Paying to improve your chances: Gambling or insurance?

Will a more risk-averse individual spend more or less to improve probabilities, say on marketing efforts that enhance the chance of a sale? For any two payoffs and starting probabilities, the answer is unfortunately indeterminate. However, interpreting gambling as increasing small chances of good outcomes and insurance as reducing small chances of bad outcomes, the more risk-averse individual will pay less (more) to gamble (insure). We find a critical switching probability that depends on the individuals and outcomes involved. If the good outcome is less (more) likely than this critical value, the expenditures represent gambling (insurance).     

Testing Prospect Theories Using Probability Tradeoff Consistency

The two versions of prospect theory, original prospect theory (OPT; Kahneman and Tversky, 1979) and cumulative prospect theory (CPT; Tversky and Kahneman, 1992), use different composition rules to combine the value function and the probability weighting function and hence value gambles with two or more non-zero outcomes differently. Previous tests of OPT and CPT have yielded mixed results, with some investigations supporting OPT and some supporting CPT. We extend the probability tradeoff consistency axiom used in Abdellaoui (2002) for CPT to OPT, and develop a critical test of the two prospect theories based on their respective probability tradeoff consistency conditions. An empirical investigation of the critical test shows that choices are consistent with OPT, but not CPT, for gambles that do not involve a certainty effect, and consistent with both CPT and OPT for gambles that do involve a certainty effect, provided that an editing operation is invoked for OPT.     

Preference towards control in risk taking: Control,no control,or randomize?

This paper experimentally investigates preference towards different methods of control in risk taking. Participants are asked to choose between different ways for choosing which numbers to bet on for a gamble. They can choose the numbers themselves (control), let the experimenter choose (no control), or randomize. Classical economic theories predict indifference among the three methods. I found that participants exhibit strict preference for control, preference for no control, and preference for randomization. These preferences are robust as participants are willing to pay a small amount of money to implement their preferred method. Most participants believe that the winning probability under different methods is the same. This result contributes to the literature by clarifying that for most participants who exhibit preference for control, their preference is not due to illusion of control, but by source preference. Participants invest less in the risky gamble when they are not offered their preferred method.     

Subjective expected utility theory revisited: A reductio ad absurdum paradox

A paradox is posed and analyzed in which people reverse their preferences for information on probabilities versus prizes once the range of the unknown probabilities is sufficiently narrowed. This reversal is shown to be incompatible with both objective expected utility (EU) as well as subjective versions in which the same probability transformation applies to all random variables. Experimental data are presented showing that the reversals occur with small, medium and large payoffs.The present paradox is compared with those of Allais and Ellsberg, and found to differ in substantive ways. It raises further questions about the normative status of expected utility theory, especially its treatment of probability and value. The paradox specifically calls into question EU's substitution and compound probability axioms.     

For whom does social comparison induce risk-taking?

A ‘bonus culture’ among financial traders has been blamed for the excessive risk-taking in the run-up to the latest financial crisis. I show that when individuals are more social gain seeking than social loss averse (i.e. gloating is stronger than envy), social comparison predicts more risk-taking as well as a preference for negatively correlated gambles. Testing these two joint propositions in a laboratory experiment, I find that preference for positively or negatively correlated outcomes is highly correlated with risk-taking in a social risky investment task. While only a third of subjects prefer negatively correlated outcomes in a peer comparison setting, in line with relatively stronger social gain seeking, those subjects invest on average 50 % more in a risky gamble in their peer comparison setting than a reference group that made the same decision in an isolated individual setting. Subjects with a preference for positively correlated outcomes, in line with relatively stronger social loss aversion, do not show a higher propensity to invest in a risky gamble compared to the individual reference group.     

Effects of Outcome and Probabilistic Ambiguity on Managerial Choices

Information ambiguity is prevalent in organizations and likely influences management decisions. This study examines, given imprecise probabilities and outcomes, how managers make choices when they are provided with single-figure benchmarks. Seventy-nine MBA students completed two experiments. We found that, in a decision framed as a decision under certainty involving an ambiguous outcome, the majority of the subjects were ambiguity prone in the loss condition and switched to ambiguity aversion in the gain condition. However, in the presence of probabilistic ambiguity in a decision under risk, this expected switching pattern was shown only when the difference in riskiness between the two choice options (in the loss condition) was perceived to be relatively small. In a companion study, we used a written protocol approach to identify factors that affect decision makers' investment choices when faced with ambiguous outcomes. Protocols frequently mentioned that the ambiguous outcome option was risky, even in the case which was framed as a decision under certainty in the problem statement. In a decision under risk with ambiguous outcomes, the combination of probabilistic risk and outcome ambiguity was seen as even more risky.     

