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On the Optimal Design of Lotteries
Abstract
In recent years, the expected utility model of choice under risk has been generalized to cope with phenomena such as probability weighting. In the present paper, one such generalized approach, the rank-dependent expected utility model, is applied to the problem of lottery gambling. The model is used to derive an optimal prize structure for lotteries, ivolving a few large prizes and a large number of small prizes. Other forms of gambling, such as racetrack betting, are discussed in the light of this result. Copyright 1991 by The London School of Economics and Political Science.
... Agents choose actions based on the expected utility, that is, the sum of the products of the utilities of outcomes and their corresponding probabilities. Although EU is a standard model, it does not satisfactorily explain the behavior of people who buy lottery tickets [16]. ...
... This study investigates the optimal lottery problem based on the CPT framework, for the first time, whereas previous studies are based on RDEU or other frameworks. There are several other differences between this study and previous studies: Quiggin [16] deals with the maximization of the buyer's expected utility under a fixed profit for the seller with RDEU. The profit maximization is studied in [6,13]. ...
... satisfy the KKT conditions (12)- (16). ...
A lottery is a popular form of gambling between a seller and multiple buyers, and its profitable design is of primary interest to the seller. Designing a lottery requires modeling the buyer decision-making process for uncertain outcomes. One of the most promising descriptive models of such decision-making is the cumulative prospect theory (CPT), which represents people's different attitudes towards gain and loss, and their overestimation of extreme events. In this study, we design a lottery that maximizes the seller's profit when the buyers follow CPT. The derived problem is nonconvex and constrained, and hence, it is challenging to directly characterize its optimal solution. We overcome this difficulty by reformulating the problem as a three-level optimization problem. The reformulation enables us to characterize the optimal solution. Based on this characterization, we propose an algorithm that computes the optimal lottery in linear time with respect to the number of lottery tickets. In addition, we provide an efficient algorithm for a more general setting in which the ticket price is constrained. To the best of the authors' knowledge, this is the first study that employs the CPT framework for designing an optimal lottery.
... Explaining participation in lotteries by risk-averse consumers represents a challenge for standard expected utility theory (Quiggin, 1991). Lottery tickets can be considered a risky financial asset, where prizes represent uncertain returns to investment, and also a consumption good providing entertainment value to ticket buyers. ...
... Furthermore, the explanatory variables included in these models do not explain why bettors accept an unfair gamble. Quiggin (1991) argued that, in regard to lottery tickets, there is no acceptable explanation that includes risk aversion and concluded that the only reason for betting is the chance of winning a large jackpot. Therefore, Forrest et al. (2002) propose an alternative model to explain the demand for lotto, the "jackpot" model. ...
... Beenstock and Haitovsky (2001) also found that jackpot size increased sales, and that sales declined with ticket price; they also found evidence that consumers prefer a number of smaller prizes, other things equal, based on the addition of a sixth small prize. Both papers extend the research of Clotfelter and Cook (1990) who analyzed the effect of changing prices and payoffs on lottery ticket sales, and Quiggin's (1991) analysis of the optimal prize structure in lottery design that questioned whether it is better to have a single prize or a multiple prizes. ...
Government sponsored lotteries operate around the world. Their popularity has grown substantially over time. Legal lottery gambling generates significant public revenue, much of it from the lower part of the income distribution. Lottery is almost always an unfair bet, so explaining the purchase of lottery tickets by risk-averse consumers has long challenged economic theory. Lotteries can be analyzed from the perspective of public finance, as source of public revenue, or consumer theory, as a consumer commodity. We survey the state of economic research on lotteries from both perspectives, focusing on the key empirical findings.
... (See Figure 1 for several examples of Prelec's S-shaped probability weighting functions.) The concept of probability weighting has been invoked to explain a number of peculiar behavioral patterns; one of the canonical examples is people's participation in gambling and lotto games [Quiggin, 1991, Kahneman, 2011. 5 We use probability weighting here to ask the following basic question. ...
... Nonetheless, people participate in these games in large numbers. Work in behavioral economics has advanced probability weighting as one explanation for this irrational behavior, via the tendency to over-weigh small probabilities-here the chance of winning the lottery[Quiggin, 1991, Kahneman, 2011. ...
Many policies allocate harms or benefits that are uncertain in nature: they produce distributions over the population in which individuals have different probabilities of incurring harm or benefit. Comparing different policies thus involves a comparison of their corresponding probability distributions, and we observe that in many instances the policies selected in practice are hard to explain by preferences based only on the expected value of the total harm or benefit they produce. In cases where the expected value analysis is not a sufficient explanatory framework, what would be a reasonable model for societal preferences over these distributions? Here we investigate explanations based on the framework of probability weighting from the behavioral sciences, which over several decades has identified systematic biases in how people perceive probabilities. We show that probability weighting can be used to make predictions about preferences over probabilistic distributions of harm and benefit that function quite differently from expected-value analysis, and in a number of cases provide potential explanations for policy preferences that appear hard to motivate by other means. In particular, we identify optimal policies for minimizing perceived total harm and maximizing perceived total benefit that take the distorting effects of probability weighting into account, and we discuss a number of real-world policies that resemble such allocational strategies. Our analysis does not provide specific recommendations for policy choices, but is instead fundamentally interpretive in nature, seeking to describe observed phenomena in policy choices.
... Logically, this approach also leads to a change of the crowdsourcer's objective: maximizing revenue becomes maximizing profit-revenue minus the (non-fixed) cost (prize)-which is also his utility. To the best of our knowledge, this paper is the first that introduces prize as a function into optimal Tullock contest design, a subject that is being pursued since the 1990's [20] following Tullock's seminal work [15] in 1980. ...
... One stream of research explores whether the prize should be allocated to a single winner or all the players in a hierarchical manner. For example, [20] applies a rank-dependent expected utility model to a lottery in which the prize was divided into, according to ranks, a few large prizes and a large number of small prizes. However, [27] proves that it is optimal to give the entire prize to a single winner in a symmetric equilibrium. ...
Incentive mechanisms for crowdsourcing have been extensively studied under the framework of all-pay auctions. Along a distinct line, this paper proposes to use Tullock contests as an alternative tool to design incentive mechanisms for crowdsourcing. We are inspired by the conduciveness of Tullock contests to attracting user entry (yet not necessarily a higher revenue) in other domains. In this paper, we explore a new dimension in optimal Tullock contest design, by superseding the contest prize—which is fixed in conventional Tullock contests—with a prize function that is dependent on the (unknown) winner's contribution, in order to maximize the crowdsourcer's utility. We show that this approach leads to attractive practical advantages: (a) it is well-suited for rapid prototyping in fully distributed web agents and smartphone apps; (b) it overcomes the disincentive to participate caused by players' antagonism to an increasing number of rivals. Furthermore, we optimize conventional, fixed-prize Tullock contests to construct the most superior benchmark to compare against our mechanism. Through extensive evaluations, we show that our mechanism significantly outperforms the optimal benchmark, by over three folds on the crowdsourcer's utility cum profit and up to nine folds on the players' social welfare.
