Coalition Of Players Research Articles

The Shapley Value of a Patent Licensing Game: the Asymptotic Equivalence to Non-cooperative Results

We consider a game in characteristic form played by firms and an outside patent holder of a cost-reducing innovation. The worth of a coalition of players is the total Cournot profit the coalition can guarantee to obtain when it operates an optimal number of its firms while the complement operates any number of its firms as to minimize the profit of the coalition. Only firms in a coalition with the patent holder are allowed to use the efficient technology. We prove that when the number of firms is large, the Shapley value of the patent holder approximates the payoff he obtains in the non-cooperative auction game traditionally studied in the literature.

Complete axiomatization and decidability of Alternating-time temporal logic

Alternating-time Temporal Logic (ATL), introduced by Alur, Henzinger and Kupferman, is a logical formalism for the specification and verification of open systems involving multiple autonomous players (agents, components). In particular, this logic allows for the explicit expression of coalition abilities in such systems, modelled as infinite transition games between the coalition and its complement. Formally, ATL is a non-normal multi-modal extension of CTL (regarded as a one-player fragment of ATL) with temporal operators indexed by coalitions of players, and thus expressing selective quantification over those paths which can be effected as outcomes of infinite transition games between the coalition and its complement. We present a sound and complete axiomatization of the logic ATL, based on Pauly's axiomatization of his Coalition Logic, augmented with axioms and rules for fixed point formulae characterizing the temporal operators. The completeness proof is by construction of a bounded branching tree model for each ATL-consistent formula. These models can be folded into finite models, thus rendering the finite model property for ATL. We also describe an automata-based decision procedure for ATL by translating the satisfiability problem to the nonemptiness problem for alternating automata on infinite trees. When considering formulae over a fixed finite set of players the decidability problem is shown to be EXPTIME-complete.

Consistent solutions for cooperative games with externalities

In order to calculate the worth of a coalition of players, the coalition needs to predict the actions of outsiders. We propose that, for a given solution concept, such predictions should be made by applying the solution concept to the “reduced society” consisting of the non-members. We illustrate by computing the r-core for the case of Bertrand competition with differentiated commodities.

Cooperative games of choosing partners and forming coalitions in the marketplace

Two games of interacting between a coalition of players in a marketplace and the residual players acting there are discussed, along with two approaches to fair imputation of gains of coalitions in cooperative games that are based on the concepts of the Shapley vector and core of a cooperative game. In the first game, which is an antagonistic one, the residual players try to minimize the coalition's gain, whereas in the second game, which is a noncooperative one, they try to maximize their own gain as a coalition. A meaningful interpretation of possible relations between gains and Nash equilibrium strategies in both games considered as those played between a coalition of firms and its surrounding in a particular marketplace in the framework of two classes of n- person games is presented. A particular class of games of choosing partners and forming coalitions in which models of firms operating in the marketplace are those with linear constraints and utility functions being sums of linear and bilinear functions of two corresponding vector arguments is analyzed, and a set of maximin problems on polyhedral sets of connected strategies which the problem of choosing a coalition for a particular firm is reducible to are formulated based on the firm models of the considered kind.

A Modal Logic for Coalitional Power in Games

We present a modal logic for reasoning about what groups of agents can bring about by collective action. Given a set of states, we introduce game frames which associate with every state a strategic game among the agents. Game frames are essentially extensive games of perfect information with simultaneous actions, where every action profile is associated with a new state, the outcome of the game. A coalition of players is effective for a set of states in a game if the coalition can guarantee the outcome of the game to lie in . We propose a modal logic (Coalition Logic) to formalize reasoning about effectivity in game frames, where expresses that coalition is effective for . An axiomatization is presented and completeness proved. Coalition Logic provides a unifying game-theoretic view of modal logic: Since nondeterministic processes and extensive games without parallel moves emerge as particular instances of game frames, normal and non-normal modal logics correspond to 1- and 2-player versions of Coalition Logic. The satisfiability problem for Coalition Logic is shown to be PSPACE-complete.

Perfect Information Leader Election in log* n+O(1) Rounds

In the leader election problem, n players wish to elect a random leader. The difficulty is that some coalition of players may conspire to elect one of its own members. We adopt the perfect information model: all communication is by broadcast, and the bad players have unlimited computational power. Protocols proceed in rounds: although players are synchronized between rounds, within each round the bad players may wait to see the inputs of the good players. A protocol is called resilient if a good leader is elected with probability bounded away from 0. We give a simple, constructive leader election protocol that is resilient against coalitions of size βn, for any β<1/2. Our protocol takes log*n+O(1) rounds, each player sending at most logn bits per round. For any constant k, our protocol can be modified to take k rounds and offer resilience against coalitions of size εn/(log(k)n)3, were ε is a small enough constant and log(k) denotes the logarithm iterated k times. This is constructive for k⩾3. The primary component of the above protocols is a new collective sampling protocol: for a set S of large enough (polynomial) size, this protocol generates an element s∈S in a single round so that for any subset T⊂S, Pr[s∈T]⩽|T||S|α(1−β) for a constant α>0.

