Von Neumann-Morgenstern Research Articles

Sequential Decision Analysis of the Traditional Deterrence Game

We examine the traditional deterrence game between Challenger and Defender. We treat two variations of the game – the complete information game and the one-sided information game where the first mover, Challenger, is the uncertain player. We employ sequential decision theory to analyze the game of incomplete information. The analysis is basic in that we employ a simplifying assumption, specifically that Challenger’s valuation of the status quo is fixed at zero. We examine the behavior of Challenger using both a von-Neumann-Morgenstern decision rule and a Kahneman-Tversky decision rule. The formal results show that given the right combination of outcome valuations and probability values and weightings, Challenger can make choices under the von Neumann-Morgenstern decision rule that are reversed from those made under the Kahneman-Tversky decision rule. We then relate these reversals to the concept of misperception found in the International Relations and Peace Science literatures. Finally, we comment on extensions of the ensuing analysis.

Expected Utility, Prospect Theory and Asset Pricing

Investors base asset valuation decisions on current day movements in asset and market returns. This is shown in the risk-return relationship of the simple asset pricing model using a four-way partition of daily returns. The partitioning is derived from current movements in individual asset and market returns. The underlying behavioral theory is a valuation framework that generalizes von Neumann-Morgenstern expected utility and allows for some commonly observed anomalies. We test key elements of Kahneman and Tversky's (1979) prospect theory and find reference dependence, asymmetric valuation of gains and losses, and nonproportional marginal sensitivity to large changes in expected returns in 1986-2000 data for 100 companies: the Dow-Jones 30 industrials and 30 and 40 companies selected at random from the S&P mid cap and small cap index lists respectively.

Probabilistic assignments of identical indivisible objects and uniform probabilistic rules

We consider a probabilistic approach to the problem of assigning k indivisible identical objects to a set of agents with single-peaked preferences. Using the ordinal extension of preferences we characterize the class of uniform probabilistic rules by Pareto efficiency, strategy-proofness, and no-envy. We also show that in this characterization no-envy cannot be replaced by anonymity. When agents are strictly risk averse von Neumann-Morgenstern utility maximizer, then we reduce the problem of assigning k identical objects to a problem of allocating the amount k of an infinitely divisible commodity.

The Efficient Market Hypothesis as a Martingale

It is well known that the efficient market hypothesis and a martingale are closely related. This paper studies the use of a martingale as a substitute for the efficient market hypothesis in a monetary overlapping generations economy where individuals maintain the von Neumann-Morgenstern expected utility proposition. Within this economic framework two results are derived. (1) A martingale is a direct consequence of equilibrium conditions in the optimisation of individuals’ expected utilities. (2) Learning is precluded in a successive transition from one state of equilibrium to the next. These two results accord fully with the definition of a martingale.

Power indices and the veil of ignorance

We provide an axiomatic foundation of the expected utility preferences over lotteries on roles in simple superadditive games represented by the two main power indices, the Shapley-Shubik index and the Banzhaf index, when they are interpreted as von Neumann-Morgenstern utility functions. Our axioms admit meaningful interpretations in the setting proposed by Roth in terms of different attitudes toward risk involving roles in collective decision procedures under the veil of ignorance. In particular, an illuminating interpretation of `efficiency', up to now missing in this set up, as well as of the corresponding axiom for the Banzhaf index, is provided.

A Smooth Model of Decision Making under Ambiguity

We propose and axiomatize a model of preferences over acts such that the decision maker evaluates acts according to the expectation (over a set of probability measures) of an increasing transformation of an act's expected utility. This expectation is calculated using a subjective probability over the set of probability measures that the decision maker thinks are relevant given her subjective information. A key feature of our model is that it achieves a separation between ambiguity, identified as a characteristic of the decision maker's subjective information, and ambiguity attitude, a characteristic of the decision maker's tastes. We show that attitudes towards risk are characterized by the shape of the von Neumann-Morgenstern utility function, as usual, while attitudes towards ambiguity are characterized by the shape of the increasing transformation applied to expected utilities. We show that the negative exponential form of this transformation is the special case of constant ambiguity aversion. Ambiguity itself is defined behaviorally and is shown to be characterized by properties of the subjective set of measures. This characterization of ambiguity is formally related to the definitions of subjective ambiguity advanced by Epstein-Zhang (2001) and Ghirardato-Marinacci (2002). One advantage of this model is that the well-developed machinery for dealing with risk attitudes can be applied as well to ambiguity attitudes. The model is also distinct from many in the literature on ambiguity in that it allows smooth, rather than kinked, indifference curves. This leads to different behavior and improved tractability, while still sharing the main features (e.g., Ellsberg's Paradox, etc.). The Maxmin EU model (e.g., Gilboa and Schmeidler (1989)) with a given set of measures may be seen as an extreme case of our model with infinite ambiguity aversion. Two illustrative applications to portfolio choice are offered.

