Weighted search games

In this paper we shall deal with search games in which the strategic situation is developed on a lattice. The main characteristic of these games is that the points in each column of the lattice have a specific associated weight which directly affects the payoff function. Thus, the points in different columns represent points of different strategic value. We solve three different types of games. The first involves search, ambush and mixed situations, the second is a search and inspection game and the last is related to the accumulative games.     

Static search games played over graphs and general metric spaces

We define a general game which forms a basis for modelling situations of static search and concealment over regions with spatial structure. The game involves two players, the searching player and the concealing player, and is played over a metric space. Each player simultaneously chooses to deploy at a point in the space; the searching player receiving a payoff of 1 if his opponent lies within a predetermined radius r of his position, the concealing player receiving a payoff of 1 otherwise. The concepts of dominance and equivalence of strategies are examined in the context of this game, before focusing on the more specific case of the game played over a graph. Methods are presented to simplify the analysis of such games, both by means of the iterated elimination of dominated strategies and through consideration of automorphisms of the graph. Lower and upper bounds on the value of the game are presented and optimal mixed strategies are calculated for games played over a particular family of graphs.     

Asymmetric equilibria in dynamic two-sided matching markets with independent preferences

A fundamental fact in two-sided matching is that if a market allows several stable outcomes, then one is optimal for all men in the sense that no man would prefer another stable outcome. We study a related phenomenon of asymmetric equilibria in a dynamic market where agents enter and search for a mate for at most n rounds before exiting again. Assuming independent preferences, we find that this game has multiple equilibria, some of which are highly asymmetric between sexes. We also investigate how the set of equilibria depends on a sex difference in the outside option of not being mated at all.     

Oligopoly models for market price of electricity under demand uncertainty and unit reliability

Several oligopoly models have been proposed for representing strategic behavior in electricity markets, which include Bertrand, Cournot, and Supply Function Equilibrium (SFE). For the most part, these models are deterministic, with the exception of the SFE originally developed by Klemperer and Meyer. However, their model does not include supply side uncertainties. In this paper, we consider both load and supply side uncertainties (resulting from generator availabilities). We obtain Nash equilibrium solutions for Cournot and SFE models, in which asymmetric firms (whose generating units have different costs and capacities) submit their bids so that each firm’s expected profit is maximized.     

Introducing competition in healthcare services: The role of private care and increased patient mobility

We study the operational implications from competition in the provision of healthcare services, in the context of national public healthcare systems in Europe. Specifically, we study the potential impact of two alternative ways through which policy makers have introduced such competition: (i) via the introduction of private hospitals to operate alongside public hospitals and (ii) via the introduction of increased patient choice to grant European patients the freedom to choose the country they receive treatment at. We use a game-theoretic framework with a queueing component to capture the interactions among the patients, the hospitals and the healthcare funders. Specifically, we analyze two different sequential games and obtain closed form expressions for the patients’ waiting time and the funders’ reimbursement cost in equilibrium. We show that the presence of a private provider can be beneficial to the public system: the patients’ waiting time will decrease and the funders’ cost can decrease under certain conditions. Also, we show that the cross-border healthcare policy, which increases patient mobility, can also be beneficial to the public systems: when welfare requirements across countries are sufficiently close, all funders can reduce their costs without increasing the patients’ waiting time. Our analysis implies that in border regions, where the cost of crossing the border is low, “outsourcing” the high-cost country’s elective care services to the low-cost country is a viable strategy from which both countries’ systems can benefit.     

Opportunistic timing and manipulation in Australian Federal Elections

In many parliamentary systems, election timing is an important decision made by governments in order to maximize their expected remaining life in power. Governments can also introduce policy or economic actions to enhance their popular standing and thus their chance of being re-elected. On the other hand, an oppositions’ natural objective is to gain power, and they will also apply controls through their own policies to reduce the governments’ chance of being re-elected. In this paper we employ a dynamic programming approach to determine the optimal timing for governments and oppositions to best utilize their limited resources. At each decision branch, the optimal control is interpreted as a Nash–Cournot equilibrium of a zero-sum political game which, in certain states, admits mixed strategy solutions. We perform a case study on the Australian Federal Election for House of Representatives.     

