On arguments from self-interest for the Nash solution and the Kalai egalitarian solution to the bargaining problem

I argue in this paper that there are two considerations which govern the dynamics of a two-person bargaining game, viz. relative proportionate utility loss from conceding to one's opponent's proposal and relative non-proportionate utility loss from not conceding to one's opponent's proposal, if she were not to concede as well. The first consideration can adequately be captured by the information contained in vNM utilities. The second requires measures of utility which allow for an interpersonal comparison of utility differences. These considerations respectively provide for a justification of the Nash solution and the Kalai egalitarian solution. However, none of these solutions taken by themselves can provide for a full story of bargaining, since, if within a context of bargaining one such consideration is overriding, the solution which does not match this consideration will yield unreasonable results. I systematically present arguments to the effect that each justification from self-interest for respectively the Nash and the Kalai egalitarian solution is vulnerable to this kind of objection. I suggest that the search for an integrative model may be a promising line of research.I am grateful to Jon Elster, A. Hylland, F. Spinnewijn, J. Verhoeven and the members of the research group for theoretical sociology in the K.U.L. for their comments on an earlier version of this paper. I also thank the Ministry for Foreign Affairs (Norway) for its financial support and the members of the department of sociology in the University of Oslo for their hospitality and interest in my work.     

Rawlsian Nash solutions

We investigate the conditions which guarantee the choice of Rawlsian solutions by players of unequal maximin positions bargaining under Nash rules. The conditions, viz., subsymmetry and the location of the Pareto set, relate to the structure of the utility set,S.     

Implementation without rationality assumptions

Hurwicz (Social Choice and public decision making. Essays in honor of Kenneth J. Arrow, Cambridge University Press, Cambridge, 1986) was the first to study an approach to implementation theory based on choice functions instead of preference relations. We argue that the solution concept used by him, the generalized Nash equilibrium, is not really compatible with the idea that individual behavior is describable by a choice function. A new solution concept that better fits the choice function framework is then introduced. Using this, we investigate what behavioral assumptions are needed for the full characterizations of Nash implementable social choice correspondences to still hold. We will show that a condition known as Property α is central.     

LEXICOGRAPHIC ADDITIVITY FOR MULTI-ATTRIBUTE PREFERENCES ON FINITE SETS

Payoff dominance, a criterion for choosing between equilibrium points in games, is intuitively compelling, especially in matching games and other games of common interests, but it has not been justified from standard game-theoretic rationality assumptions. A psychological explanation of it is offered in terms of a form of reasoning that we call the Stackelberg heuristic in which players assume that their strategic thinking will be anticipated by their co-player(s). Two-person games are called Stackelberg-soluble if the players' strategies that maximize against their co-players' best replies intersect in a Nash equilibrium. Proofs are given that every game of common interests is Stackelberg-soluble, that a Stackelberg solution is always a payoff-dominant outcome, and that in every game with multiple Nash equilibria a Stackelberg solution is a payoff-dominant equilibrium point. It is argued that the Stackelberg heuristic may be justified by evidentialist reasoning.     

A theorem for Bayesian group decisions

This paper presents a natural extension of Bayesian decision theory from the domain of individual decisions to the domain of group decisions. We assume that each group member accepts the assumptions of subjective expected utility theory with respect to the alternatives from which they must choose, but we do not assume, a priori, that the group as a whole accepts those assumptions. Instead, we impose a multiattribute utility independence condition on the preferences of the group with respect to the expected utilities of its actions as appraised by its members. The result is that the expected utility of an alternative for the group is a weighted average of the expected utilities of that alternative for its members. The weights must be determined collectively by the group. Pareto optimality is not assumed, though the result is consistent with Pareto optimality.     

Some quantitative concepts of value and utility from a utilitarian point of view

This paper clarifies and interprets some basic quantitative concepts of value, utility and utility function from a utilitarian point of view. First, I discuss the question as to whether value is objective or subjective. I hold that value is subjective and statistical in nature (although from the various subjective values of a certain object a norm can usually be obtained). Second, I emphasize the distinction between use value and exchange value in relation to utility. Third, I propose a law of diminishing incremental interest, which refers to the incremental (marginal) utility of money. Fourth, I identify the utility of money with the von Neumann-Morgenstern utility. Fifth, I question the necessity of the usual normalization of utility functions and the restricted linear transformation (and the consequent concept of strategic equivalence). Sixth, I discuss in detail the terminal values and utilities of a utility function from a philosophical rather than mathematical point of view, particularly the boundedness of a utility function and the magnitudes of V ₀ and U ₀. Finally, I conclude that, in order to be able to have interpersonal comparisons of utility, utility should have the same dimension as value rather than no dimension, and the normalization problem should be reconsidered in the light of terminal values and utilities.     

