Group efforts when performance is determined by the “best shot”

We investigate the private provision of a public good whose level is determined by the maximum effort made by a group member. Costs of effort are either commonly known or privately known. For symmetric perfect-information games, any number of players may be active and we characterize the unique (mixed-strategy) equilibrium in which active contributors use the same strategy. Increasing the number of active players leads to stochastically lower individual efforts and level of the public good. When information is private, the symmetric equilibrium is in pure strategies. Increasing the number of players yields a pointwise reduction in the equilibrium contribution strategy but an increase in equilibrium payoffs. Comparative statics with respect to costs and levels of risk aversion are derived. Finally, whether information is public or private, equilibria are inefficient—we provide mechanisms that improve efficiency.     

Large common value auctions with risk averse bidders

We analyze large symmetric auctions with conditionally i.i.d. common values and risk averse bidders. Our main result characterizes the asymptotic equilibrium price distribution for the first- and second-price auctions. As an implication, we show that with constant absolute risk aversion (CARA), the second-price auction raises significantly more revenue than the first-price auction. While this ranking seems robust in numerical analysis also outside the CARA specification, we show by counterexamples that the result does not generalize to all risk averse utility functions.     

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     

Poverty and probability: aspiration and aversion to compound lotteries in El Salvador and India

Some experimental participants are averse to compound lotteries: they prefer simple lotteries that depend on only one random event, even when the simple lotteries offer lower expected value. This paper proposes that many behavioral “investments” represent more compound risk for poorer people—who often face multiple dimensions of deprivation—than for richer people. As a result, identical aversion to compound lotteries can prevent investment among poorer people, but have no effect on richer people. The paper reports five studies: two initial studies that document that aversion to compound lotteries operates as an economic preference, two “laboratory experiments in the field” in El Salvador, and one Internet survey experiment in India. Poorer Salvadoran women who choose a compound lottery are 27 percentage points more likely to have found formal employment than those who chose a simple lottery, but lottery choice is unrelated to employment for richer women. Poorer students at the national Salvadoran university choose more compound lotteries than richer students, on average, implying that aversion to compound lotteries screened out poorer aspirants but not richer ones. Poorer and lower-caste Indian participants who choose compound lotteries are more likely than those who choose simple lotteries to have a different occupation than their parents, which is not the case for better-off participants. These findings suggest that the consequences of aversion to compound lotteries are different in the context of poverty and disadvantage.     

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.     

Beliefs and Public Good Provision with Anonymous Contributors

We analyze a static game of public good contributions where finitely many anonymous players have heterogeneous preferences about the public good and heterogeneous beliefs about the distribution of preferences. In the unique symmetric equilibrium, the only individuals who make positive contributions are those who most value the public good and who are also the most pessimistic; that is, according to their beliefs, the proportion of players who most like the public good is smaller than it would be according to any other possible belief. We predict whether the aggregate contribution is larger or smaller than it would be in an analogous game with complete information and heterogeneous preferences, by comparing the beliefs of contributors with the true distribution of preferences. A trade‐off between preferences and beliefs arises if there is no individual who simultaneously has the highest preference type and the most pessimistic belief. In this case, there is a symmetric equilibrium, and multiple symmetric equilibria occur only if there are more than two preference types.     

Lotteries, Group Size, and Public Good Provision

We analyze the effect of group size on public good provision under the Morgan (2000) lottery mechanism. For a pure public good, the lottery performs quite well as public good provision is found to increase in group size, even when the lottery prize is held constant. By contrast, for fully rival public goods, per capita provision is found to decrease in group size, even when the lottery prize is proportional to group size. Further, the per capita level of provision will approach zero when group size is sufficiently large.     

Risk aversion in symmetric and asymmetric contests

We analyze existence, uniqueness and properties of equilibria in incompletely discriminating Tullock contests with logistic contest success functions, when contestants are risk averse. We prove that a Nash equilibrium for such a contest exists, but give an example of a symmetric contest with both symmetric and asymmetric equilibria, showing that risk aversion may lead to multiple equilibria. Symmetric contests have unique symmetric equilibria but additional conditions are necessary for general uniqueness. We also study the effects on incumbents of additional competitors entering the contest under these conditions and examine the effects of risk aversion on rent dissipation in symmetric and asymmetric contests.     

The Optimal Provision of Curative Goods

We analyze the optimal contract between a risk neutral regulator providing a curative goods and a risk averse patient who learns the realized value of his/her health status after the contracting stage. Consumption of a curative good (healthcare) reduces the disutility associated with a disease. We show that the consumption of curative goods is larger than in the complete information case, that this overprovision increases with the degree of patients’ risk‐aversion and the marginal cost of treatment. Ceilings on the amount of healthcare are part of the optimal contract when risk aversion is important.     

Rotating Savings and Credit Associations When Participants are Risk Averse*

We model rotating savings and credit associations (Roscas) among risk‐averse participants who experience privately observed income shocks. A random Rosca is not advantageous, whereas a bidding Rosca is if temporal risk aversion is less pronounced than static risk aversion. The payoff scheme of a bidding Rosca facilitates risk sharing in the presence of information asymmetries. The risk‐sharing performance of a simple arrangement where a group of homogenous individuals runs several bidding Roscas simultaneously is as good as that of a linear risk‐sharing contract, and is more enforceable because it carries a fixed rather than a variable contribution.     

The Important Thing Is Not (Always) Winning but Taking Part: Funding Public Goods with Contests

This paper considers a public good game with heterogeneous endowments and incomplete information affected by extreme free riding. I overcome this problem through the implementation of a deterministic contest in which several prizes may be awarded. I identify a monotone equilibrium, in which the contribution is strictly increasing in the endowment. I prove that it is optimal for the social planner to set the last prize equal to zero, but otherwise total expected contribution is invariant to the prize structure. Finally, I show that private provision via a contest Pareto‐dominates public provision and is higher than the total contribution raised through a lottery.     

