A linear proportional effort allocation rule

This paper proposes a new class of allocation rules in network games. Like the solution theory in cooperative games of how the Harsanyi dividend of each coalition is distributed among a set of players, this new class of allocation rules focuses on the distribution of the dividend of each network. The dividend of each network is allocated in proportion to some measure of each player’s effort, which is called an effort function. With linearity of the allocation rules, an allocation rule is specified by the effort functions. These types of allocation rules are called linear proportional effort allocation rules. Two famous allocation rules, the Myerson value and the position value, belong to this class of allocation rules. In this study, we provide a unifying approach to define the two aforementioned values. Moreover, we provide an axiomatic analysis of this class of allocation rules, and axiomatize the Myerson value, the position value, and their non-symmetric generalizations in terms of effort functions. We propose a new allocation rule in network games that also belongs to this class of allocation rules.     

Weighted allocation rules for standard fixed tree games

In this paper we consider standard fixed tree games, for which each vertex unequal to the root is inhabited by exactly one player. We present two weighted allocation rules, the weighted down-home allocation and the weighted neighbour-home allocation, both inspired by the painting story in Maschler et al. (1995) . We show, in a constructive way, that the core equals both the set of weighted down-home allocations and the set of weighted neighbour allocations. Since every weighted down-home allocation specifies a weighted Shapley value (Kalai and Samet (1988)) in a natural way, and vice versa, our results provide an alternative proof of the fact that the core of a standard fixed tree game equals the set of weighted Shapley values. The class of weighted neighbour allocations is a generalization of the nucleolus, in the sense that the latter is in this class as the special member where players have all equal weights.     

Convex and exact games with non-transferable utility

We generalize exactness to games with non-transferable utility (NTU). A game is exact if for each coalition there is a core allocation on the boundary of its payoff set.     

Quitting games – An example

Quitting games are multi-player sequential games in which, at every stage, each player has the choice between continuing and quitting. The game ends as soon as at least one player chooses to quit; each player i then receives a payoff r _S ⁱ, which depends on the set S of players that did choose to quit. If the game never ends, the payoff to each player is zero.? We exhibit a four-player quitting game, where the “simplest” equilibrium is periodic with period two. We argue that this implies that all known methods to prove existence of an equilibrium payoff in multi-player stochastic games are therefore bound to fail in general, and provide some geometric intuition for this phenomenon. Received: October 2001     

Coalitional values for cooperative games withr alternatives

In this paper we consider games withn players andr alternatives. In these games the worth of a coalition depends not only on that coalition, but also on the organization of the other players in the game. We propose two coalitional values that are extensions of the Owen value (1977). We give some relations with the Owen value and an axiomatic characterization of each value introduced in this work. Finally, we compare both values. This research has been supported partially by U.P.V./E.H.U. research project 035.321-HB048/97, and the DGES of MEC project PB96-0247.     

The core of a class of non-atomic games which arise in economic applications

We study the core of a non-atomic game v which is uniformly continuous with respect to the DNA-topology and continuous at the grand coalition. Such a game has a unique DNA-continuous extension on the space B ₁ of ideal sets. We show that if the extension is concave then the core of the game v is non-empty iff is homogeneous of degree one along the diagonal of B ₁. We use this result to obtain representation theorems for the core of a non-atomic game of the form v=f^μ where μ is a finite dimensional vector of measures and f is a concave function. We also apply our results to some non-atomic games which occur in economic applications. Received May 1998/Revised version September 1998     

Cores and large cores when population varies

In a TU cooperative game with populationN, a monotonic core allocation allocates each surplusv (S) among the agents of coalitionS in such a way that agenti's share never decreases when the coalition to which he belongs expands.We investigate the property of largeness (Sharkey [1982]) for monotonic cores. We show the following result. Given a convex TU game and an upper bound on each agent' share in each coalition containing him, if the upper bound depends only upon the size of the coalition and varies monotonically as the size increases, then there exists a monotonic core allocation meeting this system of upper bounds. We apply this result to the provision of a public good problem.     