... The purchase of lottery tickets by consumers who are generally risk-averse constitutes a problem for expected utility theory (Quiggin, 1991). Lottery tickets could be considered to be financial assets with risk, where prizes are taken to be the returns to investment, and also as providing entertainment. ...
... They find a direct and positive effect on sales from increases in the announced jackpot and an inverse relationship between sales and the price of a ticket. Before, Clotfelter and Cook (1990) focused on the analysis of the effect of changing price and payoffs on lottery ticket sales, and later, Quiggin (1991) deals with the optimal prize structure in lottery design and asks whether it is better to have a single prize or a multiplicity of prizes. Concerning the prize structure a preference for multiplicity is observed 22 . ...
Lotteries operate today in many countries around the world. This type of gambling is usually run by governments and is sometimes described as regressive. Lottery is an unfair bet, so explaining the purchase of lottery tickets by risk‐averse consumers has been a challenge for economic theory. Lotteries can be analysed from either of two economic perspectives: as a source of public revenue or as a consumer commodity. In this paper the state of economic research on lotteries is reviewed, focusing on its main empirical findings.
... If wealth is in the convex region, subjects are predicted to take fifty-fifty gambles, even if the gambles are actuarially unfair to a moderate degree. As Quiggin (1991) points out, the levels of wealth in the convex region are predicted not to be found in the society, for at these levels people will gamble until they reach a level of wealth in one of the concave regions. Although this argument may seem contrived, it points at a major weakness of the Friedman-Savage hypothesis. ...
... The break even effect is enhanced by the underweighting of moderate to large probabilities. This was pointed out by Quiggin (1991), and can be illustrated by treating the zero outcomes in the two exemplary lotteries as the outcomes denoting the loss of the ticket fee. The chance of losing the ticket fee in the two prize lottery is now 1−(q+r) . ...
... Drawing on both laboratory and field observations, the economics and psychology literatures reveal a tendency for individuals to display longshot preference, i.e., risk affinity towards small chances of sizable gains, while simultaneously exhibiting aversion to risk involving moderate chances of winning (Kahneman & Tversky, 1979;Chark et al., 2020). Theoretical models offer preferential foundations for the observed concurrence of longshot preference behavior revealed in lottery purchases and pervasive risk aversion evident in capital markets (see, e.g., Quiggin 1991;Chew & Tan 2005). The literature has also developed models of preference over the timing of uncertainty resolution-or hope attitude-being distinct from attitude towards risk (Kreps & Porteus, 1978;Chew & Epstein, 1989). ...
The empirical literature points to a stylized phenomenon of increased demand for hope following adversity. Clotfelter and Cook (1989) suggest that hope is a key sentiment underpinning recreational gambling. Chew and Ho (1994, this journal) offer the view of hope being experienced in lottery products when people enjoy delaying the resolution of uncertainty. Taking air quality as an indicator of subjective well-being, we hypothesize a positive causal relationship between air pollution and lottery sales. We test this hypothesis using data from China and find that air pollution measured by particle concentration increases demand for a popular lottery for which province-level daily sales records exist. The relationship can readily be seen on combining high-frequency, spatially resolved lottery sales and particle pollution data. Our findings support the adversity-hope hypothesis in the context of air quality and lottery sales and point to further tests using other measures of adversity and proxies of demand for hope.
... Drawing on both laboratory and field observations, the economics and psychology literatures reveal a tendency for individuals to display longshot preference, i.e., risk affinity towards small chances of sizable gains, while simultaneously exhibiting aversion to risk involving moderate chances of winning (Kahneman & Tversky, 1979;Chark et al., 2020). Theoretical models offer preferential foundations for the observed concurrence of longshot preference behavior revealed in lottery purchases and pervasive risk aversion evident in capital markets (see, e.g., Quiggin 1991;Chew & Tan 2005). The literature has also developed models of preference over the timing of uncertainty resolution-or hope attitude-being distinct from attitude towards risk (Kreps & Porteus, 1978;Chew & Epstein, 1989). ...
... However, some research in this direction includes the work of Forgeard and Seligman (2012), who proposed a mechanism whereby an agent chooses options in which she is optimistic due to her biases. 5 For example, both Kahneman and Tversky (1979) and Quiggin (1991) illustrate many hypothetical situations where agents voluntarily insure against loss but still gamble due to such attitudes. 6 The result agrees with the work of Bar-Gill (2005), who showed that optimistic parties can extract better bargaining outcomes. ...
This paper uses the rank-dependent expected utility (RDEU) model to capture the effects of optimism and pessimism on the choice between a pre-trial settlement and a trial (or more generally, between a private settlement and a litigation). These two legal procedures are described as a bargaining game and a contest game, respectively. My models predict that a contest occurs if the aggregate optimism premium in a contest (AOPC) is sufficiently high. I also find that the AOPC tends to be higher for close cases. Such predictions are consistent with the Priest-Klein empirical observation that a plaintiff's winning probability is often near 50% in many areas of civil litigation. I also show that the highest levels of effort in both a bargaining game and a contest game are exerted when one is moderately optimistic. However, excessive optimism will reduce one's effort level, and hence, one's winning rate. As a result, when faced with an excessively optimistic party, a risk-neutral party may prefer a contest over bargaining.
... Not surprisingly, the size of the jackpot heavily influences lottery sales and consumer demand. Big rollovers and high jackpots have a positive impact on total lottery sales or revenues (Quiggin 1991;Thiel 1991;Scoggins 1995). Garrett and Sobel (1999) and Walker and Young (2001) found that skewness of prize distribution also positively impacted lottery sales. ...
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.
... As the pricing kernel is proportional to the product of the marginal utility and the derivative of the weighting function, the pricing kernel does not necessarily fall as we move from a low wealth state to a high wealth state. This analysis is equivalent to that used to explain the purchase of lottery tickets by investors with concave utility functions (Quiggin, 1991). ...
We show that if sophisticated institutional managers and individual investors perceive tail-risks differently, then a new explanation for the pricing kernel puzzle emerges. We show, by example, that even a tiny difference in tail-risk perception by the two investor types can explain the pricing kernel puzzle.
... In fact, several studies suggest that specific types of gambling products are more associated with socio-economic conditions (Worthington 2001;Worthington et al. 2007). Particularly, the payout rate has been identified as one of the main factors that, besides the jackpot, has an impact on consumer preferences and demand (Quiggin 1991). It is therefore presumable that low-income people willing to leapfrog the societal hierarchy may be more attracted by those games which offer a higher payout. ...
