Games In Characteristic Function Form Research Articles

Coalitional bargaining with consistent counterfactuals

We propose a new solution concept for TU cooperative games in characteristic function form, the SCOOP, which builds on the symmetric Nash Bargaining Solution (NBS) by adding a consistency requirement for negotiations inside every coalition. The SCOOP specifies the probability of success and the payoffs to each coalition. The players share the surplus of a coalition according to the NBS. The disagreement payoffs are computed as the expectation of payoffs in other coalitions, using a common probability distribution that is derived from the prior distribution. The predicted outcome can be probabilistic or deterministic, but only an efficient coalition can succeed with probability one. We discuss the necessary and sufficient conditions for an efficient solution. In either case, the SCOOP always exists, is generically unique for superadditive games, and is easy to compute. Moreover, in the spirit of the Nash program, we propose a non-cooperative protocol whose stationary equilibrium identifies the SCOOP as the limit equilibrium outcome.

A cooperative game in Asian international electric power integration

The paper examines effects of long-term international integration in the field of electric power industry. An advantage of integration is provoked by the fact that the cost of introducing new generation capacities significantly exceeds the cost of new power lines and transmission of the energy from existing power station of another country. When countries form a coalition, the problem is to allocate the coalition’s surplus over its participants. It can be solved by notions of cooperative game theory. The present investigation is based on the real data on six countries of the Northeast Asian region: Russia, Mongolia, China, North Korea, South Korea and Japan. Electric power system is described by the ORIRES model. This model optimizes power generation, power flows, and the development of generation capacities and power lines of the system. We formulate corresponding cooperative game in characteristic function form, and specify the Core and the Shapley value. Effects of international integration are discussed.

Computing optimal coalition structures in polynomial time

The optimal coalition structure determination problem is in general computationally hard. In this article, we identify some problem instances for which the space of possible coalition structures has a certain form and constructively prove that the problem is polynomial time solvable. Specifically, we consider games with an ordering over the players and introduce a distance metric for measuring the distance between any two structures. In terms of this metric, we define the property of monotonicity, meaning that coalition structures closer to the optimal, as measured by the metric, have higher value than those further away. Similarly, quasi-monotonicity means that part of the space of coalition structures is monotonic, while part of it is non-monotonic. (Quasi)-monotonicity is a property that can be satisfied by coalition games in characteristic function form and also those in partition function form. For a setting with a monotonic value function and a known player ordering, we prove that the optimal coalition structure determination problem is polynomial time solvable and devise such an algorithm using a greedy approach. We extend this algorithm to quasi-monotonic value functions and demonstrate how its time complexity improves from exponential to polynomial as the degree of monotonicity of the value function increases. We go further and consider a setting in which the value function is monotonic and an ordering over the players is known to exist but ordering itself is unknown. For this setting too, we prove that the coalition structure determination problem is polynomial time solvable and devise such an algorithm.

COALITION FORMATION AND BARGAINING PROTOCOLS: A REVIEW OF THE LITERATURE

AbstractThis paper offers a review of the vast literature regarding bargaining and coalition formation. This topic has been generally described as the attempt to provide strategic foundations to cooperative solution concepts. It can therefore be seen as the linking ring between the non‐cooperative and the cooperative game‐theoretic approach to coalition formation. Its central role in economic theory and its relatively long history that goes back to the Nash program have fostered a large academic production, including surveys. Nonetheless, this paper will focus on an aspect that is often neglected in the dedicated surveys: the specificities of the bargaining protocols leading to different outcomes. Although generally downgraded to the rank of details, the differences in bargaining protocols, even when minor, can cause significant changes in fundamental aspects such as the possibility to reach full cooperation, the distribution of final pay‐offs and the time taken to reach an agreement. Focused on externalities‐free games, therefore on bargaining protocols sustaining solution concepts for cooperative games in characteristic function form, the paper aims at providing a brief but exhaustive review of the topic that could result in a very useful tool for any researcher approaching the subject of coalitional bargaining.

