Ilia Tsetlin

The impartial culture maximizes the probability of majority cycles

Many papers have studied the probability of majority cycles, also called the Condorcet paradox, using the impartial culture or related distributional assumptions. While it is widely acknowledged that the impartial culture is unrealistic, conclusions drawn from the impartial culture are nevertheless still widely advertised and reproduced in textbooks. We demonstrate that the impartial culture is the worst case scenario among a very broad range of possible voter preference distributions. More specifically, any deviation from the impartial culture over linear orders reduces the probability of majority cycles in infinite samples unless the culture from which we sample is itself inherently intransitive. We prove this statement for the case of three candidates and we provide arguments for the conjecture that it extends to any number of candidates.

Multiattribute Utility Satisfying a Preference for Combining Good with Bad

An important challenge in multiattribute decision analysis is the choice of an appropriate functional form for the utility function. We show that if a decision maker prefers more of any attribute to less and prefers to combine good lotteries with bad, as opposed to combining good with good and bad with bad, her utility function should be a weighted average (a mixture) of multiattribute exponential utilities (“mixex utility”). In the single-attribute case, mixex utility satisfies properties typically thought to be desirable and encompasses most utility functions commonly used in decision analysis. In the multiattribute case, mixex utility implies aversion to any multivariate risk. Risk aversion with respect to any attribute decreases as that attribute increases. Under certain restrictions, such risk aversion also decreases as any other attribute increases, and a multivariate one-switch property is satisfied. One of the strengths of mixex utility is its ability to represent cases where utility independence does not hold, but mixex utility can be consistent with mutual utility independence and take on a multilinear form. An example illustrates the fitting of mixex utility to preference assessments.

Decision Making with Multiattribute Performance Targets: The Impact of Changes in Performance and Target Distributions

In many situations, performance on several attributes is important. Moreover, a decision makers utility may depend not on the absolute level of performance on each attribute, but rather on whether that level of performance meets a target, in which case the decision maker is said to be target oriented. For example, typical attributes in new product development include cost, quality, and features, and the corresponding targets might be the best performance on these attributes by competing products. Targets and performance levels typically are uncertain and often are dependent. We develop a model that allows for uncertain dependent targets and uncertain dependent performance levels, and we study implications for decision making in this general multiattribute target-oriented setting. We consider the impact on expected utility of modifying key characteristics of performance (or target) distributions: location, spread, and degree of dependence. In particular, we show that explicit consideration of dependence is important, and we establish when increasing or decreasing dependence is beneficial. We illustrate the results numerically with a normal model and discuss some extensions and implications.

Sequential vs. Single-Round Uniform-Price Auctions

We study sequential and single-round uniform-price auctions with affiliated values. We derive symmetric equilibrium for the auction in which k1 objects are sold in the first round and k2 in the second round, with and without revelation of the first-round winning bids. We demonstrate that auctioning objects in sequence generates a lowballing effect that reduces the first-round price. Total revenue is greater in a single-round, uniform auction for k=k1+k2 objects than in a sequential uniform auction with no bid announcement. When the first-round winning bids are announced, we also identify a positive informational effect on the second-round price. Total expected revenue in a sequential uniform auction with winning-bids announcement may be greater or smaller than in a single-round uniform auction, depending on the models parameters.

Modifying Variability and Correlations in Winner-Take-All Contests

We consider contests with a fixed proportion of winners based on relative performance. Special attention is paid to winner-take-all contests, which we define as contests with relatively few winners receiving relatively large awards, but we consider the full range of values of the proportion of winners. If a contestant has the opportunity to modify the distribution of her performance, what strategy is advantageous? When the proportion of winners is less than one-half, a riskier performance distribution is preferred; when this proportion is greater than one-half, it is better to choose a less risky distribution. Using a multinormal model, we consider modifications in the variability of the distribution and in correlations with the performance of other contestants. Increasing variability and decreasing correlations lead to improved chances of winning when the proportion of winners is less than one-half, and the opposite directions should be taken for proportions greater than one-half. Thus, it is better to take chances and to attempt to distance oneself from the other contestants (i.e., to break away from the herd) when there are few winners; a more conservative, herding strategy makes sense when there are many winners. Our analytical and numerical results indicate that the probability of winning can change substantially as variability and/or correlations are modified. Furthermore, in a game-theoretic setting in which all contestants can make modifications, choosing a riskier (less risky) performance distribution when the proportion of winners is low (high) is the dominant best-response strategy. We briefly consider some practical issues related to the recommended strategies and some possible extensions.