In this paper we document the income-related inequality in gambling. We employ a novel database from 2014–2017 waves of the Italian Population Survey on Alcohol and other Drugs (IPSAD) which also include information on the preferences for games of chance. Following the Erryegers Index, our findings suggest that traditional lotteries are concentrated among the richest individuals, while betting and new generation games tend to be pro-poor games. The decomposition of income-related inequalities reveals that pro-rich inequality observed in traditional games is mainly driven by gender, age, and working condition. Higher components of the pro-poor inequality observed in betting and new generation games come instead from income and age. Since the pro-poor games are also the major contributors of the growth in gambling turnover and the increase in gambling disorders, our results indicate that a relevant part of increasing social costs associated to gambling are more likely to be paid by the less-well off, and potentially most vulnerable members of the society.
... The frequent small prizes combat account holders' fatigue by reinforcing their continued interest despite the rarity of big wins. This payoff structure is also consistent with Quiggin's (1991) model, which takes as its starting point an assumption that bettors care about how their wealth compares to that of others (rank-dependent utility). Robson (1996) makes an argument that he bases on biological evolution for utility functions that have relative success as an argument of the function. ...
... For example, Quiggin [23] considers the problem of distributing a fixed amount amongst several homogeneous players with RDU preferences. He concludes that, under certain conditions on players' RDU preferences, the optimum allocation system is a lottery scheme with a few large prizes and a large number of small prizes, and is strictly preferred over distributing the total amount deterministically amongst the players. ...
We show that, in a resource allocation problem, the ex ante aggregate utility of players with cumulative-prospect-theoretic preferences can be increased over deterministic allocations by implementing lotteries. We formulate an optimization problem, called the system problem, to find the optimal lottery allocation. The system problem exhibits a two-layer structure comprised of a permutation profile and optimal allocations given the permutation profile. For any fixed permutation profile, we provide a market-based mechanism to find the optimal allocations and prove the existence of equilibrium prices. We show that the system problem has a duality gap, in general, and that the primal problem is NP-hard. We then consider a relaxation of the system problem and derive some qualitative features of the optimal lottery structure.
... How should prizes be split up? In other words, how much of the expected value of a ticket should go towards the jackpot and how much to smaller prizes, and thus, how skewed should payouts be? Quiggin (1991) developed a model suggesting that the optimal structure of a lotto game should feature a large prize (or maybe a few) and many smaller prizes, suggesting a relatively high level of skewness. Garrett and Sobel (1999) find that across all state lotto games as of 1995, bettors act as if they are risk averse and skewness seeking in that there is a preference of prizes skewed toward the jackpot. ...
Operators of state-owned lottery games generally attempt to set the parameters of the game(s) to maximize revenue to the state. This paper discusses the state of the literature on how to set the parameters of a lotto game so as to accomplish this objective.
... As the pricing kernel is proportional to the product of the marginal utility and the derivative of the weighting function, the pricing kernel does not necessarily fall as we move from a low wealth state to a high wealth state. This analysis is equivalent to that used to explain the purchase of lottery tickets by investors with concave utility functions (Quiggin 1991). ...
... Kahneman and Tversky (1979) argue that lottery players actually over-weight the low chances associated to colossal jackpots and that the popularity of lotteries (and of slot machine games) is in fact due to their low cost chance of winning huge amounts of money. Alternative explanations suggest that when smaller prizes are offered in addition to the jackpot then playing the lottery is a utility maximizing behaviour consistent with players' preferences (Quiggin, 1991). Garrett and Sobel (1999) use data about on-line lotteries in United States and find that gamblers, even if risk-averse, love the positive skewness of returns. ...
We develop an online survey to investigate the characteristics of slot/videopoker players and scratchers (individuals buying tickets of scratch-off lotteries). We find evidence of a negative relationship between gambling and financial literacy. More specifically, after controlling for gender, education, income, employment status and area of residence: (1) slot and/or videopoker players have an 8% lower probability of answering correctly to all of the three standard financial education questions; (2) scratch-off players have a 10% lower probability of answering correctly to the (third) mutual fund risk diversification question, (3) a correct answer to all financial education questions raises by 14% the probability of declaring unwillingness to play due to lack of economic convenience. Scratch-off players are as well more impatient and more likely to sacrifice expected value for positive skewness and they overestimate the probability of winning at least the price of the ticket. Our findings suggest that scratchers and slot/videopoker players may be unable to evaluate the overall consequences of gambling on their economic wellbeing due to their lower financial education.
... Long-term store contains learned sequences of information processing that may be initiated by a control process or by environmental or internal information input but that are then executed automatically with few demands upon the capacity of short-term store [21][22][23][24][25][26][27][28][29][30]. This automatic process can be activated in response to a particular input configuration, where the inputs may be externally or internally generated and include the general situational context, or it will be activated automatically without the necessity of active control or attention by the subject [31][32][33][34][35][36]. For example, a red traffic light might initiate a braking response when the perceiver is in a car, and a walking, halting, or traffic-scanning response when the perceiver is a pedestrian. ...
Research by cognitive psychologists is challenging the rationality paradigm. It has started to penetrate economic modeling with a vast experimental documentation of cognitive limitations that has been accumulated so far raising the questions: Does behavioral decision models, which capture hope or expectation to risk in experimental settings, can help us comprehend investor behavior in financial markets. We try to portrait a whole picture on the attention hypothesis we have developed around pricing patterns within the innovative pharmaceuticals sector firms. We try to encompass it from several aspects: first, we lay a general psychological framework for Attention theory describing cognitive mechanisms such as the rule of selective attention in processing information in short-term. Second we explore financial aspects of selective attention, and last we explore real life investors` behavior around a major milestone within the drug development process – advisory committee milestone. Our findings based on 78 events dating for 2002-2014, a puzzling investors` behavior. We observe that higher trading is held around regulatory event in what seems like a speculative “micro bubbles”. In this paper, we try to portrait a whole picture on the attention hypothesis and merge it with these “micro bubbles” we assume to be in the drug development industry.
... The lottery bonus was also modied in period 3. Instead of drawing in the lottery once 10 Quiggin (1991) derives some results for the design of an optimal lottery given rank-dependent preferences with overweighting of extreme positive; the results are broadly similar to the structure used in this experiment. Table 1.5 columns 5 and 6; they sizes increased from around 5% initially to around 15%-20% of weekly pay at the two kg thresholds. ...
