Probabilities and beliefs

Choice-theoretic definitions of subjective probabilities originated with the work of Ramsey and de Finetti and attained their definitive form in the work of Savage. These probabilities are intended to provide a numerical representation of a decision maker's beliefs regarding the likely realization of alternative events. In this article, I argue that the choice-theoretic definitions of subjective probabilities involve a tacit convention—namely, state-independent utility functions—that is not implied by the axioms, and, as a consequence, choice-theoretic subjective probabilities, even when they exist, do not necessarily represent the decision makers' beliefs.     

Sequential Parimutuel Betting in the Laboratory

This paper investigates parimutuel betting in the laboratory. Our experimental design relies on a simple sequential betting game where equilibrium strategies are characterized according to objective probabilities, the number of bettors, and publicly observable odds. The empirically well-documented phenomenon referred to as the favorite-longshot bias is observed in two of our three treatments. We offer a theoretical explanation of the subjects' behavior which relies on rank-dependent expected utility and pessimistic expectations about future bets.     

A test of the gambler's fallacy: Evidence from pari-mutuel games

The “gambler's fallacy” is the belief that the probability of an event is decreased when the event has occurred recently, even though the probability is objectively known to be independent across trials. Clotfelter and Cook (1991, 1993) find evidence of the gambler's fallacy in analysis of data from the Maryland lottery's“Pick 3” numbers game. In the Maryland lottery, the payout to all numbers is equal at $250 on a winning fifty-cent bet, so the gambler's fallacy betting strategy costs bettors nothing. This article looks at the importance of the gambler's fallacy in the New Jersey lottery's three-digit numbers game, a pari-mutuel game where a lower amount of total wagering on a number increases the payout to that number. Results indicate that the gambler's fallacy exists among bettors in New Jersey, although to a lesser extent than among those in Maryland.     

Biases in Assessments of Probabilities: New Evidence from Greyhound Races

This paper investigates biases in the perceptions of probabilities using data from the 1989 and 1994 seasons at the Woodlands greyhound park in Kansas City, Kansas. Results reveal consistent evidence that the gambler's fallacy exists. The results also reveal that gamblers overestimate the probability of a win by the favorite and the dog in the lucky seven position. However, the comparison also suggests some learning by bettors between the first season of operation in 1989 and the 1994 season.     

Where o’ where did my baseball cards go?: Race,performance, and placement in the Topps ERA, 1956–1980

On the surface, baseball cards appear to be little more than uninteresting object of child's play; yet they provide invaluable data for assessing unique aspects of the social order. They reveal much about the state of society when they were produced and the people who collected them. This paper examines the unique numbering system the Topps company developed to differentiate among the “royalty of the diamond” from 1956 to 1980. Their system assigned to each card in a set a number that explicitly designated its placement, while implicitly establishing a hierarchical ordering among the players, ostensibly based on performance. It is arguable whether the numbering system Topps developed was based on performance or the ascribed characteristic of race. The results of our work suggest that player performance, not race, was the primary factor in the assignment of “prestigious” card numbers in Topps’ system.     

Strategic and Sincere Voting in a One‐Sided Election: The Canadian Federal Election of 1997*

Objective. We are interested in whether and how voters make strategic decisions in a race that is, according to the polls, expected to be very one sided. Looking specifically at the choices available to ideologically right‐of‐center voters in the 1997 Canadian federal election, we argue that strategic considerations will be filtered by voters' assessments of the competitiveness of the race both locally and nationally. Methods. We estimate logistic regression models measuring support for the two right‐of‐center Canadian political parties. Our models focus on the relationship between assessments of district‐ and national‐level party prospects on voting for the Progressive Conservative Party. Results. We find that voters who consider the race competitive emphasize district‐level data in their strategic calculations. However, those who consider the election to be all but over look more closely at national‐level concerns when deciding which right‐wing party to support. Conclusions. We conclude that earlier understandings of tactical voting should be updated to take into consideration the circumstances under which voters will use national‐level evaluations of relative party viability in casting their ballots.     

