Partition Function Games Research Articles

Coalition Formation for Outsourced Spectrum Sensing in Cognitive Radio Network

We inspect an opportunistic Cognitive Radio Network formed between a Primary Network Provider (PNP) and secondary IoT operators. In our system, spectrum-sensing duty is offloaded to a Sensing Agent (SA), which allows the IoT operators to improve data rate and reduce energy consumption otherwise needed for spectrum-sensing. IoT operators pay PNP for accessing licensed spectrum and SA for providing sensing as a service. We use Nash bargaining to finalize negotiations among PNP, IoT operators, and SA. Our investigation shows that incorporating SA improves the throughput and utility of PNP and IoT operators. Coalition formation among IoT operators helps them avoid intra-coalitional interference by accessing the spectrum in time-shared fashion. As the valuation of a coalition depends on other coalitions, we analyze coalition formation as a Partition Function Game. We emphasize economic fairness among the coalitions to prevent monopolization. We define a proportional fairness-based metric based on which the final coalition structure is determined. In general, finding an optimal coalition structure is an NP-hard problem. The fairness criterion, along with the unique properties of the Nash bargaining solution allows us to transform the coalition formation problem into a more tractable Number Partitioning Problem and develop an efficiently computable coalition formation technique.

Optimal coalition structures for probabilistically monotone partition function games

For cooperative games with externalities, the problem of optimally partitioning a set of players into disjoint exhaustive coalitions is called coalition structure generation, and is a fundamental computational problem in multi-agent systems. Coalition structure generation is, in general, computationally hard and a large body of work has therefore investigated the development of efficient solutions for this problem. However, the existing methods are mostly limited to deterministic environments. In this paper, we focus attention on uncertain environments. Specifically, we define probabilistically monotone partition function games, a subclass of the well-known partition function games in which we introduce uncertainty. We provide a constructive proof that an exact optimum can be found using a greedy approach, present an algorithm for finding an optimum, and analyze its time complexity.

Marginality and convexity in partition function form games

In this paper an order on the set of embedded coalitions is studied in detail. This allows us to define new notions of superaddivity and convexity of games in partition function form which are compared to other proposals in the literature. The main results are two characterizations of convexity. The first one uses non-decreasing contributions to coalitions of increasing size and can thus be considered parallel to the classic result for cooperative games without externalities. The second one is based on the standard convexity of associated games without externalities that we define using a partition of the player set. Using the later result, we can conclude that some of the generalizations of the Shapley value to games in partition function form lie within the cores of specific classic games when the original game is convex.

A novel simulation-based method for assessment of collusion potential in wholesale electricity markets

Abstract This paper proposes a simulation-based method for assessing the collusion potential among Generating Companies (GENCOs) who participate in a wholesale electricity market. For this purpose, a game-theoretic framework is developed to model the strategic bidding behaviors of GENCOs in both non-cooperative and collusive oligopoly electricity markets. In a collusive oligopoly market, GENCOs may negotiate amongst themselves to form coalitions. The coalition formation problem in an economy with externalities can be adequately modeled in coalitional form by using partition function games (PFGs). The optimal bidding strategies of coalitions can be calculated based on the partial cooperative equilibrium concept. When GENCOs submit their coordinated bids and the Independent System Operator (ISO) clears the market, each coalition has to allocate obtained profit among its members. In favor of a stable profit allocation, the core solution concept is employed. Also, Shapley value is employed to distribute the utility of the coalition in a fairway. To investigate the partial cooperative equilibrium concept, the proposed model is applied to the IEEE 9-bus test system. As an application of Shapley value and core solution concept, the profit of the grand coalition is distributed among its members. Numerical results show that in a wholesale electricity market GENCOs have strong incentives to make collusion.