A Note on Optimal Insurance in the presence of a Nonpecuniary Background Risk

This note examines the theory of optimal insurance purchasing in the presence of a nonpecuniary background risk. The occurrence of the qualitative uninsurable background loss can increase, decrease or can leave the marginal utility of wealth unchanged, whereas a financial background loss (as in Doherty and Schlesinger, 1883a) always increases it. Existing theorems on the optimal level of insurance and the optimal form of insurance contracts are shown to hold only under restrictive assumptions on the correlation level between risks. The paper shows that sufficient conditions for the validity of the theorems depend not only on the correlation between the insurable and noninsurable losses, but also on the variation of the marginal utility of wealth with respect to the nonpecuniary variable.     

Error Propagation in the Elicitation of Utility and Probability Weighting Functions

Elicitation methods in decision-making under risk allow us to infer the utilities of outcomes as well as the probability weights from the observed preferences of an individual. An optimally efficient elicitation method is proposed, which takes the inevitable distortion of preferences by random errors into account and minimizes the effect of such errors on the inferred utility and probability weighting functions. Under mild assumptions, the optimally efficient method for eliciting utilities and probability weights is the following three-stage procedure. First, a probability is elicited whose subjective weight is one half. Second, the utility function is elicited through the midpoint chaining certainty equivalent method using the probability elicited at the first stage. Finally, the probability weighting function is elicited through the probability equivalent method.     

Buying and selling price for risky lotteries and expected utility theory with gambling wealth

I analyze two expected utility models which abandon the consequentialist assumption of terminal wealth positions. In the expected utility of gambling wealth model, in which initial wealth is allowed to be small, I show that a large WTA/WTP gap is possible and the (Rabin in Econometrica, 68(5), 1281–1292, 2000) paradox may be resolved. Within the same model the classical preference reversal which allows arbitrage is not possible, whereas preference reversal (involving buying prices in place of selling prices), which does not allow arbitrage, is possible. In the expected utility of wealth changes model, in which there is no initial wealth, I show that both a WTA/WTP gap as well as the classical preference reversal are possible due to loss aversion, both in its general as well as some specific forms.     

Risk seeking with diminishing marginal utility in a non-expected utility model

The present work takes place in the framework of a non-expected utility model under risk: the RDEU theory (Rank Dependent Expected Utility, first initiated by Quiggin under the denomination of Anticipated Utility), where the decision maker's behavior is characterized by two functionsu andf. Our first result gives a condition under which the functionu characterizes the decision maker's attitude towards wealth. Then, defining a decision maker as risk averter (respectively risk seeker) when he always prefers to any random variable its expected value (weak definition of risk aversion), the second result states that a decision maker who has an increasing marginal utility of wealth (a convex functionu) can be risk averse, if his functionf issufficiently below his functionu, hence if he is sufficientlypessimistic. Obviously, he can also be risk seeking with a diminishing marginal utility of wealth. This result is noteworthy because with a stronger definition of risk aversion/risk seeking, based on mean-preserving spreads, Chew, Karni, and Safra have shown that the only way to be risk averse (in their sense) in RDEU theory is to have, simultaneously, a concave functionu and a convex functionf.     

The Welfare Consequences of Strategic Voting in Two Commonly Used Parliamentary Agendas

This paper studies the welfare consequences of strategic voting in two commonly used parliamentary agendas by comparing the average utilities obtained in simulated voting under two behavioural assumptions: expected utility maximising behaviour and sincere behaviour. The average utility obtained in simulations is higher with expected utility maximising behaviour than with sincere voting behaviour under a broad range of assumptions. Strategic voting increases welfare particularly if the distribution of preference intensities correlates with voter types.      

Binary 2 × 2 games

The 2 × 2 game is the simplest interactive decision model that portrays concerned decision makers with genuine choices. There are two players, each of whom must choose one of two strategies, so that there are four possible outcomes. Binary 2 × 2 games are 2 × 2 games with no restrictions on the players' preference relations over the outcomes. They therefore generalize the strict ordinal 2 × 2 games and the ordinal 2 × 2 games, classes which have already been studied extensively. This paper enumerates the strategically distinct binary 2 × 2 games. It also identifies important subsets defined by the number of pure Nash equilibria and the occurrence of dominant strategies.     

Demand for health risk reductions: A cross-national comparison between the U.S. and Canada

Using a large stated preference survey conducted across the U.S. and Canada, we assess differences in individual willingness to pay (WTP) for health risk reductions between the two countries. Our utility-theoretic choice model allows for systematically varying marginal utilities for avoided future time in different adverse health states (illness-years, recovered/remission years, and lost life-years). We find significant differences between Canadian and U.S. preferences. WTP also differs systematically with age, gender, education, and marital status, as well as a number of attitudinal and subjective health-perception variables. Age profiles for WTP are markedly different across the two countries. Canadians tend to display flatter age profiles, with peak WTP realized at older ages.     

