Time-consistency of cooperative solutions

Management in complex social and economics networks includes elements of cooperative behavior or full cooperation between agents involved in decision making process. The most appropriate mathematical tool for modeling in this case is the mathematical theory of cooperative games. Unfortunately the classical cooperative game theory considers cooperation as one-shot interaction between the decision makers and for this reason can not be used for modeling of dynamic interactions arising in long term strategic management. The theory of cooperative differential games is the most adequate tool for modeling strategic management development on a given time interval. The use of this theory from the beginning poses the problems connected with dynamic stability (time-consistency) of optimal cooperative solutions. The consideration of optimality principles taken from the classical cooperative game theory shows the time-inconsistency and thus non applicability of these principles in strategic management. In this paper the methods of construction of time consistent solutions is proposed for the problems of strategic management in social and economic networks. The authors tried to present a rather complicated material on acceptable level. Theory is illustrated with a number of examples and a more comprehensive analysis of joint venture.

Axiomatic Attribution for Multilinear Functions

We study the attribution problem, that is, the problem of attributing a change in the value of a characteristic function to its independent variables. We make three contributions. First, we propose a formalization of the problem based on a standard cost sharing model. Second, we show that there is a unique attribution method that satisfies Dummy, Additivity, Conditional Nonnegativity, Affine Scale Invariance, and Anonymity for all characteristic functions that are the sum of a multilinear function and an additive function. We term this the Aumann-Shapley-Shubik method. Conversely, we show that such a uniqueness result does not hold for characteristic functions outside this class. Third, we study multilinear characteristic functions in detail; we describe a computationally efficient implementation of the Aumann-Shapley-Shubik method and discuss practical applications to pay-per-click advertising and portfolio analysis.Comment: 21 pages, 2 figures, updated version for EC '1

Finite horizon bargaining and the consistent field

This paper explores the relationships between noncooperative bargaining games and the consistent value for non-transferable utility (NTU) cooperative games. A dynamic approach to the consistent value for NTU games is introduced: the consistent vector field. The main contribution of the paper is to show that the consistent field is intimately related to the concept of subgame perfection for finite horizon noncooperative bargaining games, as the horizon goes to infinity and the cost of delay goes to zero. The solutions of the dynamic system associated to the consistent field characterize the subgame perfect equilibrium payoffs of the noncooperative bargaining games. We show that for transferable utility, hyperplane and pure bargaining games, the dynamics of the consistent fields converge globally to the unique consistent value. However, in the general NTU case, the dynamics of the consistent field can be complex. An example is constructed where the consistent field has cyclic solutions; moreover, the finite horizon subgame perfect equilibria do not approach the consistent value.Noncooperative bargaining games, consistent value, consistent

Cost Sharing, Differential Games, and the Moulin-Shenker Rule

The Moulin-Shenker rule (Sprumont (1998)) is a nonlinear solution concept for solving heterogeneous cost sharing problems. The first part of the paper shows an axiomatic characterization of this solution using bounds on cost shares and consistency. The second part is devoted to differential games for heterogeneous production problems. It is shown for 2-player games that by an appropriate choice of the game dynamics there is essentially a unique Markov perfect Nash equilibrium. An axiomatic analysis follows for the appropriate game dynamics, which leads in turn to a strategic characterization of the Moulin-Shenker rule.

Game Theory

The Special Issue “Game Theory” of the journal Mathematics provides a collection of papers that represent modern trends in mathematical game theory and its applications. The works address the problem of constructing and implementation of solution concepts based on classical optimality principles in different classes of games. In the case of non-cooperative behavior of players, the Nash equilibrium as a basic optimality principle is considered in both static and dynamic game settings. In the case of cooperative behavior of players, the situation is more complicated. As is seen from presented papers, the direct use of cooperative optimality principles in dynamic and differential games may bring time or subgame inconsistency of a solution which makes the cooperative schemes unsustainable. The notion of time or subgame consistency is crucial to the success of cooperation in a dynamic framework. In the works devoted to dynamic or differential games, this problem is analyzed and the special regularization procedures proposed to achieve time or subgame consistency of cooperative solutions. Among others, special attention in the presented book is paid to the construction of characteristic functions which determine the power of coalitions in games. The book contains many multi-disciplinary works applied to economic and environmental applications in a coherent manner