Winning coalitions in plurality voting democracies

We consider plurality voting games being simple games in partition function form such that in every partition there is at least one winning coalition. Such a game is said to be weighted if it is possible to assign weights to the players in such a way that a winning coalition in a partition is always one for which the sum of the weights of its members is maximal over all coalitions in the partition. A plurality game is called decisive if in every partition there is exactly one winning coalition. We show that in general, plurality games need not be weighted, even not when they are decisive. After that, we prove that (i) decisive plurality games with at most four players, (ii) majority games with an arbitrary number of players, and (iii) decisive plurality games that exhibit some kind of symmetry, are weighted. Complete characterizations of the winning coalitions in the corresponding partitions are provided as well.

Stability of the merger-to-monopoly and a core concept for partition function games

This paper is concerned with an old question: Will oligopolistic firms have incentives to merge to monopoly and will the monopoly, if the firms indeed merge, be stable? To answer this question, I motivate and introduce a new core concept for a general partition function game and prove stability of the merger-to-monopoly by applying the new core concept, labelled the strong-core, to Cournot oligopoly modelled as a partition function game. The paper shows that the Cournot oligopoly with any finite number of homogeneous firms without capacity constraints admits a non-empty strong-core and so does the Cournot oligopoly of not necessarily homogeneous firms with capacity constraints that are equal to their “historical” outputs. These results imply that oligopolistic firms will have incentives to merge to monopoly both in the long- and short-run and if the firms indeed merge to monopoly, the merger-to-monopoly will be stable.

Coalition formation and history dependence

Farsighted formulations of coalitional formation, for instance, by Harsanyi and Ray and Vohra, have typically been based on the von Neumann–Morgenstern stable set. These farsighted stable sets use a notion of indirect dominance in which an outcome can be dominated by a chain of coalitional “moves” in which each coalition that is involved in the sequence eventually stands to gain. Dutta and Vohra point out that these solution concepts do not require coalitions to make optimal moves. Hence, these solution concepts can yield unreasonable predictions. Dutta and Vohra restricted coalitions to hold common, history‐independent expectations that incorporate optimality regarding the continuation path. This paper extends the Dutta–Vohra analysis by allowing for history‐dependent expectations. The paper provides characterization results for two solution concepts that correspond to two versions of optimality. It demonstrates the power of history dependence by establishing nonemptyness results for all finite games as well as transferable utility partition function games. The paper also provides partial comparisons of the solution concepts to other solutions.

On horizontal cooperation in linear production processes with a supplier that controls a limited resource

In this paper we study the effect that the external management of a limited (natural) resource such as carbon dioxide or water quotas has on the behaviour of firms in a given sector. To do this, we choose a model in which all firms have the same technology and this is lineal. In the analysis of the problem games in partition function form arise in a natural way. It is proved, under certain conditions, that stable allocations exist in both cases with certainty and uncertainty.

The Risk-Based Core for Cooperative Games With Uncertainty

In coalitional games with uncertain payoffs, a deviating coalition can only form expectations regarding its post-deviation payoff. Classical approaches address the problem from the side of conservatism, expecting the worst, or by explicit assumptions of the emerging state of the world. We borrow the idea of risk from the finance literature and compare the payoff of staying with the original outcome with the risk of deviating. Employing this idea to the core leads to a new concept that we call the risk-based core. We introduce this concept and discuss its properties. We find an inclusion relation between cores of games with increasingly conservative players. The model is also suitable to study cooperative games in partition function form where the value of a coalition depends on the entire partition. For the cores of such games our main result yields many of the familiar inclusion relations as corollaries, while the inclusion of the optimistic core in the optimistic recursive core turns out to be non-robust.

Winning Coalitions in Plurality Voting Democracies

We study the issue of assigning weights to players that identify winning coalitions in plurality voting democracies. For this, we consider plurality games which are simple games in partition function form such that in every partition there is at least one winning coalition. Such a game is said to be precisely supportive if it is possible to assign weights to players in such a way that a coalition being winning in a partition implies that the combined weight of its members is maximal over all coalitions in the partition. A plurality game is decisive if in every partition there is exactly one winning coalition. We show that decisive plurality games with at most four players, majority games with an arbitrary number of players, and almost symmetric decisive plurality games with an arbitrary number of players are precisely supportive. Complete characterizations of a partition's winning coalitions are provided as well.

