Von Neumann-Morgenstern Research Articles

A connection between von Neumann-Morgenstern expected utility and symmetric potential games

AbstractThis paper establishes a previously unexplored connection between expected utility theory and potential games. Starting with a decision problem with a complete preference relation over lotteries on a finite set of alternatives, we construct a two-person symmetric game using a payoff function that represents the preference relation, and show that if the preference relation satisfies the von Neumann-Morgenstern expected utility axioms then the constructed game is a potential game. Conversely, starting with a two-player symmetric game, we (uniquely) construct a (complete) preference relation over the lotteries using the first player’s payoffs in the game, and show that if the game is a potential game then the resulting preference relation satisfies the expected utility axioms.

The mutual funds puzzle and the elusive von Neumann-Morgenstern preference relation

This research studies the Mutual Funds Puzzle from the viewpoint of an economic agent that satisfies the von Neumann-Morgenstern preference relation. We postulate that the probability of a fund manager to outperform the market decreases in a growing market. To examine this supposition, we prove that a risk-averse investor's behavior is consistent with the binary econometric model and develop a ranking methodology for mutual funds' performances. We find supporting evidence for our conjecture in examining the Israeli capital market during 1/1/2009–3/13/2022. Subsequently, we conclude that investors regard mutual funds as hedging instruments for countering unexpected events rather than investment instruments.

The quantum foundations of utility and value.

This work presents a generalization of game theory and new perspectives on utility and value. Using quantum formalism, we prove that classical game theory is a special case of quantum game theory. We show that the von Neumann entropy and von Neumann-Morgenstern utility are equivalent and that the Hamiltonian operator represents value. This article is part of the theme issue 'Thermodynamics 2.0: Bridging the natural and social sciences (Part 1)'.

A Connection between Von Neumann-Morgenstern Expected Utility and Symmetric Potential Games

A Connection between Von Neumann-Morgenstern Expected Utility and Symmetric Potential Games

A MCDA-based Approach on Corporate’s Investment Decision Process: A case study of a Small-sized Firm’s decision on Foreign Direct Investment

Small and medium-sized enterprises have frequently been facing great troubles in capital budgeting and investment decisions. Part of these troubles comes from the very nature of the modern financial decision problems. While well-known classic optimization methods can properly solve traditional financial decision problems with a single objective, they fail at solving real-world complex problems, which involves the interplay of multiple concurrent decision factors, against a backdrop of financial constraints. As a result, the corporate finance field has been demanding more sophisticated approaches, which integrate traditional quantitative methods with Multi-Criteria Decision Analysis techniques (MCDA), probabilistic models, simulations and optimization methods. This paper combines Corporate Finance theory with a MCDA approach based on the von Neumann-Morgenstern’s Utility Theory and presents a case study within a small-sized firm that needs to decide whether or not to expand into international operations, given a set of decision factors and financing constraints. The results deliver a structured decision framework to assist the firm’s managers, increasing impartiality in the investment decision process and help the decision makers (DM) reach a rightful, well-substantiated decision to their investment decision problem.

Stable cores in information graph games

In an information graph situation, a finite set of agents and a source are the set of nodes of an undirected graph with the property that two adjacent nodes can share information at no cost. The source has some information (or technology), and agents in the same component as the source can reach this information for free. In other components, some agent must pay a unitary cost to obtain the information. We prove that the core of the derived information graph game is a von Neumann-Morgenstern stable set if and only if the information graph is cycle-complete, or equivalently if the game is concave. Otherwise, whether there always exists a stable set is an open question. If the information graph consists of a ring that contains the source, a stable set always exists and it is the core of a related situation where one edge has been deleted.

A multiplayer game model to detect insiders in wireless sensor networks.

