Equilibrium Investment Strategies in Foreign Environmental Projects

This paper proposes a two-player, finite-horizon differential game model to analyze joint implementation in environmental projects, one of the flexible mechanisms considered in the Kyoto Protocol. Using a nonlinear specification for the damage costs, we show that, if joint implementation is available, both players will invest in environmental projects in their own location and one of the players will invest abroad, the identity of that player changing at most once during the planning horizon.Communicated by G. LeitmannResearch supported by FQRSC, Quéebec and NSERC, Canada. The second author’s research was partially supported by MEC under Project SEJ2005-03858 and by JCYL under Project VA045A06, confinanced by FEDER funds.     

Can international environmental cooperation be bought?

In this paper a two-stage game of international environmental agreement formation with asymmetric countries is analytically solved. The equilibrium of the game makes it possible to determine the size and composition of a stable agreement. Two cases are studied. In the first case, countries differ only in abatement costs, while in the second case, they differ in environmental damages. In both cases, two different institutional settings, one without transfers and another with transfers, are considered. The results establish that the asymmetry assumption has no important effects on the scope of cooperation in comparison with the symmetric case if transfers are not used or abatement costs represent the only difference among countries. However, when the only difference is in environmental damages, the level of cooperation that can be bought through a self-financed transfer scheme increases with the degree of asymmetry.     

Economics of collective monitoring: a study of environmentally constrained electricity generators

This paper investigates the costs of monitoring of a distributed multi-agent economic activity in the presence of constraints on the agents’ joint outputs. If the regulator monitors agents individually she calculates each agent’s optimal contribution to the constrains by solving a constrained welfare-maximisation problem. This will maximise welfare but may be expensive because monitoring cost rises with the number of agents. Alternatively, the regulator could monitor agents collectively, using a detector, or detectors, to observe if each constraint is jointly satisfied. This will ease implementation cost, but lower welfare. We define the welfare difference between each regime of monitoring for a fairly inclusive electricity generation model and formulate some predictions. The behaviour of two generators in a coupled-constrained, three-node case study reproduces these predictions. We find that the welfare loss from collective monitoring can be small if the constraints are tight. We also learn that, under either regime, the imposition of transmission and environmental restrictions may benefit the less efficient generator and shift surplus share towards the emitters, decreasing consumer surplus.     

AN INTERNATIONAL MARINE POLLUTANT SINK IN AN ASYMMETRIC ENVIRONMENTAL TECHNOLOGY GAME

Abstract A differential game with N countries as players is developed for the analysis of a marine resource with open access used by the countries as a pollutant sink and for production of an ecosystem good. Consumption, investment, abatement, and environmental research sectors may differ across countries. Under the assumption that property rights are well defined within the sovereignty of each country although there is a lack of enforcement in the international area, the game is solved for the feedback Nash equilibrium, using the Isaacs–Bellman dynamic programming technique, identifying the nonlinear feedback Nash flows of consumption, investment, harvest effort, abatement, and environmental research effort in each country, which is then compared to the Pareto optimal global solution.     

A two-stage supply chain coordination mechanism considering price sensitive demand and quantity discounts

This paper explores the coordination between a supplier and a buyer within a decentralized supply chain, through the use of quantity discounts in a game theoretic model. Within this model, the players face inventory and pricing decisions. We propose both cooperative and non-cooperative approaches considering that the product traded experiences a price sensitive demand. In the first case, we study the dynamics of the game from the supplier's side as the leader in the negotiation obtaining a Stackelberg equilibrium, and then show how the payoff of this player could still improve from this point. In the second case, a cooperative model is formulated, where decisions are taken simultaneously, emulating a centralized firm, showing the benefits of the cooperation between the players. We further formulate a pricing game, where the buyer is allowed to set different prices to the final customer as a reaction to the supplier's discount decisions. For the latter we investigate the difference between feasibility of implementing a retail discount given a current coordination mechanism and without it. Finally the implications of transportation costs are analyzed in the quantity discount schedule. Our findings are illustrated with a numerical example showing the difference in the players’ payoff in each case and the optimal strategies, comparing in each case our results with existing work.     

Common resources, optimality and taxes in dynamic games with increasing number of players

In the paper we examine the problem of exploitation of a common renewable resource. We use two kinds of models of this problem: games with finitely many players and games with a continuum of players. Equilibria in both cases are calculated and the payoffs are compared with payoffs in the situation in which each player behaves as a single owner of the resource. Various concepts of optimality are considered: optimality in the sense of property, Pareto optimality, maximization of a social welfare function of specific type and environmental sustainability. Another issue is the problem of enforcement of assumed profiles by so-called “linear” tax systems. Special attention is paid to the comparison of games with finitely many players and their continuum-of-players limit game.     

