INEQUALITY AVERSION CAUSES EQUAL OR UNEQUAL DIVISION IN ALTERNATING‐OFFER BARGAINING

A solution to Rubinstein (1982)'s open‐ended, alternating‐offer bargaining problem for two equally patient bargainers who exhibit similar degrees of inequality aversion is presented. Inequality‐averse bargainers may experience envy if they are worse off, and guilt if they are better off, but they still reach agreement in the first period under complete information. If the guilt felt is strong, then the inequality‐averse bargainers split a pie of size one equally regardless of their degree of envy. If the guilt experienced is weak, then the agreed split is tilted away from the Rubinstein division towards a more unequal split whenever the degree of envy is smaller than the discounted degree of guilt. Envy and weak guilt have opposite effects on the equilibrium division of the pie, and envy has a greater marginal impact than weak guilt. Equally inequality‐averse bargainers agree on the Rubinstein division if the degree of envy equals the discounted degree of guilt. As both bargainers' sensation of inequality aversion diminishes, the bargaining outcome converges to the Rubinstein division.     

The effect of decision weights in bargaining problems

Bargaining problems are considered where the preferences of the bargainers deviate from expected utility but can be modelled according to rank-dependent utility theory. Under rank-dependent utility both the utility function and the probability weighting function influence the risk attitude of a decision maker. The same definition of risk aversion leads to two forms of risk aversion: utility risk aversion and probabilistic risk aversion. The main finding is that these two forms can have surprisingly opposite consequences for bargaining solutions that exhibit a weak monotonicity property. In particular, in a large class of bargaining problems both increased utility risk aversion and decreased probabilistic risk aversion of the opponent are advantagous for a player. This is demonstrated for the Kalai-Smorodinsky bargaining solution. The Nash bargaining solution does not behave regularly in this respect.     

Self-enforcing collusion in large dynamic markets

The “folk theorem” in game theory implies that any outcome that is better for all players than some single period Nash outcome can be achieved through noncooperative equilibrium in repeated games with discounting. Whether the folk theorem holds for a repeated Cournot oligopoly as the number of firms, N, increases without bound, is investigated. It is shown that the folk theorem holds in the limit iff demand increases at the same rate as the number of firms and the Cournot price sequence is bounded strictly above by the supremum of marginal cost for large N.     

Misrepresentation and stability in the marriage problem

It has been shown previously that, for two-sided discrete markets of the kind exemplified by the “marriage problem,” no strategy-proof procedure for aggregating preferences into stable outcomes exists. Here it is shown that (Nash) equilibrium misrepresentation of preferences nevertheless results in a stable outcome in terms of the true preferences when the aggregation procedure yields the optimal stable outcome in terms of the stated preferences for one side of the market.     

Risk sensitivity in bargaining with more than two participants

In two-person bargaining over riskless outcomes where the outcome is chosen according to the Nash solution or the Kalai-Smorodinsky solution, an increase in a player's risk aversion is to the advantage of the opponent. It is shown in this note that for the n-person Nash solution, an increase in risk aversion is to the player's own disadvantage but needs not be advantageous to all the opponents. For the n-person Kalai-Smorodinsky solution, the increase is to the player's own disadvantage and to the advantage of all the opponents.     

Whose opinion counts? Implementation by sortition

Sortition is the process of selecting decision makers or senators by a lottery. We introduce sortition in implementation theory by augmenting a mechanism with a kleroterion or lottery machine p that selects the senators. An outcome is implemented after consulting only the opinions of the senators. We call the corresponding notion of implementation as “p-implementation”, and provide necessary and sufficient conditions for p-implementation. Our main result is that in “economic” environments, every Nash implementable social choice rule (SCR) is also p-implementable if p selects every quartet of players with positive probability and always selects at least three senators. We apply this result to two kleroteria: “oligarchic democracy” and “random sampling”. In economic environments, every Nash implementable SCR can be implemented by oligarchic democracy of three oligarchs. In economic environments, every Nash implementable SCR can be implemented by randomly selecting four senators.     

Stable Tariffs and Retaliations*

A two‐country, two‐commodity model of trade is considered to reformulate the tariff retaliations. It is known that tariff retaliations lead to a Nash‐equilibrium, a non‐free‐trade outcome. However, the negotiation process underlying the Nash equilibrium does not capture the notion of retaliation properly. We use the “contingent threat situation” to reformulate tariff retaliations. In this context, we show that the free trade is a stable outcome. More surprisingly, this interesting result is also valid for the “Johnson case,” where one country is better off under the tariff‐ridden Nash equilibrium compared to free trade.     