Optimal cost sharing for capacitated facility location games

We consider cost sharing for a class of facility location games, where the strategy space of each player consists of the bases of a player-specific matroid defined on the set of resources. We assume that resources have nondecreasing load-dependent costs and player-specific delays. Our model includes the important special case of capacitated facility location problems, where players have to jointly pay for opened facilities. The goal is to design cost sharing protocols so as to minimize the resulting price of anarchy and price of stability. We investigate two classes of protocols: basic protocols guarantee the existence of at least one pure Nash equilibrium and separable protocols additionally require that the resulting cost shares only depend on the set of players on a resource. We find optimal basic and separable protocols that guarantee the price of stability/price of anarchy to grow logarithmically/linearly in the number of players. These results extend our previous results (cf. von Falkenhausen & Harks, 2013), where optimal basic and separable protocols were given for the case of symmetric matroid games without delays.     

Evolutionary games

This paper deals with a mathematical game. As the name implies, the game concept is formulated with biological evolution in mind. An evolutionary game differs from the usual game concepts in that the players cannot choose their strategies. Rather, the strategies used by the players are handed down from generation to generation. It is the survival characteristics of a strategy that determine the outcome of the evolutionary game. Players interact and receive payoffs according to the strategies they are using. These interactions, in turn, determine the fitness of players using a given strategy. The survival characteristics of strategy are determined directly from the fitness functions. Necessary conditions for determining an evolutionarily stable strategy are developed here for a continuous game. Results are illustrated with an example.Dedicated to G. LeitmannThis work was supported by NSF Grant No. INT-82-10803 and The University of Western Australia (Visiting Fellowship, Department of Mathematics, 1983).     

Modeling cooperation on a class of distribution problems

In this paper we study models of cooperation between the nodes of a network that represents a distribution problem. The distribution problem we propose arises when, over a graph, a group of nodes offers certain commodity, some other nodes require it and a third group of nodes neither need this material nor offer it but they are strategically relevant to the distribution plan. The delivery of one unit of material to a demand node generates a fixed profit, and the shipping of the material through the arcs has an associated cost. We show that in such a framework cooperation is beneficial for the different parties. We prove that the cooperative situation arising from this distribution problem is totally balanced by finding a set of stable allocations (in the core of an associated cooperative game). In order to overcome certain fairness problems of these solutions, we introduce two new solution concepts and study their properties.     

Searching for equilibrium positions in a game of political competition with restrictions

This paper considers a problem of political economy in which a Nash equilibrium study is performed in a proposed game with restrictions where the two major parties in a country vary their position within a politically flexible framework to increase their number of voters. The model as presented fits the reality of many countries. Moreover, it avoids the uniqueness of equilibrium positions. The problem is stated and solved from a geometric point of view.     

Depreciation games

The main aim of this paper is to show how Game Theory can be used in the day-to-day running of a company. No precedent, as far as we know, in the literature of the application of Cooperative Game Theory in order to obtain new depreciation methods for a fixed asset which is owned by a company. To this end, we will first illustrate the usual depreciation methods employed in accountancy. Then, with the objective of finding new depreciation methods, we will establish the relationship between Game Theory and Accountancy. On the basis of this relationship we introduce some new depreciation methods. Finally, a family of solutions is defined which improves the properties of traditional and previously introduced methods. Financial support from the Generalitat Valenciana through project GV05/077 is gratefully acknowledged by J. Aparicio. Financial support from the Generalitat Valenciana through project GRUPOS04/79 and from the Ministerio de Educación y Ciencia through project HI2002/0032 are gratefully acknowledged by J. Sánchez-Soriano.     

Complete stability of noncooperative games

This paper is concerned with a class of noncooperative games ofn players that are defined byn reward functions which depend continuously on the action variables of the players. This framework provides a realistic model of many interactive situations, including many common models in economics, sociology, engineering, and political science. The concept of Nash equilibrium is a suitable companion to such models.A variety of different sufficient conditions for existence, uniqueness, and stability of a Nash equilibrium point have been previously proposed. By sharpening the noncooperative aspect of the framework (which is really only implicit in the original framework), this paper attempts to isolate one set of natural conditions that are sufficient for existence, uniqueness, and stability. It is argued thatl quasicontraction is such a natural condition. The concept of complete stability is introduced to reflect the full character of noncooperation. It is then shown that, in the linear case, the condition ofl quasicontraction is both necessary and sufficient for complete stability.This research was supported by the Air Force Office of Scientific Research under Grant No. AFOSR 77-3141 and by the National Science Foundation under Grant No. GK18748.     

