Continuity properties of majority rule with intermediate preferences

This paper shows that certain conditions which have previously been shown (by Grandmont) to be sufficient for a society's majority rule relation to be transitive or acyclic are also sufficient for the map from distributions of voter preferences to indices identified with their majority rule relations to be continuous. Applications of this result to societies with certain classical assumptions on preferences reveal that, in such societies, the map from distributions of voter preferences to their majority rule equilibria is also continuous.     

The give-up problem for blocked regional lists with multi-winners

The current electoral law for the Italian Parliament prescribes blocked, linearly ordered lists of candidates for each party within each constituency. The peculiarity of the Italian electoral system is that a party can present the same candidate in different constituencies. There are several seats at stake in each constituency; these seats are allocated to the parties proportionally to the total number of votes they get. If the blocked list mechanism-which assigns the seats obtained by a party in a constituency to the first candidates of the corresponding ordered list-causes some candidates to win in more than one constituency, they may retain only one of the seats, giving up all the remaining ones. Thus, the problem arises for a party to find a suitable “schedule of give-ups” that produces the final set of winners for that party. In order to do this, we assume that such decision is centralized and based on some models of global (inter-regional) preferences over the set of candidates. In this paper, we introduce two classes of models to formulate the “give-up problem”, i.e., utility and ordinal models, and we show that for both of them some natural formulations of the problem can be efficiently solved by network flows techniques.     

Maximizing political efficiency via electoral cycles: An optimal control model

The exchange between an elected politician, such as a president, prime minister or a local governor and interest groups is analyzed as an optimization problem. The optimal control model shows the conditions required from regulatory policy and resource investment in order to maximize the politician's utility from the interest group's support. Given one interest group, such a policy includes two time intervals: Well in advance of the elections the politician in office should invest a constant level of resources, while for a certain period close to the elections the politician increases or decreases investment, depending on the electoral significance of that interest group. This proves that electoral cycles not only empirically exist, but also maximize the politician's utility from interest groups' support. Given several interest groups, at each point in time, the politician should invest in the group that contribute the most for his or her political interests.     

Analysis and design of selection committees: a game theoretic secretary problem

Firms often delegate important decisions to committees which are set up specifically for that purpose; for example selection committees. We analyze the equilibrium behavior of a game in which committee members (the players) interview candidates sequentially, either hiring or going on to the next one. The players have differing evaluations of candidates (e.g. one cares about typing skills; the other about IT skills), which become their utilities if the candidate is hired. We then consider the optimal design (rules of the game) of such a committee, from the point of view of the firm. That is, which rules hire candidates which maximize the firm’s utility. Our committee game has a first round in which the members sequentially, by order of player number, say ‘yea’ or ‘nea’ to the candidate. If there are sufficient ‘yeas’ then she is tentatively hired; otherwise she is rejected. In the former case, members who said nea can veto the candidate in the second round. Thus the candidate is either hired, rejected, or vetoed. In the last case, the member casting a veto has one less to use on later candidates. We analyze equilibria where a player may say ‘yea’ to a candidate he would prefer not to hire, in order to force the other player to use up a valuable veto. We show that for the uniform candidate distribution there is a unique equilibrium and better candidates for the firm are hired when there are more vetoes. However we exhibit a candidate distribution where increasing the numbers of vetoes results in hiring worse candidates.     