Collusive Agreements in Vertically Differentiated Markets

This paper introduces a number of game-theoretic tools to model collusive agreements among firms in vertically differentiated markets. I firstly review some classical literature on collusion between two firms producing goods of exogenous different qualities. I then extend the analysis to a n-firm vertically differentiated market to study the incentive to form either a whole market alliance or partial alliances made of subsets of consecutive firms in order to collude in prices. Within this framework I explore the price behaviour of groups of colluding firms and their incentive to either pruning or proliferating their products. It is shown that a selective pruning within the cartel always occurs. Moreover, by associating a partition function game to the n-firm vertically differentiated market, it can be shown that a sufficient condition for the cooperative (or coalitional) stability of the whole industry cartel is the equidistance of firms’ products along the quality spectrum. Without this property, and in presence of large quality differences, collusive agreements easily lose their stability. In addition, introducing a standard infinitely repeated-game approach, I show that an increase in the number of firms in the market may have contradictory effects on the incentive of firms to collude: it can make collusion easier for bottom and intermediate firms and harder for the top quality firm. Finally, by means of a three-firm example, I consider the case in which alliances can set endogenously qualities, prices and number of variants on sale. I show that, in every formed coalition, (i) market pruning dominates product proliferation and (ii) partial cartelisation always arises in equilibrium, with the bottom quality firm always belonging to the alliance.

Coalition Structure Generation Utilizing Graphical Representation of Partition Function Games

Forming effective coalition is a central research challenge in AI and multi-agent systems. The Coalition Structure Generation (CSG) problem is well-known as one of major research topics in coalitional games. The CSG problem is to partition a set of agents into coalitions so that the sum of utilities is maximized. This paper studies a CSG problem for partition function games (PFGs), where the value of a coalition differs depending on the formation of other coalitions generated by non-member agents. Traditionally, in PFGs, the input of a coalitional game is a black-box function called a partition function that maps an embedded coalition (a coalition and the coalition structure) to its value. Recently, a novel concise representation scheme called the Partition Decision Trees (PDTs) has been proposed. The PDTs is a graphical representation based on multiple rules. In this paper, we propose new algorithms that can solve a CSG problem by utilizing PDTs representation. More specifically, we modify PDTs representation to effectively handle negative value rules and apply the depth-first branch and bound algorithm. We experimentally show that our algorithm can solve a CSG problem well.

International Fisheries Management and Recreational Benefits: The Case of Baltic Salmon

AbstractThis article studies how accounting for the benefits of recreational fisheries affects the formation and stability of an international fisheries agreement (IFA) on the management of Baltic salmon stocks. The interaction between four countries is modelled through a partition function game, under two scenarios. In the first scenario, countries take their participation decision for the IFA based only on the net present value of profits from commercial fisheries. In the second scenario, the net present value of the recreational benefits from angling is also considered. The results show that accounting for recreational benefits leads to the formation of the grand coalition, whereas only partial cooperation occurs when payoffs are confined to profits from commercial fisheries.

Nash bargaining solution under externalities

We define a Nash bargaining solution (NBS) of partition function games. Based on a partition function game, we define an extensive game, which is a propose–respond sequential bargaining game where the rejecter of a proposal exits from the game with some positive probability. We show that the NBS is supported as the expected payoff profile of any stationary subgame perfect equilibrium (SSPE) of the extensive game such that in any subgame, a coalition of all active players forms immediately. We provide a necessary and sufficient condition for such an SSPE to exist. Moreover, we consider extensions to the cases of nontransferable utilities, time discounting and multiple-coalition formation.

International Fisheries Management and Recreational Benefits: The Case of Baltic Salmon

This article studies how accounting for the benefits of recreational fisheries affects the formation and stability of an international fisheries agreement (IFA) on the management of Baltic salmon stocks. The interaction between four countries is modelled through a partition function game, under two scenarios. In the first scenario, countries take their participation decision for the IFA based only on the net present value of profits from commercial fisheries. In the second scenario, the net present value of the recreational benefits from angling is also considered. The results show that accounting for recreational benefits leads to the formation of the grand coalition, whereas only partial cooperation occurs when payoffs are confined to profits from commercial fisheries.