Insiders might have incentives and objectives opposed to those of the belonging organization. It is hard to detect them because of their privileges that partially protect them. In Wireless Sensor Networks (WSNs), significant security issues arise, including compromised nodes by insiders that disrupt the normal network operation. Immediate defensive actions to isolate malicious nodes would mitigate any related impacts. A multiplayer game model is proposed as a solution to the problem of insider attacks in WSNs, the Game of Wireless Sensor Networks (GoWiSeN). It is an imperfect information game, formulated with the use of non-cooperative game theory, holding the assumption that all players are rational. The model consists of several Local Intrusion Detection Systems (LIDSs), which are located to different nodes and communicate with a Global Intrusion Detection System (GIDS). Each LIDS gives suggestions whether the monitoring node is trusted or not. The game is being played between a potential attacker, the nodes and the GIDS. The GIDS is responsible for making a final decision and for isolating a compromised node in case of an internal attack. The theoretical model represents these interactions in an extensive form game. The formal elements of the game are specified, the outcomes of the game are quantified by first specifying players’ preferences, and then, by using the von Neumann-Morgenstern utility function, and payoffs are obtained. The game is constructed and solved, by locating NE in pure and mixed strategies. Experimental evaluations conducted on real network datasets, using IDSs of different capabilities, simulate special cases and compromised nodes in a WSN, verify the model efficiency, and show how the game should be played.

Correlation Made Simple: Applications to Salience and Regret Theory

Abstract I offer an axiomatization of risk models where the choices of the decision maker are correlation sensitive. By extending the techniques of conjoint measurement to the nondeterministic case, I show that transitivity is the von Neumann-Morgenstern axiom that has to be relaxed to allow for these richer patterns of behavior. To illustrate the advantages of the modeling choice, we provide a simple axiomatization for the salience theory model in our general framework. This approach leads to clear comparison to popular preexisting models, such as regret and reference dependence, and allows one to single out the ordering property as the feature that brings salience theory outside the prospect theory realm.

Generalized Shortfall Risk Measure Based on Insurance Premium

Utility-based shortfall risk (SR) measure proposed by (F\”ollmer and Schied, 2002) has been well studied in risk management and finance. In this paper, we revisit the concept from insurance premium perspective. We show under some moderate conditions that the indifference equation-based insurance premium calculation can be equivalently formulated as an optimization problem similar to the definition of SR and subsequently call the premium functional as a generalized shortfall risk measure (GSR). We then use the latter formulation to investigate the properties of the GSR with a focus on the case that the preference functional is a distorted expected value function based on prospect theory. Specifically, we exploit Weber's methods and techniques (Weber, 2006) for characterization of the shortfall risk measure to derive a relationship between properties of GSR based on the prospect theory and the underlying value function and weighting function in terms of convexity/concavity and positive homogeneity. We also investigate the GSR as a functional of cumulative distribution function of random loss/liability and derive local and global Lipschitz continuity of the function under Wasserstein metric, a property which is related to statistical robustness of the GSR. The results cover the premium risk measures based on the von Neumann-Morgenstern's expected utility and Yaari's dual theory of choice as special cases. Finally, we propose a computational scheme for calculating GSR.

Експертний аналіз при проведенні подієвих заходів в туризмі

Introduction. The article considers the process of forming decisions aimed to ensure the quality and effectiveness of events, which should be considered as a system of interconnected elements (factors) that can affect the holding of mass events. Carrying out of actions taking into account the factor signs influencing processes of advancement, the organization and carrying out of event actions on system of economic, information and social indicators, allows to establish interdependencies and to define in quantitative expression separate relations and combinations of influential factors. Analytical methods and economic-mathematical models are used for factor analysis. The purpose of the study is to increase the efficiency of event tourism events, by implementing the correct choice and variation of factors in the implementation of organizational processes. The methodological and informational basis of the article are scientific and theoretical works of scientists on research and analysis of expert evaluation methods. During the research we analyzed the methods of examination that can be used in assessing the processes of creating and promoting events. All these methods include: the method of expert evaluation “psychological experiment”, the method of survey, quantitative and qualitative evaluations of experts, direct quantitative evaluation, intermediate method, Churchman-Akoff method, von Neumann-Morgenstern method, lottery method and more. The study did not identify examples of the use of factor analysis in assessing the processes of creating and promoting events. In order to form decisions on the creation and implementation of tourist events, a roadmap of phased decision-making has been developed, and this allows to form the most effective solutions with rational use of resources and determine the conditions for reducing the risks of tourist events. Having studied the above methods, it was for the first time proposed to combine two, method namely, the method of expert assessments and the method of survey, all this will provide a complete detailed description of the factor influence on the creation and passage of the event.

