Models for conflict resolution in ecosystem management

Tools developed in the fields of decision analysis and game theory that have potential for use in public sector conflict resolution are reviewed. The strengths and weaknesses, as well as the axioms defining principles of fairness, are examined for multiobjective optimization, Nash and Nash–Harsanyi solutions, voting models, and the Shapley value. The case of conflict between stakeholders over proposed oil and gas leasing on National Forest System lands is presented as a sample application. We conclude that the Shapley value is the appropriate approach for determining the “fairness” of alternative conflict solutions, at least in those situations where cardinal utilities can be estimated.     

Targeted campaign competition,loyal voters,and supermajorities

We consider campaign competition in which candidates compete for votes among a continuum of voters by engaging in persuasive efforts that are targetable. Each individual voter is persuaded by campaign effort and votes for the candidate who targets more persuasive effort to this voter. Each candidate chooses a level of total campaign effort and allocates his effort among the set of voters. We completely characterize equilibrium for the majoritarian objective game and compare that to the vote-share maximizing game. If the candidates are symmetric ex ante, both types of electoral competition dissipate the rents from office in expectation. However, the equilibria arising under the two electoral objectives qualitatively differ. In majoritarian elections, candidates randomize over their level of total campaign effort, which provides support for the puzzling phenomenon of the emergence of supermajorities in majoritarian systems. Vote-share maximization leads to an equilibrium in which both candidates make deterministic budget choices and reach a precise fifty–fifty split of vote shares. We also study how asymmetry between the candidates affects the equilibrium. If some share of the voters is loyal to one of the candidates, then both candidates expend the same expected efforts in equilibrium, but the advantaged candidate wins with higher probability for majoritarian voting or a higher share of voters for vote-share maximization.     

Determinacy of equilibrium in outcome game forms

We show the generic finiteness of the number of probability distributions on outcomes induced by Nash equilibria for two-person game forms such that either (i) one of the players has no more than two strategies or (ii) both of the players have three strategies, and (iii) for outcome game forms with three players, each with at most two strategies. Finally, we exhibit an example of a game form with three outcomes and three players for which the Nash equilibria of the associated game induce a continuum of payoffs for an open non-empty set of utility profiles.     

Convergence in games with continua of equilibria

In game theory, the question of convergence of dynamical systems to the set of Nash equilibria has often been tackled. When the game admits a continuum of Nash equilibria, however, a natural and challenging question is whether convergence to the set of Nash equilibria implies convergence to a Nash equilibrium. In this paper we introduce a technique developed in Bhat and Bernstein (2003) as a useful way to answer this question. We illustrate it with the best-response dynamics in the local public good game played on a network, where continua of Nash equilibria often appear.     

Bargaining solutions with non-standard objectives

I examine the pure-strategy solutions of the sealed-bid bargaining game with incomplete information, when the buyer's and seller's objectives are other than the standard objective, namely maximization of expected profit. The motivation for this exploration lies in three problems of the standard formulation: the necessity of assuming common priors, the existence of uncountably many Nash equilibria, with no means for the players to coordinate on any one of them, and the uncertain relationship between these equilibria and observed behavior in bargaining experiments. Specifically, I consider two alternative objectives: minimization of maximum regret, and maximization of maximum profit. The solution concept here is not Nash equilibrium, but rather -individually rational strategy bundle. For that reason, I shall, where appropriate, use the word “solution” in place of “equilibrium.” Yet we find that the notion of Nash Equilibrium reappears, in a sense to be explained. In the minimax-regret case I find (in contrast to the case of expected profit) a unique solution; this solution reduces, for priors with coincident support, to the linear equilibrium of Chatterjee-Samuelson. In the maximum-profit case there are many solutions; they turn out to be slight generalizations of the one-step equilibria of Leininger-Linhart-Radner.     

