A non-cooperative approach to the cost spanning tree problem

We associate to each cost spanning tree problem a non-cooperative game, which is inspired by a real-life problem. We study the Nash equilibria and subgame perfect Nash equilibria of this game. We prove that these equilibria are closely related with situations where agents connect sequentially to the source.Finicial support from the Ministerio de Ciencia y Tecnologia and FEDER, and Xunta de Galicia through grants BEC2002-04102-C02-01 and PGIDIT03PXIC30002PN is gratefully acknowledged.     

Realizing fair outcomes in minimum cost spanning tree problems through non-cooperative mechanisms

In the context of minimum cost spanning tree problems, we present a bargaining mechanism for connecting all agents to the source and dividing the cost among them. The basic idea is very simple: we ask each agent the part of the cost he is willing to pay for an arc to be constructed. We prove that there exists a unique payoff allocation associated with the subgame perfect Nash equilibria of this bargaining mechanism. Moreover, this payoff allocation coincides with the rule defined in Bergantiños and Vidal-Puga [Bergantiños, G., Vidal-Puga, J.J., 2007a. A fair rule in minimum cost spanning tree problems. Journal of Economic Theory 137, 326–352].     

Optimal Equilibria in the Non-Cooperative Game Associated with Cost Spanning Tree Problems

We study the Pareto optimal equilibria payoffs of the non-cooperative game associated with the cost spanning tree problem. We give two characterisations of these payoffs: one based on the tree they induce and another based on the strategies played by agents. Moreover, an algorithm for computing all these payoffs is provided.     

Truth-telling and Nash equilibria in minimum cost spanning tree models

In this paper we consider the minimum cost spanning tree model. We assume that a central planner aims at implementing a minimum cost spanning tree not knowing the true link costs. The central planner sets up a game where agents announce link costs, a tree is chosen and costs are allocated according to the rules of the game. We characterize ways of allocating costs such that true announcements constitute Nash equilibria both in case of full and incomplete information. In particular, we find that the Shapley rule based on the irreducible cost matrix is consistent with truthful announcements while a series of other well-known rules (such as the Bird-rule, Serial Equal Split, and the Proportional rule) are not.     

Semidefinite complementarity reformulation for robust Nash equilibrium problems with Euclidean uncertainty sets

Consider the N-person non-cooperative game in which each player’s cost function and the opponents’ strategies are uncertain. For such an incomplete information game, the new solution concept called a robust Nash equilibrium has attracted much attention over the past several years. The robust Nash equilibrium results from each player’s decision-making based on the robust optimization policy. In this paper, we focus on the robust Nash equilibrium problem in which each player’s cost function is quadratic, and the uncertainty sets for the opponents’ strategies and the cost matrices are represented by means of Euclidean and Frobenius norms, respectively. Then, we show that the robust Nash equilibrium problem can be reformulated as a semidefinite complementarity problem (SDCP), by utilizing the semidefinite programming (SDP) reformulation technique in robust optimization. We also give some numerical example to illustrate the behavior of robust Nash equilibria.     

Equilibria in a competitive model arising from linear production situations with a common-pool resource

In this paper we deal with linear production situations in which there is a limited common-pool resource, managed by an external agent. The profit that a producer can attain depends on the amount of common-pool resource obtained through a certain procedure. We contemplate a competitive process among the producers and study the corresponding non-cooperative games, describing their (strict) Nash equilibria in pure strategies. It is shown that strict Nash equilibria form a subset of strong Nash equilibria, which in turn form a proper subset of Nash equilibria.     

A two-player competitive discrete location model with simultaneous decisions

In this paper we will describe and study a competitive discrete location problem in which two decision-makers (players) will have to decide where to locate their own facilities, and customers will be assigned to the closest open facilities. We will consider the situation in which the players must decide simultaneously, unsure about the decisions of one another, and we will present the problem in a franchising environment. Most problems described in the literature consider sequential rather than simultaneous decisions. In a competitive environment, most problems consider that there is a set of known and already located facilities, and new facilities will have to be located, competing with the existing ones. In the presence of more than one decision-maker, most problems found in the literature belong to the class of Stackelberg location problems, where one decision-maker, the leader, locates first and then the other decision-maker, the follower, locates second, knowing the decisions made by the first. These types of problems are sequential and differ significantly from the problem tackled in this paper, where we explicitly consider simultaneous, non-cooperative discrete location decisions. We describe the problem and its context, propose some mathematical formulations and present an algorithmic approach that was developed to find Nash equilibria. Some computational tests were performed that allowed us to better understand some of the features of the problem and the associated Nash equilibria. Among other results, we conclude that worsening the situation of a player tends to benefit the other player, and that the inefficiency of Nash equilibria tends to increase with the level of competition.     

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.     