A sufficiency condition for coalitive pareto-optimal solutions

In ak-player, nonzero-sum differential game, there exists the possibility that a group of players will form a coalition and work together. If allk players form the coalition, the criterion usually chosen is Pareto optimality whereas, if the coalition consists of only one player, a minmax or Nash equilibrium solution is sought.In this paper, games with coalitions of more than one but less thank players are considered. Coalitive Pareto optimality is chosen as the criterion. Sufficient conditions are presented for coalitive Pareto-optimal solutions, and the results are illustrated with an example.     

Potential,value, and coalition formation

In this paper, a simple probabilistic model of coalition formation provides a unified interpretation for several extensions of the Shapley value. Weighted Shapley values, semivalues, weak (weighted or not) semivalues, and the Shapley value itself appear as variations of this model. Moreover, some notions that have been introduced in the search of alternatives to Shapley’s seminal characterization, as ‘balanced contributions’ and the ‘potential’ are reinterpreted from this point of view. Natural relationships of these conditions with some mentioned families of ‘values’ are shown. These reinterpretations strongly suggest that these conditions are more naturally interpreted in terms of coalition formation than in terms of the classical notion of ‘value.’      

Strategy-proof coalition formation

We analyze coalition formation problems in which a group of agents is partitioned into coalitions and agents’ preferences only depend on the coalition to which they belong. We study rules that associate to each profile of preferences a partition of the society. We focus on strategy-proof rules on restricted domains of preferences, as the domains of additively representable or separable preferences. In such domains, the only strategy-proof and individually rational rules that satisfy either a weak version of efficiency or non-bossiness and flexibility are single-lapping rules. Single-lapping rules are characterized by severe restrictions on the set of feasible coalitions that are consistent with hierarchical organizations. These restrictions are necessary and sufficient for the existence of a unique core-stable partition. In fact, single-lapping rules always select the associated unique core-stable partition. Thus, our results highlight the relation between the non-cooperative concept of strategy-proofness and the cooperative concept of uniqueness of core-stable partitions.     

A fundamental study for partially defined cooperative games

The payoff of each coalition has been assumed to be known precisely in the conventional cooperative games. However, we may come across situations where some coalitional values remain unknown. This paper treats cooperative games whose coalitional values are not known completely. In the cooperative games it is assumed that some of coalitional values are known precisely but others remain unknown. Some complete games associated with such incomplete games are proposed. Solution concepts are studied in a special case where only values of the grand coalition and singleton coalitions are known. Through the investigations of solutions of complete games associated with the given incomplete game, we show a focal point solution suggested commonly from different viewpoints.     

Profit maximizing coalitions with shared capacities in distribution networks

We study the distribution network structure of multiple firms in the context of demand sensitivity to market offers. The problem consists in determining the profitability of horizontal collaboration between firms in a collaborative distribution schema. It considers the case of a set of regional distribution centers (DCs) where each DC is initially dedicated solely to one firm’s distribution activities and studies when it is beneficial that the DC owners collaborate through sharing their storage-throughput capacity. Such strategic decisions are made in order to improve the distribution capabilities of firms in terms of response time and cost-efficiency compared to the stand-alone situation. The problem is modeled as a coalition formation game in a cooperative framework, and we propose a collaborative distribution game with profit maximization. Three sharing mechanisms are modeled and tested: egalitarian allocation, proportional allocation, and Shapley value. The collaboration decision conditions for a given firm are analytically derived according to the sharing method considered and used to enhance the solution approach. Our numerical results clearly highlight the impact of this innovative collaboration opportunity on the firms’ performance in terms of distribution cost savings and revenue increases. An observed behavior is that the formation of several sub-coalitions prevails over the formation of a grand coalition, and that different cost sharing methods can lead to different sub-coalitions. We also provide managerial insights on the appropriate size of a coalition in various business instances tested, and on the key drivers that foster horizontal collaborative behavior among firms.     

