Lottery pricing under time pressure

This article investigates how subjects determine minimum selling prices for lotteries. We design an experiment where subjects have at every moment an incentive to state their minimum selling price and to adjust the price, if they believe that the price that they stated initially was not optimal. We observe frequent and sizeable price adjustments. We find that random pricing models cannot explain the observed price patterns. We show that earlier prices contain information about future price adjustments. We propose a model of Stochastic Pricing that offers an intuitive explanation for these price adjustment patterns.     

唐代物价制定及其作用

在唐代,市估不但对官方行为具有强制性作用,是官方买卖、收付折算、平赃定罪、控制物价必须依据的价格标准;对一般民间(私人)交易也有着不同程度的约束(指导、调节)作用,且在有人请求、发生争执、立券公证等情况下也不排除依市估而定。笔者以为对唐代国家的物价管理政策及“官市依估,私但两和”的交易原则,不宜做绝对性理解。     

The Becker-DeGroot-Marschak mechanism and nonexpected utility: A testable approach

The Becker-DeGroot-Marschak mechanism is widely used to elicit decision makers' selling prices of lotteries. This mechanism leads, however, to the preference reversal phenomenon, which seemed to indicate nontransitive preferences. To solve this puzzle, Karni and Safra (1987) introduced a new interpretation of this mechanism based on two-stage lotteries without the independence axiom. In this article, we suggest a set of empirically testable hypotheses based on their interpretation of the mechanism. One of these tests can be used to find the utility and the probability transformation functions of an anticipated utility maximizer.This article consists of an earlier paper with a similar title (University of Toronto WP #8809) and the paper Elicitation of Certainty Equivalents and the Becker-DeGroot-Marschak Mechanism.     

Buying and selling price for risky lotteries and expected utility theory with gambling wealth

I analyze two expected utility models which abandon the consequentialist assumption of terminal wealth positions. In the expected utility of gambling wealth model, in which initial wealth is allowed to be small, I show that a large WTA/WTP gap is possible and the (Rabin in Econometrica, 68(5), 1281–1292, 2000) paradox may be resolved. Within the same model the classical preference reversal which allows arbitrage is not possible, whereas preference reversal (involving buying prices in place of selling prices), which does not allow arbitrage, is possible. In the expected utility of wealth changes model, in which there is no initial wealth, I show that both a WTA/WTP gap as well as the classical preference reversal are possible due to loss aversion, both in its general as well as some specific forms.     

Are Preference Reversals Errors? An Experimental Investigation

This paper investigates whether some part of the preference reversal phenomenon can be attributed to errors in the responses of subjects in experiments. Such errors have been well documented in other investigations of behaviour in risky decision problems, but their relevance to the preference reversal phenomenon has not been explored. Building on earlier work, we develop an extended error model and apply it to the results of an experiment in which subjects tackle risky choice problems on five separate occasions. In this experiment subjects had to answer choice questions in three occasions and to state selling and buying prices in the remaining two occasions. Our results indicate that scale compatibility can be ruled out as a significant sole explanation of the preference reversal phenomenon. Moreover, we can show that a considerable fraction of observed preference reversals can be classified as pricing errors, whereas choice errors turn out to play a minor role.     

A possibilistic hierarchical model for behaviour under uncertainty

Hierarchical models are commonly used for modelling uncertainty. They arise whenever there is a `correct' or `ideal' uncertainty model but the modeller is uncertain about what it is. Hierarchical models which involve probability distributions are widely used in Bayesian inference. Alternative models which involve possibility distributions have been proposed by several authors, but these models do not have a clear operational meaning. This paper describes a new hierarchical model which is mathematically equivalent to some of the earlier, possibilistic models and also has a simple behavioural interpretation, in terms of betting rates concerning whether or not a decision maker will agree to buy or sell a risky investment for a specified price. We give a representation theorem which shows that any consistent model of this kind can be interpreted as a model for uncertainty about the behaviour of a Bayesian decision maker. We describe how the model can be used to generate buying and selling prices and to make decisions.     

Stochastic Choice and Consistency in Decision Making Under Risk: An Experimental Study

This paper reports the results of an experiment designed to uncover the stochastic structure of individual preferences over lotteries. Unlike previous experiments, which have presented subjects with pair-wise choices between lotteries, our design allowed subjects to choose between two lotteries or (virtually) any convex combination of the two lotteries. We interpret the mixtures of lotteries chosen by subjects as a measure of the stochastic structure of choice. We test between two alternative interpretations of stochastic choice: the random utility interpretation and the deterministic preferences interpretation. The main findings of the experiment are that the typical subject prefers mixtures of lotteries rather than the extremes of a linear lottery choice set. The distribution of choices does not change between a first and second asking of the same question. We argue that this provides support for the deterministic preferences interpretation over the random utility interpretation of stochastic choice. As a subsidiary result, we find a small proportion of subjects make choices that violate transitivity, but the level of intransitive choice falls significantly over time.     