Voting power in municipal annexation elections

In many American states, municipal annexation and consolidation require concurrent majority votes of all affected jurisdictions. The effectiveness and fairness of this voting procedure have been criticized on the grounds that a small minority of voters can frustrate the preferences of the overall majority. This paper investigates the extent to which the relative ability of voters in large and small jurisdictions to influence voting outcomes in procedures requiring concurrent majorities is influenced by jurisdictional size. The Banzhaf index, which counts the number of case4s in which a given voter could reverse the overall group decision by changing positions on an issue, is applied to this problem of concurrent voting majorities. Mathematical analysis indicates that the ratio of power between voters in small and large jurisdictions approximately equals the inverse of the square root of the ratio of their population size.     

On pure-strategy Nash equilibria in price–quantity games

This paper examines the existence and characteristics of pure-strategy Nash equilibria in oligopoly models in which firms simultaneously set prices and quantities. Existence of a pure-strategy equilibrium is proved for a class of price–quantity games. If the demand function is continuous, then the equilibrium outcome is similar to that of a price-only model. With discontinuous demand and limited spillover, there are rationing equilibria in which combined production falls short of market demand. Moreover, there might again be an equilibrium reflecting the outcome of a price game. Competition in price and quantity thus yields Bertrand outcomes under a variety of market conditions.     

Simple collective equilibria in stopping games

At each moment in time, an alternative from a finite set is selected by a stochastic process. Players observe the selected alternative and sequentially cast a yes or a no vote. If the set of players casting a yes vote is decisive for the selected alternative, it is accepted and the game ends. Otherwise the next period begins. We refer to this class of problems as stopping games. Collective choice games, quitting games, and coalition formation games are particular examples. When the core of a stopping game is non-empty, a subgame perfect equilibrium in pure stationary strategies is shown to exist. But in general, even subgame perfect equilibria in mixed stationary strategies may not exist. We show that aggregate voting behavior can be summarized by a collective strategy. We insist on pure strategies, allow for simple forms of punishment, and provide a constructive proof to show that so-called two-step simple collective equilibria always exist. This implies the existence of a pure strategy subgame perfect equilibrium. We apply our approach to the case with three alternatives exhibiting a Condorcet cycle and to a model of redistributive politics.     

Stable matchings and the small core in Nash equilibrium in the college admissions problem

Both rematching proof and strong equilibrium outcomes are stable with respect to the true preferences in the marriage problem. We show that not all rematching proof or strong equilibrium outcomes are stable in the college admissions problem. But we show that both rematching proof and strong equilibrium outcomes in truncations at the match point are all stable in the college admissions problem. Further, all true stable matchings can be achieved in both rematching proof and strong equilibrium in truncations at the match point. We show that any Nash equilibrium in truncations admits one and only one matching, stable or not. Therefore, the core at a Nash equilibrium in truncations must be small. But examples exist such that the set of stable matchings with respect to a Nash equilibrium may contain more than one matching. Nevertheless, each Nash equilibrium can only admit at most one true stable matching. If, indeed, there is a true stable matching at a Nash equilibrium, then the only possible equilibrium outcome will be the true stable matching, no matter how different are players' equilibrium strategies from the true preferences and how many other unstable matchings are there at that Nash equilibrium. Thus, we show that a necessary and sufficient condition for the stable matching rule to be implemented in a subset of Nash equilibria by the direct revelation game induced by a stable mechanism is that every Nash equilibrium profile in that subset admits one and only one true stable matching. Received: 30 December 1998 / Accepted: 12 October 2001 This paper is a revision of the paper “Manipulation and Stability in a College Admissions Problem” circulated since 1994. I thank Rich McLean, Abraham Neyman, Mark Satterthwaite, Sang-Chul Suh, and Tetsuji Yamada for helpful discussions. I thank the associate editor and the two anonymous referees for their helpful comments that have greatly improved the paper. I am grateful to the Kellogg G.S.M. at the Northwestern University for the hospitality for my visit. Any errors are mine.     