An interval-valued utility theory for decision making with Dempster-Shafer belief functions

The main goal of this paper is to describe an axiomatic utility theory for Dempster-Shafer belief function lotteries. The axiomatic framework used is analogous to von Neumann-Morgenstern's utility theory for probabilistic lotteries as described by Luce and Raiffa. Unlike the probabilistic case, our axiomatic framework leads to interval-valued utilities, and therefore, to a partial (incomplete) preference order on the set of all belief function lotteries. If the belief function reference lotteries we use are Bayesian belief functions, then our representation theorem coincides with Jaffray's representation theorem for his linear utility theory for belief functions. We illustrate our representation theorem using some examples discussed in the literature, and we propose a simple model for assessing utilities based on an interval-valued pessimism index representing a decision-maker's attitude to ambiguity and indeterminacy. Finally, we compare our decision theory with those proposed by Jaffray, Smets, Dubois et al., Giang and Shenoy, and Shafer.

Stable Cores in Information Graph Games

In an information graph situation, some agents that are connected by an undirected graph can share with no cost some information or technology that can also be obtained from a source. If an agent is not connected to an informed player, this agent pays a unitary cost to obtain this technology. A coalitional cost game can be defined from this situation, and the core of this game is known to be non- empty. We prove that the core of an information graph game is a von Neumann-Morgenstern stable set if and only if the graph is cycle- complete, or equivalently if the information graph game is concave. When the graph is not cycle-complete, whether there always exists a stable set is an open question. In this regard, we show that if the information graph consists of a ring that contains the source, then a stable set always exists and it is the core of a related information graph situation where one edge has been deleted.

The Strong Consistency of Neutral and Monotonic Binary Social Decision Rules

The purpose of this paper is to investigate the strong consistency of neutral and monotonic binary social decision rules. Individuals are assumed to satisfy von Neumann-Morgenstern axioms of individual rationality. The main result of the paper shows that there does not exist any neutral and monotonic non-null non-dictatorial binary social decision rule which is strongly consistent. The relationship between restricted preferences and the existence of strong equilibria is also investigated. It is shown that for every non-dictatorial social decision function satisfying the conditions of independence of irrelevant alternatives, neutrality, monotonicity and weak Pareto-criterion there exists a profile of individual orderings satisfying value-restriction corresponding to which there is no strong equilibrium.

Conditional Pure Strategy Dominance

Given a dynamic game with ordinal preferences, we deem a strategy sequentially rational if there exist a von Neumann-Morgenstern utility function that agrees with the assumed ordinal preferences and a conditional probability system with respect to which the strategy is a maximizer. We prove that this notion of sequential rationality is characterized by a notion of dominance, called Conditional B-Dominance, that extends Pure Strategy Dominance of Borgers (1993) to dynamic games represented in their extensive form. Additionally, we introduce an iterative procedure based on Conditional B-Dominance with a forward induction reasoning flavour, called Iterative Forward B-Dominance, that we prove satisfies nonemptiness. Finally, we show that Iterative Forward B-Dominance selects the unique backward induction outcome in generic dynamic games with perfect information.

Minimal Enforceability and Indirect Domination Relations in the Shapley-Scarf Economy

We consider the von Neumann-Morgenstern (vNM) stable sets based on a farsighted version of weak domination relation for the barter model with indivisible goods of Shapley and Scarf (1974). Recent research on farsighted stable sets has focused on coalitional sovereignty, a term used by Ray and Vohra (2015) regarding what coalitions can do to other coalitions, and they note the importance of this issue and how they may affect the results in characteristic function form games. However, defining a farsighted version of indirect weak domination introduces an additional issue of what coalitions can or cannot do from the inside, an issue which needed not be analyzed for the stronger versions of indirect domination nor for myopic domination relations in the literature. We first impose a minimality condition on the coalitions that can deviate in just one step, as coalitions in this model represent trading cycles, where such a minimality condition is often imposed. Under the domain of preferences defined by Klaus, Klijn, and Walzl (2010), each Pareto efficient allocation in the core and its Pareto indifferent allocations constitute a vNM stable set defined by a farsighted weak domination that respects the minimal enforceability condition, and those are the only essentially singleton such vNM stable sets.