A Note on a Class of Solutions for Games with Externalities Generalizing the Shapley Value

In this paper we study a family of extensions of the Shapley value for games in partition function form with n players. In particular, we provide a complete characterization for all linear, symmetric, efficient and null solutions in these environments. Finally, we relate our characterization result with other ways to extend the Shapley value in the literature.

International Agreements in the Multispecies Baltic Sea Fisheries

This study analyses a multispecies (cod, herring and sprat) partition function game among three asymmetric countries in the Baltic Sea. Through an empirical application, the study highlights the importance of dynamic multispecies models in coalition formation. We show that the grand coalition between the three countries can be stabilised when the most efficient country compensates the other countries. This solution is not feasible when using a single-species model for the cod population; thus, taking into account the multispecies nature of the Baltic Sea fisheries, the scope of cooperation increases compared with a single-species simulation. In contrast to the predominant conception, climate change may improve the likelihood of a binding agreement because with a lower cod biomass, countries will gain more from cooperation.

Sharing a Fish Stock When Distribution and Harvest Costs are Density Dependent

We studied cooperative and competitive solutions for managing a trans-boundary fish stock that has a stock-size dependent distribution and harvesting costs, using Norwegian spring-spawning (NSS) herring (Clupea harengus) as a case study. This is among the first studies to investigate the effect of a dynamic distribution in a partition function game framework, i.e., including externalities. The special feature of NSS herring is that it comes under the sole ownership of Norway when the stock size is sufficiently low. The three players in our game are differentiated by stock ownership, cost per unit effort and stock elasticity of harvest. We find that cost asymmetry improves the likelihood of forming a stable grand coalition as harvesting load can be assigned to each fishing zone according to a player’s relative cost level and the density of fish. A dynamic distribution increases the bargaining power of Norway because of her ability to control both the total stock and the stock shares in different zones, especially during the stock transition years. However, the bargaining power of Norway may disappear if her cost of harvesting exceeds that of the other players. This loss of bargaining power can take on two forms: first, a dynamic distribution may encourage a minor player to free ride because Norway, needing a coalition partner to improve her cost-effectiveness, would be less inclined to retaliate as retaliation hurts her ally as much as the free-rider; second, in a stable grand coalition, Norway is not guaranteed to receive an allocation of benefit shares in proportion to (not to say in excess of) her stock ownership. At the current stock condition, the strategy where Norway reduces stock abundance below the level where the stock becomes non-migratory is found attractive only if the discounting rate is sufficiently high.

International Fisheries Agreements: A Game Theoretical Approach

This paper surveys the application of game theory to the economic analysis of international fisheries agreements. The relevance of this study comes not only from the existence of a vast literature on the topic but especially from the specific features of these agreements. The emphasis of the survey is on coalition games, an approach that has become prominent in the fisheries economics literature over the last decade. It is shown that coalition games were first applied to international fisheries agreements in the late 1990s addressing cooperative issues under the framework of characteristic function games. Then, progressively, this cooperative approach was combined with non-cooperative elements such as the stability analysis of the agreements. Finally, partition function games, which model coalition formation endogenously, were introduced and became the standard approach to study the formation and stability of international fisheries agreements. A key message that emerges from this literature strand is that self-enforcing cooperative management of internationally shared fish stocks is generally difficult to achieve. Hence, the international legal framework and regulations play a decisive role on ensuring cooperation over the use of these resources.

Expectation Formation Rules and the Core of Partition Function Games

This paper proposes axiomatic foundations of expectation formation rules, by which deviating players anticipate the reaction of external players in a partition function game. The axioms single out the projection rule among the rules that depend on the current partition and the pessimistic rule among the ones that are independent of the current partition. This analysis suggests that the projection core and the pessimistic core are natural candidates to study the stability of games in partition function form, and we compute these cores in two standard applications of coalition formation with externalities, namely cartels and public goods.