The influence of variables on pseudo-Boolean functions with applications to game theory and multicriteria decision making

The power of players in a collective decision process is a central issue in game theory. For this reason, the concept of influence of players on a simple game has been introduced. More generally, the influence of variables on Boolean functions has been defined and studied. We extend this concept to pseudo-Boolean functions, thus making it possible to appraise the degree of influence of any coalition of players in cooperative games. In the case of monotone games, we also point out the links with the concept of interaction among players. Although they do not have the same meaning at all, both influence and interaction functions coincide on singletons with the so-called Banzhaf power index. We also define the influence of variables on continuous extensions of pseudo-Boolean functions. In particular, the Lovász extension, also called discrete Choquet integral, is used in multicriteria decision making problems as an aggregation operator. In such problems, the degree of influence of decision criteria on the aggregation process can then be quite relevant information. We give the explicit form of this influence for the Choquet integral and its classical particular cases.

Nonsymmetric Values of Nonatomic and Mixed Games

This paper presents a new unifying approach to the study of nonsymmetric (or quasi-) values of nonatomic and mixed games. A family of path values is defined, using an appropriate generalization of Mertens diagonal formula. A path value possesses the following intuitive description: consider a function (path) γ attaching to each player a distribution function on [0, 1]. We think of players as arriving randomly and independently to a meeting when the arrival time of a player is distributed according to γ. Each player's payoff is defined as his marginal contribution to the coalition of players that have arrived earlier. Under certain conditions on a path, different subspaces of mixed games (pNA, pM, bv′FL) are shown to be in the domain of the path value. The family of path values turns out to be very wide—we show that on pNA, pM and their subspaces the path values are essentially the basic construction blocks (extreme points) of quasi-values.

Easy weighted majority games

In a weighted majority game each player has a positive integer weight and there is a positive integer quota. A coalition of players is winning (losing) if the sum of the weights of its members exceeds (does not exceed) the quota. A player is pivotal for a coalition if her omission changes it from a winning to a losing one. Most game theoretic measures of the power of a player involve the computation of the number of coalitions for which that player is pivotal. Prasad and Kelly [Prasad, K., Kelly, J.S., 1990. NP-completeness of some problems concerning voting games. International Journal of Game Theory 19, 1–9] prove that the problem of determining whether or not there exists a coalition for which a given player is pivotal is NP-complete. They also prove that counting the number of coalitions for which a given player is pivotal is #P-complete. In the present paper we exhibit classes of weighted majority games for which these problems are easy.

The Compromise Value For Cooperative Games With Random Payoffs

The compromise value is introduced for cooperative games with random payoffs, that is, for cooperative games where the payoff to a coalition of players is a random variable. It is a compromise between utopia payoffs and minimal rights. This solution concept is based on the compromise value for NTU games and the 1/2 -value for TU games. It is shown that the nonempty core of a game is bounded by the utopia payoffs and the minimal rights. Further, we show that the compromise value of a cooperative game with random payoffs is determined by the 1/2 -value of a related TU game if the players have special types of preferences. Finally, the compromise value and the marginal value, which is defined as the average of the marginal vectors, coincide on the class of one- and two-person games.

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.

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.

On the rate of taxation in a cooperative bin packing game

We investigate a cooperative game with two types of players envolved: Every player of the first type owns a unit size bin, and every player of the second type owns an item of size at most one. The value of a coalition of players is defined to be the maximum overall size of packed items over all packings of the items owned by the coalition into the bins owned by the coalition. We prove that fore=1/3 this cooperative bin packing game ise-balanced in the taxation model of Faigle and Kern (1993).

The Shapley Value for n-Person Games in Generalized Characteristic Function Form

This paper is a first study of n-person games in coalitional form in which the characteristic functions are defined on all possible orders in coalitions of players. Such an approach seems to be useful in modeling some economic situations as well as some subtle relations between players in coalitions. Our main objective in this paper is to introduce axiomatically a unique value on the more general class of games indicated above. For the classical cooperative TU games our value coincides with the well-known Shapley value. We also discuss probabilistic values on the space of generalized games.

The Shapley value for cooperative games under precedence constraints

Cooperative games are considered where only those coalitions of players are feasible that respect a given precedence structure on the set of players. Strengthening the classical symmetry axiom, we obtain three axioms that give rise to a unique Shapley value in this model. The Shapley value is seen to reflect the expected marginal contribution of a player to a feasible random coalition, which allows us to evaluate the Shapley value nondeterministically. We show that every exact algorithm for the Shapley value requires an exponential number of operations already in the classical case and that even restriction to simple games is #P-hard in general. Furthermore, we outline how the multi-choice cooperative games of Hsiao and Raghavan can be treated in our context, which leads to a Shapley value that does not depend on pre-assigned weights. Finally, the relationship between the Shapley value and the permission value of Gilles, Owen and van den Brink is discussed. Both refer to formally similar models of cooperative games but reflect complementary interpretations of the precedence constraints and thus give rise to fundamentally different solution concepts.