Value and perfection in stochastic games

A stochastic game isvalued if for every playerk there is a functionr ^k:S→R from the state spaceS to the real numbers such that for every ε>0 there is an ε equilibrium such that with probability at least 1−ε no states is reached where the future expected payoff for any playerk differs fromr ^k(s) by more than ε. We call a stochastic gamenormal if the state space is at most countable, there are finitely many players, at every state every player has only finitely many actions, and the payoffs are uniformly bounded and Borel measurable as functions on the histories of play. We demonstrate an example of a recursive two-person non-zero-sum normal stochastic game with only three non-absorbing states and limit average payoffs that is not valued (but does have ε equilibria for every positive ε). In this respect two-person non-zero-sum stochastic games are very different from their zero-sum varieties. N. Vieille proved that all such non-zero-sum games with finitely many states have an ε equilibrium for every positive ε, and our example shows that any proof of this result must be qualitatively different from the existence proofs for zero-sum games. To show that our example is not valued we need that the existence of ε equilibria for all positive ε implies a “perfection” property. Should there exist a normal stochastic game without an ε equilibrium for some ε>0, this perfection property may be useful for demonstrating this fact. Furthermore, our example sews some doubt concerning the existence of ε equilibria for two-person non-zero-sum recursive normal stochastic games with countably many states. This research was supported financially by the German Science Foundation (Deutsche Forschungsgemeinschaft) and the Center for High Performance Computing (Technical University, Dresden). The author thanks Ulrich Krengel and Heinrich Hering for their support of his habilitation at the University of Goettingen, of which this paper is a part.     

Simple centrifugal incentives in spatial competition

This paper studies the effects of introducing centrifugal incentives in an otherwise standard Downsian model of electoral competition. First, we demonstrate that a symmetric equilibrium is guaranteed to exist when centrifugal incentives are induced by any kind of partial voter participation (such as abstention due to indifference, abstention due to alienation, etc.) and, then, we argue that: (a) this symmetric equilibrium is in pure strategies, and it is hence convergent, only when centrifugal incentives are sufficiently weak on both sides; (b) when centrifugal incentives are strong on both sides (when, for example, a lot of voters abstain when they are sufficiently indifferent between the two candidates) players use mixed strategies—the stronger the centrifugal incentives, the larger the probability weight that players assign to locations near the extremes; and (c) when centrifugal incentives are strong on one side only—say for example only on the right—the support of players’ mixed strategies contain all policies except from those that are sufficiently close to the left extreme.     

On a Parameterized System of Nonlinear Equations with Economic Applications

We study a parameterized system of nonlinear equations. Given a nonempty, compact, and convex set, an affine function, and a point-to-set mapping from the set to the Euclidean space containing the set, we constructively prove that, under certain (boundary) conditions on the mapping, there exists a connected set of zero points of the mapping, i.e., the origin is an element of the image for every point in the connected set, such that the connected set has a nonempty intersection with both the face at which the affine function is minimized and the face at which that function is maximized. This result generalizes and unifies several well-known existence theorems including Browder??s fixed point theorem and Ky Fan??s coincidence theorem. An economic application with constrained equilibria is also discussed.     

Non-cooperative stochastic dominance games

A non-cooperative stochastic dominance game is a non-cooperative game in which the only knowledge about the players' preferences and risk attitudes is presumed to be their preference orders on the set ofn-tuples of pure strategies. Stochastic dominance equilibria are defined in terms of mixed strategies for the players that are efficient in the stochastic dominance sense against the strategies of the other players. It is shown that the set of SD equilibria equals all Nash equilibria that can be obtained from combinations of utility functions that are consistent with the players' known preference orders. The latter part of the paper looks at antagonistic stochastic dominance games in which some combination of consistent utility functions is zero-sum over then-tuples of pure strategies.     

A model of far-sighted electoral competition

A model is described in which candidates adopt positions in sequences of election contests against opponents randomly drawn from a large population. Symmetric steady-state equilibria of the model require rational selection of positions by all candidates against the aggregated behavior of the population, taking into account constraints which an individual's current selections impose on his future selections.     

Finding location equilibria for competing firms under delivered pricing

We address the problem of finding location equilibria of a location-price game where firms first select their locations and then set delivered prices in order to maximize their profits. Assuming that firms set the equilibrium prices in the second stage, the game is reduced to a location game for which a global minimizer of the social cost is a location equilibrium if demand is completely inelastic and marginal production cost is constant. The problem of social cost minimization is studied for both a network and a discrete location space. A node optimality property when the location space is a network is shown and an Integer Linear Programming (ILP) formulation is obtained to minimize the social cost. It is also shown that multiple location equilibria can be found if marginal delivered costs are equal for all competitors. Two ILP formulations are given to select one of such equilibria that take into account the aggregated profit and an equity criterion, respectively. An illustrative example with real data is solved and some conclusions are presented.     