A consistent value for games with n players and r alternatives

In Bolger [1993], an efficient value was obtained for a class of games called games with n players and r alternatives. In these games, each of the n players must choose one and only one of the r alternatives. This value can be used to determine a player’s “a priori” value in such a game. In this paper, we show that the value has a consistency property similar to the “consistency” for TU games in Hart/Mas-Colell [1989] and we present a set of axioms (including consistency) which characterizes this value.  The games considered in this paper differ from the multi-choice games considered by Hsiao and Raghavan [1993]. They consider games in which the actions of the players are ordered in the sense that, if i >j, then action i carries more “weight” than action j.  These games also differ from partition function games in that the worth of a coalition depends not only on the partitioning of the players but also on the action chosen by each subset of the partition. Received: April 1994/final version: June 1999     

The Egalitarian Solution for Multichoice Games

In a multichoice game a coalition is characterized by the level at which each player is acting, and to each coalition a real number is assigned. A multichoice solution assigns, for each multichoice game, a numerical value to each possible activity level of each player, intended to measure the contribution of each such level to reaching the grand coalition in which each player is active at the maximal level. The paper focuses on the egalitarian multichoice solution, characterized by the properties of Efficiency, Zero Contribution, Additivity, Anonymity, and Level Symmetry. The egalitarian solution is also shown to satisfy the property of marginalism: it measures the effect of lowering, ceteris paribus, a certain activity level by one. The solution is compared to a multichoice solution studied in Klijn, Slikker, and Zarzuelo (1999). Finally, it is discussed how the formalism of this paper can be applied to the different framework of multi-attribute utilities.     

Two-step coalition values for multichoice games

We introduce and compare several coalition values for multichoice games. Albizuri defined coalition structures and an extension of the Owen coalition value for multichoice games using the average marginal contribution of a player over a set of orderings of the player’s representatives. Following an approach used for cooperative games, we introduce a set of nested or two-step coalition values on multichoice games which measure the value of each coalition and then divide this among the players in the coalition using either a Shapley or Banzhaf value at each step. We show that when a Shapley value is used in both steps, the resulting coalition value coincides with that of Albizuri. We axiomatize the three new coalition values and show that each set of axioms, including that of Albizuri, is independent. Further we show how the multilinear extension can be used to compute the coalition values. We conclude with a brief discussion about the applicability of the different values.     

A dynamic solution in N-person cooperative game theory

We formulate a cooperative game as an extended form game in which each player in turn proposes payoffs to a coalition over M steps. Payoffs at time t are discounted by a penalty function f(t). If all players in a coalition agree to their payoffs, they receive them. Under a convergence hypothesis verified by computer for three players in many cases, we compute the payoffs resulting from a coalition pattern and give necessary conditions for particular patterns. The resulting solution is related to the Nash bargaining solution and the competitive solution.     

Dynamic realization games in newsvendor inventory centralization

Consider a set N of n (> 1) stores with single-item and single-period nondeterministic demands like in a classic newsvendor setting with holding and penalty costs only. Assume a risk-pooling single-warehouse centralized inventory ordering option. Allocation of costs in the centralized inventory ordering corresponds to modelling it as a cooperative cost game whose players are the stores. It has been shown that when holding and penalty costs are identical for all subsets of stores, the game based on optimal expected costs has a non empty core (Hartman et al. 2000, Games Econ Behav 31:26–49; Muller et al. 2002, Games Econ Behav 38:118–126). In this paper we examine a related inventory centralization game based on demand realizations that has, in general, an empty core even with identical penalty and holding costs (Hartman and Dror 2005, IIE Trans Scheduling Logistics 37:93–107). We propose a repeated cost allocation scheme for dynamic realization games based on allocation processes introduced by Lehrer (2002a, Int J Game Theor 31:341–351). We prove that the cost subsequences of the dynamic realization game process, based on Lehrer’s rules, converge almost surely to either a least square value or the core of the expected game. We extend the above results to more general dynamic cost games and relax the independence hypothesis of the sequence of players’ demands at different stages.     

Sequencing games without initial order

In this note we study uncertainty sequencing situations, i.e., one-machine sequencing situations in which no initial order is specified. We associate cooperative games with these sequencing situations, study their core, and provide links with the classic sequencing games introduced by Curiel et al. (Eur J Oper Res 40:344–351, 1989). Moreover, we propose and characterize two simple cost allocation rules for uncertainty sequencing situations with equal processing times.