Common ratio using delay

We present an experiment in which we add a common delay in a choice between two risky prospects. The results show that delay produces the same change in preferences as in the well-documented common ratio effect in risky lotteries. The added common delay acts as if the probabilities were divided by some common ratio. Moreover, we show that there is a strong magnitude effect, in the sense that the effect of delay depends on the magnitude of the outcome. The results are consistent with the recently introduced probability time trade-off (PTT) model by Baucells and Heukamp. We present a parameterization of the model based on the experimental results, showing that the value function exhibits increasing relative risk aversion, the weighting function is s-shaped, and the intrinsic discount rate is decreasing.     

Risky investment decisions: How are individuals influenced by their groups?

We investigate the effect of group versus individual decision-making in the context of risky investment decisions in which all subjects are fully informed of the probabilities and payoffs. Although there is full information, the lottery choices pose cognitive challenges so that people may not be sure of their expected utility-maximizing choice. Making such decisions in a group context provides real-time information in which group members can observe others’ choices and revise their own decisions. Our experimental results show that simply observing what others in the group do has a significant impact on behavior. Coupling real-time information with group decisions based on the median value, i.e., majority rule, makes the median investment choice focal, leading people with low values to increase investments and those with high values to decrease investments. Group decisions based on the minimum investment amount produce more asymmetric effects.     

(Sub) Optimality and (non) optimal satisficing in risky decision experiments

We implement a risky choice experiment based on one-dimensional choice variables and risk neutrality induced via binary lottery incentives. Each participant confronts many parameter constellations with varying optimal payoffs. We assess (sub)optimality, as well as (non)optimal satisficing by eliciting aspirations in addition to choices. Treatments differ in the probability that a binary random event, which are payoff—but not optimal choice—relevant is experimentally induced and whether participants choose portfolios directly or via satisficing, i.e., by forming aspirations and checking for satisficing before making their choice. By incentivizing aspiration formation, we can test satisficing, and in cases of satisficing, determine whether it is optimal.     

Violations of dominance in pricing judgments

The dominance principle states that the judged price of gamble A should be equal to or greater than the judged price of gamble B whenever A's outcomes are equal to or better than the corresponding outcomes of B, holding everything else constant. Subjects often violate the dominance principle by assigning a higher price to a gamble with some probability of winning a positive amount, Y, otherwise zero, than to a superior gamble with the same chances of winning Y, otherwise winning X. Violations also occur with losses. Results are consistent with a configural-weight theory in which the decision weight for each outcome depends on the rank of the outcome with respect to the other outcomes in the lottery and the value of the outcome (zero vs. nonzero).The authors thank Shi-jie Chang, Duncan Luce, and Lisa Ordóñez for comments on an earlier draft. This research was supported by a National Science Foundation grant to the first author (BNS-8451368). Requests for reprints should be sent to Barbara Mellers, Dept. of Psychology, University of California, Berkeley, CA 94720.     

Some implications of common consequences in lotteries

This work studies the implications of some aspects of preferences toward risk in the choice between two binary lotteries exhibiting a common consequence. The results obtained are then applied to two different problems: the choice between two risky challenges characterized by different rewards in the case of success and different probabilities of success and the choice between self-protection and self-insurance in the presence of the risk of incurring financial loss.

     

Is Probability Weighting Sensitive to the Magnitude of Consequences? An Experimental Investigation on Losses

Experimental investigations of the probability weighting function over losses are scarce and all involve small payoffs. The paper aims to give new insight into the probability weighting function for losses, by eliciting it through a simple two-stage semi-parametric procedure over more realistic losses, and by investigating its sensitivity to the magnitude of the payoffs. Current data confirm previous evidence of convex utility functions and inverse-S-shaped weighting functions. Still, at least for small probabilities, probability weighting appears to be affected by the size of consequences: the larger the losses, the more aversive the gambles and the more pessimistic the subjects are.     

The Micro and Macro Dimensions of Gambling in the United States

A national probability sample of 1736 respondents and a Nevada State probability sample of 296 respondents were surveyed during the summer of 1975 to determine the extent of gambling activity in the United States, to estimate the amount of government revenue that could result from various changes in the gambling laws, and to predict the social consequences of these changes. While there were large variations among regions and demographic groups, a majority of all adult Americans gamble. Wagers on commercial games amounted to $22.4 billion in 1974 of which $17 billion was wagered legally and approximately $5 billion illegally.     

