Population Monotonic Allocation Research Articles

Population Lorenz-monotonic allocation schemes for TU-games

Sprumont (Games Econ Behav 2:378–394, 1990) introduces population monotonic allocation schemes (PMAS) and proves that every assignment game with at least two sellers and two buyers, where each buyer-seller pair derives a positive gain from trade, lacks a PMAS. In particular glove games lacks PMAS. We propose a new cooperative TU-game concept, population Lorenz-monotonic allocation schemes (PLMAS), which relaxes some population monotonicity conditions by requiring that the payoff vector of any coalition is Lorenz dominated by the corresponding restricted payoff vector of larger coalitions. We show that every TU-game having a PLMAS is totally balanced, but the converse is not true in general. We obtain a class of games having a PLMAS, but no PMAS in general. Furthermore, we prove the existence of PLMAS for every glove game and for every assignment game with at most five players.Additionally, we also introduce two new notions, PLMAS-extendability and PLMAS-exactness, and discuss their relationships with the convexity of the game.

Cooperation in spare parts systems with penalty cost per unit backlogged

Inventory pooling is well known to reduce inventory costs and improve customer service. Although potential benefits are well understood, one important prerequisite to form a stable coalition between independent companies, who like to cooperate to benefit from inventory pooling, is agreeing on a cost allocation such that no group of companies would better off leaving the coalition. Existence of such stable allocations would depend on the scenarios that these companies are cooperating in.In this paper, we study cooperation between a group of companies, who keep spare parts to maintain their expensive equipment, by pooling their inventory in several practically motivated scenarios. To address the cost allocation problem, we borrow concepts from cooperative game theory and examine the core of the resulting cooperative games in each scenario. We especially focus on systems with high penalty cost per unit backlogged, which is modelled as a well known (S−1,S) inventory system with backlogging.We study cooperation under four scenarios, namely cooperation under penalty cost, cooperation with joint capacity investment, cooperation under service level constraints and cooperation by inventory pooling. The first three scenarios consider cooperation by investing into a central pool of inventory optimally, whereas in the last scenario, companies cooperate by pooling their available inventory only. In most of the scenarios, we show the existence of stable cost allocations by identifying a core allocation, which is easy to compute and, in some, a population monotonic allocation (PMAS). In other scenarios, we present examples explaining why finding a stable allocation might not be possible by showing the core of the resulting games can be empty. To establish these results, we also derive several interesting properties (e.g. subhomogeneity of degree one) of the main performance measure (i.e., average number of backorders).

Assignment games with population monotonic allocation schemes

AbstractWe characterize the assignment games which admit a population monotonic allocation scheme (PMAS) in terms of efficiently verifiable structural properties of the nonnegative matrix that induces the game. We prove that an assignment game is PMAS-admissible if and only if the positive elements of the underlying nonnegative matrix form orthogonal submatrices of three special types. In game theoretic terms it means that an assignment game is PMAS-admissible if and only if it contains either a veto player or a dominant veto mixed pair, or the game is a composition of these two types of special assignment games. We also show that in PMAS-admissible assignment games all core allocations can be extended to a PMAS, and the nucleolus coincides with the tau-value.

Average monotonic cooperative games with nontransferable utility

A non-negative transferable utility (TU) game is average monotonic if there exists a non-negative vector according to which the relative worth is not decreasing when enlarging the coalition. We generalize this definition to the nontransferable utility (NTU) case. It is shown that an average monotonic NTU game shares several properties with an average monotonic TU game. In particular it has a special core element and there exists a population monotonic allocation scheme. We show that an NTU bankruptcy game is average monotonic with respect to the claims vector.

Population monotonic allocation schemes for the two-period economic lot-sizing games

It is an open question whether a population monotonic allocation scheme (PMAS) exists for the economic lot-sizing games (ELSGs) with holding and backlogging. In this paper, we contribute in showing that a PMAS exists for the two-period ELSGs with holding and backlogging by constructing a closed-form PMAS. Our results partially answer the open question and shed some light on how to construct a closed-form PMAS for ELSGs.

Quasi-Owen Value for Games on Augmenting Systems with a Coalition Structure

This paper focuses on games on augmenting systems with a coalition structure that can be seen as an extension of games with a coalition structure and games on augmenting systems. Considering the player payoffs, the quasi-Owen value is defined. To show the rationality of this payoff index, five representative axiomatic systems are established. The population monotonic allocation scheme (PMAS) and the core are introduced. Moreover, the relationships between the PMAS and quasi-Owen value as well as the core and quasi-Owen value are discussed. Finally, an illustrative example is given to show the concrete application of the new payoff indices.