Large symmetric games are characterized by completeness of the desirability relation

The paper presents a characterization of continuous cooperative games (set functions) which are monotonic functions of countably additive non-atomic measures. The characterization is done through a natural desirablity relation defined on the set of coalitions of players. A coalition S is at least as desirable as a coalition T (with respect to a given game υ (in colational form)), if for each coalition U that is disjoint from S ∪ T, υ( S ∪ U) ⩾ υ( T ∪ U). The characterization asserts, that a game υ is of the form υ = f · μ, where μ is a non-atomic signed measure and f is a monotonic and continuous function on the range of μ, if, and only if, it is in pNA′ (i.e., it is a uniform limit of polynomials in non-atomic measures or equivalently it is uniformly continuous function in the NA-topology) and has a complete desirability relation.

Communication and its cost in graph-restricted games

We consider cooperative games which are modified by subordination to communication networks. Two players i and j will be able to cooperate only if they pay for the cost w ij of their communication link (i,j). Coalitions of players are treated similarly and a new characteristic function form game is developed thus. We also examine incentive for players to cooperate in such situations as well as a related index of a player's communicative strength.

Necessary and sufficient conditions for stability of effectivity functions

An effectivity functionE assigns to every coalitionS of players a familyE (S) of subsetsB of an outcome setA such thatS can force the outcome to belong to any of the setsB inE (S). The effectivity functionE is stable if for every preference profile there is an outcomex with the property that there is no coalitionS and subsetB ofA such thatB eE (S) and each player inS prefers everyy eB tox. The paper gives a necessary and sufficient condition for an effectivity function to be stable.

The Cooperative as a Coalition: A Game‐Theoretic Approach

Most formal models of the economic behavior of farmer cooperatives picture that behavior as deriving from the optimization of a single objective function determined by a single agent (Helmberger and Hoos 1962), by a group of agents with identical goals (Phillips), or by simple (nonstrategic) majority-rule voting of the membership on a single issue. Models incorporating voting assume single-peaked preferences and no logrolling (interdependent voting) between issues; therefore, no voting paradoxes arise and the cooperative's objective is determined by the preferences of its median member (Knoeber and Baumer, Zusman). With few exceptions (Zusman) previous models thus abstract from the thorny issue of group choice in cooperatives when members have at least partially divergent goals and engage in strategic behavior. The lack of attention to group choice in previous models reflects the primary interest of earlier theoreticians in industrial organization issues. These authors concentrated on how the pricing and output decisions of cooperatives might differ from those of investorowned firms (IOFs) and affect the competitive balance of an industry. Theoretical analysis of how cooperatives allocate costs and benefits among a heterogenous membership, an issue that has become increasingly exigent as cooperatives have developed larger and more diverse memberships, was largely ignored. Cooperatives face many decisions, however, in which members' preferences cannot be assumed to be homogenous. Examples include the pricing of different services to members, including the possibility of differential pricing based on members' patronage; the location of facilities; and the allocation of overhead costs and pool receipts. Furthermore, the prefernces of management and the board of directors on many of these issues may differ from those of the rank-and-file membership. Although both the management literature and many theorists have informally discussed decision making in the context of heterogenous preferences (e.g., Helmberger and Hoos 1962, p. 290; Helmberger; Trifon), there is a need to develop models that address this issue explicitly and in so doing suggest alternative ways of structuring cooperatives to deal with group choice. The main thesis of this paper is that many decisions involving the allocation of costs and benefits within cooperatives can be usefully conceptualized as n-person cooperative games.' In the parlance of game theory, games are games in which players are allowed to communicate and make binding commitments with one another. The theory of games is usually used to model situations in which there are gains from joint action by a potential coalition of players, but where the players must bargain among themselves about how the net benefits of the joint action are to be shared. Failure to agree on an allocation of net benefits among players prevents the coalition from forming (Roth).

Quota vs positional power in four-person apex games

In coalition formation experiments, which coalitions will form and how players in coalitions will allocate their jointly gained rewards are typically predicted as a function of the players' relative power. In this paper, we isolated two logically independent sources of a player's power: the player's contribution to the rewards obtainable by coalitions of which he may be a member (“quota power”), and the number of distinct coalitions which a player may join (“positional power”). The separation of the two types of power is clearly shown in apex games, where a single player (Apex) attempts to lure any of the other players (Base) from the coalition of all Bases. A series of four-person computer-controlled apex and nonapex characteristic function games, varying with respect to the relationship of quota power to positional power, were played by 15 quartets of male players. Quota power was strongly manifested in all outcome measures, while positional power appeared only as a bias toward equal allocations of reward when coalitions among players of equal positional power formed. Neither of two social psychological theories that are applicable, pivotal power and weighted probability theory, predicted coalition frequencies. In a test of five solution concepts, the competitive bargaining set predicted payoff allocations better than either of the two aforementioned theories, the kernel, or an equal-split model.