Strategies as states

We define rationality and equilibrium when states specify agents’ actions and agents have arbitrary partitions over these states. Although some suggest that this natural modeling step leads to paradox, we show that Bayesian equilibrium is well defined and puzzles can be circumvented. The main problem arises when player j's partition informs j of i's move and i knows j's strategy. Then i's inference about j's move will vary with i's own move, and i may consequently play a dominated action. Plausible conditions on partitions rule out these scenarios. Equilibria exist under the same conditions, and more generally ε equilibria usually exist.     

Price characterization of stable allocations in exchange economies with externalities

A pure exchange economy where the consumers have utility functions U_i(v₁(x₁),…, v_m(x_m)) for i = 1,…, m and where x_j is the consumption of consumer j, is studied. U_i may be nonincreasing or nondecreasing in v_j for j ≠ i. i is said to be nonbenevolent or nonmalevolent towards j, accordingly.An allocation is stable if no coalition can redistribute what it receives in the allocation to get an allocation which is preferred, given the consumptions of the consumers in the complementary coalition. Results concerning the relation among the Paretooptimal, stable and equilibrium allocations (under different definitions of equilibrium) are given. In particular, it turns out that in case every consumer is non-benevolent towards every other consumer, the classical results, concerning the relation between Paretooptimal allocations and equilibrium allocations, can be generalized in a satisfactory way.     

An efficiency theorem for incompletely known preferences

There are n agents who have von Neumann-Morgenstern utility functions on a finite set of alternatives A. Each agent i's utility function is known to lie in the nonempty, convex, relatively open set Ui. Suppose L is a lottery on A that is undominated, meaning that there is no other lottery that is guaranteed to Pareto dominate L no matter what the true utility functions are. Then, there exist utility functions ui∈Ui for which L is Pareto efficient. This result includes the ordinal efficiency welfare theorem as a special case.     

Envy,wealth, and class hierarchies

A persion i is said to not envy another person j if he likes his own bundle of goods as well as he would like j's bundle. This paper explores the social structure defined by the non-envy relation, and relates it to the social structure defined by market values of bundles, or wealth.     

Pareto-optimal Nash equilibria are competitive in a repeated economy

Consider a finite exchange economy first as a static, 1 period, economy and then as a repeated economy over T periods when the utility of each agent is the mean utility over T. A family of strategic games is defined via a set of six general properties the most distinct of which is the ability of agents to move commodities forward in time. Now consider Pareto optimal allocations in the T period economy which are also Nash equilibria in this family of strategic games. We prove that as T becomes large this set converges to the set of competitive utility allocations in the one period economy. The key idea is that a repetition of the economy when agents can move commodities forward in the time acts as a convexification of the set of individually feasible outcomes for player i holding all other strategies fixed.     

Conservative belief and rationality

Playersʼ beliefs may be incompatible, in the sense that player i can assign probability 1 to an event E to which player j assigns probability 0. One way to block incompatibility is to assume a common prior. We consider here a different approach: we require playersʼ beliefs to be conservative, in the sense that all players must ascribe the actual world positive probability. We show that common conservative belief of rationality (CCBR) characterizes strategies in the support of a subjective correlated equilibrium where all playersʼ beliefs have common support. We also define a notion of strong rationalizability, and show that it is characterized by CCBR.     

ON ADDITIVE METHODS TO SHARE JOINT COSTS*

The Shapley value theory is extended to cost functions with multiple outputs (or to production functions with multiple inputs) where each output is demanded by a different agent and the level of demand varies. Beyond the Additivity and Dummy axioms (Shapley's original axioms) we insist that the cost-share of an agent should not decrease when she increases her demand (Demand Monotonicity). This property rules out the Aumann-Shapley pricing formula, as well as any method charging average cost for homogeneous goods. We characterize the class of cost sharing methods satisfying Additivity, Dummy, Demand Monotonicity and Cross Monotonicity. The last says that when outputs i and j are cost complements (resp-cost substitutes) the cost share of i is non decreasing (resp-non increasing) in the demand of j. Two prominent methods in the class are the Shapley-Shubik method (i.e. the Shapley value of the Stand Alone cost game) and serial cost sharing (which extends to multiple goods a formula due to Moulin and Shenker). They are characterized respectively by a lower bound and by an upper bound on individual cost shares.     