Strong Players and Stable Coalition Structures in PMAS Profit Game

In a non-negative profit game that possesses a Population Monotonic Allocation Scheme (PMAS), being a member of a larger coalition implies that your profit cannot decrease. In this paper, we refer to such games as PMAS profit games. As population monotonicity is a nice and desirable property that encourages formation of larger coalitions and implies stability of the grand coalition, we explore if this special feature of PMAS games can help in identifying additional stable coalition structures under different stability concepts in cooperative games—namely, core partitions, the von Neumann–Morgenstern (vNM) stable set, the largest consistent set, and the equilibrium process of coalition formation (EPCF)—and in developing relationships between coalition structures that are stable under these different stability concepts. We first define two special classes of players for PMAS profit games—extreme and strong players—and use them to develop an algorithm for construction of stable (core) partitions. We also use extreme players to identify absorbing states for equilibrium processes of coalition formation with high level of farsightedness. We then explore the impact of population monotonicity on the relationship between stable coalition structures under abovementioned stability concepts. While we are able to obtain some results related to stability of the grand coalition and to establish relationships between stable coalition structures under different stability notions that are consistent with the existing body of knowledge, population monotonicity in general does not add enough for strengthening of the existing results. However, we are able to show a couple of more general result that hold for arbitrary cooperative TU profit games. That is, we show that the members of vNM farsighted stable sets are core partitions, and that core partitions are members of a vNM stable sets. Moreover, we show that the members of vNM farsighted stable sets are EPCF-stable partitions.

On the population monotonicity of independent set games

An independent set game is a cooperative game dealing with profit sharing in the maximum independent set problem. A population monotonic allocation scheme is a rule specifying how to share the profit of each coalition among its participants such that every participant is better off when the coalition expands. In this paper, we provide a necessary and sufficient characterization for independent set games admitting population monotonic allocation schemes. Moreover, our characterization can be verified efficiently.

Population monotonicity in matching games

The matching game is a cooperative profit game defined on an edge-weighted graph, where the players are the vertices and the profit of a coalition is the maximum weight of matchings in the subgraph induced by the coalition. A population monotonic allocation scheme is a collection of rules defining how to share the profit among players in each coalition such that every player is better off when the coalition expands. In this paper, we study matching games and provide a necessary and sufficient characterization for the existence of population monotonic allocation schemes. Our characterization implies that whether a matching game admits population monotonic allocation schemes can be determined efficiently.

A Combinatorial Characterization for Population Monotonic Allocations in Convex Independent Set Games

Independent set games are cooperative games defined on graphs, where players are edges and the value of a coalition is the maximum size of independent sets in the subgraph defined by the coalition. In this paper, we study population monotonic allocation schemes for independent set games. For independent set games introduced by Deng et al. [X. Deng, T. Ibaraki and H. Nagamochi (1999). Algorithmic aspects of the core of combinatorial optimization games. Mathematics of Operations Research, 24(3), 751–766], we provide a combinatorial characterization for population monotonic allocation schemes in convex instances. For independent set games introduced by Xiao et al. [H. Xiao, Y. Wang and Q. Fang (2021). On the convexity of independent set games. Discrete Applied Mathematics, 291, 271–276], we prove the equivalence of convexity, population monotonicity and balancedness.

Population monotonic allocation schemes for vertex cover games

For vertex cover games (introduced by Deng et al. (1999) [2]), we investigate population monotonic allocation schemes (introduced by Sprumont (1990) [11]). We show that the existence of a population monotonic allocation scheme (PMAS for short) for vertex cover games can be determined efficiently and that a PMAS, if exists, can be constructed accordingly. We also show that integral PMAS-es for vertex cover games can be characterized with stable matchings and be enumerated by employing Gale-Shapley algorithm (introduced by Gale and Shapley (1962) [4]).

Monotonic core allocation paths for assignment games

We introduce a modification of Sprumont (Games Econ Behav 2:378–394, 1990) population monotonic allocation scheme (PMAS), called monotonic core allocation path (MCAP) for assignment games, which is a sequence of allocations along an order on the set of players satisfying that (1) each allocation is in the core of the subgame of the corresponding players at that step, and (2) the payoffs for each player are non-decreasing through the sequence. The notion of MCAP preserves the population monotonicity of PMAS while avoids the difficulty that PMAS does not exist in many market games. We show that for every assignment game, there is an order of players along which a MCAP exists. The terminals of MCAP form a refinement of the core. We also show that the terminals of MCAP coincide with the extreme core allocations in two subclasses of assignment games: gloves games and Bohm-Bawerk games. The strong connection of MCAP with extreme core allocations suggests some conflict between the stability of a coalition formation process and the fairness of the resulting outcomes.

Population monotonic allocation schemes for games with externalities

This paper provides conditions for a game with externalities to have a population monotonic allocation scheme (PMAS). We observe that the notion of convexity defined by Hafalir [Games Econ Behav 61:242–258, 2007] does not guarantee the existence of a PMAS in the presence of externalities. We introduce a new notion of convexity and show that while our convexity is not a stronger condition than Hafalir’s [Games Econ Behav 61:242–258, 2007] , it is a sufficient condition for a game to have a PMAS. Moreover, we show that the Aumann-Dreze value, which is defined for games with coalition structures, explicitly constructs a PMAS. In addition, we offer two necessary and sufficient conditions to guarantee a PMAS in the presence of externalities.

