Games of fixed rank: a hierarchy of bimatrix games

We propose and investigate a hierarchy of bimatrix games (A, B), whose (entry-wise) sum of the pay-off matrices of the two players is of rank k, where k is a constant. We will say the rank of such a game is k. For every fixed k, the class of rank k-games strictly generalizes the class of zero-sum games, but is a very special case of general bimatrix games. We study both the expressive power and the algorithmic behavior of these games. Specifically, we show that even for k = 1 the set of Nash equilibria of these games can consist of an arbitrarily large number of connected components. While the question of exact polynomial time algorithms to find a Nash equilibrium remains open for games of fixed rank, we present polynomial time algorithms for finding an ε-approximation.     

Manipulation and the Pareto rule

This paper examines rules that map preference profiles into choice sets. There are no agendas other than the entire set of alternatives. A rule is said to be “manipulable” if there is a person i, and a preference profile, such that i prefers the choice set obtained when he is dishonest to the one obtained when he is honest. It is “nonmanipulable” if this can never happen. The paper indicates how preferences over choice sets might be sensibly derived from preferences over alternatives, and discusses seven different notions of manipulability associated with seven different assumptions about preferences over sets of alternatives. The paper has two sections of results. In the first I show that the Pareto rule, that is, the rule that maps preference profiles into corresponding sets of Pareto optima, is nonmanipulable in four of the seven senses of manipulability, and manipulable in three of them. In the second section, I examine this conjecture: If an arbitrary rule is nonmanipulable and nonimposed, and if indifference is disallowed, then every choice set must be contained in the set of Pareto optima. The conjecture is true under the strongest definition of nonmanipulability.     

Tops-only domains

In this paper we consider the standard voting model with a finite set of alternatives A and n voters and address the following question: what are the characteristics of domains \({\mathcal D}\) that induce the property that every strategy-proof social choice function \({f: {\mathcal D}^n \rightarrow A}\) satisfying unanimity, has the tops-only property? We first impose a minimal richness condition which ensures that for every alternative a, there exists an admissible ordering where a is maximal. We identify conditions on \({\mathcal D}\) that are sufficient for strategy-proofness and unanimity to imply tops onlyness in the general case of n voters and in the special case, n = 2. We provide an algorithm for constructing tops-only domains from connected graphs with elements of A as nodes. We provide several applications of our results. Finally, we relax the minimal richness assumption and partially extend our results.     

Characterization of certain choice functions

We characterize rationality of certain choice functions C by C(A) ∩ C(B) ? C(A ∪ B) ? C(A) ∪ C(B). We also characterize choice functions coming from preference relations which are partial orders and some other types of relations. Furthermore, we also characterize the admissible set of a binary relation by the fact it is invariant under certain changes in the relation.     

Budget preferences and direct utility

This paper extends a result of Sakai, who presented conditions for indirect preferences from which a utility function can be deduced if demand is single valued. It will be shown that—adding a “partial Lipschitz condition”—Sakai's theorem can be extended to multi-valued demand. Our result follows from the extension of a theorem of Hurwicz and Richter, who have shown that, under certain hypotheses on demand correspondences, an upper semicontinuous utility function f exists, so that the set of utility maximal elements μ_f(B) is contained in the choice set h(B) for every budget B. By our partial Lipschitz condition h(B) ?- μ_f(B) also follows.     

A characterization of the distributions that imply mean—Variance utility functions

If there is a riskless asset, then the distribution of every portfolio is determined by its mean and variance if and only if the random returns are a linear transformation of a spherically distributed random vector. If there is no riskless asset, then the spherically distributed random vector is replaced by a random vector in which the last n ? 1 components are spherically distributed conditional on the first component, which has an arbitrary distribution. If the number of assets is infinite, then there must exist random variables m, v, y, where the distribution of y conditional on m and v is standard normal, such that every portfolio is distributed as some linear combination of m and vy. If there is a riskless asset, then m has zero variance. These distributions exhibit two-fund separability even if the utility function is not concave.     