On demand responsiveness in additive cost sharing

Under partial responsibility, the ranking of cost shares should never contradict that of demands.The Solidarity axiom says that if agent i demands more, j should not pay more if k pays less. It characterizes the quasi-proportional methods, sharing cost in proportion to `rescaled' demands.Full responsibility rules out cross-subsidization for additively separable costs. Restricting solidarity to submodular cost characterizes the fixed-flow methods, containing the Shapley-Shubik and serial methods.The quasi-proportional methods meet—but most fixed-flow methods fail—Group Monotonicity: if a group of agents increase their demands, not all of them pay less. Serial cost sharing is an exception.     

A recursion relation for the probability of the paradox of voting

For alternatives x_i, i = 1,…, m, giving rise to m! linear preference orderings of which one is selected independently by each of N voters, a recursion relation is developed which expresses the probability that x_i is the Condorcet winner when there are N voters in terms of the probability of this event when there are N ? 1 voters. Hence the probabilities of the paradox of voting when N is odd, and of Condorcet indecision when N is even may be obtained. The relationship holds for any set of probabilities, or culture, governing the selection of the preference orderings by the voters.     

The separability axiom and equal-sharing methods

A quasi-linear social choice problem is defined as selecting one (among finitely many) indivisible public decision and a vector of monetary transfers among agents to cover the cost of this decision. This decision is based upon individual preferences, which are assumed to be additively separable and linear in money. The Separability axiom is a consistency property for choice methods on societies with variable size: the decision is not affected if we remove an arbitrary agent under the condition that he be guaranteed his original utility level and the cost to the remaining agents is modified accordingly. Thus the utility level assigned by the social choice function to agent i is the price at which the other agents are unanimously willing to buy agent is share of the decision power. A general characterization of choice methods satisfying this axiom is provided. Three subclasses of particular interest are characterized by additional milder axioms. Those are: (i) equal sharing of the surplus left over some reference utility (e.g., the utility at a status quo decision), (ii) utilitarian methods that merely select the efficient public decision and perform no monetary transfers, and (iii) equal allocation of nonseparable costs, which divides equally the surplus left over from the utility derived from the pivotal mechanism (also known as the Vickrey-Clarke-Groves mechanism).     

Imitation processes with small mutations

This note characterizes the impact of adding rare stochastic mutations to an “imitation dynamic,” meaning a process with the properties that absent strategies remain absent, and non-homogeneous states are transient. The resulting system will spend almost all of its time at the absorbing states of the no-mutation process. The work of Freidlin and Wentzell [Random Perturbations of Dynamical Systems, Springer, New York, 1984] and its extensions provide a general algorithm for calculating the limit distribution, but this algorithm can be complicated to apply. This note provides a simpler and more intuitive algorithm. Loosely speaking, in a process with K strategies, it is sufficient to find the invariant distribution of a K×K Markov matrix on the K homogeneous states, where the probability of a transit from “all play i” to “all play j” is the probability of a transition from the state “all agents but 1 play i, 1 plays j” to the state “all play j”.     

Uncertainty and the Voluntary Provision of a Pure Public Good in a Two‐Moment Decision Model

In this paper, we explore the potential benefits of uncertainty that may arise in a two‐moment model of the voluntary provision of a pure public good. We find that an increase in a given contributor i’s risk associated with the aggregate contribution level of the other contributors (i.e., an increase in social uncertainty) induces that contributor to increase his own contribution level if and only if the uncertainty's incremental effect on the expected value of his net marginal utility is negative. Contributor i’s welfare likewise increases when a closely related condition is met, namely that the uncertainty's marginal effect on his expected marginal utility value of the public good exceeds its countervailing effect on the numeraire. Further, the corresponding aggregate contribution to the public good increases in the presence of free‐riding if and only if the incremental effect of contributor i’s contribution on the aggregate expected value of all other contributors’ net marginal utilities is small‐enough positive. We derive similar conditions for the case of private uncertainty, where the increase in contributor i’s risk is associated with his own marginal valuation of the public good. A simple example illustrates these conceptual results. Numerical analysis demonstrates that an increase in private uncertainty can have a nonmonotonic impact on contributor i’s welfare.     