In the first chapter of this dissertation, I study the introduction of a bonus scheme with lottery payoffs at a large agricultural firm in Malawi. The ???lottery bonus??? was motivated by evidence from lab studies and by real world behavior that suggests that people often become risk-seeking when faced with small chances of large gains. Attendance- and performance-based bonuses were introduced for over 1,600 piece-rate workers. Attendance at work was higher under both schemes and that the effect of the lottery bonus on the probability of full weekly attendance was about twice as large as the effect of the fixed bonus. The larger effect on attendance appears to persist over time as workers gain experience with the lottery distribution.
The second chapter, written jointly with Jason Kerwin, reports results from a field experiment with casual workers in Malawi designed to investigate how income timing and temptation affects spending decisions. Despite anecdotal evidence and suggestive survey data that in this study???s context market days increase the temptation to overspend, being paid at the site of the local market on Saturday compared to Friday did not strongly matter for expenditure levels or temptation spending. However, workers in the monthly group are much more likely than the weekly payment group to invest in a risk-free short-term ???bond??? that required a large payment and that was offered by the project in the week after the last payday.
The third chapter, written jointly with Xavier Gin??, Jessica Goldberg and Dean Yang, describes the results of a field experiment among Malawian farmers aimed at facilitating formal savings for agricultural inputs. Treated farmers were offered the opportunity to have their cash crop harvest proceeds deposited directly into new bank accounts in their own names, while farmers in the control group were paid in cash. The treatment led to higher savings and raised agricultural input usage. Because the treatment effect on savings was only a small fraction of the treatment effect on the value of agricultural inputs, mechanisms other than alleviation of savings constraints per se are needed to explain the treatment???s impact on input utilization.
... The fi scal aspects of these compacts appear to be completely ad hoc with no evidence of fundamental public fi nance principles being applied in their design by states and tribes, much less consideration being given to the design of optimal fi scal compacts in any sense. While state lotteries have benefi ted from the economic analysis of the optimal design of the prize structure (e.g., see Quiggin (1991)), there is as yet no economic analysis of optimal compacts for tribes to share the revenues with state and local governments. One area in need of additional research, consequently, is the question of designing appropriate state-tribal compacts. ...
This paper provides an overview of the forms of taxation that are applied to casinos by state and local governments, and analyzes those taxes and fees from a policy perspective. First, the paper contains a comprehensive review of the taxes and fees applied to commercial casinos in the 11 states where casinos are legal. The two most common forms of taxation include a tax on the net amount gambled (AGR, adjusted gross receipts, or gross receipts minus prizes paid) and admission taxes charged on riverboat casinos. A wide range of tax rates are applied to AGR by the states. Second, economic analysis of the efficiency and equity issues related to casino taxes is presented. Included in the analysis is consideration of the revenue offsets involved with other state and local taxes and the uses of the funds. Finally, a summary of our current knowledge of casino taxation and suggestions for needed research are presented.
... Friedman and Savage (1952) had posited that in order to incorporate this simultaneous risk aversion and risk seeking displayed by an economic agent in a utility function, his/her utility function of wealth should consist of a concave segment, followed by a convex segment, followed yet again by a concave segment. However Quiggin (1991) has shown that the conclusion of Friedman and Savage about the third concave segment in the utility function is erroneous. The utility function of an economic agent showing simultaneous risk aversion and risk seeking behavior would therefore consist of a single concave segment and a single convex segment. ...
Traditional finance theories assume that investors are risk averse whereas in reality investors exhibit both risk averse and risk seeking behaviors. For example the same individual could be purchasing insurance (risk averse) and lottery ticket (risk seeking) simultaneously. We propose that the utility function of an economic agent is concave (risk averse) for wealth below his/her current wealth and is convex (risk seeking) for wealth above his/her current wealth. This type of utility function allows the agent to display simultaneous risk averse and risk seeking behavior. We show that for an agent with such a utility function, the mean-skewness space is relevant in understanding his/her behavior. More specifically, we prove that in equilibrium, the efficient frontier in the mean skewness space is concave and downward sloping. We test our equilibrium relationship empirically with stocks listed on the NYSE from 2002 to 2007 and find that our results support our theory.
... On peut noter que certains auteurs attribuent une « valeur d'usage » au fait même de jouer, en interprétant la participation à un jeu comme le loto ou l'Euromillions comme l'achat d'un « droit au rêve » (Conlisk, 1993, Forrest et al., 2002, Quiggin, 1991. Dans cette optique, Kearney (2005) analyse les effets de substitution entre les loteries et d'autres biens de consommation et montre que les dépenses consacrées aux loteries se substituent, non pas à celles effectuées dans d'autres jeux de hasard, mais bien à celles d'autres biens de consommation. ...
The demand of Euromillions lottery tickets: an international comparison
We analyze the demand of Euromillions lottery tickets, a European lotto-like game launched in 2004 and played simultaneously in nine countries with the same rules and the same draws. Using first the effective price methodology, we show that price elasticities are very different across countries. Especially, Spain and Portugal exhibit a low price elasticity and high mean sales, meaning a low sensitivity to jackpot increases. On the contrary, Ireland and the United Kingdom exhibit a very high long-run elasticity and a large sensitivity to jackpot variations. The interpretation of these results is linked to lower per capita
Gdp
in the two former countries, to the large development of syndication play in Spain and to the special tax regime in Portugal. For
Uk
and Ireland, bookmaking activities and the highly competitive betting market partly explain the results. We then show that cumulative prospect theory (Tversky and Kahneman [1992]), though allowing to explain participation to the lottery, does not improve the estimation of the demand function.
Classification JEL : D81, H71
... Quiggin, 1982, Tversky and Wakker, 1995, Gonzalez and Wu, 1999 has at its core the use of decision weights. Modelling the optimal lottery design for individuals under this framework is the subject of Quiggin (1991). ...
We present a model in which a risk-netural monopolist designs a lottery and sells a ticket to a regret averse consumer. We show that, because he pays disproportionate attention to states in which there are large outcome di¤erences across acts, the regret averse consumer may display a preference for a long shot bet that has a negative ex-pected value. For a general preference speci…cation we determine the lottery contract that maximises expected revenue for the monopolist and we derive conditions under which the lottery is neither trivial nor unrealistic. We use a measure of regret aver-sion to interpret the results, highlighting the trade-o¤ between the consumer's regret aversion and risk aversion in the determination of the optimal contract. With this measure of regret aversion we are able to parameterise regret preferences and calculate the optimal contract for consumers with varying degrees of regret aversion. Thus we are able to con…rm the prediction that the monopolist's expected pro…ts are increasing in the consumer's regret aversion.