The Gender Gap

The author explains the “gender gap” as a Nash equilibrium of a game with incomplete information about women's work at home and in the marketplace. Expectations about women's lower wages leads to the overutilization of women in the household, and this, in turn, leads to lower productivity and lower wages for women in the marketplace. The situation is rational but generally Pareto inferior. With logistic learning by doing, at high levels of skill there is a Pareto‐superior equilibrium, where men and women share efforts equally at home and receive the same pay in the marketplace, firms enhance their profits, and there is more welfare at home. Inequity at home breeds inequity in the marketplace and, reciprocally, inequity in the marketplace leads to inequity at home, causing a persistent gender gap. Appropriate contracts may be needed to implement the superior solution, since generally governments do not intervene in family matters.     

Implementation,Elimination of Weakly Dominated Strategies and Evolutionary Dynamics

This paper studies convergence and stability properties of T. Sjöström's (1994, Games Econom. Behav.6, 502–511) mechanism, under the assumption that boundedly rational players find their way to equilibrium using monotonic evolutionary dynamics and best-reply dynamics. This mechanism implements most social choice functions in economic environments using as a solution concept one round of deletion of weakly dominated strategies and one round of deletion of strictly dominated strategies. However, there are other sets of Nash equilibria, whose payoffs may be very different from those desired by the social choice function. With monotonic dynamics, all these sets of equilibria contain limit points of the evolutionary dynamics. Furthermore, even if the dynamics converge to the “right” set of equilibria (i.e., the one which contains the solution of the mechanism), it may converge to an equilibrium which is worse in welfare terms. In contrast with this result, any interior solution of the best-reply dynamics converges to the equilibrium whose outcome the planner desires. Journal of Economic Literature Classification Numbers: C72, D70, D78.     

Evolutionary Stability in the Finitely Repeated Prisoner 's Dilemma Game

The dynamic evolutionary stability of mutual defection is proven for the repeated prisoner 's dilemma game where payoffs are cumulative and the number of repetitions is known. This agrees with the classical result that the only Nash equilibrium outcome is to defect at all stages of this repeated game. Moreover, it is shown that, for any initial polymorphic population, the evolutionary dynamic converges to a unique Nash equilibrium strategy that depends on the original polymorphism. Both these results confirm earlier conjectures concerning the application of evolutionary game theory to the repeated prisoner 's dilemma.Journal of Economic LiteratureClassification Numbers: C70, C72.     

On the Irrelevance of Risk Attitudes in Repeated Two-Outcome Games

We study equilibrium and maximin play in supergames consisting of the sequential play of a finite collection of stage games, where each stage game has two outcomes for each player. We show that for two-player supergames in which each stage game is strictly competitive, in any Nash equilibrium of the supergame, play at each stage is a Nash equilibrium of the stage game provided preferences over certain supergame outcomes satisfy a natural monotonicity condition. In particular, equilibrium play does not depend on risk attitudes. We establish an invariance result for games with more than two players when the solution concept is subgame perfection. Journal of Economic Literature Classification Numbers: C72, C9.     

On the evolution of conditional cooperation

A long series of laboratory and field experiments, as well as conventional empirical studies, has established that (1) individuals voluntarily provide themselves with public goods at levels exceeding those predicted by the Nash voluntary contributions mechanism, and (2) agents reciprocate increases in the contributions of their counterparts in such settings (conditional cooperation). This paper presents a simple model of the evolution of preferences for conditional cooperation in the presence of a public good, which explains these two empirical findings without employing reputational or group selection arguments. In this model, individuals inherit preferences to match other agents' contributions to the provision of a public good, at some specified “matching rate.” Agents whose preferences induce them to be relatively successful – in material terms – increase in number, from one generation to the next. Under complete information and with randomly matched groups of N agents who have quasilinear preferences over the public good and a private good, the unique evolutionarily stable matching rate is 1, leading to Pareto optimal voluntary provision of the public good, regardless of group size N. The evolutionarily stable matching rate can be viewed as an endogenous social norm.     