Reinterpreting mixed strategy equilibria: a unification of the classical and Bayesian views

We provide a new interpretation of mixed strategy equilibria that incorporates both von Neumann and Morgenstern's classical concealment role of mixing, as well as the more recent Bayesian view originating with Harsanyi. For any two-person game, G, we consider an incomplete information game, in which each player's type is the probability he assigns to the event that his mixed strategy in G is “found out” by his opponent. We show that, generically, any regular equilibrium of G can be approximated by an equilibrium of in which almost every type of each player is strictly optimizing. This leads us to interpret i's equilibrium mixed strategy in G as a combination of deliberate randomization by i together with uncertainty on j's part about which randomization i will employ. We also show that such randomization is not unusual: for example, i's randomization is nondegenerate whenever the support of an equilibrium contains cyclic best replies.     

Unequal inequalities. II

This paper analyzes properties of measures of inequality, applied to income inequalities but meaningful for practically any measure of dispersion in economics. We call n the number of persons, i the person's index, x_i person i's income, $\text{x}\text{= Σ(}\text{x}{}_{\mathrm{i}}\text{n}\text{)}$ the average income, x the vector of the x_i's or income distribution, I(x) a real-valued function of x which is the measure (or index) of inequality.Part I (Sects. I–V), which appeared in the last issue of this journal, analyzed several structures or properties, and specific forms, of I. We distinguished several I's: the measures of inequality per person (or “absolute”) I^a, per pound (or “relative”) $\text{I}{}^{\mathrm{r}}\text{=}\text{I}{}^{\mathrm{a}}\text{x}$, and total nI^a. We presented several possible properties of inequality measures, such as: I = 0 if all x_i's are equal (“zero at equality”), I > 0 otherwise (“positivity out of equality”), symmetry of I for x (“impartiality”), $\text{((}\text{?I}\text{?x}{}_{\mathrm{i}}\text{) ? (}\text{?I}\text{?x}{}_{\mathrm{i}}\text{))(x}{}_{\mathrm{i}}\text{? x}{}_{\mathrm{j}}\text{) > 0}$ for x_i ≠ x_j (“rectifiance” of the function I, or “transfers principle,” this being the strict form whereas the weak one is with sign ?), the fact that $\text{(}\text{?(}\text{x}\text{? I}{}^{\mathrm{a}}\text{)}\text{?}{}_{\mathrm{i}}\text{)}\text{(}\text{?(}\text{x}\text{? I}{}^{\mathrm{a}}\text{)}\text{?}{}_{\mathrm{j}}\text{)}$ does not depend upon x_k for k ≠ i,j (“welfare independence,” or, for short, “independence”). Rectifiance plus symmetry is Schur-convexity. Independence plus symmetry plus zero at equality implies that $\text{x}\text{≡}\text{x}\text{? I}{}^{\mathrm{a}}\text{= ?}{}^{\mathrm{?1}}\text{[(}\text{1}\text{n}\text{) Σ ?(x}{}_{\mathrm{i}}\text{)]}$ where $\text{x}$ is the “equal equivalent income”; and we will show that, these three properties being satisfied, the following ones are equivalent to each other: positivity out of equality, rectifiance, quasi-convexity, ?'s concavity.Part I largely focused on the study of six related specific measures of inequality, which in particular possess all the above properties: ?, α, and Ξ being positive parameters, they are $\text{I}{}_{\mathrm{c}}{}^{\mathrm{a}}\text{=}\text{x}\text{+ξ ? [(}\text{1}\text{n}\text{∑ (x}{}_{\mathrm{i}}\text{+ ξ)}{}^{\mathrm{1?epsi;}}\text{]}{}^{\mathrm{<mtext>1</mtext><mtext>1</mtext><mtext>??</mtext>}}$, $\text{I}{}_{\mathrm{c}}{}^{\mathrm{a}}\text{=}\text{x}\text{+ξ ? ∏ (x}{}_{\mathrm{i}}\text{+ ξ)}{}^{\mathrm{<mtext>1</mtext><mtext>n</mtext>}}$, $\text{I}{}_{\mathrm{c}}{}^{\mathrm{r}}\text{=}\text{I}{}_{\mathrm{c}}{}^{\mathrm{a}}\text{x}$, I_r=I_c^r for ξ=O, $\text{I}{}_{\mathrm{r}}{}^{\mathrm{a}}\text{=}\text{x}\text{I}{}_{\mathrm{r}}\text{=I}{}_{\mathrm{c}}{}^{\mathrm{a}}$ for ξ=, $\text{I}{}_{\mathrm{l}}\text{=(}\text{1}\text{α}\text{)}\text{log}\text{[(}\text{1}\text{n}\text{) ∑ eα·(x?x}{}_{\mathrm{i}}\text{)]}$ and $\text{I}{}_{\mathrm{l}}{}^{\mathrm{r}}\text{=}\text{I}{}_{\mathrm{l}}\text{x}$. Lower indices c, r, l respectively stand for “centrist,” “rightist,” and “leftist” measures of inequality. I_r and I_l are invariant under respectively equiproportional variation in, or equal addition to, all incomes; measures which have the first of these two properties are said to be “intensive.”We now consider different and more general measures, and other properties. We first reconcile the last two properties by dropping the “indepencence” one (Section VI.). Then, we analyze another mildly equalitarian property, the “principle of diminishing transfers” (Section VII). Section VIII turns to the relations between inequality measures and Lorenz and concentration curves. We then consider the effect on inequality of additions of incomes, and we analyze the properties of “diminishing equality” (Section IX). The effect of unions of populations is the topic of Section X. Finally, the last section (XI) presents the more general relations between the various structural properties of inequality measures.¹     