... This model has been extended by several researchers (Flemming, 1969;Kim, 1973;Ng, 1975;Crossley, Low and Smith, 2011) to argue that since agents' budget sets are discontinuous, either due to large indivisible expenditures such as university tuition, or due to imperfect credit markets, lottery tickets are purchased to 'convexify' agents' budget sets. However, a number of important objections to the Friedman-Savage model have been made (Machina, 1982;Quiggin, 1991); the most important one is that the individuals' observed gambling behaviour does not change radically in response to changes in their initial wealth, which is consistent with the stylised facts about gambling in the US reported by Kearney et al. (Forthcoming). The Friedman-Savage utility function, however, implies that behaviour will be highly sensitive to changes in initial wealth, with only individuals near the inflexion points displaying propensities to both gamble and insure. ...
... While the jackpot prize may be of primary interest to purchasers of lottery tickets, it is not the only characteristic, and the prizes offered as well as the percentage of sales used to fund the prizes offered (the payout rate) may have an impact on consumer preferences and consumer demand. Quiggin (1991) provides a mathematical model of lottery demand which shows that, although smaller prizes do not have much impact on the overall expected value of a ticket, they do reduce the expected losses so that consumers may prefer lottery games with multiple prizes and prize levels. The model also suggests that product differentiation of lottery tickets is particularly important when consumers of lottery products have very different risk preferences. ...
Lotteries represent an important source of government revenues in many states and countries, so they are of interest to public finance economists. In addition, lotteries provide researchers interested in microeconomic theory and consumer behavior with a type of experimental lab that allows economists to explore these topics. This paper surveys the existing literature on lotteries organized around these two central themes. The first section examines the microeconomic aspects of lotteries including consumer decision-making under uncertainty, price and income elasticities of demand for lottery tickets, cross-price elasticities of lottery ticket to each other and to other gambling products, consumer rationality and gambling, and the efficiency of lottery markets. The second section covers topics related to public finance and public choice including the revenue potential of lotteries, the tax efficiency and dead-weight loss of lottery games, the horizontal and vertical equity of lotteries, earmarking and the fungibility of lottery revenues, and individual state decisions to participate in participate in public lotteries.
... With respect to the public good motive (3), however, Proposition 3 is valid in such a situation as well. 24 Quiggin (1991) applies the rank dependent expected utility model to a lottery with multiple prizes. It is argued that the optimal structure of prizes is close to that observed in games like lotto. ...
Lotteries are often used as a method for financing public goods. Based on the voluntary contribution model, recent research has explained this phe- nomenon by demonstrating that fixed-prize lotteries alleviate the free-rider problem. The present paper shows that this lottery mechanism works better for small groups than in a large economy and that pari-mutuel lotteries lead to a higher surplus than fixed-prize lotteries provided that public good provision is only a minor motive among the bettors. These findings may explain why fixed-prize lotteries are mostly used for financing small-scale projects whereas large lotteries usually rely on a pari-mutuel prize structure.
... The multiple-prizes design reduces the probability of losing (i.e., not winning any prize) from near certainty (in a single large prize design) to some smaller probability for which significant underweighting applies. The anticipated loss associated with the gamble thus falls and the gamble seems much more attractive (Quiggin, 1991). 2 A similar explanation can be proposed based on Prospective Reference theory (Viscusi, 1989). ...
Mainstream explanations to gambling specify conditions under which human agents are locally risk loving. Such theories, however, fail to explain the typically observed prize distribution of a few large prizes and a large number of medium ones—hence the medium prizes paradox. In the current study we show that adaptive learning models recently proposed in the literature offer a solution. Simulations of such models predict that multiple medium prizes will slow down the decrease (over time) in agents' inclination to gamble. We run a laboratory experiment that supports this explanation and shows that the positive effect of medium prizes on the inclination to gamble increases with time.
... Other economists, including Walker and Young (2001), Garrett and Sobel (1999), and Forrest, et al. (2002) indirectly support the presence of scale economies by determining that lottery participants are attracted to games with greater skewness rather than higher overall expected returns. Quiggin (1991) confirms this result by finding that lottery players prefer games with high jackpot prizes in addition to smaller prizes. This paper extends the analysis on the scale economies of lotto in two ways. In the original paper, Cook and Clotfelter look for evidence of scale economies in U.S. state lotto games by multiplying the probability of ...
It has been 15 years since Cook and Clotfelter described the scale economies associated with state-run lotto games in an American Economic Review article entitled “The Peculiar Scale Economies of Lotto.” U.S. states with larger populations are identified as having the ability to offer games with larger jackpots to attract higher sales per capita. The current paper extends this analysis to all current U.S. state and multi-state lotto-style lottery games as well as to a sample of international lotto games for comparative purposes. The development of the two major U.S. multi-state games over time is also examined to illustrate that changes in the structure of those games can be explained by an application of the scale economies concept offered by Cook and Clotfelter.
... roving one's standard of living could induce risk-averse individuals to play unfair gambles. Kahneman and Tversky (1979) argued that in placing decision weights on the probability of each possible outcome, people tended to overweight low probabilities and underweight high probabilities. Overweighting low probability makes unfair gambles attractive. Quiggin (1991) employs a rank-dependent utility function to explain why risk-averse people might play unfair gambles such as lotteries. If the lottery is comprised of a large number of smaller prizes and a few large prizes, risk-averse individuals could find it worthwhile playing. Analysis of actual gambling behavior has been undertaking by a number o ...
To examine the effect of increased skewness on risk taking, we conduct controlled laboratory experiments. Our instrument is an adaption of the Eckel and Grossman (2002, 2008) risk measure (with six gamble choices). The Eckel and Grossman measure is a simplest, easy to understand, exercise that gives sufficient heterogeneity in choices while at the same time minimizing errors. Its simplicity also makes it easy to adapt. The adapted gamble choices are designed to have the same expected payoffs and risk as the original gamble choices, but to exhibit increasing degrees of right skewness. The adapted instrument is used to address the question; does skewness encourage greater or less risk taking? Does the possibility of winning a high-earnings, long shot encourage people into greater or less overall risk taking? It also permits us to confirm, in a controlled environment, that people prefer positive skewness?
Many policies allocate harms or benefits that are uncertain in nature: they produce distributions over the population in which individuals have different probabilities of incurring harm or benefit. Comparing different policies thus involves a comparison of their corresponding probability distributions, and we observe that in many instances the policies selected in practice are hard to explain by preferences based only on the expected value of the total harm or benefit they produce. In cases where the expected value analysis is not a sufficient explanatory framework, what would be a reasonable model for societal preferences over these distributions? Here we investigate explanations based on the framework of probability weighting from the behavioral sciences, which over several decades has identified systematic biases in how people perceive probabilities. We show that probability weighting can be used to make predictions about preferences over probabilistic distributions of harm and benefit that function quite differently from expected-value analysis, and in a number of cases provide potential explanations for policy preferences that appear hard to motivate by other means. In particular, we identify optimal policies for minimizing perceived total harm and maximizing perceived total benefit that take the distorting effects of probability weighting into account, and we discuss a number of real-world policies that resemble such allocational strategies. Our analysis does not provide specific recommendations for policy choices, but is instead interpretive in nature, seeking to describe observed phenomena in policy choices.