Majority committees

Suppose an m-member committee is to be selected by a set of voters from a set X of M > m nominees. If A is an m-member committee, and if as many voters prefer A to B as prefer B to A for every other committee B of m nominees, then A is a majority committee of size m. Although the existence of majority nominees (m = 1) has been extensively analyzed, little attention has been given to conditions that imply the existence of majority committees of size m ? 2. Known restrictions on voters' preferences on X that guarntee the existence of a majority nominee could be applied directly to voters' preferences on m-member committees. However, this definitional exercise lacks intuitive appeal, and a different approach is taken in this paper. The paper presumes that profiles of voters' preferences on X are either dichotomous or single peaked. Both cases ensure the existence of a majority nominee. Independence-monotonicity assumptions are then used to connect voters' preferences on X to their preferences on committees of equal size. Although these assumptions guarantee the existence of majority committees when m = 1 and when m = M ? 1, they do not generally do so when 1 < m < M ? 1. The latter observation motivates additional restrictions on profiles. In the dichotomous case, we consider profiles in which all voters have the same number k of nominees in their preferred subsets, and show that this restriction guarantees the existence of a majority committee of size m for 1 < m < M ? 1 only when k = 1 or k = M ? 1. In the single-peaked case, we consider profiles in which all voters have the same most-preferred (peak) nominee, and prove that this guarantees the existence of a majority committee of size m for every m between 1 and M ? 1.     

Universality of Nash components

We show that Nash equilibrium components are universal for the collection of connected polyhedral sets. More precisely for every polyhedral set we construct a so-called binary game—a game where all players have two pure strategies and a common utility function with values either zero or one—whose success set (the set of strategy profiles where the maximal payoff of one is indeed achieved) is homeomorphic to the given polyhedral set. Since compact semi-algebraic sets can be triangulated, a similar result follows for the collection of connected compact semi-algebraic sets.We discuss implications of our results for the strategic stability of success sets, and use the results to construct a Nash component with index k for any fixed integer k.     

Increases in risk aversion and the distribution of portfolio payoffs

Oliver Hart proved the impossibility of deriving general comparative static properties in portfolio weights. Instead, we derive new comparative statics for the distribution of payoffs: A is less risk averse than B iff A?s payoff is always distributed as B?s payoff plus a non-negative random variable plus conditional-mean-zero noise. If either agent has nonincreasing absolute risk aversion, the non-negative part can be chosen to be constant. The main result also holds in some incomplete markets with two assets or two-fund separation, and in multiple periods for a mixture of payoff distributions over time (but not at every point in time).     

Designer path independent choice functions

Summary. This paper provides an algorithm for the construction of all PICFs on a finite set of alternatives, V, designed by an a priori given set I of initial choices as well as the determination of whether the initial set I is consistent with path independence. The algorithm is based on a new characterization result for path independent choice functions (PICF) on finite domains and uses that characterization as the basis of the algorithm. The characterization result identifies two properties of a partition of the Boolean algebra as necessary and sufficient for a choice function C to be a PICF: (i): For every subset A of V the set is an interval in the Boolean algebra 2 ^V . (ii): If A/B is an interval in the Boolean algebra such that C(A) = C(B) and if M/N is an upper transpose of A/B then C(M) = C(N). The algorithm proceeds by expanding on the implications of these two properties.Received: 5 November 2003, Revised: 20 July 2004, JEL Classification Numbers: D00, D70.     

Strategic approximations of discontinuous games

An infinite game is approximated by restricting the players to finite subsets of their pure strategy spaces. A strategic approximationof an infinite game is a countable subset of pure strategies with the property that limits of all equilibria of all sequences of approximating games whose finite strategy sets eventually include each member of the countable set must be equilibria of the infinite game. We provide conditions under which infinite games admit strategic approximations.     

A note on the fundamental theorem of exact aggregation

If individual i's demands for a commodity are a function of prices, p, income M_i and a vector of attributes A_i, then aggregate demand is

This paper derives the necessary and sufficient conditions of f_i, F and a system of functions g_k(M₁,…,M_N,A₁,…,A_N) symmetric in the M's and A's such that F can be written in the form F(p, g, (M₁,…,M_n,A₁,…,A_N),…, g_n(M₁,…,M_n,A₁,… ,A_n)) for all values of its arguments.     

Priorities in the location of multiple public facilities

A collective decision problem is described by a set of agents, a profile of single-peaked preferences over the real line and a number of public facilities to be located. We consider public facilities that do not suffer from congestion and are non-excludable. We characterize the class of rules satisfying Pareto-efficiency, object-population monotonicity and sovereignty. Each rule in the class is a priority rule that selects locations according to a predetermined priority ordering among “interest groups”. We characterize the subclasses of priority rules that respectively satisfy anonymity, avoid the no-show paradox, strategy-proofness and population-monotonicity. In particular, we prove that a priority rule is strategy-proof if and only if it partitions the set of agents into a fixed hierarchy. Any such rule can also be viewed as a collection of generalized peak-selection median rules, that are linked across populations, in a way that we describe.     

