The ‘Troubling Tradeoffs’ Paradox and a Resolution

Ravallion ( 2012a ) argues that the Human Development Index (HDI) embeds questionable tradeoffs between the dimensions used to compute the index. To alleviate these problems he proposes the adoption of one of the indices developed by Chakravarty ( 2003 ). In this paper I identify the following paradox: while the Chakravarty indices clearly exhibit more sensible tradeoffs than the HDI, the HDI produces more sensible rankings than the Chakravarty indices. To solve the paradox I identify the axioms behind each methodology responsible for the unintuitive tradeoffs and rankings and illustrate how to develop an index with these questionable axioms removed. This approach can result in methodologies that exhibit more intuitive tradeoffs by design, as it seeks inputs from the public as to what those tradeoffs ought to be, and produces rankings that are more in line with what the HDI wishes to measure: human development and capabilities, as conceptualized by Sen ( 1985 ).     

Epsilon Cores of Games with Limited Side Payments: Nonemptiness and Equal Treatment

We introduce the concept of a parameterized collection of games with limited side payments, ruling out large transfers of utility, and demonstrate conditions ensuring that a game with limited side payments has a nonempty -core. Our main result is that, when some degree of side-paymentness within nearly-effective small groups is assumed and large transfers are prohibited, then all payoffs in the -core treat similar players similarly. A bound on the distance between -core payoffs of any two similar players is given in terms of the parameters describing the game. These results add to the literature showing that games with many players and small effective groups have the properties of competitive markets. Journal of Economic Literature Classification Numbers: C71, C78, D71.     

Cooperative Outcomes through Noncooperative Games

This paper shows the equivalence between the stable solution set of any cooperative game in characteristic form (G1) and the subgame perfect Nash equilibria in pure strategies of a certain noncooperative game (G2). Players of G1 are named "agents." G2 is played by different players ("principals") who compete in wages to attract agents. The equivalence result holds when there are enough principals (if the game is superadditive, two principals suffice). Finally, another related cooperative game (G3) is constructed with both principals and agents as players. For G2 and G3 the same result is then proven, for any number of principals. Journal of Economic Literature Classification Numbers: C71 and C72.     

Lexicographic composition of simple games

Certain voting bodies can be modeled as a simple game where a coalition's winning depends on whether it wins, blocks or loses in two smaller simple games. There are essentially five such ways to combine two proper games into a proper game. The most decisive is the lexicographic rule, where a coalition must either win in G₁, or block in G₁ and win in G₂. When two isomorphic games are combined lexicographically, a given role for a player confers equal or more power when held in the first game than the second, if power is assessed by any semi-value. A game is lexicographically separable when the players of the two components partition the whole set. Games with veto players are not separable, and games of two or more players with identical roles are separable only if decisive. Some separable games are egalitarian in that they give players identical roles.     

Convex Games and Stable Sets

It is well known that the core of a convex coalitional game with a finite set of players is the unique von Neumann–Morgenstern stable set of the game. We extend the definition of a stable set to coalitional games with an infinite set of players and give an example of a convex simple game with a countable set of players which does not have a stable set. But if a convex game with a countable set of players is continuous at the grand coalition, we prove that its core is the unique von Neumann–Morgenstern stable set. We also show that a game with a countable (possibly finite) set of players which is inner continuous is convex iff the core of each of its subgames is a stable set.Journal of Economic LiteratureClassification Numbers: C70, C71.     

On the Least Core and the Mas-Colell Bargaining Set

We show that the least core of a TU coalitional game with a finite set of players is contained in the Mas-Colell bargaining set. This result is extended to games with a measurable space of players in which the worth of the grand coalition is at least that of any other coalition in the game. As a consequence, we obtain an existence theorem for the Mas-Colell bargaining set in TU games with a measurable space of players. Journal of Economic Literature Classification Number: C71.     

Coalition Formation and Potential Games

In this paper we study the formation of coalition structures in situations described by a cooperative game. Players choose independently which coalition they want to join. The payoffs to the players are determined by an allocation rule on the underlying game and the coalition structure that results from the strategies of the players according to some formation rule. We study two well-known coalition structure formation rules and show that for both formation rules there exists a unique component-efficient allocation rule that results in a potential game. Journal of Economic Literature Classification Numbers: C71, C72.     