Experimental payment protocols and the Bipolar Behaviorist

If someone claims that individuals behave as if they violate the independence axiom (IA) when making decisions over simple lotteries, it is invariably on the basis of experiments and theories that must assume the IA through the use of the random lottery incentive mechanism (RLIM). We refer to someone who holds this view as a Bipolar Behaviorist, exhibiting pessimism about the axiom when it comes to characterizing how individuals directly evaluate two lotteries in a binary choice task, but optimism about the axiom when it comes to characterizing how individuals evaluate multiple lotteries that make up the incentive structure for a multiple-task experiment. We reject the hypothesis about subject behavior underlying this stance: we find that preferences estimated with a model that assumes violations of the IA are significantly affected when one elicits choices with procedures that require the independence assumption, as compared to choices elicited with procedures that do not require the assumption. The upshot is that one cannot consistently estimate popular models that relax the IA using data from experiments that assume the validity of the RLIM.     

证券投资基金羊群效应实证研究

我国基金市场中存在显著的“羊群效应”,“羊群效应”随着股票历史收益率的增加呈现逐步减弱的趋势;基金卖出的“羊群效应”强于买入的“羊群效应”,“羊群效应”与基金投资绩效之间具有正相关性;基金在买卖小规模公司股票时的“羊群效应”明显强于买卖大规模公司股票时的“羊群效应”。     

The Lottery and Income Inequality in the States*

Objective. Since the early 1970s, income inequality in the United States has increased dramatically. We examine the impact of state lotteries on income inequality in the American states from 1976–1995. Methods. We use cross‐sectional time‐series data to evaluate the effect of lotteries as well as those of other state tax policies, redistributive programs, and demographic factors on income inequality. Results. We find that state lotteries foster income concentration. Ceteris paribus, states with lotteries have higher levels of income inequality than those states without a lottery. We also find that additional demographic and policy factors have an impact on income inequality in the states. Conclusions. One of the most important policy‐oriented determinants of income inequality is the lottery and a significant portion of the increase in income inequality over our two‐decade time period is attributable to the increasing prevalence and popularity of state lotteries.     

Ambiguity aversion in the small and in the large for weighted linear utility

The widely observed preference for lotteries involving precise rather than vague of ambiguous probabilities is called ambiguity aversion. Ambiguity aversion cannot be predicted or explained by conventional expected utility models. For the subjectively weighted linear utility (SWLU) model, we define both probability and payoff premiums for ambiguity, and introduce alocal ambiguity aversion function a(u) that is proportional to these ambiguity premiums for small uncertainties. We show that one individual's ambiguity premiums areglobally larger than another's if and only if hisa(u) function is everywhere larger. Ambiguity aversion has been observed to increase 1) when the mean probability of gain increases and 2) when the mean probability of loss decreases. We show that such behavior is equivalent toa(u) increasing in both the gain and loss domains. Increasing ambiguity aversion also explains the observed excess of sellers' over buyers' prices for insurance against an ambiguous probability of loss.     

中国股指期货对股票市场的流动性影响

随着中国股指期货的推出,股指期货对股票市场流动性的影响又成为一个新的课题。文章运用向量自回归(VAR)模型、Granger因果检验分析了中国股指期货对股票市场的流动性影响,检测了流动性溢价理论在中国股指期货市场的应用。最终得出结论,股票市场的买卖价差和期货现货基差之间存在双向的因果关系,即股票市场的流动性波动对投资股指期货的预期收益变化有重要的作用。     

Loss Averse Behavior

A behavioral condition of loss aversion is proposed and tested. Forty-nine students participated in experiments on binary choices among lotteries involving small scale real gains and losses. At the aggregate level, a significant proportion of the choices are in the direction predicted by loss aversion. Individuals can be classified as loss averse (28 participants), gain seeking (12), and unclassified (9). A comparison with risk behavior for binary choices on lotteries involving only gains shows that risk attitudes vary across these domains of lotteries. A gender effect is also observed: proportionally more women are loss averse. In contrast to the predictions of comonotonic independence, the size of common outcomes has systematic influence on choice behavior. JEL Classification: D81, C91     