Social enterprise tree network games

We investigate the cost allocation strategy associated with the problem of providing service among network users residing at nodes of a social network. We assume that the social network platform is established in a symmetric complete network. There is a cost associated with each link and the service between any pair of nodes can be delivered via a directed path. The example of a cost efficient solution for such network is a (non-rooted) minimum cost directed spanning tree. The network cost should be distributed among users who might have conflicting interests. The objective of this paper is to formulate the above cost allocation problem as a cooperative game, to be referred to as a Social Enterprise Tree Network (SETN) game, and develop a “fair” and efficient cost allocation scheme. The SETN game is related to the minimum cost spanning tree game. The profound difference is that in the minimum cost spanning tree game the service is delivered from some common source node to the rest of the network, while under the social network platform there is no source and the service is established through the interaction among all participating nodes. The input to our cost allocation problem is the optimal non-rooted directed spanning tree SETN. We formulate several associated SETN games in characteristic function form. Then we construct a couple of efficient cost allocation algorithms that find some points in the core of those SETN games and thus result in subsidy-free cost allocations.

English

We investigate the cost allocation strategy associated with the problem of providing some network service from source to a number of users, via the Minimum Cost Steiner Tree Network that spans the source and all the receivers. cost of such a Steiner tree network, is distributed among its receivers. objective of this paper is to develop a reasonably fair and computationally efficient cost allocation rule associated with the above cost allocation problem. Since finding the optimal Steiner tree is an NP-hard problem, the input to our cost allocation problem is the best known solution obtained using some heuristic. In order to allocate the cost of this Steiner tree to the users (receiver nodes), we formulate the associated Steiner Tree Network (STN) game in characteristic function form. It is well known that the core of the general STN game might be empty. We propose a new cost allocation rule for the modified STN game which might be attractive to network users due to its monotonic properties, associated with network growth. ACM CCS (2012) Classification : Networks→Network Algorithms→Network Economics; Mathematics of computing→Discretemathematics→Combinatorics→Combinatorial optimization; Theory of computation → Theory and algorithms for application domains → Algorithmic game theory and mechanism design→Algorithmic game theory; Applied computing→Operations research→Decision analysis → Multi-criterion optimization and decisionmaking *To cite this article: D. Skorin-Kapov , The Monotonic Cost Allocation Rule in Steiner Tree Network Games, CIT. Journal of Computing and Information Technology , vol. 24, no. 1, pp. 17-29, 2016.

A hybrid exact algorithm for complete set partitioning

In the Complete Set Partitioning problem we are given a finite set of elements where every subset is associated with a value, and the goal is to partition this set into disjoint subsets so as to maximise the sum of subset values. This abstract problem captures the Coalition Structure Generation problem in cooperative games in characteristic function form, where each subset, or coalition, of agents can make a profit when working together, and the goal is to partition the set of agents into coalitions to maximise the total profit. It also captures the special case of the Winner Determination problem in combinatorial auctions, where bidders place bids on every possible bundle of goods, and the goal is to find an allocation of goods to bidders that maximises the profit of the auctioneer.The main contribution of this article is an extensive theoretical analysis of the search space of the Complete Set Partitioning problem, which reveals that two fundamentally different exact algorithms can be significantly improved upon in terms of actual runtime. These are (1) a dynamic programming algorithm called “DP” [48,36] and (2) a tree-search algorithm called “IP” [32]. We start by drawing a link between DP and a certain graph describing the structure of the search space. This link reveals that many of DP's operations are in fact redundant. Consequently, we develop ODP—an optimal version of DP that avoids all of its redundant operations. Since ODP and IP are based on different design paradigms, each has its own strengths and weaknesses compared to the other. Thus, one has to trade off the advantages of one algorithm for the advantages of the other. This raises the following question: Is this trade-off inevitable? To answer this question, we develop a new representation of the search space, which links both algorithms, and allows for contrasting the workings of the two. This reveals that ODP and IP can actually be combined, leading to the development of ODP-IP—a hybrid algorithm that avoids the limitations of its constituent parts, while retaining and significantly improving upon the advantages of each part.We benchmark our algorithm against that of Björklund et al. (2009) [8], which runs in O(2n) time given n agents. We observe that the algorithm of Björklund et al. relies on performing arithmetic operations with very large integers, and assumes that any such operation has unit cost. In practice, however, working with large integers on a modern PC is costly. Consequently, when implemented, our O(3n) algorithm outperforms that of Björklund et al. by several orders of magnitude on every problem instance, making ours the fastest exact algorithm for complete set partitioning to date in practice.

DETERMINING THE NUCLEOLUS OF COMPROMISE STABLE GAMES

The main goal is to illustrate that the so-called indirect function of a cooperative game in characteristic function form is applicable to determine the nucleolus for a subclass of coalitional games called compromise stable transferable utility (TU) games. In accordance with the Fenchel–Moreau theory on conjugate functions, the indirect function is known as the dual representation of the characteristic function of the coalitional game. The key feature of a compromise stable TU game is the coincidence of its core with a box prescribed by certain upper and lower core bounds. For the purpose of the determination of the nucleolus, we benefit from the interrelationship between the indirect function and the prekernel of coalitional TU games. The class of compromise stable TU games contains the subclasses of clan games, big boss games and $1$- and $2$-convex $n$-person TU games. As an adjunct, this paper reports the indirect function of clan games for the purpose of determining their nucleolus.