Choosing the best objects by a set of the qualitative features

It is generally accepted that the choice of the best technical solutions in the design of complex technical systems must be carried out on the basis of a multi-criteria approach. However, it is known that at the initial design stages, developers, as a rule, do not have exact values of basic technical indicators. Moreover, many (if not all) indicators are qualitative in nature, and the probability of making a non-optimal decision increases due to the uncertainty of the initial data. Therefore, it is advisable to try to restrict the choice domain, eliminating the deliberately “uncompetitive” objects. In the well-known ELECTRE method, for this purpose, some “preference” relation is introduced, indirectly determined by the choice of “weights” of particular quality indicators and threshold values of the so-called concordance and discordance indices. Unfortunately this approach is characterized by the lack of a clear operational sense of the weights and thresholds introduced, as well as the possible occurrence of oriented cycles in the preference relation graph. This makes it difficult to interpret the results of applying the procedure for excluding “worst” objects by the ELECTRE method. In this regard, the article proposes a new approach to the problem of choice domain restriction. On the set of the objects under consideration, some irreflexive and asymmetric dominance relation is constructed, allowing a solution according to von Neumann-Morgenstern, which simultaneously is also the so-called C-core. At the same time, the considered dominance relation – unlike the preference relation in the ELECTRE method already mentioned – has a clear “physical” meaning and does not require the introduction of any directly immeasurable parameters like “weights” and “thresholds,” which always question the adequacy of the model to actual decision maker’s preferences system.

On R. Muth’s Impossible Dream: Trying to turn Subjective Probabilities into Objective Probabilities

R. Muth’s 1961 Econometrica paper is based on a fundamental and basic confusion on his part about the nature of subjective and objective probability that is impossible to remedy. A bifurcation occurred among scientists, mathematicians, and statisticians, starting in the mid 17th century with the work of Pascal, Fermat,and Cardano, about whether probability was subjective or objective. It could not be both. It had to be one or the other. Subjective probability always dealt with probability as being inside or internal to the human mind. It was a degree of belief of an individual based on his mental, personal, and psychological assessment of the data and factors he believed were involved in determining the behavior of the particular phenomena. On the other hand, Objective probability was always outside or external to the human mind. It was a part of the empirical, real world of phenomena that could be observed and was completely independently from any personal, emotional, mental, psychological or subjective factors. Objective probability was always a relative frequency in occurrence that would approach the true or correct answer in the limit as the number of observations of the particular phenomena approached infinity or some very large number that was an approximation of infinity. Historically, there are no adherents of the subjective approach to probability who accept the objective approach to probability and there are no adherents of the objective approach to probability who accept a subjective approach to probability. There is not, nor could there ever be, a subjective –objective hybrid theory because the definitions of subjective and objective probability directly contradict and conflict with each other. Muth had no historical knowledge of these two completely different definitions, forms, concepts and approaches to probability. Neither did the referees of his paper for Econometrica nor any proponent, advocate, supporter or user of what Muth claimed was the rational expectations hypothesis, which was that, for a given information set, the subjective estimates of probability (subjective probability distributions) were distributed around one single, objective, true, correct objective probability distribution. Muth apparently never realized that his definition was an oxymoron. It is impossible by definition. Nowhere in any of his references are there any that deal with the completely different theories of subjective and objective probability; there are no references to any sources in the history of science, history of probability, history of statistics, philosophy of science, philosophy of probability, or philosophy of statistics. There are no references to methodology or epistemology or to the philosophy of social science. Muth used the 1944 Von Neumann-Morgenstern approach to Expected Utility as the foundation for his theory of consumer and producer maximization of utility and profit in his 1961 article. However, it was made very clear by Von Neumann and Morgenstern that they were going to assume the use of objective frequencies only, because they were only going to deal with risk, where risk was defined to be based on known, objective probability. No subjective probability was allowed. Von Neumann had a completely different theory to deal with (complete) uncertainty when no probabilities were known, which he called min-max. Thus, the individual consumer-producer probability distributions would have to be objective, not subjective, in Muth’s paper in order to be consistent with the theory of risk analyzed by Von Neumann-Morgenstern in 1944. Out of nowhere, Muth starts talking about subjective probabilities (distributions) being distributed, for a given information set, around a true, correct, objective probability distribution. This is impossible and can never occur. There is absolutely no foundation at all in any theory of probability for Muth’s definition of rational expectations. The entire literature on the rational expectations hypothesis needs to be completely and totally reformulated and rewritten so as to be consistent with a foundation in either the subjective theory of probability or the objective theory of probability. Otherwise, rational expectationists need to state that they are not dealing with probability and statistics at all. They are simply assuming the very extreme position of complete knowledge from the Greenspan or Keynesian Continuum or spectrum that ranges from situations of complete ignorance to situations of complete knowledge. Currently, there simply is no foundation in any theory of probability for the rational expectations hypothesis, as defined by Muth in 1961 and accepted in the literature in economics since the Robert Lucas paper of 1972, which simply repeats all of the unsupportable claims made by Muth in his definition of rational expectations in 1961 that confused subjective and objective probability.