Using a multiple indicators and multiple causes (MIMIC) model, this paper estimates the extent of illegal gambling in Italian regions over the period 2013–18. By treating illegal gambling as an unobserved latent variable directly related to its causes and effects, this model gives information about the relationship between cause and indicator variables and the latent variable from covariance structures. From the analysis, it emerges that the share of illegal gambling increases with the value of the winnings paid; it decreases when the number of authorized machines increases. We also find that individuals with low levels of education and low income show a greater propensity to gamble illegally.
We show that, in a resource allocation problem, the ex ante aggregate utility of players with cumulative-prospect-theoretic preferences can be increased over deterministic allocations by implementing lotteries. We formulate an optimization problem, called the system problem, to find the optimal lottery allocation. The system problem exhibits a two-layer structure comprised of a permutation profile and optimal allocations given the permutation profile. For any fixed permutation profile, we provide a market-based mechanism to find the optimal allocations and prove the existence of equilibrium prices. We show that the system problem has a duality gap, in general, and that the primal problem is NP-hard. We then consider a relaxation of the system problem and derive some qualitative features of the optimal lottery structure.
We examine the coexistence of insurance and gambling in the context of limited liability. We develop a model where actuarially fair insurance is available to a risk-averse decision maker for a liability risk with non-bankrupting severity. The remaining wealth may be invested in a zero expected value risky project (i.e., gambled). The risk of bankruptcy is endogenous since either fully insuring or forgoing the project will guarantee solvency. We show that, for a range of parameters, it is optimal to both insure and gamble. The amounts insured and invested are chosen to create the potential for bankruptcy. Our results are robust to the cases where the risky project can cause bankruptcy without a liability loss and where the risky project’s expected return is nonzero.
An implicit goal of Bitcoin's reward structure is to diffuse network influence over a diverse, decentralized population of individual participants. Indeed, Bitcoin's security claims rely on no single entity wielding a sufficiently large portion of the network's overall computational power. Unfortunately, rather than participating independently, most Bitcoin miners join coalitions called mining pools in which a central pool administrator largely directs the pool's activity, leading to a consolidation of power. Recently, the largest mining pool has accounted for more than half of network's total mining capacity. Relatedly, "hosted mining" service providers offer their clients the benefit of economies-of-scale, tempting them away from independent participation. We argue that the prevalence of mining coalitions is due to a limitation of the Bitcoin proof-of-work puzzle -- specifically, that it affords an effective mechanism for enforcing cooperation in a coalition. We present several definitions and constructions for "nonoutsourceable" puzzles that thwart such enforcement mechanisms, thereby deterring coalitions. We also provide an implementation and benchmark results for our schemes to show they are practical.
This work elicits the utility functions of financial practitioners and measures their loss aversion coefficients under prospect theory (1992) using the parameter-free method of Abdellaoui et al. (2006). The measurements in the field corroborate the latter's measurements in the laboratory regarding the concavity of the utility function for gains and convexity for losses. However, although loss aversion exists in the aggregate, the median practitioner is found to be less loss averse than the median student. Conditions that characterize a real market experience but are difficult to realize in the artificial context of the laboratory may account for the behavioral difference. Among them are the schooling in the assessment of prospects, the volatility of the market and the Wall Street's compensation incentives. An important proviso is that the preferences of the students/practitioners analyzed following another method reflect consistent preferences. The qualitative investigation of the preferences of MBA students using the parameter-free method developed by Baucells and Heukamp (2006) supports the results of Abdellaoui et al.'s (2006) for students. A noteworthy result is the strong tendency to shift from loss aversion to gain seeking for the higher overall probability of gain or the higher probability of maximal gain combined with a limited extreme loss.
High quality gambling participation and prevalence studies are important to monitor and manage risks associated with an ever changing gambling environment. Such studies are expensive and choices must be made to balance precision and cost constraints. Common approaches include not screening infrequent gamblers and/or gamblers with low gambling expenditure, on the assumption that these gamblers are unlikely to be problem gamblers. Some studies combine moderate risk and problem gamblers to obtain robust groups for analysis. The resultant methodological variations compromise comparability and interpretation of changes over time and across jurisdictions. The Victorian Gambling Study 2008-2012 (VGS) and the Swedish Longitudinal Gambling Study 2008-2014 (Swelogs) are large jurisdiction-wide surveys with sample sizes of 15,000 and 8,165 respectively. Both studies screened all past-year gamblers. This paper reports on the modelling of some common methodological variants. Understanding the impact of these different variations enables better design of future studies and facilitates comparisons between existing prevalence studies within and across jurisdictions.
The present study proposes a two-stage procedure to collect data about semi-compensatory choice processes in a digital economy environment. At the first stage, individuals form their viable choice set by specifying a set of tolerated criteria thresholds. At the second stage, individuals choose their most preferred alternative. The digital economy environment allows seamless tracking of the two-stage choice protocols without interfering with the natural choice process and without introducing problems related to comprehension bias, narrative inconsistency and misinterpretation of the choice protocols. The procedure is accompanied by a questionnaire collecting personal information that can be associated to threshold selection. The procedure is applied to off-campus apartment rental choices by students in order to demonstrate its capability to retrieve tolerated criteria thresholds and choice outcomes from cross-sectional data in a choice situation entailing many alternatives and multiple criteria. Results show the importance of elimination-based choice set formation, the distribution of threshold selection across the population, the characteristics of the considered choice sets, and the linkage between the choice set formation and the choice.
This article proposes an equilibrium approach to lottery markets in which a firm designs an optimal lottery to rank-dependent expected utility (RDU) consumers. We show that a finite number of prizes cannot be optimal, unless implausible utility and probability weighting functions are assumed. We then investigate the conditions under which a probability density function can be optimal. With standard RDU preferences, this implies a discrete probability on the ticket price, and a continuous probability on prizes afterwards. Under some preferences consistent with experimental literature, the optimal lottery follows a power-law distribution, with a plausibly extremely high degree of prize skewness.
In this article, we determine the optimal design of lottery-linked savings (LLS) programmes. LLS vehicles, such as lottery bonds, are financial instruments that preserve depositors’ principal but provide randomized variable returns to these depositors through periodic lottery drawings, in lieu of the regular coupon payments traditionally received by bondholders. This type of asset is very popular in Europe. We note that the framing of LLS instruments is linked to their popularity, and we attempt to elucidate that link and thereby determine the optimal structure of payments for LLS programmes. Our goal is to frame LLS programmes in a manner that both maximizes their investors’ satisfaction and preserves their ability to serve as inexpensive sources of capital for their issuers. We demonstrate that the optimal pay-off structure is characterized by a high level of skewness, as investors are willing to accept a decrease in the small and medium prizes of the lottery in exchange for an increase in the grand prize. Thus, to attract a maximum number of depositors, the issuers of LLS programmes should prioritize the jackpot size relative to the other lottery pay-offs and insist on high skewness for the lottery pay-offs relating to the LLS assets.