Unit-Consistent Poverty Indices

This paper characterizes unit-consistent poverty indices. The unit consistency axiom requires that poverty rankings (not poverty indices) remain unaffected when all incomes and the poverty lines are expressed in different measuring units. We consider two general frameworks of poverty measurement: the semi-individualistic framework that includes all decomposable indices and all rank-based indices; and the Dalton–Hagenaars framework that contains a subset of decomposable indices. Within the semi-individualistic framework, classes of unit-consistent poverty indices can be characterized for different value judgements about poverty measurement. Within the Dalton-Hagenaars framework, unit-consistent poverty indices are completely characterized without invoking any value judgement a priori. I thank Peter Lambert, Mike Hoy, Thesia Garner and an anonymous referee for their very helpful comments and suggestions.     

Forming and Dissolving Partnerships in Cooperative Game Situations

A group of players in a cooperative game are partners (e.g., as in the form of a union or a joint ownership) if the prospects for cooperation are restricted such that cooperation with players outside the partnership requires the accept of all the partners. The formation of such partnerships through binding agreements may change the game implying that players could have incentives to manipulate a game by forming or dissolving partnerships. The present paper seeks to explore the existence of allocation rules that are immune to this type of manipulation. An allocation rule that distributes the worth of the grand coalition among players is called partnership formation‐proof if it ensures that it is never jointly profitable for any group of players to form a partnership and partnership dissolution‐proof if no group can ever profit from dissolving a partnership. The paper provides results on the existence of such allocation rules for general classes of games as well as more specific results concerning well‐known allocation rules.     

The power of ESS: An experimental study

Abstract. Our experimental design mimics a traditional evolutionary game framework where players are matched pairwise to play a symmetric 33 bimatrix game that has two Nash equilibria. One equilibrium is an evolutionary stable state, or ESS; the other is an equilibrium in dominated strategies. Our primary experimental result is the observation that the ESS becomes extremely attractive when subjects have minimal information about the payoff functions, although the dominated equilibrium assures the highest equilibrium payoff. The attractiveness of the ESS is only moderate when players are completely informed about the 33 payoff matrix. Correspondence to: S.K. Berninghaus     

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.     

A “Super” Folk Theorem for dynastic repeated games

We analyze dynastic repeated games. These are repeated games in which the stage game is played by successive generations of finitely-lived players with dynastic preferences. Each individual has preferences that replicate those of the infinitely-lived players of a standard discounted infinitely-repeated game. Individuals live one period and do not observe the history of play that takes place before their birth, but instead create social memory through private messages received from their immediate predecessors. Under mild conditions, when players are sufficiently patient, all feasible payoff vectors (including those below the minmax of the stage game) can be sustained by sequential equilibria of the dynastic repeated game with private communication. In particular, the result applies to any stage game with n  ≥  4 players for which the standard Folk Theorem yields a payoff set with a non-empty interior. We are also able to characterize fully the conditions under which a sequential equilibrium of the dynastic repeated game can yield a payoff vector not sustainable as a subgame perfect equilibrium of the standard repeated game. For this to be the case it must be that the players’ equilibrium beliefs violate a condition that we term “inter-generational agreement.” A previous version of this paper was circulated as Anderlini et al. (2005). We are grateful to Jeff Ely, Leonardo Felli, Navin Kartik, David Levine, Stephen Morris, Michele Piccione, Andrew Postlewaite, Lones Smith and to seminar audiences at Bocconi, Cambridge, CEPR-Guerzensee, Chicago, Columbia, Edinburgh, Essex, Georgetown, Leicester, LSE, Northwestern, Oxford, Rome (La Sapienza), Rutgers, SAET-Vigo, Stanford, SUNY-Albany, UCL, UC-San Diego, Venice and Yale for helpful feedback.     

In Defense of DEFECT

The one-state machine that always defects is the only evolutionarily stable strategy in the machine game that is derived from the prisoners' dilemma, when preferences are lexicographic in complexity. This machine is the only stochastically stable strategy of the machine game when players are restricted to choosing machines with a uniformly bounded complexity. Journal of Economic Literature Classification Numbers: C70, C72.     