The effect of residence in enclaves on civic concern: An initial exploration

Will those residing in urban enclaves—enclosed socially homogenous residential communities protected by private security—have a lower level of civic concern as compared to those who do not reside in such enclaves? Reich's (1991) work suggests that this is to be expected. To conduct an empirical test, two random samples were drawn from two areas of New York City, one a residential enclave—Battery Park City—the other a non-enclave—Chelsea. The analysis establishes that those who do not live in the enclave have a statistically significant higher level of civic concern. Sample respondent years of education, age, income, gender, and career (arts/business) differences were employed as control variables; after having introduced each control variable independently, the main research hypothesis was sustained.     

Known, Unknown, and Unknowable Uncertainties

In normative decision theory, the weight of an uncertain event in a decision is governed solely by the probability of the event. A large body of empirical research suggests that a single notion of probability does not accurately capture peoples' reactions to uncertainty. As early as the 1920s, Knight made the distinction between cases where probabilities are known and where probabilities are unknown. We distinguish another case –- the unknowable uncertainty –- where the missing information is unavailable to all. We propose that missing information influences the attractiveness of a bet contingent upon an uncertain event, especially when the information is available to someone else. We demonstrate that the unknowable uncertainty –- falls in preference somewhere in between the known and the known uncertainty.     

The political orientation of Blacks and Whites: Converging,diverging, or remaining constant?

This paper examines the extent to which patterns of public opinion between Blacks and Whites have converged, diverged, or remained constant since the late1970s. It explores how trends in public opinion reflect a decline in the role of race in the formation of pubic opinion. The findings show a slight convergence of public opinion between Blacks and Whites on some issues and stabilization on others depending of the nature of the issue. The data also showed that the magnitude of difference on most issues, other than those related to race, rarely constituted anything more than a gap in public opinion and not a gulf of chasm. The fact that significant differences continue to exist between Blacks’ and Whites’ public opinions suggest that race is still a significant indicator of opinion position.     

Realism and antirealism in social science

Up until recently social scientists took it for granted that their task was to account for the social world as objectively as possible: they were realists in practice if not always in their methodological sermons. This situation started to change in the 1960s, when a number of antirealist philosophies made inroads into social studies.This paper examines critically the following kinds of antirealism: subjectivism, conventionalism, fictionism, social constructivism, relativism, and hermeneutics. An attempt is made to show that these philosophies are false and are causing serious damage to social studies.Next the subjective interpretation of probability is analyzed as a case of subjectivism. An approach to the subjective perception of justice is sketched as an example of the objective study of subjective experience.Finally, the three main varieties of realism — naive, critical, and scientific — are outlined. It is argued that the scientific attitude involves scientific realism, which is put in practice even by scholars who, like Weber and Simmel, called themselves antirealists.     

The puzzle of cooperation in a game of chicken: an experimental study

The objective of this article is to investigate the impact of agent heterogeneity (as regards their attitude towards cooperation) and payoff structure on cooperative behaviour, using an experimental setting with incomplete information. A game of chicken is played considering two types of agents: ‘unconditional cooperators’, who always cooperate, and ‘strategic cooperators’, who do not cooperate unless it is in their interest to do so. Overall, our data show a much higher propensity to cooperate than predicted by theory. They also suggest that agent heterogeneity matters: the higher the proportion of ‘strategic cooperators’ in the population, the higher their probability to cooperate. Finally, our data confirm that higher rewards to cooperation (embedded in the payoff structure) tend to lower defection. Taken together, our results suggest that the subjects might be non-expected utility maximizers, dealing with both outcomes and probabilities in a non-linear manner.     

The Diversification Theorem Restated: Risk-pooling Without Assignment of Probabilities

Bayesian decision theory assumes that agents making choices assign subjective probabilities to outcomes, even in cases where information on probabilities is obviously absent. Here we show that agents that presume that they are equal risks can share risks mutually beneficially, even if the probabilities of losses are unpredictable or genuinely uncertain. We show also that different risk aversions among pool members do not exclude mutually beneficial loss sharing at uncertainty. Sharing when individuals’ losses differ in probabilities or in amount may still make individuals better off. Our findings are related to the theory of the insurance firm, to the management of development risks, and to the theory of justice.JEL Classification: D8, G22, L10, L30     

Coalitional Interval Games for Strategic Games in Which Players Cooperate

We propose a method to associate a coalitional interval game with each strategic game. The method is based on the lower and upper values of finite two-person zero-sum games. Associating with a strategic game a coalitional interval game we avoid having to take either a pessimistic or an optimistic approach to the problem. The paper makes two contributions to the literature: It provides a theoretical foundation for the study of coalitional interval games and it also provides, studies, and characterizes a natural method of associating coalitional interval games with strategic games.      