Coalitional games with veto players: Consistency, monotonicity and Nash outcomes

Following Dagan et al. [Dagan, N., Volij, O., Serrano, R. (1997). A non-cooperative view on consistent bankruptcy rules, Games Econ. Behav. 18, 55–72], we construct an extensive form game for veto-balanced TU games in which a veto player is the proposer and the other players are responders. The set of Nash outcomes of this extensive form game is described, and compared to solutions of TU games such as the nucleolus, kernel and egalitarian core. We find necessary and sufficient conditions under which the nucleolus of the game is a Nash outcome.     

Elections and strategic positioning games

We formalize the interplay between expected voting behavior and strategic positioning behavior of candidates as a common agency problem in which the candidates (i.e., the principals) compete for voters (i.e., agents) via the issues they choose and the positions they take. A political situation is defined as a feasible combination of candidate positions and expected political payoffs to the candidates. Taking this approach, we are led naturally to a particular formalization of the candidates' positioning game, called a political situation game. Within the context of this game, we define the notion of farsighted stability (introduced in an abstract setting by Chwe 1994) and apply Chwe's result to obtain existence of farsightedly stable outcomes. We compute the farsightedly stable sets for several examples of political situations games, with outcomes that conform to real-world observations.Received: 18 May 2001, Accepted: 22 January 2002, JEL Classification: C7, D7, D8Myrna H. Wooders: http://www.warwick.ac.uk/fac/soc/Economics/wooders/An earlier version of this paper was completed while the first author was visiting the Department of Economics, University of Exeter. The first author gratefully acknowledges Exeter's support and hospitality. Similarly, the second author gratefully acknowledges the support and hospitality of the Centre for Operations Research and Econometrics (CORE) of the University of Louvain-la-Neuve, Belgium and of the University of Cergy-Pontoise, France. Both authors are indebted to Amrita Dhillon, John Duggan and Gilat Levy for helpful comments about references. Both authors thank an anonymous referee for helpful comments.     

A social procedurial approach to the Pareto optimization problematique

This article applies a variant of game theory to the Pareto multi-value problematique, that is situations where members of a group, community or society are faced with alternative allocations, institutional arrangements, or states of the world and may collectively choose an allocation, institutional arrangement or state of the world if they can agree on it. This type of multiple value decision situation is increasingly prevalent not only on the level of societal and political issues but on the level of many enterprises, particularly those advocating corporate social responsibility. Because actors hold and apply values from different perspectives, there are potential contradictory value judgments and incompatible equilibria. In a world of contradiction, incommensurability, and disequilibrium, to what extent can conflicts be resolved and social equilibrium accomplished? Force works but it is inherently unstable. Drawing on an extension of classical game theory, generalized game theory (GGT), this article addresses the multi-value problematique in terms of collective “resolution procedures.” These regulative procedures—or social algorithms—are applied to problems of conflict and suboptimality in a multiple value world such as Pareto envisioned. This paper (the first of two) outlines key elements of GGT, defines the Pareto multi-value problematique, pointing out several of the critical weaknesses, theoretical as well as empirical, of the Pareto approach. GGT is then applied in defining and analyzing several major procedures to realize improvements in a multi-value world characterized by conflict and sub-optimality. A second article conceptualizes a complex of societal games making up a social system with 2-phase multi-level game processes; it applies the conceptualization to the different societal procedures for multi-value choice under conditions of conflict. Procedures such as democratic voting, adjudication and administrative decision-making, and multi-lateral negotiation are capable of producing outcomes that in many cases are widely accepted as legitimate and become social equilibria (at least within some range of conditions). These procedures and the conditions for their activation and implementation are modelled and explicated through a generalized game approach.     

Cost sharing: efficiency and implementation

We study environments where a production process is jointly shared by a finite group of agents. The social decision involves the determination of input contribution and output distribution. We define a competitive solution when there is decreasing-returns-to-scale which leads to a Pareto optimal outcome. Since there is a finite number of agents, the competitive solution is prone to manipulation. We construct a mechanism for which the set of Nash equilibria coincides with the set of competitive solution outcomes. We define a marginal-cost-pricing equilibrium (MCPE) solution for environments with increasing returns to scale. These solutions are Pareto optimal under certain conditions. We construct another mechanism that realizes the MCPE.     