Characterization of the private alternatives domains admitting arrow social welfare functions

An n-person social choice problem is considered in which the alternatives are n dimensional vectors with the ith component of such a vector being the part of the alternative effecting individual i alone. Assuming that individuals are selfish (i is indifferent between any two alternatives that have the same ith component) we characterize all the families of permissible individual preferences that admit nondictatorial Arrow-type social welfare functions. We also show that the existence of such a function for a given family of preferences is independent of n provided that is greater than one.     

On the hellwig method of feasible portfolio construction

The work feasible portfolio is built into the work, that is, the k-dimensional Q column vector with components q_i where q_i 0 for i=1,...,k and q₁+...+q_k=1. We define i=1,...,k in the following way:

The effect on optimal consumption of increased uncertainty in labor income in the multiperiod case

We consider a multiperiod, additive utility, optimal consumption model with a riskless investment and a stochastic labor income. The main result is that for utility functions belonging to the set F, consumption decreases when we go from any sequence of distribution functions representing labor income to a more risky sequence. A concave utility function belongs to F if its first derivative exists everywhere and is convex.     

Cheap talk in games with incomplete information

The paper studies Bayesian games which are extended by adding pre-play communication. Let Γ be a Bayesian game with full support and with three or more players. The main result is that if players can send private messages to each other and make public announcements then every communication equilibrium outcome, q, that is rational (i.e., involves probabilities that are rational numbers) can be implemented in a sequential equilibrium of a cheap talk extension of Γ, provided that the following condition is satisfied: There exists a Bayesian Nash equilibrium s in Γ such that for each type t_i of each player i the expected payoff of t_i in q is larger than the expected payoff of t_i in s.     

Von Neumann-Morgenstern stable sets, discounting, and Nash bargaining

We establish a link between von Neumann-Morgenstern stable set and the Nash solution in a general n-player utility set. The stable set-solution is defined with respect to a dominance relation: payoff vector u dominates v if one player prefers u even with one period delay. We show that a stable set exists and, if the utility set has a smooth surface, any stable set converges to the Nash bargaining solution when the length of the period goes to zero.     

Essential aggregation procedures on restricted domains of preferences

The restricted domains of individuals' preferences that permit the construction of Arrow social welfare functions and nonmanipulable voting procedures in which each of n voters has some power are characterized. In this context a domain is the Cartesian product of n sets of strict preference orderings. Variants of this result are obtained under the additional requirement of neutrality and in the case when alternatives are vectors whose ith components affect only the ith voter. Kalai and Muller's analogous result (J. Econ. Theory16 (1977), 457–469) concerning nondictatorial procedures is discussed and proved as a corollary to the main theorem.