Collusion properties of coalition values for games in characteristic function form

This paper provides some new properties of coalition values which are blockability value and viability value. The investigated properties are related to coalition collusion within games in characteristic function form. Coalition proxy agreement and coalition association agreement of games in characteristic function form are defined. It turns out that the sum of blockability value of two coalitions which do not have common members and one coalition is a dummy coalition does not decrease if there exists a coalition proxy agreement between these two coalitions. It is also verified that the sum of blockability value of two coalitions which have no common members with coalition association agreement is equal to or less than the value with coalition proxy agreement in a convex game. Some propositions which show that the collusion properties of viability value are opposite to the collusion properties of blockability value are provided. A new axiom relevant to coalition collusion is proposed, and it is confirmed that blockability value satisfies the proposed axiom. A counterexample which shows that viability value does not always satisfy the proposed axiom is given.

Sufficient conditions for unique stable sets in three agent pillage games

Pillage games (Jordan, 2006a) have two features that make them richer than cooperative games in either characteristic or partition function form: they allow power externalities between coalitions; they allow resources to contribute to coalitions’ power as well as to their utility. Extending von Neumann and Morgenstern’s analysis of three agent games in characteristic function form to anonymous pillage games, we characterise the core for any number of agents; for three agents, all anonymous pillage games with an empty core represent the same dominance relation. When a stable set exists, and the game also satisfies a continuity and a responsiveness axiom, it is unique and contains no more than 15 elements, a tight bound. By contrast, stable sets in three agent games in characteristic or partition function form may not be unique, and may contain continua. Finally, we provide an algorithm for computing the stable set, and can easily decide non-existence. Thus, in addition to offering attractive modelling possibilities, pillage games seem well behaved and analytically tractable, overcoming a difficulty that has long impeded use of cooperative game theory’s flexibility.

Equitable Distribution in a Three Players Problem

Abstract Jazz band is a 3 player superadditive game in characteristic function form. Three players have to divide the payoff they can get, while being in a grand coalition, provided their individual and duo coalitions payoffs are known. Assumptions of individual and collective rationality lead to the notion of the core of the game. We discuss offers that cannot readily be refused [OCRR] as the solutions of the game in case of an empty core, when duo coalitions are the best options but only for two out of three players. The experiment shows that even in case of an empty core the most probable results are three-way coalitions and the share of the weakest player usually exceeds his OCRR. The Shapley value is introduced and its fairness is discussed as it lies at the side of the core while, on the other hand, the nucleolus lies exactly at the center of the core. We conclude that, in spite of that, the Shapley value is the best candidate for a fair sharing solution of the jazz band game and other similar games as, opposite to the other values, it is dependent both on individual and duo coalitions payoffs.

On connections between individual values and coalition values for games in characteristic function form

This paper investigates some connections between individual values, such as the Shapley value and the Banzhaf value, and coalition values, such as blockability value, viability value and profitability value, for games in characteristic function form. It turns out that the Shapley value, the Banzhaf value, blockability value, viability value and profitability value assign the same real number to each coalition where the characteristic function is superadditive and inessential. It is also clarified that there is equalization of the Banzhaf value and profitability value of each player in constant-sum game. A new class of games in characteristic function form, which is called equally average coalition solidarity, is proposed in this paper. A proposition which shows that every player’s real number evaluated by the Shapley value corresponds with the real number evaluated by the Banzhaf value of the player in the proposed class of games in characteristic function form is provided.

A new axiom and interrelationships among coalition values for games in characteristic function form

A new axiom which is called dummy coalitions axiom within games in characteristic function form is proposed in this paper. Propositions which show that existing coalition values satisfy the proposed axiom are given. This paper also provides some interrelationships among coalition values. It is shown that there exists complementary relationship between blockability value and viability value. The author also clarifies that blockability value corresponds with profitability value of every coalition in constant-sum game.