A cognitive hierarchy theory of one-shot games: Some preliminary results

Strategic thinking, best-response, and mutual consistency (equilibrium) are three key modelling principles in noncooperative game theory. This paper relaxes mutual consistency to predict how players are likely to behave in in one-shot games before they can learn to equilibrate. We introduce a one-parameter cognitive hierarchy (CH) model to predict behavior in one-shot games, and initial conditions in repeated games. The CH approach assumes that players use k steps of reasoning with frequency f (k). Zero-step players randomize. Players using k (≥ 1) steps best respond given partially rational expectations about what players doing 0 through k - 1 steps actually choose. A simple axiom which expresses the intuition that steps of thinking are increasingly constrained by working memory, implies that f (k) has a Poisson distribution (characterized by a mean number of thinking steps τ ). The CH model converges to dominance-solvable equilibria when τ is large, predicts monotonic entry in binary entry games for τ < 1:25, and predicts effects of group size which are not predicted by Nash equilibrium. Best-fitting values of τ have an interquartile range of (.98,2.40) and a median of 1.65 across 80 experimental samples of matrix games, entry games, mixed-equilibrium games, and dominance-solvable p-beauty contests. The CH model also has economic value because subjects would have raised their earnings substantially if they had best-responded to model forecasts instead of making the choices they did

THE NORWEGIAN SPRING-SPAWNING HERRING FISHERY: A STYLIZED GAME MODEL

This paper presents an empirically based, game-theoretic model of the exploitation of the Norwegian spring-spawning herring stock, also known as the Atlanto-Scandian herring stock. The model involves five exploiters; Norway, Iceland, the Faroe Islands, the EU, and Russia and an explicit, stochastic migratory behavior of the stock. Under these conditions Markov Perfect (Nash) equilibrium game strategies are calculated and compared to the jointly optimal exploitation pattern. Not surprisingly, it turns out that the solution to the competitive game is hugely inefficient, leading very quickly to the virtual exhaustion of the resource. The scope for cooperative agreements involving the calculation of Shapley values is investigated. Although the grand coalition of all players maximizes overall benefits, such a coalition can hardly be stable over time unless side payments are possible.Resource /Energy Economics and Policy,

Cooperative game theory and its application to natural, environmental, and water resource issues : 2. application to natural and environmental resources

This paper provides a review of various applications of cooperative game theory (CGT) to issues of natural and environmental resources. With an increase in the level of competition over environmental and natural resources, the incidents of disputes have been at the center of allocation agreements. The paper reviews the cases of common pool resources such as fisheries and forests, and cases of environmental pollution such as acid rain, flow, and stock pollution. In addition to providing examples of cooperative solutions to allocation problems, the conclusion from this review suggests that cooperation over scarce environmental and natural resources is possible under a variety of physical conditions and institutional arrangements. CGT applications to international fishery disputes are especially useful in that they have been making headway in policy-related agreements among states and regions of the world. Forest applications are more local in nature, but of great relevance in solving disputes among communities and various levels of governments.Environmental Economics&Policies,Fisheries&Aquaculture,Common Property Resource Development,Economic Theory&Research,Ecosystems and Natural Habitats

Cooperative game theory and its application to natural, environmental, and water resource issues : 3. application to water resources

This paper reviews various applications of cooperative game theory (CGT) to issues of water resources. With an increase in the competition over various water resources, the incidents of disputes have been in the center of allocation agreements. The paper reviews the cases of various water uses, such as multi-objective water projects, irrigation, groundwater, hydropower, urban water supply, wastewater, and transboundary water disputes. In addition to providing examples of cooperative solutions to allocation problems, the conclusion from this review suggests that cooperation over scarce water resources is possible under a variety of physical conditions and institutional arrangements. In particular, the various approaches for cost sharing and for allocation of physical water infrastructure and flow can serve as a basis for stable and efficient agreement, such that long-term investments in water projects are profitable and sustainable. The latter point is especially important, given recent developments in water policy in various countries and regional institutions such as the European Union (Water Framework Directive), calling for full cost recovery of investments and operation and maintenance in water projects. The CGT approaches discussed and demonstrated in this paper can provide a solid basis for finding possible and stable cost-sharing arrangements.Town Water Supply and Sanitation,Environmental Economics&Policies,Water Supply and Sanitation Governance and Institutions,Water Supply and Systems,Water and Industry