Location game on disjoint line segments

Two players are endowed with resources for setting up N locations on K line segments of identical length, with N > K ≥ 1. The players alternately choose these locations (possibly in batches of more than one in each round) in order to secure the area closer to their locations than that of their rival’s. The player with the highest secured area wins the game and otherwise the game ends in a tie. Earlier research has shown that, if an analogous game is played on disjoint circles, the second mover advantage is in place only if K = 1, while for K > 1 both players have a tying strategy. It was also shown that these results hold for line segments of identical length when rules of the game additionally require players to take exactly one location in the first round. In this paper we show that the second mover advantage is still in place for K ≥ 1 and 2K − 1 ≤ N, even if the additional restriction is dropped, while K ≤ N < 2K − 1 results in the first mover advantage. Our results allow us to draw conclusions about a natural variant of the game, where the resource mobility constraint is more stringent so that in each round each player chooses a single location and we show that the second mover advantage re-appears for K ≤ N < 2K − 1 if K is an even number. In all the cases the losing player has a strategy guaranteeing him arbitrarily small loss.     

Multilayers in a modulated stochastic game

We are concerned with an antagonistic stochastic game between two players A and B which finds applications in economics and warfare. The actions of the players are manifested by a series of strikes of random magnitudes at random times exerted by each player against his opponent. Each of the assaults inflicts a random damage to enemy's vital areas. In contrast with traditional games, in our setting, each player can endure multiple strikes before perishing. Predicting the ruin time (exit) of player A, along with the total amount of casualties to both players at the exit is a main objective of this work. In contrast to the time sensitive analysis (earlier developed to refine the information on the game) we insert auxiliary control levels, which both players will cross in due game before the ruin of A. This gives A (and also B) an additional opportunity to reevaluate his strategy and change the course of the game. We formalize such a game and also allow the real time information about the game to be randomly delayed. The delayed exit time, cumulative casualties to both players, and prior crossings are all obtained in a closed-form joint functional.     

Activity optimization games with complementarity

This paper presents a general class of cooperative games, activity optimization games with complementarity, and explores their attributes and diverse applications. These games involve interactions by a number of players controlling complex systems of complementary activities. Examples include a general welfare game; a game with procurement of inputs for production; a game with investment and production; and an activity selection game. Every game in this class is a convex game and hence has a nonempty core. Optimal activity levels and certain game solutions have well-behaved qualitative properties. Attractive computational features are noted.     

Variational inequality formulation for the games with random payoffs

We consider an n-player non-cooperative game with random payoffs and continuous strategy set for each player. The random payoffs of each player are defined using a finite dimensional random vector. We formulate this problem as a chance-constrained game by defining the payoff function of each player using a chance constraint. We first consider the case where the continuous strategy set of each player does not depend on the strategies of other players. If a random vector defining the payoffs of each player follows a multivariate elliptically symmetric distribution, we show that there exists a Nash equilibrium. We characterize the set of Nash equilibria using the solution set of a variational inequality (VI) problem. Next, we consider the case where the continuous strategy set of each player is defined by a shared constraint set. In this case, we show that there exists a generalized Nash equilibrium for elliptically symmetric distributed payoffs. Under certain conditions, we characterize the set of a generalized Nash equilibria using the solution set of a VI problem. As an application, the random payoff games arising from electricity market are studied under chance-constrained game framework.     

Nash equilibria in the two-player kidney exchange game

Kidney exchange programs have been set in several countries within national, regional or hospital frameworks, to increase the possibility of kidney patients being transplanted. For the case of hospital programs, it has been claimed that hospitals would benefit if they collaborated with each other, sharing their internal pools and allowing transplants involving patients of different hospitals. This claim led to the study of multi-hospital exchange markets. We propose a novel direction in this setting by modeling the exchange market as an integer programming game. The analysis of the strategic behavior of the entities participating in the kidney exchange game allowed us to prove that the most rational game outcome maximizes the social welfare and that it can be computed in polynomial time.     

Cooperation in dividing the cake

This paper defines models of cooperation among players partitioning a completely divisible good (such as a cake or a piece of land). The novelty of our approach lies in the players’ ability to form coalitions before the actual division of the good with the aim to maximize the average utility of the coalition. A social welfare function which takes into account coalitions drives the division. In addition, we derive a cooperative game which measures the performance of each coalition. This game is compared with the game in which players start cooperating only after the good has been portioned and has been allocated among the players. We show that a modified version of the game played before the division outperforms the game played after the division.     

The core of a game with a continuum of players and finite coalitions: The model and some results

In this paper we develop a new model of a cooperative game with a continuum of players. In our model, only finite coalitions - ones containing only finite numbers of players - are permitted to form. Outcomes of cooperative behavior are attainable by partitions of the players into finite coalitions: this is appropriate in view of our restrictions on coalition formation. Once feasible outcomes are properly defined, the core concept is standard - no permissible coalition can improve upon its outcome. We provide a sufficient condition for the nonemptiness of the core in the case where the players can be divided into a finite number of types. This result is applied to a market game and the nonemptiness of the core of the market game is stated under considerably weak conditions (but with finite types). In addition, it is illustrated that the framework applies to assignment games with a continuum of players.     