Based on the rank-dependent expected utility model (Quiggin, 1991), hypotheses are formed in this study regarding optimal prize promotion structure with reference to associated probabilistic aspects. Referencing both modeling works and related behavioral theories, we compared several design schemes. Using purchase scenarios of a low-value product (bread) and a high-value product (cell phone) we determined the optimal design among promotion schemes that differ by winning probability, prize amount, and number of prize value levels. We found that a promotion offering a combination of high-value prizes plus some low-value prizes was invariably preferred over a promotion offering only high- or low-value prizes. We also explored whether these high- or low-value prizes should be in a series of ascending value or prizes at each value level should be of the same value and whether or not some moderately valuable prizes should also be included.
The paper presents a unifying approach for analyzing the microfounda-tions of gambling. On basis of Normal Randomness Expected Utility (in-tegrating expected utility of wealth, elation and disappointment in an in-tertemporal framework) optimal wager and riskiness in the "share-holder" and "roulette type" gambles are compared. Predictions are derived, how individual gamblers react with regard to wager and "riskiness" to alter-native methods of increasing the house advantage, to di¤erent preferences (e.g. time preferences) or levels of wealth.
The purpose in this letter is to demonstrate, employing two parametric forms of the Markowitz model of utility, that heterogeneity of preferences of Markowitz agents can contribute towards an explanation of why lotteries typically have multiple rather than single prizes.
Popular lotteries typically give a very small probability to win a large prize and a moderate chance to win a smaller prize. In this paper a rank dependent model is axiomatized with an S-shaped weighting function, capable of giving an account for the popularity of these lotteries. Also, the role of utility, loss aversion and scale compatibility in explaining the buying of lotteries is discussed. Correctie: http://www1.fee.uva.nl/creed/wakker/pdf/heiancor.pdf
The estimation of semi-compensatory models is gaining momentum in transport planning in recent years. However, traditional survey methodologies focus on collecting solely compensatory choice data, which leads to information loss when semi-compensatory models are estimated. The present study proposes a novel web-based survey that enables collecting data about the entire semi-compensatory choice process. The web-based environment allows seamless tracking of semi-compensatory choice protocols without interfering with the natural choice process and without introducing problems related to comprehension bias, narrative inconsistency and misinterpretation of the choice protocols. The procedure is applied to rental apartment choice by students and results shed light on semi-compensatory choice by: (1) demonstrating the importance of choice set formation; (2) unravelling the distribution of threshold selection across the population; (3) revealing the linkage between the viable choice-set and the choice.
We describe a coupled hydrologic–economic spreadsheet model for the Murray-Darling Basin that allows analyses of water allocation and use by different sectors including agriculture and environment under alternative policy scenarios. The model is a simple, lumped optimisation model which includes partitioning of rainfall into runoff and evapotranspiration, a reach by reach water balance of the river system, irrigation demand and revenue generation. Groundwater is not considered because groundwater use is a small part of the overall water use. The model is used to optimize profit, diversions and flows subject to hydrological and economic constraints determined by the policy scenario.We use the model to examine approaches of acquiring water for reallocation to the environment, and their impacts on irrigation water use and regional income from agriculture. We show that the optimal approach for acquisition depends on: economic factors, including the cost of water and the profits generated by its use; institutional factors, such as restrictions to trade between regions; and, hydrological factors, particularly the connectivity of and losses within the river network.The volume of water to be acquired does not, in general, equal the volume to be allocated. For a downstream site, water must be acquired from upstream, and more water must be acquired than is to be allocated: the volume acquired is that to be allocated plus transmission losses between the locations of acquisition and allocation. For upstream sites, it is optimal to acquire some water from downstream, and less water must be acquired than is to be allocated: the volume acquired is that to be allocated, less the transmission losses no longer incurred between the locations of acquisition and allocation.The volumes of water that must be acquired to satisfy an allocation target and maintain flows in the river system are affected by restrictions on trade between sub-catchments.
Any jackpot building game is designed to have a negative expected return for the gambler, but it can be profitable under certain circumstances. Previous studies have shown that the purchase of a single ticket of US-American state lotteries is sometimes a gamble with a positive expected value. Lottery winnings are not taxed in Europe, which suggests that the profitability of European games may be even higher. We present an exact formula for the calculation of the expected value of a single lotto ticket and find European lottery drawings to be far less profitable for the gambler compared to the US-American lotto market. Those US lotteries that generate profitable drawings are not characterized by higher redistribution rates or by their specific rules, but by the purchasing behavior of the gamblers. These gamblers buy far fewer tickets (per capita) and they barely react to increasing jackpots, even though the jackpots are large enough to cause positive expected winnings.
Psychologists trained in psychophysics tend to think that subjective probability is related to objective probability in more or less the same way that the subjective loudness of a tone is related to its objective intensity. The purpose of this paper is to examine the content, merits, and limitations of such an approach. The discussion will focus on two closely related matters. The first is the idea of a set of functions relating subjective to objective probability. The second is whether or not the subjective probabilities of a set of mutually exclusive events, one of which must happen, should add up to one. The paper begins by denning two classes of decision theories. After some preliminary discussion of utility and subjective probability functions, it next considers the class of theories which result when subjective probabilities are assumed to add up to one. This class turns out to have some serious difficulties. A brief review of experimental evidence provides empirical reasons for avoiding these difficulties by rejecting additivity. Then the paper examines models which do not require subjective probabilities to add up to one. Such models require a reformulation of the concept of utility; utility scales must have true zero points. Finally the paper derives two experimentally testable properties which any plausible utility-subjective probability model, additive or otherwise, must have, and presents a model which I find attractive.
Experimental studies have shown that the key behavioral assumption of expected utility theory, the so-called "independence axiom," tends to be systematically violated in practice. Such findings would lead us to question the empirical relevance of the large body of literature on the behavior of economic agents under uncertainty which uses expected utility analysis. The first purpose of this paper is to demonstrate that the basic concepts, tools, and results of expected utility analysis do not depend on the independence axiom, but may be derived from the much weaker assumption of smoothness of preferences over alternative probability distributions. The second purpose of the paper is to show that this approach may be used to construct a simple model of preferences which ties together a wide body of observed behavior toward risk, including the Friedman-Savage and Markowitz observations, and both the Allais and St. Petersburg Paradoxes.