Economic lot-sizing games

In this paper we introduce a new class of OR games: economic lot-sizing (ELS) games. There are a number of retailers that have a known demand for a fixed number of periods. To satisfy demand the retailers order products at the same manufacturer. By placing joint orders instead of individual orders, costs can be reduced and a cooperative game arises. In this paper we show that ELS games are balanced. Furthermore, we show that two special classes of ELS games are concave.     

Shortest path games

We study cooperative games that arise from the problem of finding shortest paths from a specified source to all other nodes in a network. Such networks model, among other things, efficient development of a commuter rail system for a growing metropolitan area. We motivate and define these games and provide reasonable conditions for the corresponding rail application. We show that the core of a shortest path game is nonempty and satisfies the given conditions, but that the Shapley value for these games may lie outside the core. However, we show that the shortest path game is convex for the special case of tree networks, and we provide a simple, polynomial time formula for the Shapley value in this case. In addition, we extend our tree results to the case where users of the network travel to nodes other than the source. Finally, we provide a necessary and sufficient condition for shortest paths to remain optimal in dynamic shortest path games, where nodes are added to the network sequentially over time.     

Partnership formation with age-dependent preferences

We analyze a model of partnership formation in which players’ preferences are based on the age of a prospective partner. There are two classes of individuals, called for convenience here male and female. Males and females are fertile for the same length of time, normalized to one unit. A male enters the mating pool at age 0 and meets prospective partners according to a Poisson process. At equilibrium, he accepts a female if the utility from mating exceeds the expected utility from future search, which depends on the acceptance strategies of all males and females and the corresponding steady-state distribution of age in the pool of unmated individuals. Females face an analogous problem. Mating pairs are only formed by mutual consent and individuals leave the pool of unmated individuals on finding a mating partner or reaching the age of 1. A policy iteration algorithm is used to determine the equilibrium acceptance strategies and the corresponding steady-state distribution of the age of individuals in the mating pool. Two examples are presented.     

Selfish routing in public services

It is well observed that individual behaviour can have an effect on the efficiency of queueing systems. The impact of this behaviour on the economic efficiency of public services is considered in this paper where we present results concerning the congestion related implications of decisions made by individuals when choosing between facilities. The work presented has important managerial implications at a public policy level when considering the effect of allowing individuals to choose between providers. We show that in general the introduction of choice in an already inefficient system will not have a negative effect. Introducing choice in a system that copes with demand will have a negative effect.     

Mature or emerging markets: Competitive duopoly investment decisions

We develop a competitive investment model wherein two competing firms consider investing into two projects targeting, separately, a mature and an emerging market. The returns firms obtain from investments into these markets are assumed to follow an S-shaped curve and depend on both firms’ actions. Considering symmetric environments (in terms of investment opportunities), we find that different forms of interactions may arise (e.g., Prisoner’s Dilemma and Game of Chicken) and outline corresponding strategies that offer higher returns by exploiting first-mover advantages, cooperation opportunities and aggressive choices. We also discuss the market conditions that can lead to these outcomes. Finally, considering non-symmetric environments, we show that a firm may be better off when its competitor’s budget increases.     

An inspection game with multiple inspectees

This paper deals with a non-zero-sum inspection game with one inspector and several inspectees. Each inspectee country makes a decision from the standpoint of his national interest while the inspector distributes staff to inspectee facilities to detect illegal behavior. We propose a method to find a Nash equilibrium for the game, which could help the inspector make an effective plan about how to assign staff to individual suspect facilities in those countries. Avenhaus et al. studied this kind of two-inspectee model. This paper extends Avenhaus’ research to a multi-inspectee model taking account of optimal dispatching of inspection staff.     

Constrained cost-coupled stochastic games with independent state processes

We study non-cooperative constrained stochastic games in which each player controls its own Markov chain based on its own state and actions. Interactions between players occur through their costs and constraints which depend on the state and actions of all players. We provide an example from wireless communications.