probabilities of winning at (0, 0) are sufficiently small, neither player expends effort whether they move simultaneously or sequentially.     

Multi-player contests with asymmetric information

We consider imperfectly discriminating, common-value, all-pay auctions (or contests) in which some players know the value of the prize, others do not. We show that if the prize is always of positive value, then all players are active in equilibrium. If the prize is of value zero with positive probability, then there is some threshold number of informed players such that if there are less, all uninformed players are active, and otherwise all uninformed players are inactive.     

Expected utility theory from the frequentist perspective

We present an axiomatization of expected utility from the frequentist perspective. It starts with a preference relation on the set of infinite sequences with limit relative frequencies. We consider three axioms parallel to the ones for the von Neumann–Morgenstern (vN–M) expected utility theory. Limit relative frequencies correspond to probability values in lotteries in the vN–M theory. This correspondence is used to show that each of our axioms is equivalent to the corresponding vN–M axiom in the sense that the former is an exact translation of the latter. As a result, a representation theorem is established: The preference relation is represented by an average of utilities with weights given by the relative frequencies.     

On the uniqueness of subjective probabilities

Summary The purpose of this paper is twofold: First, within the framework of Savage (1954), we suggest axiomatic foundations for the representation of event-dependent preference relations over acts. This representation has the form of expectation of event-debendent utility with respect to non-unique subjective probabilities on the set of states. Second, we give an economic-theoretic motivation for selecting a unique probability distribution as an appropriate concept of subjective probabilities. However, unlike in Savage's theory, this notion of subjective probabilities does not necessarily represent the decisions-maker's belief regarding the likelihood of events.Our approach involves a departure from Savage's postulate P4, which guarantees the completeness of Savage's likelihood relation on the set of all events. Instead, we assume the existence of a finite partition of the set of states, {S ₁,...S _n}, such that, for events within each element of this partition P4 is satisfied. This weakening of Savage's axioms suffices for the existence of an expected event-dependent utility representation, but not for the uniqueness of the subjective probabilities.In many economic problems involving decision-making under uncertainty the existence of a unique probability is presumed and, in fact, is essential for the statement of the result. An example is Arrow's (1965) finding that all risk averse decision-makers will invest in a risky asset provided its expected rate of return exceeds that of an alternative risk-free asset. We show that a unique probability distribution can be chosen so as to render such results meaningful. Namely, any risk averse decision-maker will hold a positive position in the risky asset if and only if its expected rate of return with respect to the chosen probability exceeds that of the riskless asset.The research described in this paper began while the authors visited the Mathematisches Forschungsinstitut Oberwolfach, Germany. It was carried out in part while the second author visited the Santa Fe Institute in Sante Fe, New Mexico, USA and the Autonomous University of Barcelona, Spain. The second author would also like to acknowledge the financial support by NSF grant 911873.     

Exactly what happens after the Anscombe–Aumann race?

This paper derives a representation of preferences for a choice theory with vague environments; vague in the sense that the agent does not know the precise lotteries over outcomes conditional on states. Instead, he knows only a possible set of these lotteries for each state. Thus, this paper’s main departure from the standard subjective expected utility model is to relax an assumption about the environment, rather than weakening the axiomatic structure. My model is consistent with the behavior observed in the Ellsberg experiment. It can capture the same type of behavior as the multiple priors models, but can also result in behavior that is different from both the behavior implied by standard subjective expected utility models and the behavior implied by the multiple priors models. This paper is a revised chapter of my Ph.D. dissertation at Cornell University. I am very grateful to David Easley for extensive comments and suggestions. I also thank Larry Blume, Ani Guerdjikova, Edi Karni, Ted O’Donoghue, Maureen O’Hara, Jacob Sagi, seminar participants at the University of Copenhagen, Cornell University, ESEM 2006, Fuqua—Duke University, FUR 2006, Johns Hopkins University, Purdue University, Queen’s University, the University of Warwick, the associate editor, and an anonymous referee for comments and suggestions.     

Consuming your way to efficiency: Public goods provision through non-distortionary tax lotteries

We revisit the classical result that taxation of private consumption is distortionary and therefore precludes the efficient provision of public goods. We introduce a nonlinear consumption tax which we call a ‘tax lottery’. Under this scheme, an ad-valorem consumption tax is supplemented with a lottery in which consumers can win cash prizes. The winning probabilities in this lottery depend on all consumers' private good consumption decisions. We show that for a given ad-valorem tax, an appropriately designed lottery can implement an efficient allocation in pure-strategy Nash equilibrium. The lottery component corrects the distortion in private consumption due to the ad-valorem tax, while the resulting tax revenue is sufficient to efficiently provide the public good and pay out the lottery prize.     

Past experience of uncertainty affects risk aversion

In an experiment with more than 500 participants we study how past experience of uncertainty (imperfect knowledge of the state space) affects risk preferences. Participants in our experiment choose between a sure outcome and a lottery in 32 periods. All treatments are exactly identical in periods 17–32 but differ in periods 1–16. In the early periods of the risk treatment there is perfect information about the lottery; in the ambiguity Treatment participants perfectly know the outcome space but not the associated probabilities; in the unawareness treatment participants have imperfect knowledge about both outcomes and probabilities. We observe strong treatment effects on behavior in periods 17–32. In particular, participants who have been exposed to an environment with very imperfect knowledge of the state space subsequently choose lotteries with high (low) variance less (more) often compared to other participants. Estimating individual risk attitudes from choices in periods 17–32 we find that the distribution of risk attitude parameters across our treatments can be ranked in terms of first order stochastic dominance. Our results show how exposure to environments with different degrees of uncertainty can affect individuals’ subsequent risk-taking behavior.