The “Allais Paradox” is often cited as an example of the limited descriptive ability of the Expected Utility model. It is shown here that the Allais Paradox does not induce a contradiction in the Subjectively Weighted Utility (SWU) model proposed elsewhere as a descriptive extension of the expected utility framework. This approach is compared with the CE and Prospect theories of decision making in which a “subcertainty” property explains the Allais Paradox. The related issues of independence of outcomes and the aggregation of outcomes are also discussed briefly.
I. Are there uncertainties that are not risks? 643. — II. Uncertainties that are not risks, 647. — III. Why are some uncertainties
not risks? — 656.
One of the most notable counterexamples to expected utility theory is the “Allais paradox” (M. Allais, 1953, Econometrica, 31, 503–546). A number of alternative theories have been proposed in an attempt to resolve this paradox, notably including and , 24, 67–72). It is shown that SWU theory necessarily involves violations of dominance, but that the theory can be modified to avoid these violations. The result is a special case of J. Quiggin's anticipated utility theory (1982, Journal of Economic Behaviour and Organisation, 3, 323–343).
A parsimonious extension to the Expected Utility (EU) model is proposed involving one additional parameter. The model is an example of a Subjectively Weighted Utility (SWU) model, which differs from the EU model only in the way probabilities are incorporated. The probability transformation is of a form that has appeared in the literature on human information processing and the additional parameter α has been interpreted as an index of information processing performance.The SWU model is shown to have considerable descriptive ability. It is discussed in the context of the substitutability axiom, distinct utility curves, insurance purchasing and gambling behavior, and probability variance preferences.
Subjective and estimated objective winning probabilities are obtained from 20,247 harness horse races. It is shown that subjectively a horse with a low winning probability is exaggerated and one with a high probability of winning is depressed. Various hypotheses characterizing the bettors' behavior to explain the observed subjective-objective probability relation are explored. Under some simplified assumptions, a utility of wealth function of a decision maker is derived, and a quantitative summary measure of his risk attitude is defined. Attitude toward risk of a representative bettor is examined. It is found that he is a risk lover and tends to take more risk as his capital dwindles.
This paper presents a set of certainty-equivalence (CE) axioms which allow the individual's preferences to be expressed by a cardinal utility index in the case of quantifiable single-good, uncertain (or certain) prospects. This axiom set differs from the von Neumann-Morgenstern (NM) axiom set. The analysis is extended to the multigood case, and its implications for risk taking are derived. Various tests reported in the literature on experimental psychology show the CE theory to outperform the von Neumann-Morgenstern one. Further, the CE theory satisfactory explains the Allais paradox, which contradicts the NM theory, and is thus to be preferred to the latter as a positive theory for decisions under uncertainty.
I. Introduction, 148. — II. The model, 150. — III. The nature of market imperfections, 152. — IV. Summary and implications,
153.
The efficiency and profitability of exotic racetrack bets such as exactas and daily doubles are examined. Efficiency is understood to mean that above average returns cannot be made in the long run once risk is appropriately controlled for. The markets in question are found not to be efficient; the inefficiencies, however, are insufficient to permit simple strategies to show a consistent profit. Some evidence of "smart money" exists in that holders of inside information may bet on exactas rather than equivalent standard bets in order to avoid signaling their actions to the betting public. Copyright 1987 by The Review of Economic Studies Limited.
This paper attempts to answer the question: When is a random variable Y “more variable” than another random variable X?
This remark proves that Quiggin's anticipated utility function may solve the Allais paradox and the common ratio effect. For some generalizations of these it is needed to assume that the decision-weight function is concave.
A new theory of cardinal utility, with an associated set of axioms, is presented. It is a generalization of the von Neumann-Morgenstern expected utility theory, which permits the analysis of phenomena associated with the distortion of subjective probability.
This paper shows that: (1) the "preference reversal" phenomenon can be consistent with transitive preferences if these preferences violate the independence axiom of expected utility theory and (2) for the class of experiments that were used to produce the evidence concerning "preference reversal," the elicitation of certainty equivalents is possible if, and only if, the respondent's preferences can be represented by functionals that are linear in the probabilities. Furthermore, a more general class of experiments is not immune to "preference reversal" if nonexpected utility preferences are admitted. Copyright 1987 by The Econometric Society.
This paper investigates the consequences of the following modification of Expected Utility theory: instead of requiring independence with respect to probability mixtures of risky prospects, require independence with respect to direct mixing of payments o f risky prospects. A new theory of choice under risk- a so-called Dual theory-is obtained. Within this new theory, the following questions are considered: (1) numerical representation of preferences; (2) properties of the utility function; ( 3) the possibility for resolving the "paradoxes" of Expected Utilit y theory; ( 4) the characterization of risk aversion; and (5) comparative statics. The paper ends with a discussion of other non-Expected Utility theories proposed recently. Copyright 1987 by The Econometric Society.
Analysis of decision making under risk has been dominated by expected utility theory, which generally accounts for people's actions. Presents a critique of expected utility theory as a descriptive model of decision making under risk, and argues that common forms of utility theory are not adequate, and proposes an alternative theory of choice under risk called prospect theory. In expected utility theory, utilities of outcomes are weighted by their probabilities. Considers results of responses to various hypothetical decision situations under risk and shows results that violate the tenets of expected utility theory. People overweight outcomes considered certain, relative to outcomes that are merely probable, a situation called the "certainty effect." This effect contributes to risk aversion in choices involving sure gains, and to risk seeking in choices involving sure losses. In choices where gains are replaced by losses, the pattern is called the "reflection effect." People discard components shared by all prospects under consideration, a tendency called the "isolation effect." Also shows that in choice situations, preferences may be altered by different representations of probabilities. Develops an alternative theory of individual decision making under risk, called prospect theory, developed for simple prospects with monetary outcomes and stated probabilities, in which value is given to gains and losses (i.e., changes in wealth or welfare) rather than to final assets, and probabilities are replaced by decision weights. The theory has two phases. The editing phase organizes and reformulates the options to simplify later evaluation and choice. The edited prospects are evaluated and the highest value prospect chosen. Discusses and models this theory, and offers directions for extending prospect theory are offered. (TNM)
We identify incentives generated by the Bretton Woods II system that may have contributed to the sub-prime liquidity crisis now working its way through the international monetary system. We then evaluate the persistent conjecture that the liquidity crisis is or will become a balance of payments crisis for the United States. Given that it happens, the additional costs associated with a sudden stop of net capital flows to the United States could be quite substantial. But we observe that emerging market governments have continued to acquire US assets even as yields have fallen, and the incentives for continuing to do so remain strong. Moreover, the Bretton Woods II system, which has clearly been the most resilient of the forces driving current markets, continues to generate low real interest rates in industrial countries and growth in emerging markets that will help limit the damage from the liquidity crisis. Copyright © 2009 Blackwell Publishing Ltd.
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