Manipulation of Preferences and Relative Utilitarianism

Given n agents with von Neumann-Morgenstern utility functions who wish to divide m commodities, consider the n-person noncooperative game with strategies consisting of concave, increasing von Neumann-Morgenstern utility functions, and whose outcomes are the relative utilitarian solution. It is shown that any constrained equal-income competitive equilibrium allocation for the true utilities is a Nash equilibrium outcome for the noncooperative game. Conditions are presented under which these are the only pure strategy equilibrium outcomes.

Approximate CAPM When Preferences Are CRRA

In general equilibrium models of financial markets, the capital asset pricing formula does not hold when agents have von Neumann-Morgenstern utility with constant relative risk aversion. In this paper we examine under which conditions on endowments and dividends the pricing formula provides a good benchmark for equilibrium returns. While it is easy to construct examples where equilibrium returns are arbitrarily far from those predicted by CAPM, we show that there is a large class of economies where CAPM provides a very good approximation. Although the pricing formula does not hold exactly for the chosen specification, it turns out that pricing-errors are extremely small.

On the existence and efficiency of the von Neumann-Morgenstern stable set in a n-player prisoners' dilemma

We show that there exist von Neumann-Morgenstern (vN-M) stable sets in a n-player version of the prisoners' dilemma game with preplay negotiations in which every player can deviate unilaterally from the currently proposed combination of actions but can not do so jointly with other players, and that every vN-M stable set includes at least one Pareto-efficient outcome. The negotiation among the players is formulated as the “individual contingent threats situation” within the framework of the theory of social situations due to Greenberg (1990). The method of proving the existence also provides us with a step-by-step method of constructing the vN-M stable set.

Assignment games with stable core

We prove that the core of an assignment game (a two-sided matching game with transferable utility as introduced by Shapley and Shubik, 1972) is stable (i.e., it is the unique von Neumann-Morgenstern solution) if and only if there is a matching between the two types of players such that the corresponding entries in the underlying matrix are all row and column maximums. We identify other easily verifiable matrix properties and show their equivalence to various known sufficient conditions for core-stability. By these matrix characterizations we found that on the class of assignment games, largeness of the core, extendability and exactness of the game are all equivalent conditions, and strictly imply the stability of the core. In turn, convexity and subconvexity are equivalent, and strictly imply all aformentioned conditions.

Pseudo-Analysis

Based on semiring structure there is presented briefly the so called pseudoanalysis in an analogous way as classical analysis, introducing ⊕-measure, pseudointegral, pseudo-convolution, pseudo-Laplace transform, etc. There are presented two further generalizations related the pseudo-analysis. First is related to further generalization of the real semiring structure to the case when the operations ⊕ and ⊙ are noncommutative and non-associative, and the (left) right distributivity of ⊙ over ⊕ plays a crucial rule. The obtained results are applied on nonlinear PDE. The second generalization is related to relaxation of the distributivity law, and an application to a generalization of von Neumann-Morgenstern utility theory.

Stable sets and standards of behaviour

In this paper we present a constructive, behavioural and axiomatic approach to the notion of a stable set as a model of the standard of behaviour of a social organisation. The socially stable set we introduce is a generalisation of the von Neumann-Morgenstern stable set. In contrast with the original version, our stability concept is always solvable. The standard of behaviour, reflecting the established conceptual order of a society or organisation, emerges from a dominance relation on alternative conceptions that are relevant with regard to a certain issue. This common social choice phenomenon, that permeates our societies and organisations, we have tried to clarify. Two axiomatic characterisations as well as a construction algorithm for socially stable sets are presented. These characterisations are based on behavioural postulates regarding the individual or collective strategic behaviour of effective sets. Relations between socially stable sets and other notions of stability are discussed.

On Professor Kohn and Expected Utility: Correction and Clarification-Rejoinder

Professor Horowitz correctly identifies the limitation of my assuming separable utility functions to derive a marginal condition for efficiency under uncertainty. Correction this limitation, he provides a simple but powerful condition that encompasses the nonseparable as well as the separable case. This condition replaces the dubious Equation (14) derived in Kohn (1999). In a departure from von Neumann-Morgenstern theory, for cases in which the decisions of a risk-averse community are compared with those it would make were it risk-neutral, it is proposed here that the same utility function holds for risk-neutrality as for risk-aversion, but that the stochastic quantities be replaced by their expected value in the former. [Q25]

Expected Qualitative Utility Maximization

A model for decision making that generalizes Expected Utility Maximization is presented. This model, Expected Qualitative Utility Maximization, encompasses the Maximin criterion. It relaxes both the Independence and the Continuity postulates. Its main ingredient is the definition of a qualitative order on nonstandard models of the real numbers and the consideration of nonstandard utilities. Expected Qualitative Utility Maximization is characterized by an original weakening of von Neumann-Morgenstern's postulates. Subjective probabilities may be defined from those weakened postulates, as Anscombe and Aumann (1963, Annals of Mathematical Statistics 34, 199–205) did from the original postulates. Subjective probabilities are numbers, not matrices as in the Subjective Expected Lexicographic Utility approach. Journal of Economic Literature Classification Number: D81.

