THEORIES OF CHOICE UNDER IGNORANCE AND UNCERTAINTY

Abstract. In this paper, Knight's distinction between risk and uncertainty, and its significance for economic analysis are examined. The paper consists of a survey of some recent developments on the theory of choice under uncertainty and some applications of these theories to problems for which Bayesian Decision Theory has not proved entirely satisfactory. Two problems are examined in detail. The first is that of finance and insurance and the second is that of risktaking behaviour with special emphasis on lotteries.     

Adam Smith on lotteries: an interpretation and formal restatement

The paper concerns a neglected aspect of the Wealth of Nations (with the notable exception of D. Levy 1999 Levy, D. M. 1999. “Adam Smith’s Katallactic Model of Gambling: Approbation from the Spectator.” Journal of the History of Economic Thought 21 (1): 81–91. doi:10.1017/S1053837200002868.[Crossref] , [Google Scholar]), dealing directly with decision under risk. In a few pages from book I, chapter 10, Adam Smith explicitly named “lotteries” various objects of choice (possible occupations, or investment opportunities, for instance) and provided an analysis which standard expected utility glasses would hardly fit. Taking this into account allows a better understanding of the part played by typical characters like the “projector” or the “sober man”, in such matters as Smith’s conception of entrepreneurship or of the credit market. The use of some modern concepts in decision analysis (inverse stochastic dominance, rank dependent utility, prudence toward risk), is a means to show the existence, in Smith’s work, of an original theory from decision under risk, where his analysis of lotteries in the Wealth of Nations is consistent with statements from his moral philosophy on asymmetric sensitivity to gains and losses and to the regulating part played by the impartial spectator.     

Extreme events and the cumulative distribution of net gains in gambling and structured products

We argue that ethical principles in advertising and market communication cannot be properly discovered and applied to gambling without a deep understanding of its probabilistic implications, in particular when extreme events are influential. We carry out a probabilistic analysis of lottery games with lifetime prizes in order to derive sound recommendations about the pertinent information that should be communicated to nudge gamblers. We propose to focus on the cumulative distribution of net gains, for which there is currently no information available to gamblers. This holds true for structured products in which extreme events matter as well.     

Lottery pricing and price elasticity dispersion in the UK and Canada

We estimate effective price elasticities for different quantiles of the demand distribution of the UK National Lottery and the Canadian Lotto 649. We show that price elasticities vary significantly from draw to draw and have a tendency to increase with lottery participation and jackpot size. Our findings indicate that setting lottery rules on the basis of mean effective price elasticities should be faced with caution because expected profits are negatively related to the evident variation of elasticities among lottery draws. We also simulate alternative active rollover distributions and show that limiting the rollover accumulation by withholding portions and ploughing them back in future nonrollover draws is potentially profitable.     

Explaining Overbidding in First Price Auctions Using Controlled Lotteries

In this paper, we study the behavior of individuals when facing two different, but incentive-wise identical, institutions. We pair the first price auction with an equivalent lottery. Once a subject is assigned a value for the auctioned object, the first price auction can be modeled as a lottery in which the individual faces a given probability of winning a certain payoff. This set up allows us to explore to what extent the misperception of the probability of winning in the auction is responsible for bidders in a first price auction to bidding above the risk neutral Nash equilibrium prediction. The first result we obtain is that individuals, even though facing the same choice over probability/payoff pairs, behave differently depending on the type of choice they are called to make. When facing an auction, subjects with high values tend to bid significantly above the bid they choose in the corresponding lottery environment. We further find that in both the lottery and the auction environments, subjects tend to bid in excess of the bid predicted by the risk neutral model, at least for intermediate range values. Finally, we find that the difference between the lottery behavior and the auction behavior is substantially, but not totally, eliminated by showing the subjects the probability of winning the auction.     

Bayesian beliefs with stochastic monotonicity: An extension of Machina and Schmeidler

Machina and Schmeidler show that the probabilistic sophistication can be obtained in an Anscombe–Aumann setting without imposing expected utility by maintaining stochastic monotonicity and adding a new axiom loosely analogous to Savage's P4. This analogous axiom, however, is very strong. In this note, we obtain probabilistic sophistication using a weaker (and more natural) analog of Savage's P4. Stochastic monotonicity is sufficient to bridge the gap, where Anscombe and Aumman use independence twice, we use stochastic monotonicity twice.     

RETESTING THE UNCERTAINTY EFFECT USING LOTTERIES WITH REAL PRODUCTS AND MONEY

In the current study, several experiments re‐examine the uncertainty effect using lotteries that include real products, monetary outcomes and electronic gift cards in a between‐subjects design. The study also takes the selling position into consideration, in addition to the buying position considered by all previous works on the uncertainty effect. The results indicate that for all types of lotteries, the bids are higher than the bids for the worst possible realization. These findings are consistent with the internality axiom and do not support the uncertainty effect.     

Economic shocks and lottery sales: an examination of Maine State lottery sales

ABSTRACT

Most U.S. states earn significant amount of revenues from lottery sales. However, they are also criticized for promoting the lotteries because they have been seen as taking advantage of poor populations. The purpose of this study is to identify the impact of various economic factors on lottery sales by using zip-code level sales within the state of Maine. The results show that an increase of 1% in unemployment rate results in a 0.38% increase in draw lottery sales, but it has no significant impact on instant lottery sales. This highlights the importance of differentiating between two major types of lotteries.     

Inducing risk preferences in multi-stage multi-agent laboratory experiments

Though there has been some debate over the practical efficacy of using binary lotteries for controlling risk preferences in experimental environments, the question of its theoretical validity within the contexts it is often used, namely multi-stage multi-agent settings, has not been addressed. Whilst the original proof of its validity featured a single-agent single-stage context, its practical use has seen a wide range of implementations. Practitioners have implicitly assumed that whenever the setting and form of implementation they have chosen deviates from the original single-agent single-period proof, it remains theoretically valid. There has been virtually no debate in the practitioner literature on the theoretical validity of binary lotteries in a more general context, or on whether the form of implementation matters. The current article addresses these questions, establishes limitations on validity and suggests some design principles for future implementation of binary lotteries for the purpose of controlling risk preferences.     

Optimal pricing of public lotteries and comparison of competing mechanisms

This article establishes optimal pricing rules for rationing indivisible units of rival and otherwise nonexcludable goods by lottery or a hybrid of a lottery and outright sale by posted price. Given the distributional objective of maximizing expected consumer surplus, the solutions to unconstrained and constrained versions of the pricing problem may be expressed in classic inverse elasticity form, with the lottery price appearing as an entry fee, user fee or a combination of the two. Numerical analysis of a rich class of private value distributions indicates that sizable gains in expected consumer surplus can be realized over competitive pricing and zero pricing.