COLLECTIVE DECISION MAKING IN AN INTERNATIONAL RIVER POLLUTION MODEL

In this paper, we study international river pollution problems. We introduce a model in which countries located along a river from upstream to downstream derive benefits from causing pollution, but also incur environmental costs from experiencing its own pollution and the pollution of all its upstream countries. The total welfare, being the sum of all benefits minus the sum of all costs, is maximized when all countries cooperate. Several principles from international water law are applied to find reasonable and fair distributions of the total welfare that can be obtained under full cooperation. Such a distribution of the welfare at efficient pollution levels can be implemented by monetary compensations.     

The core of an economy with multilateral environmental externalities

When environmental externalities are international — i.e. transfrontier — they most often are multilateral and embody public good characteristics. Improving upon inefficient laissez-faire equilibria requires voluntary cooperation for which the game-theoretic core concept provides optimal outcomes that have interesting properties against free riding. To define the core, however, the characteristic function of the game associated with the economy (which specifies the payoff achievable by each possible coalition of players—here, the countries) must also reflect in each case the behavior of the players which are not members of the coalition. This has been for a long time a disputed issue in the theory of the core of economies with externalities. Among the several assumptions that can be made as to this behaviour, a plausible one is defined in this paper, for which it is shown that the core of the game is nonempty. The proof is constructive in the sense that it exhibits a strategy (specifying an explicit coordinated abatement policy and including financial transfers) that has the desired property of nondomination by any proper coalition of countries, given the assumed behavior of the other countries. This strategy is also shown to have an equilibrium interpretation in the economic model.     

Node-Consistent Shapley Value for Games Played over Event Trees with Random Terminal Time

We consider a class of dynamic games played over an event tree, with random terminal time. We assume that the players wish to jointly optimize their payoffs throughout the whole planning horizon and adopt the Shapley value to share the joint cooperative outcome. We devise a node-consistent decomposition of the Shapley value, which means that in any node of the event tree, the players prefer to stick to cooperation and to continue implementing the Shapley value rather than switching to noncooperation. For each node and each player, we provide two payment values, one that applies if the game terminates at that node and the other if the game continues. We illustrate our results with an example of pollution control.     

On the role of chance moves and information in two-person games

The value of information has been the subject of many studies in a strategic context. The central question in these studies is how valuable the information hidden in the chance moves of a game is for one or more of the players. Generally speaking, only the extra possibilities that are beneficial for the players have been considered so far. In this note we study the value of information for a special class of two-person games. For these games we also investigate how “badly” the players can do, both with and without knowing the result of the chance move. In this way one can determine to what extent the players are restricted in their possibilities by the fact that some information is hidden in the chance moves of the games. This allows for a comparison of the influence of the chance move to the control that the players have over the game result.     

On finite tetraprimary groups

A three-player game is considered in which the first and second players have dynamic superiority over the third player. Two fixed time points are specified. The game ends if either the first player captures the third player at the first time point, or the second player captures the third player at the second time point. We analyze a situation when the initial positions in the game are such that neither the first nor the second player alone can capture the third player at the specified points of time. We propose sufficient conditions on the parameters of the game under which, for given initial states of the players, the first and second players by applying some controls can guarantee that one of them will meet the third player at the prescribed moment. Simulation results for a model example are also presented.     

An extension of the Basic Constraint Qualification to nonconvex vector optimization problems

In this paper a Basic Constraint Qualification is introduced for a nonconvex infinite-dimensional vector optimization problem extending the usual one from convex programming assuming the Hadamard differentiability of the maps. Corresponding KKT conditions are established by considering a decoupling of the constraint cone into half-spaces. This extension leads to generalized KKT conditions which are finer than the usual abstract multiplier rule. A second constraint qualification expressed directly in terms of the data is also introduced, which allows us to compute the contingent cone to the feasible set and, as a consequence, it is proven that this condition is a particular case of the first one. Relationship with other constraint qualifications in infinite-dimensional vector optimization, specially with the Kurcyuscz-Robinson-Zowe constraint qualification, are also given.     

An axiomatic characterization of the Owen–Shapley spatial power index

We present an axiomatic characterization of the Owen–Shapley spatial power index for the case where issues are elements of two-dimensional space. This characterization employs a version of the transfer condition, which enables us to unravel a spatial game into spatial games connected to unanimity games. The other axioms include two conditions concerned particularly with the spatial positions of the players, besides spatial versions of anonymity and dummy. The last condition says that dummy players can be left out in a specific way without changing the power of the other players. We show that this condition can be weakened to requiring dummies to have zero power if we add a condition of positional continuity. We also show that the axioms in our characterization(s) are logically independent.