Non-Additive Beliefs in Solvable Games

This paper studies how the introduction of non-additive probabilities (capacities) affects the solvability of strategic games.     

A note on Savage's theorem with a finite number of states

This article gives a preference-based characterization of subjective expected utility for the general equilibrium model with a finite number of states. The characterization follows Savage (1954) as closely as possible but has to abandon his axiom (P6), atomlessness of events, since this requires an infinite state space. To introduce continuity we replace (P6) with a continuity assumption on the set of consequences and assume the preferences are smooth. Then we apply Savage's sure-thing principle and his state-independence axiom to get an additively separable utility representation. Finally, to separate subjective probabilities from basic tastes, we apply a new axiom, which states that for each pair of states the marginal rate of substitution is constant along the certainty line.     

Separating marginal utility and probabilistic risk aversion

This paper is motivated by the search for one cardinal utility for decisions under risk, welfare evaluations, and other contexts. This cardinal utility should have meaningprior to risk, with risk depending on cardinal utility, not the other way around. The rank-dependent utility model can reconcile such a view on utility with the position that risk attitude consists of more than marginal utility, by providing a separate risk component: a probabilistic risk attitude towards probability mixtures of lotteries, modeled through a transformation for cumulative probabilities. While this separation of risk attitude into two independent components is the characteristic feature of rank-dependent utility, it had not yet been axiomatized. Doing that is the purpose of this paper. Therefore, in the second part, the paper extends Yaari's axiomatization to nonlinear utility, and provides separate axiomatizations for increasing/decreasing marginal utility and for optimistic/pessimistic probability transformations. This is generalized to interpersonal comparability. It is also shown that two elementary and often-discussed properties — quasi-convexity (aversion) of preferences with respect to probability mixtures, and convexity (pessimism) of the probability transformation — are equivalent.     

ETHNICITY NOT RACE: A PUBLIC HEALTH PERSPECTIVE

This article presents an argument, from a public health perspective, against the use of the term ‘race’ and for its replacement by the term ‘ethnicity’. Historically, the rise of the race concept in society was dependent on its undeserved status as an objective scientific and biological category and was associated with strategies of exclusion and political domination. Mainstream science played a key role in the rise of the race concept but has since largely abandoned it in face of evidence from population genetics. Similarly, the public health movement has historically been concerned with race/ethnicity as a determinant of unequal health status, but the race term has now all but disappeared from the Australian public health literature, where it has been replaced by the concept of ethnicity. Ethnicity is a complex social variable, with cultural and political dimensions, but no biological dimension. Adopting a public health perspective on ethnicity which recognises the fluid and contested nature of this socio‐political variable, whilst seeking to make explicit its relevance and definitional limits, allows us to dispense with the race concept altogether, since race has no additional explanatory or strategic value above that of ethnicity. The race term is still commonly used, however, in general conversation and in the media. The persistence of the race concept and of racism is difficult to explain but may be related historically to the politics of nationalism, and in modern times to the politics of difference and identity that characterise the modern multicultural nation‐state. Abandoning the terminology of race leaves racism without any logical basis, and may contribute to a process of social change, although it cannot be expected to eliminate the phenomenon of racism.     

A note on the decidability of de Finetti's coherence

To check the de Finetti coherence of a putative probability assigned to a classA of events, we must know the possible combinations of truth values (constituents) of any finite class of events inA. Even for a very simple, finite,A this can be impossible. In this case the notion of DF coherence cannot be applied to some or all the putative probabilities on this class of events.     

Dutch books and agent rationality

According to the Dutch Book Argument (DBA), if an agent's subjective probabilities fail to satisfy the axioms of the probability calculus and so make the agent vulnerable to a Dutch Book, the agent's subjective probabilities are incoherent and the agent is therefore irrational. Critics of DBA have argued, however, that probabilistic incoherence is compatible with various kinds of rationality – logico-semantic, epistemic, instrumental and prudential. In this paper, I provide an interpretation of DBA on which it is true that probabilistic incoherence entails agent irrationality. Articulating this interpretation requires the specification of some of the connections among the varieties of rationality. Once this is done, it becomes possible to vindicate a modest version of DBA.