Risk aversion elicitation: reconciling tractability and bias minimization

Risk attitude is known to be a key determinant of various economic and financial choices. Behavioral studies that aim to evaluate the role of risk attitudes in contexts of this type, therefore, require tools for measuring individual risk tolerance. Recent developments in decision theory provide such tools. However, the methods available can be time consuming. As a result, some practitioners might have an incentive to prefer “fast and frugal” methods to clean but more costly methods. In this article, we focus on a tractable procedure initially proposed by Holt and Laury (2002) to elicit risk attitude. We generalize this method to measure utility and risk aversion as follows. First, we allow measurement of probabilistic risk attitude through violations of expected utility due to probability weighting. Second, we use the outcome scale rather than the probability scale in the menu of choices. Third, we compare sure payoffs with lotteries instead of comparing non-degenerate lotteries. A within-subject experimental study illustrates the gains in tractability and bias minimization that can result from such an extension.     

Anxiety and Decision Making with Delayed Resolution of Uncertainty

In many real-world gambles, a non-trivial amount of time passes before the uncertainty is resolved but after a choice is made. An individual may have a preference between gambles with identical probability distributions over final outcomes if they differ in the timing of resolution of uncertainty. In this domain, utility consists not only of the consumption of outcomes, but also the psychological utility induced by an unresolved gamble. We term this utility anxiety. Since a reflective decision maker may want to include anxiety explicitly in analysis of unresolved lotteries, a multiple-outcome model for evaluating lotteries with delayed resolution of uncertainty is developed. The result is a rank-dependent utility representation (e.g., Quiggin, 1982), in which period weighting functions are related iteratively. Substitution rules are proposed for evaluating compound temporal lotteries. The representation is appealing for a number of reasons. First, probability weights can be interpreted as the cognitive attention allocated to certain outcomes. Second, the model disaggregates strength of preference from temporal risk aversion and thus provides some insight into the old debate about the relationship between von Neumann–Morgenstern utility functions and strength of preference value functions.     

Lottery qualities

The aim of this paper is to propose a model of decision-making for lotteries. Lottery qualities are the key concepts of the theory. Qualities allow the derivation of optimal decision-making processes and are taken explicitly into account for lottery evaluation. Our contribution explains the major violations of the expected utility theory for decisions on two-point lotteries and shows the necessity of giving explicit consideration to lottery qualities. Judged certainty equivalent and choice certainty equivalent concepts are discussed in detail along with the comparison of lotteries. Examples are provided by using different test results in the literature. JEL Classification D81     

Social comparison and risky choices

Theories (and experiments) on decision making under risk typically ignore (and exclude) a social context. We explore whether this omission is detrimental. To do so we experimentally investigate the simplest possible situation with both social comparison and risk: participants choose between two lotteries while a referent faces a fixed payoff. Participants are more risk averse when they can earn at most as much as their referent (loss situation) than when they are ensured they will earn at least as much as their referent (gain situation). Prospect theory with a social reference point would predict the exact opposite behavior. These results show that straightforward extensions of existing theories to allow for social comparison do not provide accurate predictions.     

Signed orders in linear and nonlinear utility theory

Self-reflecting signed orders on a set A and its anti-set A ^* were introduced previously as a way to account for negative as well as positive feelings about the inclusion of items in A in potential subsets of choice. The present paper extends the notion of signed orders to lotteries on A A ^*, describes reflection axioms for the lottery context, and shows how these axioms simplify utility representations for preference between lotteries. The simplified representations are then used to guide procedures for extending preferences from A A ^* and its lotteries to preferences between subsets of items.     

Reconsidering the common ratio effect: the roles of compound independence,reduction, and coalescing

Common ratio effects should be ruled out if subjects’ preferences satisfy compound independence, reduction of compound lotteries, and coalescing. In other words, at least one of these axioms should be violated in order to generate a common ratio effect. Relying on a simple experiment, we investigate which failure of these axioms is concomitant with the empirical observation of common ratio effects. We observe that compound independence and reduction of compound lotteries hold, whereas coalescing is systematically violated. This result provides support for theories which explain the common ratio effect by violations of coalescing (i.e., configural weight theory) instead of violations of compound independence (i.e., rank-dependent utility or cumulative prospect theory).     

Judged knowledge and ambiguity aversion

Competence has recently been proposed as an explanation for the degree of ambiguity aversion. Using general knowledge questions we presented subjects with simple lotteries in which they could bet on an event and against the same event. We show that the sum of certainty equivalents for both bets depends on the judged knowledge of the class of events. We also elicited the decision weights for events and complementary events. We found a similar effect of knowledge on the sum of decision weights.