The curse of hope

In K?szegi and Rabin’s (Q J Econ 1133–1165, 2006, Am Econ Rev 97:1047–1073, 2007) reference-dependent model of preferences, the chance of obtaining a better outcome can reduce an agent’s expected utility through an increase in the stochastic reference point. This means that individuals may prefer stochastically dominated lotteries. In this sense, hope, understood as a small probability of a better outcome, can be a curse. While K?szegi and Rabin focus on a linear specification of the utility function, we show that this effect occurs more broadly. Using fairly plausible assumptions and parameter values, we specify the conditions under which it occurs, as well as the type of lotteries in which this should be expected. We then show that while a simple subjective transformation of probability into weights of the reference point may in some cases mitigate the issue, in others, it can intensify it or even generate new ones. Finally, we extend the model by adding the individual’s current reference point (status quo) to the stochastic reference point. We show that this modification can reconcile K?szegi and Rabin’s model with the apparent empirical infrequency of stochastically dominated choices while maintaining its main qualitative results.     

An Ethical Interpretation of the Nash Choice Rule

This paper provides an ethical intepretation of the Nash choice rule. In a setting in which (cardinal) utilities are interpersonally comparable, this procedure is characterised by an impartiality requirement and by the assumption that choices are not responsive to the agents' relative ability to convert resources into utility.     

Individual fairness in Harsanyi’s utilitarianism: operationalizing all-inclusive utility

Fairness can be incorporated into Harsanyi’s utilitarianism through all-inclusive utility. This retains the normative assumptions of expected utility and Pareto-efficiency, and relates fairness to individual preferences. It makes utilitarianism unfalsifiable, however, if agents’ all-inclusive utilities are not explicitly specified. This note proposes a two-stage model to make utilitarian welfare analysis falsifiable by specifying all-inclusive utilities explicitly through models of individual fairness preferences. The approach is applied to include fairness in widely discussed allocation examples.     

The impact of risk sharing on efficient decision

A group of risk-averse members must choose among monetary risks and payoff-sharing rules. Departure from the status quo requires unanimous consent. Such groups drill for oil, bail out nations, and make hostile takeover bids. Assume agreement on probabilities. As is well known, if all members have identically shaped HARA utility functions, efficient group act-choices follow another such function independently of payoff sharing. We show that all other groups inevitably have complex efficient behavior, accepting gambles among individually unacceptable lotteries in almost every status quo position. We also develop proper risk aversion for groups, and treat disagreement on probabilities.Support of the Associates of the Harvard Business School (Pratt) and the Business and Government and Energy and Environmental Policy Centers (Zeckhauser) is gratefully acknowledged. We received helpful comments from Scott Johnson, Mark Machina, and a referee. Jay Patel provided assistance beyond the call of collegiality.     

A Simple Model of Secure Public Communication

Public communication is secure if a hostile third-party cannot decode the messages exchanged by the communicating parties. In Nash equilibrium, communication by computationally unbounded players cannot be secure. We assume complexity averse players, and show that a simple, secure, and costless communication protocol becomes available as the marginal complexity cost tends to zero.      

Indecisiveness aversion and preference for commitment

We present an axiomatic model of preferences over menus that is motivated by three assumptions. First, the decision maker is uncertain ex ante (i.e., at the time of choosing a menu) about her ex post (i.e., at the time of choosing an option within her chosen menu) preferences over options, and she anticipates that this subjective uncertainty will not resolve before the ex post stage. Second, she is averse to ex post indecisiveness (i.e., to having to choose between options that she cannot rank with certainty). Third, when evaluating a menu she discards options that are dominated (i.e., inferior to another option whatever her ex post preferences may be) and restricts attention to the undominated ones. Under these assumptions, the decision maker has a preference for commitment in the sense of preferring menus with fewer undominated alternatives. We derive a representation in which the decision maker’s uncertainty about her ex post preferences is captured by means of a subjective state space, which in turn determines which options are undominated in a given menu, and in which the decision maker fears, whenever indecisive, to choose an option that will turn out to be the worst (undominated) one according to the realization of her ex post preferences.     

Average willingness to pay for disease prevention with personalized health information

Personal health related information modifies individuals’ willingness to pay for disease prevention programs inasmuch as it allows health status assessment based on intrinsic (instead of average) characteristics. In this paper, we examine the effect that personalized information about the baseline probability of disease has on the average willingness to pay for programs reducing either the probability of disease (self-protection) or the severity of disease (self-insurance). We show that such information raises the average willingness to pay for self-protection while it increases the average willingness to pay for self-insurance if health and wealth are complements (i.e. the marginal utility of wealth rises with health).