Proper belief revision and rationalizability in dynamic games

In this paper we develop an epistemic model for dynamic games in which players may revise their beliefs about the opponents’ utility functions as the game proceeds. Within this framework, we propose a rationalizability concept that is based upon the following three principles: (1) at every instance of the game, a player should believe that his opponents are carrying out optimal strategies, (2) a player, at information set h, should not change his belief about an opponent’s relative ranking of two strategies s and s′ if both s and s′ could have led to h, and (3) the players’ initial beliefs about the opponents’ utility functions should agree on a given profile u of utility functions. Common belief in these events leads to the concept of persistent rationalizability for the profile u of utility functions. It is shown that for a given game tree with observable deviators and a given profile u of utility functions, every properly point-rationalizable strategy is a persistently rationalizable strategy for u. This result implies that persistently rationalizable strategies always exist for all game trees with observable deviators and all profiles of utility functions. We provide an algorithm that can be used to compute the set of persistently rationalizable strategies for a given profile u of utility functions. For generic games with perfect information, persistent rationalizability uniquely selects the backward induction strategy for every player.     

Negotiation,preferences over agreements,and the core

We analyze conditions under which negotiated agreements are efficient from the point of view of every possible coalition of negotiators. The negotiators have lexicographic preferences over agreements they reach. Their utility is the first criterion. The coalition reaching an agreement is the second criterion. In the analyzed non-cooperative discrete time bargaining game Γ the players bargain about the choice from the sets of utility vectors feasible for coalitions in a given NTU game (N, V). If Γ has a Markov perfect equilibrium, then the set of equilibrium utility vectors in Markov perfect equilibria in it equals the core of (N, V). I thank an anonymous referee, an anonymous Associate Editor, and the Editor for their comments that helped me to improve the paper. The research reported in this paper was supported by the Grant VEGA 1/1223/04 of the Ministry of Education of the Slovak Republic.     

Hybrid Elections Broaden Complexity‐Theoretic Resistance to Control

Electoral control refers to attempts by an election's organizer (“the chair”) to influence the outcome by adding/deleting/partitioning voters or candidates. The important paper of Bartholdi, Tovey, and Trick [1] that introduces (constructive) control proposes computational complexity as a means of resisting control attempts: Look for election systems where the chair's task in seeking control is itself computationally infeasible. We introduce and study a method of combining two or more candidate‐anonymous election schemes in such a way that the combined scheme possesses all the resistances to control (i.e., all the NP‐hardnesses of control) possessed by any of its constituents: It combines their strengths. From this and new resistance constructions, we prove for the first time that there exists a neutral, anonymous election scheme (whose winner problem is computable in polynomial time) that is resistant to all twenty standard types of electoral control (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Slightly Altruistic Equilibria

We introduce a refinement concept for Nash equilibria (slightly altruistic equilibrium) defined by a limit process and which captures the idea of reciprocal altruism as presented in Binmore (Proceedings of the XV Italian Meeting on Game Theory and Applications, [2003]). Existence is guaranteed for every finite game and for a large class of games with a continuum of strategies. Results and examples emphasize the (lack of) connections with classical refinement concepts. Finally, it is shown that, under a pseudomonotonicity assumption on a particular operator associated to the game, it is possible, by selecting slightly altruistic equilibria, to eliminate those equilibria in which a player can switch to a strategy that is better for the others without leaving the set of equilibria. Part of the results in this paper have been presented at: First Spain, Italy, Netherlands Meeting on Game Theory, Maastricht, 2005; Fifth International ISDG Workshop, Segovia, 2005; GATE, Université Lumière Lyon 2, 2005; XXX AMASES Workshop, Trieste 2006; CSEF, Università di Salerno, 2006.     