COPING WITH UNCERTAINTY IN n-PERSON GAMES

A solution of n-person games is proposed, based on the minimum deviation from statistical equilibrium subject to the constraints imposed by the group rationality and individual rationality. The new solution is compared with the Shapley value and von Neumann-Morgenstern's core of the game in the context of the 15-person game of passing and defeating resolutions in the UN Security Council involving five permanent members and ten nonpermanent members. A coalition classification, based on the minimum ramification cost induced by the characteristic function of the game, is also presented.

Coping with Uncertainty in n-Person Games

A solution of n-person games is proposed, based on the minimum deviation from statistical equilibrium subject to the constraints imposed by the group rationality and individual rationality. The new solution is compared with the Shapley value and von Neumann-Morgenstern's core of the game in the context of the 15-person game of passing and defeating resolutions in the UN Security Council involving five permanent members and ten nonpermanent members. A coalition classification, based on the minimum ramification cost induced by the characteristic function of the game, is also presented.

The Endowment Effect and Expected Utility

The endowment effect, which is well documented in the contingent valuation literature, alters people’s preferences according to a reference point established in an elicitation question. In particular, the utility that people place on a bundle is both a positive function of the quantities of the goods comprising the bundle, and a negative function of any loss (real or hypothetical) that the elicitation question asks them to incur. Biases such as this have lead some to reject the contingent valuation method as a means of quantifying costs and benefits in favour of other methods of preference elicitation such as standard gambles. But, most preference elicitation methods used by economists require people to express their preferences for one good in terms of their willingness to forego some of another good. Consequently, it is reasonable to expect that, and prudent to check whether, an endowment effect is also evident in other methods of preference elicitation such as von Neumann‐Morgenstern’s standard gambles. Internal inconsistencies in the standard gamble method from the experimental economics literature and from a study into the value of non‐fatal road injuries are shown to be evidence that an endowment effect is also at work in standard gambles.

Experimental Evidence for Attractions to Chance

Abstract Divide the decision-maker's future into: (i) a pre-outcome period (lasting from the decision until the outcome of that decision is known), and (ii) a sequel postoutcome period (beginning when the outcome becomes known). Anticipated emotions in both periods may influence the decision, in particular, with regard to an outcome that matters to the person, the enjoyable tension from not yet knowing what this outcome will be. In the experiments presented, lottery choice can be explained by this attraction to chance, and cannot be explained by either convex von Neumann-Morgenstern utility, or by rank-dependent risk-loving weights: attraction to chance is a separate motivator.

On comparing equilibrium and optimum payoffs in a class of discrete bimatrix games

In an m 1× m 2 bimatrix game, consider the case where payoffs to each player are randomly drawn without replacement, independently of payoffs to the other player, from the set of integers 1,2,…, m 1 m 2. Thus each player’s payoffs represent ordinal rankings without ties. In such ‘ordinal randomly selected’ games, assuming constraints on the relative sizes of m 1 and m 2 and ignoring any implications of mixed strategies, it is shown that payoffs to pure Nash equilibria (second-degree) stochastically dominate payoffs to pure Pareto optimal outcomes. Thus in such games where pure strategy sets do not differ much in size and payoffs conform with concave von Neumann-Morgenstern utility functions over ordinally ranked outcomes, players would prefer (ex ante) a ‘random pure strategy Nash equilibrium payoff’ to a ‘random pure Pareto optimal outcome payoff’.

Utility Functions for Wealth

We specify all utility functions on wealth implied by four special conditions on preferences between risky prospects in four theories of utility, under the presumption that preference increases in wealth. The theories are von Neumann-Morgenstern expected utility (EU), rank dependent utility (RDU), weighted linear utility (WLU), and skew-symmetric bilinear utility (SSBU). The special conditions are a weak version of risk neutrality, Pfanzagl's consistency axiom, Bell's one-switch condition, and a contextual uncertainty condition. Previous research has identified the functional forms for utility of wealth for all four conditions under EU, and for risk neutrality and Pfanzagl's consistency axiom under WLU and SSBU. The functional forms for the other condition-theory combinations are derived in this paper.