On the Very Severe Logical, Epistemological, Methodological and Philosophical Deficiencies in J. Muth’s Concept of Rational Expectations

J. Muth’s 1961 article on rational expectations was written without any knowledge on Muth’s part about what a subjective theory of probability entails and is based on, or what an objective theory of probability entails and is based on. Muth’s disbelief, that the individual, subjective probability distributions of firms and consumers are distributed around an objective, true, probability distribution, is not supported by any acknowledged scholar who has written on the application of the theory of probability to conduct from either a subjectivist theory of probability perspective (Ramsey, de Finetti, Savage) or an objectivist theory of probability perspective (Venn, R. von Mises, H. Reichenbach, K Popper) in history. Muth’s confusions extend to his analysis of optimizing behavior at the level of the firm and consumer. His choice of exposition used expected utility maximization. Muth implicitly must have been using the Von Neumann-Morgenstern approach under risk, which was based on the application of the relative-limiting frequency interpretation of probability, which is an objective theory of probability. The use of expected utility directly conflicts with Muth’s initial claim that the probability distributions were subjective. Muth needed to have used the Ramsey-de Finetti-Savage Subjective Expected Utility (SEU) approach to support his claims about subjective probability distributions being used by individual consumers and producers. Savage’s principle of stable estimation then shows that, as a sufficient amount of relevant evidence becomes available over time, the individual, different subjective probability distributions will gradually converge toward one subjective, probability distribution. However, there is no such “thing” in the Subjectivist theory of probability as convergence to an objective, true probability distribution or a situation where the subjective probability distributions are distributed around a true, objective, probability distribution. R. Muth’s paper is incomprehensible, as well as being incoherent and contradictory. It is inconsistent with any known theory of objective or subjective probability.

The Allais Paradox: What It Became, What It Really Was, What It Now Suggests to Us

Whereas many others have scrutinized Allais paradox from a theoretical angle, we study paradox from an historical perspective and link our findings to a suggestion as to how decision theory could make use of it today. We emphasize that Allais proposed paradox as a normative argument, concerned with the rational man and not real man, to use his words. Moreover, and more subtly, we argue that Allais had an unusual sense of normative, being concerned not so much with rationality of choices as with rationality of agent as a person. These two claims are buttressed by a detailed investigation – first of its kind – of 1952 Paris conference on risk, which set context for invention of paradox, and a detailed reconstruction – also first of its kind – of Allais's specific normative argument from his numerous but allusive writings. The paper contrasts these interpretations of what paradox historically represented, with how it generally came to function within decision theory from late 1970s onwards: that is, as an empirical refutation of expected utility hypothesis, and more specifically of condition of von Neumann-Morgenstern independence that underlies that hypothesis. While not denying that this use of paradox was fruitful in many ways, we propose another use that turns out also to be compatible with an experimental perspective. Following Allais's hints on the experimental definition of rationality, this new use consists in letting experiment itself speak of rationality or otherwise of subjects. In 1970s, a short sequence of papers inspired by Allais implemented original ways of eliciting reasons guiding subjects' choices, and claimed to be able to draw relevant normative consequences from this information. We end by reviewing this forgotten experimental avenue not simply historically, but with a view to recommending it for possible use by decision theorists today.

A Refinement of Logit Quantal Response Equillibrium

Unlike the Nash equilibrium, logit quantal response equilibrium is affected by positive affine transformations of players’ von Neumann-Morgenstern utility payoffs. This paper presents a modification of a logit quantal response equilibrium that makes this equilibrium solution concept invariant to arbitrary normalization of utility payoffs. Our proposed modification can be viewed as a refinement of a logit quantal response equilibria: instead of obtaining a continuum of equilibria (for different positive affine transformations of utility function) we now obtain only one equilibrium for all possible positive affine transformations of utility function. We define our refinement for simultaneous-move noncooperative games in the normal form. An interpretation of our refinement in terms of the implicit model of relative random errors is provided.