Generalized Almost Stochastic Dominance

Almost stochastic dominance allows small violations of stochastic dominance rules to avoid situations where most decision makers prefer one alternative to another but stochastic dominance cannot rank them. While the idea behind almost stochastic dominance is quite promising, it has not caught on in practice. Implementation issues and inconsistencies between integral conditions and their associated utility classes contribute to this situation. We develop generalized almost second-degree stochastic dominance and almost second-degree risk in terms of the appropriate utility classes and their corresponding integral conditions, and extend these concepts to higher degrees. We address implementation issues and show that generalized almost stochastic dominance inherits the appealing properties of stochastic dominance. Finally, we define convex generalized almost stochastic dominance to deal with risk-prone preferences. Generalized almost stochastic dominance could be useful in decision analysis, empirical research (e.g., in finance), and theoretical analyses of applied situations.

Approval voting and positional voting methods: Inference, relationship, examples

Abstract.Approval voting is the voting method recently adopted by the Society for Social Choice and Welfare. Positional voting methods include the famous plurality, antiplurality, and Borda methods. We extend the inference framework of Tsetlin and Regenwetter (2003) from majority rule to approval voting and all positional voting methods. We also establish a link between approval voting and positional voting methods whenever Falmagne et al.’s (1996) size-independent model of approval voting holds: In all such cases, approval voting mimics some positional voting method. We illustrate our inference framework by analyzing approval voting and ranking data, with and without the assumption of the size-independent model. For many of the existing data, including the Society for Social Choice and Welfare election analyzed by Brams and Fishburn (2001) and Saari (2001), low turnout implies that inferences drawn from such elections carry low (statistical) confidence. Whenever solid inferences are possible, we find that, under certain statistical assumptions, approval voting tends to agree with positional voting methods, and with Borda, in particular.

Strategic Choice of Variability in Multiround Contests and Contests with Handicaps

Variability can be an important strategic variable in a contest. We study optimal strategies involving choice of variability in contests with fixed and probabilistic targets, one-round and multiround contests, contests with and without handicaps, and situations where one contestant can modify variability as well as those in which all contestants have this opportunity. A contestant should maximize variability in a weak position (low mean, high handicap, or low previous performance) and minimize variability in a strong position. In some cases, only these extremes should be used. In other cases, intermediate levels of variability are optimal when the contestants position is neither too weak nor too strong.

Sophisticated approval voting, ignorance priors, and plurality heuristics: A behavioral social choice analysis in a Thurstonian framework.

This project reconciles historically distinct paradigms at the interface between individual and social choice theory, as well as between rational and behavioral decision theory. The authors combine a utility-maximizing prescriptive rule for sophisticated approval voting with the ignorance prior heuristic from behavioral decision research and two types of plurality heuristics to model approval voting behavior. When using a sincere plurality heuristic, voters simplify their decision process by voting for their single favorite candidate. When using a strategic plurality heuristic, voters strategically focus their attention on the 2 front-runners and vote for their preferred candidate among these 2. Using a hierarchy of Thurstonian random utility models, the authors implemented these different decision rules and tested them statistically on 7 real world approval voting elections. They cross-validated their key findings via a psychological Internet experiment. Although a substantial number of voters used the plurality heuristic in the real elections, they did so sincerely, not strategically. Moreover, even though Thurstonian models do not force such agreement, the results show, in contrast to common wisdom about social choice rules, that the sincere social orders by Condorcet, Borda, plurality, and approval voting are identical in all 7 elections and in the Internet experiment.

Multivariate Concave and Convex Stochastic Dominance

Stochastic dominance permits a partial ordering of alternatives (probability distributions on consequences) based only on partial information about a decision maker’s utility function. Univariate stochastic dominance has been widely studied and applied, with general agreement on classes of utility functions for dominance of different degrees. Extensions to the multivariate case have received less attention and have used different classes of utility functions, some of which require strong assumptions about utility. We investigate multivariate stochastic dominance using a class of utility functions that is consistent with a basic preference assumption, can be related to well-known characteristics of utility, and is a natural extension of the stochastic order typically used in the univariate case. These utility functions are multivariate risk averse, and reversing the preference assumption allows us to investigate stochastic dominance for utility functions that are multivariate risk seeking. We provide insight into these two contrasting forms of stochastic dominance, develop some criteria to compare probability distributions (hence alternatives) via multivariate stochastic dominance, and illustrate how this dominance could be used in practice to identify inferior alternatives. Connections between our approach and dominance using different stochastic orders are discussed.