Choice and individual welfare

We propose an abstract method of systematically assigning a “rational” ranking to non-rationalizable choice data. Our main idea is that any method of ascribing welfare to an individual as a function of choice is subjective, and depends on the economist undertaking the analysis. We provide a simple example of the type of exercise we propose. Namely, we define an individual welfare functional as a mapping from stochastic choice functions into weak orders. A stochastic choice function (or choice distribution) gives the empirical frequency of choices for any possible opportunity set (framing factors may also be incorporated into the model). We require that for any two alternatives x and y, if our individual welfare functional recommends x over y given two distinct choice distributions, then it also recommends x over y for any mixture of the two choice distributions. Together with some mild technical requirements, such an individual welfare functional must weight every opportunity set and assign a utility to each alternative x which is the sum across all opportunity sets of the weighted probability of x being chosen from the set. It therefore requires us to have a “prior view” about how important or representative a choice of x at a given situation is.     

On Forward Induction and Evolutionary and Strategic Stability

We analyze which normal form solution concepts capture the notion of forward induction, as defined by E. van Damme (1989, J. Econ. Theory48, 476-496) in the class of generic two player normal form games preceded by an outside option. We find that none of the known strategic stability concepts (including Mertens stable sets and hyperstable sets) capture this form of forward induction. On the other hand, we show that the evolutionary concept of EES set (J. Swinkels, 1992, J. Econ. Theory57, 306-332) is always consistent with forward induction. Journal of Economic Literature Classification Number: C72.     

On incentive structures of a socialist economy

This paper considers an incentive structure B_T as an alternative to the incentive structure B_D recently suggested by Domar. Both of these incentive structures induce managers in a planned economy to provide the socially desirable output level while allowing them to set prices. In two aspects B_T is better than B_D. First, B_D works only if demand is elastic at the optimal output level while B_T works whether demand is elastic or inelastic at that level. Second, even when demand is elastic at the optimal level, there are circumstances for which output converges to the optimal level faster under B_T than under B_D.     

Reciprocity and Giving in a Consecutive Three‐Person Dictator Game with Social Interaction

We study pure indirect reciprocity by setting up a modified dictator game with three players A, B, and C acting sequentially. Subject A takes a share of a pie and passes the rest to subject B, while B divides the rest between herself and C. We find that this consecutive three‐person dictator game increases generosity compared with the traditional two‐person dictator game. We analyze the influence of social interaction and uncertainty. In treatments with certainty we observe pure indirect reciprocity: B indirectly reciprocates for A's behavior in the decision on how generous to be to C.     

Enforcement and environmental quality in a decentralized emission trading system

Should the powers of monitoring compliance and allocating tradeable emissions allowances be appointed to a unique supranational regulator or decentralized to several local regulators? To answer this question, we develop a two stage-two country game where environmental regulators set the amount of emission allowances and the level of monitoring effort to achieve full compliance while the regulated firms choose actual emissions and the number of permits to be held. Various, possibly conflicting, spillovers between countries arise in a decentralized setting. We show that decentralization is socially harmful if no asymmetry among institutional settings is introduced and can be suboptimal even when decentralization features lower monitoring costs than a centralized setting. Lower monitoring costs are therefore necessary, but not sufficient, to justify decentralization. Also, our analysis reveals that welfare can be higher under decentralization even if the corresponding environmental quality is worse than under centralization. Indeed, better environmental quality is sufficient but not necessary for higher welfare under decentralization. Finally, we discuss how these results can provide a theoretical rationale for the recent evolution of the EU ETS design.     

Technology adoption in markets with network effects: Theory and experimental evidence

We examine a technology-adoption game with network effects in which coordination on either technology A or technology B constitutes a Nash equilibrium. Coordination on technology B is assumed to be payoff dominant. We define a technology’s critical mass as the minimal share of users, which is necessary to make the choice of this technology the best response for any remaining user. We show that the technology with the lower critical mass implies risk dominance and selection by the maximin criterion. We present experimental evidence that both payoff dominance and risk dominance explain participants’ choices in the technology-adoption game. The relative riskiness of a technology can be proxied using either technologies’ critical masses or stand-alone values absent any network effects.     

Minimal stable sets in tournaments

We propose a systematic methodology for defining tournament solutions as extensions of maximality. The central concepts of this methodology are maximal qualified subsets and minimal stable sets. We thus obtain an infinite hierarchy of tournament solutions, encompassing the top cycle, the uncovered set, the Banks set, the minimal covering set, and the tournament equilibrium set. Moreover, the hierarchy includes a new tournament solution, the minimal extending set, which is conjectured to refine both the minimal covering set and the Banks set.