The minimum cost shortest-path tree game

A minimum cost shortest-path tree is a tree that connects the source with every node of the network by a shortest path such that the sum of the cost (as a proxy for length) of all arcs is minimum. In this paper, we adapt the algorithm of Hansen and Zheng (Discrete Appl. Math. 65:275?C284, 1996) to the case of acyclic directed graphs to find a minimum cost shortest-path tree in order to be applied to the cost allocation problem associated with a cooperative minimum cost shortest-path tree game. In addition, we analyze a non-cooperative game based on the connection problem that arises in the above situation. We prove that the cost allocation given by an ??à la?? Bird rule provides a core solution in the former game and that the strategies that induce those payoffs in the latter game are Nash equilibrium.     

Limiting distributions of the number of pure strategy Nash equilibria in n-person games

We study the number of pure strategy Nash equilibria in a “random” n-person non-cooperative game in which all players have a countable number of strategies. We consider both the cases where all players have strictly and weakly ordinal preferences over their outcomes. For both cases, we show that the distribution of the number of pure strategy Nash equilibria approaches the Poisson distribution with mean 1 as the numbers of strategies of two or more players go to infinity. We also find, for each case, the distribution of the number of pure strategy Nash equilibria when the number of strategies of one player goes to infinity, while those of the other players remain finite.     

“Costless” regulation of monopolies with large entry cost: A game theoretic approach

A major issue within the realm of Antitrust policy is the regulation of existing monopolies. We describe a new potential indirect scheme for regulating a natural monopoly that arises from high entry cost. The approach involves minimal government intervention, and it is based on encouraging entry by offering to subsidize entry cost for potential competitors. We pose this issue as a four-stage non-cooperative game. Analysis of sub-game perfect Nash equilibria of the game reveals that the first best outcome is achieved as the unique equilibrium in which the monopolist prices at marginal cost and there is no entry. The regulation is “costless,” since no entry will occur and hence no subsidy will be paid. Received June 1995     

Approximate fixed point theorems in Banach spaces with applications in game theory

In this paper some new approximate fixed point theorems for multifunctions in Banach spaces are presented and a method is developed indicating how to use approximate fixed point theorems in proving the existence of approximate Nash equilibria for non-cooperative games.     

Game theoretic approach for fertilizer application: looking for the propensity to cooperate

This paper continues the research implemented in previous work of (Schreider et al. in Environ. Model. Assess. 15(4):223–238, 2010) where a game theoretic model for optimal fertilizer application in the Hopkins River catchment was formulated, implemented and solved for its optimal strategies. In that work, the authors considered farmers from this catchment as individual players whose objective is to maximize their objective functions which are constituted from two components: economic gain associated with the application of fertilizers which contain phosphorus to the soil and environmental harms associated with this application. The environmental losses are associated with the blue-green algae blooming of the coastal waterways due to phosphorus exported from upstream areas of the catchment. In the previous paper, all agents are considered as rational players and two types of equilibria were considered: fully non-cooperative Nash equilibrium and cooperative Pareto optimum solutions. Among the plethora of Pareto optima, the solution corresponding to the equally weighted individual objective functions were selected. In this paper, the cooperative game approach involving the formation of coalitions and modeling of characteristic value function will be applied and Shapley values for the players obtained. A significant contribution of this approach is the construction of a characteristic function which incorporates both the Nash and Pareto equilibria, showing that it is superadditive. It will be shown that this approach will allow each players to obtain payoffs which strictly dominate their payoffs obtained from their Nash equilibria.     

On the solution of affine generalized Nash equilibrium problems with shared constraints by Lemke’s method

Affine generalized Nash equilibrium problems (AGNEPs) represent a class of non-cooperative games in which players solve convex quadratic programs with a set of (linear) constraints that couple the players’ variables. The generalized Nash equilibria (GNE) associated with such games are given by solutions to a linear complementarity problem (LCP). This paper treats a large subclass of AGNEPs wherein the coupled constraints are shared by, i.e., common to, the players. Specifically, we present several avenues for computing structurally different GNE based on varying consistency requirements on the Lagrange multipliers associated with the shared constraints. Traditionally, variational equilibria (VE) have been amongst the more well-studied GNE and are characterized by a requirement that the shared constraint multipliers be identical across players. We present and analyze a modification to Lemke’s method that allows us to compute GNE that are not necessarily VE. If successful, the modified method computes a partial variational equilibrium characterized by the property that some shared constraints are imposed to have common multipliers across the players while other are not so imposed. Trajectories arising from regularizing the LCP formulations of AGNEPs are shown to converge to a particular type of GNE more general than Rosen’s normalized equilibrium that in turn includes a variational equilibrium as a special case. A third avenue for constructing alternate GNE arises from employing a novel constraint reformulation and parameterization technique. The associated parametric solution method is capable of identifying continuous manifolds of equilibria. Numerical results suggest that the modified Lemke’s method is more robust than the standard version of the method and entails only a modest increase in computational effort on the problems tested. Finally, we show that the conditions for applying the modified Lemke’s scheme are readily satisfied in a breadth of application problems drawn from communication networks, environmental pollution games, and power markets.     