Coalition-proof equilibria in a voluntary participation game

We examine the coalition-proof equilibria of a participation game in the provision of a (pure) public good. We study which Nash equilibria are achieved through cooperation, and we investigate coalition-proof equilibria under strict and weak domination. We show that under some incentive condition, (i) a profile of strategies is a coalition-proof equilibrium under strict domination if and only if it is a Nash equilibrium that is not strictly Pareto-dominated by any other Nash equilibrium and (ii) every strict Nash equilibrium for non-participants is a coalition-proof equilibrium under weak domination.     

Downsian competition with four parties

Our purpose in this article is to study a unidimensional model of spatial electoral competition with four political parties. We assume that the voters are distributed along [0,1] in such a way that the density δ of this distribution is continuous on [0,1] and strictly positive on (0,1). The parties engage in a Downsian competition which is modeled as a non-cooperative four-person game with [0,1] as the common strategy set. If ξ_i stands for the ith quartile of the above-mentioned distribution, then we prove that has a pure Nash equilibrium, if and only if for every t (ξ₁,ξ₃). Moreover, if this condition is satisfied, then has exactly six pure Nash equilibria, which are characterized by the fact that two of the parties put forward the policy that corresponds to ξ₁ and the other two of them put forward the policy that corresponds to ξ₃.     

Stationary and nonstationary strategies in Hotelling's model of spatial competition with repeated pricing decisions

This paper examines Hotelling's model of location with linear transportation cost. Existence of pure strategy subgame perfect equilibria in the infinitely repeated price game with fixed locations is proved. These subgame perfect equilibria have a stick and carrot structure. Given firm locations, there are discount factors sufficiently high that there is a subgame perfect equilibrium with a two-phase structure. Given the discount factors, there are stationary subgame perfect equilibria for a wide range of locations. However, for some pairs of location, no symmetric simple penal code exists, all subgame perfect profiles are nonstationary, and there is only one seller in the market in infinitely many periods. Received July 1996/Final version May 1998     

A Dynamic Analysis of Schelling’s Binary Corruption Model: A Competitive Equilibrium Approach

Schelling (in Micromotives and Macrobehavior, Norton, New York, 1978) suggested a simple binary choice model to explain the variation of corruption levels across societies. His basic idea was that the expected profitability of engaging in corruption depends on its prevalence. The key result of the so-called Schelling diagram is the existence of multiple equilibria and a tipping point. The present paper puts Schelling’s essentially static approach into an intertemporal setting. We show how the existence of an unstable interior steady state leads to thresholds such that history alone or history in addition to expectations (or coordination) is necessary to determine the long-run outcome. In contrast to the related literature, which classifies these two cases according to whether the unstable equilibrium is a node or a focus, the actual differentiation is more subtle because even a node can lead to an overlap of solution paths such that the initial conditions alone are insufficient to uniquely determine the competitive equilibrium. Another insight is that a (transiently) cycling competitive equilibrium can dominate the direct and monotonic route to a steady state, even if the direct route is feasible.     

QUOPLAN: A System for Optimizing Sales Quota-Bonus Plans

This paper describes an IBM-PC menu-driven and user-friendly procedure which can help sales management in setting up optimal quota-bonus plans. For a given sales force, the resulting quotabonus plan maximizes the firm's current profits subject to (1) keeping every individual sales-force member at least at his/her current level of satisfaction (or eventually, increasing this level), and (2) being consistent and harmonious across sales representatives. The QUOPLAN system is composed of two submodels. The first submodel is used by salespersons for eliciting their own utility functions, and is essentially based upon the principles of conjoint analysis. The second submodel is used by management for reconciling all the individuals' judgements and the company's objectives into a consistent quota-bonus plan.     

An agreeable collusive equilibrium in differential games with asymmetric players

We study a class of collusive equilibria in differential games with asymmetric players discounting the future at different rates. For such equilibria, at each moment, weights of players can depend on the state of the system. To fix them, we propose using a bargaining procedure according to which players can bargain again at every future moment. By choosing as threat point the feedback noncooperative outcome, the corresponding solution, if it exists, is agreeable. An exhaustible resource game illustrates the results.