Social Welfare Functionals on Restricted Domains and in Economic Environments

Arrow's “impossibility” and similar classical theorems are usually proved for an unrestricted domain of preference profiles. Recent work extends Arrow's theorem to various restricted but “saturating” domains of privately oriented, continuous, (strictly) convex, and (strictly) monotone “economic preferences” for private and/or public goods. For strongly saturating domains of more general utility profiles, this paper provides similar extensions of Wilson's theorem and of the strong and weak “welfarism” results due to d'Aspremont and Gevers and to Roberts. Hence, for social welfare functionals with or without interpersonal comparisons of utility, most previous classification results in social choice theory apply equally to strongly saturating economic domains.     

A Public Choice Approach to Strategic and Nonstrategic Environmental Policy

In this paper we take a public choice perspective on strategic environmental policy and international environmental agreements. We examine cooperative and noncooperative environmental policies under governments that are either welfare maximizers (“good dictators”) or tax revenue maximizers (“Leviathans”). We show that Leviathans can perform better in terms of welfare and that good dictators can set higher taxes. We then analyze international environmental agreements and show that the breakdown of environmental cooperation can indeed lead to a welfare gain for all signatory countries. Considering a delegation game between governments, we find that a Pareto‐superior Leviathan outcome can be the unique Nash equilibrium.     

INCOMPLETE INFORMATION GAMES WITH MULTIPLE PRIORS*

We present a model of incomplete information games, where each player is endowed with a set of priors. Upon arrival of private information, it is assumed that each player “updates” his set of priors to a set of posterior beliefs, and then evaluates his actions by the most pessimistic posterior beliefs. So each player's preferences may exhibit aversion to ambiguity or uncertainty. We define a couple of equilibrium concepts, establish existence results for them, and demonstrate by examples how players’ views on uncertainty about the environment affect the strategic outcomes.     

Existence and Uniqueness of Ordinal Nash Outcomes

In this paper we present necessary and sufficient conditions for existence and uniqueness of ordinal Nash outcomes. These outcomes are derived from the ordinal Nash solution—a reinterpretation and an extension of the Nash bargaining solution that allows bargainers to have preference relations that are more general than expected utility. Our task is undertaken by the construction of a new notion called “induced utilities”. Journal of Economic Literature Classification Number: C78.     

Bargaining with subjective mixtures

This paper reconsiders the Bargaining Problem of Nash (Econometrica 28:155–162, 1950). I develop a new approach, Conditional Bargaining Problems, as a framework for measuring cardinal utility. A Conditional Bargaining Problem is the conjoint extension of a Bargaining Problem, conditional on the fact that the individuals have agreed on a “measurement event”. Within this context, Subjective Mixture methods are especially powerful. These techniques are used to characterise versions of the Nash and the Kalai–Smorodinsky solutions. This approach identifies solutions based only on the individuals’ tastes for the outcomes. It is therefore possible to do Bargaining theory in almost complete generality. The results apply to Biseparable preferences, so are valid for almost all non-expected utility models currently used in economics.     

The individual and the market: Paul Samuelson on (homothetic) Santa Claus economics

Paul Samuelson often used the term “Santa Claus economics” for mathematical models with empirically unrealistic assumptions. I focus on one particular member of the Santa Claus family that Samuelson was very sceptical about: homothetic general equilibrium models (where all agents have identical homothetic preferences). I argue that Samuelson's concerns about these models provide insights into how he viewed the relationship between the individual and the market, a relationship that has implications for not only his economic theorising, but also his broader political–economic vision. His criticisms are also relevant to some ongoing debates within contemporary economic theory.     

Local perfection

We investigate solution concepts for normal-form games with large strategy spaces. “Trembling hand” perfection (THP) is extended to games with compact metric strategy sets. A Nash equilibrium is THP if it is the limit of equilibria in which agents face vanishingly small uncertainty. We refine THP by imposing some structure on the kind of uncertainty that agents face: a THP equilibrium is locally perfect if the uncertainty is “predominantly local” in nature, i.e., if agents' probability assessments assign, in the limit, an overwhelmingly lage proportion of the total probability mass to events “in the vicinity of the truth.”     

Coalition-Proof Nash Equilibria II. Applications

In Bernheim, Peleg, and Whinston (“Coalition-Proof Equilibria. I. Concepts,” J. Econ. Theory 42 (1987), 1–12), we proposed the notion of Coalition-Proof Nash equilibrium and Perfectly Coalition-Proof Nash equilibrium as solution concepts for strategic environments in which players can freely discuss their strategies, but cannot make binding commitments. This paper undertakes applications to several economic problems, including the behavior of Cournot oligopolists, oligopolistic entry deterrence, cooperation in finite horizon games, and social choice rule implementation.