Cooperative game analysis of retail space-exchange problems

We analyze retail space-exchange problems where two or more retailers exchange their excess retail spaces to improve the utilization of their space resource. We first investigate the two-retailer space exchange problem. In order to entice both retailers with different bargaining powers to exchange their spaces, we use the generalized Nash bargaining scheme to allocate the total profit surplus between the two retailers. Next, we consider the space-exchange problem involving three or more retailers, and construct a cooperative game in characteristic function form. We show that the game is essential and superadditive, and also prove that the core is non-empty. Moreover, in order to find a unique allocation scheme that ensures the stability of the grand coalition, we propose a new approach to compute a weighted Shapley value that satisfies the core conditions and also reflects retailers’ bargaining powers. Our analysis indicates that the space exchange by more retailers can result in a higher system-wide profit surplus and thus a higher allocation to each retailer under a fair scheme.

VALUES FOR GAMES ON THE CYCLES OF A DIGRAPH

In this paper we introduce a new class of cooperative games. We define a characteristic function over the cycles of a digraph. We present a mathematical model for this situation and an axiomatic characterization of a solution for this class of cooperative games. This is introduced as a method to measure the importance of the nodes in a digraph, and can be related with the Shapley value of a game in characteristic function form. Also, we extend the modeling by applying coalitional structures for the nodes and r-efficient solutions, where the allocation amount is a real number r, showing axiomatic solutions in both cases.

Su fficient Conditions for Unique Stable Sets in Three Agent Pillage Games

Pillage games (Jordan, 2006, “Pillage and property”, JET) have two features that make them richer than cooperative games in either characteristic or partition function form: they allow power externalities between coalitions; they allow resources to contribute to coalitions’ power as well as to their utility. Extending von Neumann and Morgenstern’s analysis of three agent games in characteristic function form to anonymous pillage games, we find: when the core is non-empty, it must take one of five forms; all such games with an empty core represent the same dominance relation. When a stable set exists, and the game also satisfies a continuity and a responsiveness axiom, it is unique and contains no more than 15 elements, a tight bound. By contrast, stable sets in three agent games in characteristic or partition function form may not be unique, and may contain continua. Finally, we provide an algorithm for computing the stable set, and can easily decide non-existence. Thus, in addition to offering attractive modelling possibilities, pillage games seem well behaved and analytically tractable, overcoming a difficulty that has long impeded use of cooperative game theory’s flexibility.

Coalition values derived from methods for comparison of coalition influence for games in characteristic function form

The authors introduce new values which express coalition influence in the situation of coalition formation. Some examples which show how introduced values work are given. Propositions in this paper provide some properties that proposed values satisfy in the framework of games in characteristic function form.

Core-stable rings in second price auctions with common values

In a common value auction in which the information partitions of the bidders are connected, all rings are core-stable. More precisely, the ex ante expected utilities of rings, at the (noncooperative) sophisticated equilibrium proposed by Einy et al. [Einy, E., Haimanko, O., Orzach, R., Sela, A., 2002. Dominance solvability of second-prices auctions with differential information. Journal of Mathematical Economics 37, 247–258], describe a cooperative games in characteristic function form, in spite of the underlying strategic externalities. A ring is core-stable if the core of this characteristic function is not empty. Furthermore, every ring can implement its sophisticated equilibrium strategy by means of an incentive compatible mechanism. An example shows that, if the bidders’ information partitions are not connected, rings may no longer be core-stable.

A note on Steiner tree games

AbstractWe investigate the cost allocation strategy associated with the problem of providing some service of common interest from some source to a number of network users, via the minimum cost directed Steiner tree (ST) network that spans the source and all the receivers. The cost of such ST is distributed among its receivers who may be individuals or organizations with possibly conflicting interests. The objective of this article is to develop a reasonably fair and efficient cost allocation scheme associated with the above cost allocation problem. Since finding the optimal Steiner tree is an NP‐hard problem, the input to our cost allocation problem is the best known Steiner tree obtained using some heuristic. To allocate the cost of this Steiner tree to the users (receiver nodes), we formulate the associated Modified Steiner Tree Network (MSTN) game in characteristic function form. Then we construct a primal‐dual cost allocation algorithm which finds some points in the core of the MSTN game and thus results in subsidy‐free cost allocations. Moreover, for the given Steiner tree, our cost allocation scheme is efficient and finds the above “fair” cost allocations in polynomial time. © 2011 Wiley Periodicals, Inc. NETWORKS, 2012

An existence result for farsighted stable sets of games in characteristic function form

In this paper we show that every finite-player game in characteristic function form (not necessarily with side payments) obeying an innocuous condition (that the set of individually rational pay-off vectors is bounded) possesses a farsighted von-Neumann–Morgenstern stable set.