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.     

Contests with Alternative Public‐Good Prizes

I study contests in which a society of players compete, by expending irreversible effort, over which one of alternative prizes should be awarded to them by the decision maker. The prizes are public goods and/or public bads for the players. The players choose their effort levels simultaneously and independently. I define each player's valuation spread as the difference between his valuations for the two public‐good/public‐bad prizes. I establish that the players' equilibrium effort levels depend solely on their valuation spreads, and that the players never expend positive effort for both prizes in equilibrium. Further, I establish that in equilibrium only players with the widest positive valuation spread and players with the widest negative valuation spread expend positive effort. Finally, I establish that the equilibrium effort level expended for each prize and the equilibrium total effort level are determined only by the widest positive valuation spread and the widest negative valuation spread.     

The Market for Sweepstakes

This paper studies the market for monopolistically supplied sweepstakes. We derive equilibrium demands for fixed-prize and variable-prize sweepstakes and determine the profit-maximizing prize level and pay-out ratio respectively. It can be profitable to offer each type of sweepstake when there is a large enough number of weighted utility consumers who have constant absolute risk attitudes, are strictly averse to small as well as symmetric risks, and display longshot preference behaviour. Moreover, for the variable-prize sweepstake, the supplier will generally find it profitable to combine sweepstakes targeting two smaller populations, and offer a single sweepstake to the combined population. This implication is corroborated by the recent spate of mergers of smaller state lotteries into larger ones.     

The Private Provision of Public Goods under Uncertainty: A Symmetric-Equilibrium Approach

Various studies have examined whether increased uncertainty about the non‐Nash response of others to an individual's voluntary contribution to a public good affects that individual's contribution so as to mitigate the free‐rider problem. We extend this single‐agent approach to the analysis of a symmetric equilibrium. We provide conditions on group size and endogenous relative risk aversion that imply increased equilibrium contributions in response to greater uncertainty about the productivity of each individual's contribution to the actual level of the public good. These results enable us to broaden the circumstances in which the theory predicts that increased uncertainty reduces free riding.     

RISK AVERSION AND FIELD OF STUDY CHOICE: THE ROLE OF INDIVIDUAL ABILITY

Does the choice of field of study depend on individual risk aversion? The direction of the relationship between individual risk attitudes and type of university degree chosen is potentially ambiguous. On the one hand, risk averse individuals may prefer degree courses which allow high returns in the labour market; on the other hand, if these degrees expose students to a higher probability of dropping out, those who are more risk averse may be induced to choose less challenging fields. Using data from a sample of students enrolled at a middle‐sized Italian public university in 2009, we find that, controlling for a large number of individual characteristics, more risk averse students are more likely to choose any other field (Humanities, Engineering, and Sciences) rather than Social Sciences. We interpret this result bearing in mind that some of these fields, such as Humanities, involve a reduction in the risk of dropping out, while others (such as Engineering and Sciences) involve a lower risk in the labour market. It also emerges that the effect of risk aversion on degree choice is related to student ability. Risk averse students characterized by high abilities tend to prefer Engineering, while the propensity of risk averse students to enrol in Humanities decreases when ability increases, suggesting that the attention paid to labour market risks and drop‐out risk varies according to student skills.     

Lotteries, Sunspots, and Incentive Constraints

We study a prototypical class of exchange economies with private information and indivisibilities. We establish an equivalence between lottery equilibria and sunspot equilibria and show that the welfare and existence theorems hold. To establish these results, we introduce the concept of the stand-in consumer economy, which is a standard, convex, finite consumer, finite good, pure exchange economy. With decreasing absolute risk aversion and no indivisibilities, we prove that no lotteries are actually used in equilibrium. We provide a simple numerical example with increasing absolute risk aversion in which lotteries are necessarily used in equilibrium. We also show how the equilibrium allocation in this example can be implemented in a sunspot equilibrium. Journal of Economic Literature Classification Numbers: D11, D50, D82.     

Overbidding and overspreading in rent-seeking experiments: Cost structure and prize allocation rules

We study experimentally the effects of cost structure and prize allocation rules on the performance of rent-seeking contests. Most previous studies use a lottery prize rule and linear cost, and find both overbidding relative to the Nash equilibrium prediction and significant variation of efforts, which we term ‘overspreading.’ We investigate the effects of allocating the prize by a lottery versus sharing it proportionally, and of convex versus linear costs of effort, while holding fixed the Nash equilibrium prediction for effort. We find the share rule results in average effort closer to the Nash prediction, and lower variation of effort. Combining the share rule with a convex cost function further enhances these results. We can explain a significant amount of non-equilibrium behavior by features of the experimental design. These results contribute towards design guidelines for contests based on behavioral principles that take into account implementation features of a contest.     

The Determinants of Risk Aversion: The Role of Intergenerational Transmission

This article studies the relationship between risk attitudes and individual characteristics focusing on the intergenerational transmission of risk preferences. We use a dataset of a sample of Italian students which allows us to build different measures of risk aversion based, respectively, on a survey asking students about their willingness to invest in a risky asset and about their preferences for job security and on the results of an entry test using explicit penalty points in the case of incorrect answers. In line with the findings highlighted by the existing literature, we find that women are more risk averse than men, more patient subjects are more risk averse, while high‐ability students are less risk averse. As far as intergenerational transmission of preferences is concerned, it emerges that students whose fathers are entrepreneurs have a higher propensity to take risks, while students whose fathers are employed in the public sector are more risk averse. Only fathers matter with regards to their children's risk attitudes. These results are robust to different measures of risk aversion and to different specifications of our model.     

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.