The influence of probabilities on the response mode bias in utility elicitation

The response mode bias, in which subjects exhibit different risk attitudes when assessing certainty equivalents versus indifference probabilities, is a well-known phenomenon in the assessment of utility functions. In this empirical study, we develop and apply a cardinal measure of risk attitudes to analyze not only the existence, but also the strength of this phenomenon. Since probability levels involved in decision problems are already known to have a strong impact on behavior, we use this approach to study the impact of probabilities on the extent of the response mode bias. We find that the direction in which probabilities influence measured risk aversion is the opposite in the certainty equivalence (CE) method versus in the probability equivalence (PE) method. Utilizing the CE elicitation approach leads to an increase of risk seeking for gambles involving high probabilities. For the PE method, subjects tend to behave risk averse with gambles of high probabilities. This behavior is reversed in the gain domain. This “tailwhip” effect is consistently replicated in several experiments, involving both loss and gain domains of lotteries.     

Performance of the similarity hypothesis relative to existing models of risky choice

Experimental studies have discovered behavior that is inconsistent with the expected utility model (EU) of risky choice (von Neumann and Morgenstern, 1953). The two approaches to address these paradoxes are tested: generalized expected utility models (GEU) and models incorporating decision-making limits or costs through question similarity. Tests are carried out over risky pairs related to well-known examples from Kahneman and Tversky's (1979) influential work. Statistical analysis reveals that GEU models of choice are significantly violated for choice patterns consistent with the similarity hypothesis. Additional tests point to shortcomings in the similarity approach that are consistent with fanning out behavior.     

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.     

Subjective Probability Weighting and the Discovered Preference Hypothesis

Numerous studies have convincingly shown that prospect theory can better describe risky choice behavior than the classical expected utility model because it makes the plausible assumption that risk aversion is driven not only by the degree of sensitivity toward outcomes, but also by the degree of sensitivity toward probabilities. This article presents the results of an experiment aimed at testing whether agents become more sensitive toward probabilities over time when they repeatedly face similar decisions, receive feedback on the consequences of their decisions, and are given ample incentives to reflect on their decisions, as predicted by Plott’s Discovered Preference Hypothesis (DPH). The results of a laboratory experiment with N = 62 participants support this hypothesis. The elicited subjective probability weighting function converges significantly toward linearity when respondents are asked to make repeated choices and are given direct feedback after each choice. Such convergence to linearity is absent in an experimental treatment where respondents are asked to make repeated choices but do not experience the resolution of risk directly after each choice, as predicted by the DPH. I thank Peter P. Wakker for useful comments and suggestions.     

Preferences for information on probabilities versus prizes: The role of risk-taking attitudes

Many real-world decisions entail choices between information on either probabilities or payoffs (i.e., prizes). Simplified versions of such decisions are examined to gain insight into preferences for different types of information as a function of risk-attitudes. General and simple decision rules are derived for cases where the utility function is concave (or convex) over the relevant payoff interval.The article further describes several experiments to test business students' intuitions concerning these optimal decision rules. In general, risk-taking attitudes did not correlate significantly with subjects' preferences for information, in violation of theorems regarding mean-preserving spreads of risk. Other tests, e.g., narrowing certain probability ranges, also resulted in preferences contrary to expected utility (EU) theory.     

Rules of behavior and expected utility theory. Compatibility versus dependence

Conclusions We have seen that many decision rules which are intuitively and/or empirically supported and compatible with MEU, are compatible with it but not dependent on it.There are of course rules of behavior which are implied in MEU and also depend on it like this:If the hope of winning any of the prizes in a lottery motivates you to buy a ticket, and if you win half the amount of the highest prize, you should play double or nothing with your prize.Suppose you would prefer a one in a million chance of winning $2 million to a two in a million chance of winning $1 million, but your first choice is not available so you buy a ticket for $1 million. If you win, you should play 50–50 double or nothing with your prize. Generalize: put x in the place of $1 million and p in the place of 1/1 000000, and test yourself against this principle (Pf = preferred to): (p, 2x)Pf(2p, x) (0.5, 2x)Pf(1, x).See Friedmann and Savage (1968).In the real world lotteries are multiprize, i.e., composite games of elements like these. The same applies: If you would not have preferred the highest prize exchanged for a higher probability of some lower prize, then winning a lower prize would put you in the market for some simple bet like above. If you stand up to this test, you are a unique person because, as we know, such bets are not made.While in the process of finishing the final draft, I got hold of (Samuelson, 1983). He expresses grave doubts as to what he calls the dogma of Expected Utility maximizing. In a somewhat apologetic way, he preserves some formulations deriving behavior from EUM because, as he states in a general way, they do not depend on that particular dogma. More specifically: many models incompatible with EUM imply risk aversion, which would result also from maximizing the expectation of a concave utility function,In view of the authoritarian disposition of some of the strongest defenders of EUM and of Samuelson's (well deserved) authority and his leading role in the school of EUM theory, his open expression of doubt may well mark the beginning of the last chapter in the history of the rise and fall of the most powerful school that has so far been active in 20th century decision theory.