Finding all pure strategy Nash equilibria in a planar location game

In this paper, we deal with a planar location-price game where firms first select their locations and then set delivered prices in order to maximize their profits. If firms set the equilibrium prices in the second stage, the game is reduced to a location game for which pure strategy Nash equilibria are studied assuming that the marginal delivered cost is proportional to the distance between the customer and the facility from which it is served. We present characterizations of local and global Nash equilibria. Then an algorithm is shown in order to find all possible Nash equilibrium pairs of locations. The minimization of the social cost leads to a Nash equilibrium. An example shows that there may exist multiple Nash equilibria which are not minimizers of the social cost.     

Computing equilibria of Cournot oligopoly models with mixed-integer quantities

We consider Cournot oligopoly models in which some variables represent indivisible quantities. These models can be addressed by computing equilibria of Nash equilibrium problems in which the players solve mixed-integer nonlinear problems. In the literature there are no methods to compute equilibria of this type of Nash games. We propose a Jacobi-type method for computing solutions of Nash equilibrium problems with mixed-integer variables. This algorithm is a generalization of a recently proposed method for the solution of discrete so-called “2-groups partitionable” Nash equilibrium problems. We prove that our algorithm converges in a finite number of iterations to approximate equilibria under reasonable conditions. Moreover, we give conditions for the existence of approximate equilibria. Finally, we give numerical results to show the effectiveness of the proposed method.     

Several extensions of the Abian–Brown Fixed Point Theorem and their applications to extended and generalized Nash equilibria on chain-complete posets

In this paper, we provide several extensions of the Abian–Brown Fixed Point Theorem from single-valued mappings to set-valued mappings on chain-complete posets. Then we examine some non-monetized, non-cooperative games where both the collections of the strategies and the ranges of the utilities for the players are posets. By applying the extensions of the Abian–Brown Fixed Point Theorem and by applying the order-preserving property of mappings, we prove some existence theorems of extended and generalized Nash equilibria for non-monetized, non-cooperative games on chain-complete posets.     

A Shared-Constraint Approach to Multi-Leader Multi-Follower Games

Multi-leader multi-follower games are a class of hierarchical games in which a collection of leaders compete in a Nash game constrained by the equilibrium conditions of another Nash game amongst the followers. The resulting equilibrium problem with equilibrium constraints is complicated by nonconvex agent problems and therefore providing tractable conditions for existence of global or even local equilibria has proved challenging. Consequently, much of the extant research on this topic is either model specific or relies on weaker notions of equilibria. We consider a modified formulation in which every leader is cognizant of the equilibrium constraints of all leaders. Equilibria of this modified game contain the equilibria, if any, of the original game. The new formulation has a constraint structure called shared constraints, and our main result shows that if the leader objectives admit a potential function, the global minimizers of the potential function over this shared constraint are equilibria of the modified formulation. We provide another existence result using fixed point theory that does not require potentiality. Additionally, local minima, B-stationary, and strong-stationary points of this minimization problem are shown to be local Nash equilibria, Nash B-stationary, and Nash strong-stationary points of the corresponding multi-leader multi-follower game. We demonstrate the relationship between variational equilibria associated with this modified shared-constraint game and equilibria of the original game from the standpoint of the multiplier sets and show how equilibria of the original formulation may be recovered. We note through several examples that such potential multi-leader multi-follower games capture a breadth of application problems of interest and demonstrate our findings on a multi-leader multi-follower Cournot game.     

Minimizing maximum risk for fair network connection with interval data

In this paper we introduce a minimax model for network connection problems with interval parameters. We consider how to connect given nodes in a network with a path or a spanning tree under a given budget, where each link is associated with an interval and can be established at a cost of any value in the interval. The quality of an individual link （or the risk of link failure, etc.） depends on its construction cost and associated interval. To achieve fairness of the network connection, our model aims at the minimization of the maximum risk over all links used. We propose two algorithms that find optimal paths and spanning trees in polynomial time, respectively. The polynomial solvability indicates salient difference between our minimax model and the model of robust deviation criterion for network connection with interval data, which gives rise to NP-hard optimization problems.     

Optimization reformulations of the generalized Nash equilibrium problem using Nikaido-Isoda-type functions

We consider the generalized Nash equilibrium problem which, in contrast to the standard Nash equilibrium problem, allows joint constraints of all players involved in the game. Using a regularized Nikaido-Isoda-function, we then present three optimization problems related to the generalized Nash equilibrium problem. The first optimization problem is a complete reformulation of the generalized Nash game in the sense that the global minima are precisely the solutions of the game. However, this reformulation is nonsmooth. We then modify this approach and obtain a smooth constrained optimization problem whose global minima correspond to so-called normalized Nash equilibria. The third approach uses the difference of two regularized Nikaido-Isoda-functions in order to get a smooth unconstrained optimization problem whose global minima are, once again, precisely the normalized Nash equilibria. Conditions for stationary points to be global minima of the two smooth optimization problems are also given. Some numerical results illustrate the behaviour of our approaches.