Characterizations of a Shapley value for multichoice games

ABSTRACTThe main focus of this paper is on a Shapley value for multichoice games introduced by van den Nouweland et al. (ZOR–Math. Meth. Oper. Res. 41 : 289–311, 1995). Here we provide several characterizations from traditional game theory and redefine them in the framework of multichoice games. Meanwhile, the relationship between core and this Shapley value for multichoice games is discussed. When multichoice games are convex, this Shapley value is a multichoice population monotonic allocation scheme (MPMAS).

Pooling of critical, low-utilization resources with unavailability

We consider an environment in which several independent service providers can collaborate by pooling their critical, low-utilization resources that are subject to unavailability. We examine the allocation of the joint profit for such a pooled situation by studying an associated cooperative game. For this game, we will prove non-emptiness of the core, present a population monotonic allocation scheme and show convexity under some conditions. Moreover, four allocation rules will be introduced and we will investigate whether they satisfy monotonicity to availability, monotonicity to profit, situation symmetry and game symmetry. Finally, we will also investigate whether the payoff vectors resulting from those allocation rules are members of the core.

Population Monotonicity in Newsvendor Games

A newsvendor game studies whether players can collaborate on inventory pooling, where the cost allocation is usually analyzed by the notion of core in cooperative game theory. It is known that the core of the newsvendor game is non-empty and one can use duality theory in stochastic programming to construct an allocation - referred to as the dual-based allocation scheme - belonging to the core. Yet, an allocation that lies in the core does not necessarily guarantee the unhindered formation of a coalition, as some existing members' allocated costs may increase when new members are added in the process. In this work, we use the concept of population monotonic allocation scheme (PMAS), which requires the cost allocated to every member of a coalition to decrease as the coalition grows, to study allocation rules in a growing population. We show that when the demands faced by the newsvendors are independent, log-concavity of their distributions is sufficient to guarantee the existence of PMAS. Specifically, for continuous demands, log-concavity ensures that the game is convex, which in turn implies PMAS exists. We also show that under the same condition the dual-based allocation scheme is a PMAS. For discrete and log-concave demands, however, the game may no longer be convex but we manage to show that the dual-based allocation scheme is a PMAS. When the demands are dependent, the game is in general not convex and we derive a sufficient condition based on the dependence structure, measured by the copula, between each player and the coalition to secure that the dual-based allocation scheme is still a PMAS.

Fuzzy Multichoice Games with Fuzzy Characteristic Functions

In this paper, a generalized form of fuzzy multichoice games with fuzzy characteristic functions is proposed, which can be seen as an extension of traditional fuzzy games. Based on the extension Hukuhara difference, fuzzy multichoice games with fuzzy characteristic functions are studied, and a Shapley function is discussed. The notion of fuzzy multichoice population monotonic allocation scheme (FMPMAS) is defined. When the given fuzzy multichoice game with fuzzy characteristic functions is convex, we show that the proposed Shapley function is a FMPMAS. Furthermore, two special kinds of fuzzy multichoice games with fuzzy characteristic functions called fuzzy multichoice games with multilinear extension form and fuzzy characteristic functions and fuzzy multichoice games with Choquet integral form and fuzzy characteristic functions are researched.

A coalitional value for fuzzy games with a coalition structure

In this paper, we shall study another coalitional value for fuzzy games with a coalition structure, which is named the proportional coalitional Shapley value. A generalized form of the proportional coalitional Shapley value for fuzzy games with a coalition structure is considered, which can be regarded as a generalization of crisp case. The axiomatic systems of this value are researched. When the original fuzzy game is fuzzy convex, the relationship between the proportional coalitional Shapley value, the fuzzy quotient core and the fuzzy quotient population monotonic allocation scheme (FQPMAS) is researched.

Cooperative fuzzy games with interval characteristic functions

In this paper, a generalized form of fuzzy games with interval characteristic functions is proposed, which can be seen as an extension of games with crisp characteristic functions. Based on the extended Hukuhara difference, the interval Shapley function for interval fuzzy games is studied. Then, the concept of interval population monotonic allocation function (IPMAF) is defined. When interval fuzzy games are convex, we prove that the interval Shapley function is an IPMAF. Furthermore, two special types of interval fuzzy games are researched, and the associated interval Shapley function is studied.

Resource Pooling and Cost Allocation Among Independent Service Providers

We study a situation where several independent service providers collaborate by complete pooling of their resources and customer streams into a joint service system. These service providers may represent such diverse organizations as hospitals that pool beds or maintenance firms that pool repairmen. We model the service systems as Erlang delay systems (M/M/s queues) that face a fixed cost rate per server and homogeneous delay costs for waiting customers. We examine rules to fairly allocate the collective costs of the pooled system amongst the participants by applying concepts from cooperative game theory. We consider both the case where players’ numbers of servers are exogenously given and the scenario where any coalition picks an optimal number of servers. By exploiting new analytical properties of the continuous extension of the classic Erlang delay function, we provide sufficient conditions for the games under consideration to possess a core allocation (i.e., an allocation that gives no group of players an incentive to split off and form a separate pool) and to admit a population monotonic allocation scheme (whereby adding extra players does not make anyone worse off). This is not guaranteed in general, as illustrated via examples.