Nash-implementation of competitive equilibria via a bounded mechanism

We deal with the problem of providing incentives for the implementation of competitive outcomes in a pure-exchange economy with finitely many households. We construct a feasible price-quantity mechanism, which fully implements Walras equilibria via Nash equilibria in fairly general environments. Traders’ preferences need neither to be ordered nor continuous. In addition, the mechanism is such that no pure strategy is weakly dominated, hence is bounded (in the sense of Jackson 1992). In particular it makes no use of any integer game.     

The stochastic lake game: A numerical solution

In this paper, we numerically solve a stochastic dynamic programming problem for the solution of a stochastic dynamic game for which there is a potential function. The players select a mean level of control. The state transition dynamics is a function of the current state of the system and a multiplicative noise factor on the control variables of the players. The particular application is for lake water usage. The control variables are the levels of phosphorus discharged (typically by farmers) into the watershed of the lake, and the random shock is the rainfall that washes the phosphorus into the lake. The state of the system is the accumulated level of phosphorus in the lake. The system dynamics are sufficiently nonlinear so that there can be two Nash equilibria. A Skiba-like point can be present in the optimal control solution.We analyze (numerically) how the dynamics and the Skiba-like point change as the variance of the noise (the rain) increases. The numerical analysis uses a result of Dechert (1978. Optimal control problems from second order difference equations. Journal of Economic Theory 19, 50–63) to construct a potential function for the dynamic game. This greatly reduces the computational burden in finding Nash equilibria solutions for the dynamic game.     

A note on the symmetry of all Nash equilibria in games with increasing best replies

It is well known that a symmetric game has only symmetric (pure strategy) Nash equilibria if its best-reply correspondences admit only increasing selections and its strategy sets are totally ordered. Several nonexamples of the literature show that this result is generally false when the totality condition of the relation that orders the strategy sets is simply dropped. Making use of the structure of interaction functions, this note provides sufficient conditions for the symmetry of all (pure strategy) Nash equilibria in symmetric games where best-reply correspondences admit only increasing selections, but strategy sets are not necessarily totally ordered.     

Identity, reputation and social interaction with an application to sequential voting

We analyze binary choices in a random utility model assuming that agent’s preferences are affected by conformism (with respect to the behavior of the society) and coherence (with respect to identity). We show that multiple stationary equilibria may arise and that the outcome looks very different from a society where all the agents take their decisions in isolation. We quantify the fraction of agents that behave coherently. We apply the analysis to sequential voting when voters “like to win”. Compared to the present literature, we enrich the setting assuming that each voter is endowed with an ideology and we consider the interplay between coherence and the desire to vote with the (perceived) majority.     

Maximizing political vote in multiple districts

This paper models a resource allocation problem in the political context. Voters and political candidates of two parties are positioned in each of n given electoral districts. We assume that each voter will vote for the candidate he is more attracted to. This attraction is modeled by an attraction function. Each of the parties now attempts to allocate a finite budget to maximize their objective, which is either the popular vote or the number of districts, in which the party has a majority. Individual scenarios are examined with respect to leader-follower solutions and Nash equilibria. The paper then describes a dynamic model that successively allocates existing funds plus additional donations to candidates in different electoral districts.     

Essential stability in games with endogenous sharing rules

We prove that essential games with endogenous sharing rules form a dense residual set and that every game with endogenous sharing rules has at least one minimal essential set of solutions. Furthermore, we establish that essential continuous games form a dense residual set and that every continuous game has at least one minimal essential set of Nash equilibria.     

Strategy-proofness on restricted separable domains

We study a problem in which a group of voters must decide which candidates are elected from a set of alternatives. The voters’ preferences on the combinations of elected candidates are represented by linear orderings. We propose a family of restrictions of the domain of separable preferences. These subdomains are generated from a partition that identifies the friends, enemies and unbiased candidates for each voter. We characterize the family of social choice functions that satisfy strategy-proofness and tops-onlyness properties on each of the subdomains. We find that these domain restrictions are not accompanied by an increase in the family of social choice functions satisfying the two properties.