Sowing Doubt Optimally in Two-Person Repeated Games

Consider a two-person repeated game, where one of the players, P1, can sow doubt, in the mind of his opponent, as to what P1's payoffs are. This results in a two-person repeated game with incomplete information. By sowing doubt, P1 can sometimes increase his minimal equilibrium payoff in the original game. We prove that this minimum is maximal when only one payoff matrix, the negative of the payoff matrix of the opponent, is added (the opponent thus believes that he might play a zero-sum game). We obtain two formulas for calculating this maximal minimum payoff. Journal of Economic Literature Classification Numbers: C7, D8.     

Large newsvendor games

We consider a game, called newsvendor game, where several retailers, who face a random demand, can pool their resources and build a centralized inventory that stocks a single item on their behalf. Profits have to be allocated in a way that is advantageous to all the retailers. A game in characteristic form is obtained by assigning to each coalition its optimal expected profit. We consider newsvendor games with possibly an infinite number of newsvendors. We prove in great generality results about balancedness of the game, and we show that in a game with a continuum of players, under a nonatomic condition on the demand, the core is a singleton. For a particular class of demands we show how the core shrinks to a singleton when the number of players increases.     

Commitment Robust Equilibria and Endogenous Timing

This paper investigates which equilibria of a game are still viable when players have the opportunity to commit themselves. To that end we study a model of endogenous timing in which players face the trade-off between committing early and moving late. It is shown that mixed (resp. pure) equilibria of the original game are subgame perfect (resp. persistent) in the timing game only when no player has an incentive to move first. Consequently, mixed equilibria are viable only if no player has an incentive to move first. One needs strong evolutionary solution concepts to draw that conclusion for pure equilibria.Journal of Economic LiteratureClassification Number: C72.     

Infinite horizon common interest games with perfect information

We consider infinite horizon common interest games with perfect information. A game is a K-coordination game if each player can decrease other players' payoffs by at most K times his own cost of punishment. The number K represents the degree of commonality of payoffs among the players. The smaller K is, the more interest the players share. A K-coordination game tapers off if the greatest payoff variation conditional on the first t periods of an efficient history converges to 0 at a rate faster than K^−t as t→∞. We show that every subgame perfect equilibrium outcome is efficient in any tapering-off game with perfect information. Applications include asynchronously repeated games, repeated games of extensive form games, asymptotically finite horizon games, and asymptotically pure coordination games.     

The Envelope Theorem and Comparative Statics of Nash Equilibria

The Envelope Theorem for Nash equilibria shows that the strategic reaction of the other players in the game is important for determining how parameter perturbations affect a given player's indirect objective function. The fundamental comparative statics matrix of Nash equilibria for theithplayer in anN-player static game includes the equilibrium response of the otherN−1players in the game to the parameter perturbation and is symmetric positive semidefinite subject to constraint. This result is fundamental in that it holds for all sufficiently smooth Nash equilibria and is independent of any curvature or stability assumptions imposed on the game.Journal of Economic LiteratureClassification Numbers: C72, C61.     

PEER TRANSPARENCY IN TEAMS: DOES IT HELP OR HINDER INCENTIVES?*

In a joint project involving two players of a two‐round effort investment game with complementary efforts, transparency, by allowing players to observe each other’s efforts, achieves at least as much, and sometimes more, collective and individual efforts relative to a nontransparent environment. Without transparency multiple equilibria can arise, and transparency eliminates the inferior equilibria. When full cooperation arises only under transparency, it occurs gradually: No worker sinks in the maximum amount of effort in the first round, preferring instead to smooth out contributions over time. If the players’ efforts are substitutes, transparency makes no difference to equilibrium efforts.     

Coordination and Information in Critical Mass Games: An Experimental Study

We present experimental results on a repeated coordination game with Pareto-ranked equilibria in which a payoff from choosing an action is positive only if a critical mass of players choose that action. We design a baseline version of the game in which payoffs remain constant for values above the critical mass, and an increasing returns version in which payoffs keep increasing for values above the critical mass. We test the predictive power of security and payoff-dominance under different information treatments. Our results show that convergence to the payoff-dominant equilibrium is the modal limit outcome when players have full information about others' previous round choices, while this outcome never occurs in the remaining treatments. The paths of play in some groups reveal a tacit dynamic coordination by which groups converge to the efficient equilibrium in a step-like manner. Moreover, the frequency and speed of convergence to the payoff-dominant equilibrium are higher, ceteris paribus, when increasing returns are present. Finally, successful coordination seems to crucially depend on players' willingness to signal to others the choice of the action supporting the efficient equilibrium.