In this paper, we introduce a new approach to measure the dissatisfaction for coalitions of players in cooperative transferable utility games. This is done by considering affine (and convex) combinations of the classical excess and the proportional excess. Based on this so-called alpha-excess, we defi ne new solution concepts for cooperative games, such as the alpha-prenucleolus and the alpha-prekernel. The classical prenucleolus and prekernel are a special case. We characterize the alpha-prekernel by strong stability and the alpha-balanced surplus property. Also, we show that the payoff vector generated by the alpha-prenucleolus belongs to the alpha-prekernel.

An important operational aspect in the management of tolled highways is how the collected tolls should be allocated over the different highway segments, either operated by different operators or by different units of one operator. This paper analyzes this toll allocation problem both from an axiomatic and a game theoretic perspective. Based on different toll charging systems, specifically the distance-based toll system and the fixed toll system, we propose three allocation or sharing methods: the Segments Equal Sharing method, the Exits Equal Sharing method, and the Entrances Equal Sharing method. After direct and game theoretic characterizations of these methods, we apply them to several real-life highways.

In this paper, we analyze priority coalitional games as an extension of balanced games with transferable utility. Here, conditions imposed by a stable allocation used in the past need to be respected when deciding on a new allocation of the revenues obtained by the grand coalition. Using claims problems as a tool, we define weak and strong priority games with respect to a given allocation and a priority coalition, and study their properties and core. In the weak priority game, the value of a coalition reflects that coalitional members with priority should not be harmed, i.e., the coalitional value should never be below their total benefits according to the initial allocation. In the strong priority game, in addition to the previous condition, players outside the coalition that are not in the priority group should not harbour aspirations exceeding those benefits given by the initial allocation. Priority games are also extended to analyze cooperation restricted by a hierarchical structure. These games restrict the core so that desirable monotonicity properties are satisfied when there is a hierarchy. This allows for reconciling core selection and monotonicity over a hierarchy.

This work deals with the weighted excesses of players in cooperative games which are obtained by summing up all the weighted excesses of all coalitions to which they belong. We first show that the resulting payoff vector is the corresponding least square value by lexicographically minimizing the individual weighted excesses of players over the preimputation set, and thus give an alternative characterization of the least square values. Second, we show that these results give rise to lower and upper bounds for the core payoff vectors and, using these bounds, we show that the least square values can be seen as the center of a polyhedron defined by these bounds. This provides a second new characterization of the least square values. Third, we show that the individually rational least square value is the solution that lexicographically minimizes the individual weighted excesses of players over the imputation set.

This paper axiomatically studies bankruptcy problems with nontransferable utility by focusing on generalizations of consistency and the contested garment principle. On the one hand, we discuss several consistency notions and introduce the class of parametric bankruptcy rules which contains the proportional rule, the constrained relative equal awards rule, and the constrained relative equal losses rule. On the other hand, we introduce the class of adjusted bankruptcy rules and characterize the relative adjustment principle by truncation invariance, minimal rights first, and a weak form of relative symmetry.

This paper studies bankruptcy problems with nontransferable utility as a generalization of bankruptcy problems with monetary estate and claims. Following the theory on TU-bankruptcy, we introduce a duality notion for NTU-bankruptcy rules and derive several axiomatic characterizations of the proportional rule and the constrained relative equal awards rule.

This paper analyses Chinese postman games with repeated players, which generalize Chinese postman games by dropping the one-to-one relation between edges and players. In our model, we allow players to own more than one edge, but each edge belongs to at most one player. The one-to-one relation between edges and players is essential for the equivalence between Chinese postman-totally balanced and Chinese postman-submodular graphs shown in Granot et al. (1999). We illustrate the invalidity of this result in our model. Besides, the location of the post office has a relevant role in the submodularity and totally balancedness of Chinese postman games with repeated players. Therefore, we focus on sufficient conditions on the assignment of players to edges to ensure submodularity of Chinese postman games with repeated players, independently of the associated travel costs. Moreover, we provide some insights on the difficulty of finding necessary conditions on assignment functions to this end.

This paper is a reaction on Ginsburgh and Zang (2003).It reconsiders the problem where a group of museums offer a pass such that the owner can visit these museums an unlimited number of times during a fixed period of time.The problem addressed is how to share the total joint income of this pass system among the museums.Ginsburgh and Zang propose to use the Shapley value of an associated cooperative game.Arguments are provided to model this problem within the framework of bankruptcy problems.

This paper analyzes bankruptcy games with nontransferable utility as a generalization of bankruptcy games with monetary payoffs. Following the game theoretic approach to NTU-bankruptcy problems, we study some appropriate properties and the core of NTU-bankruptcy games. Generalizing the core cover and the reasonable set to the class of NTU-games, we show that NTU-bankruptcy games are compromise stable and reasonable stable. Moreover, we derive a necessary and sufficient condition for an NTU-bankruptcy rule to be game theoretic.

In this paper, we analyze sequential bankruptcy problems, which generalize bankruptcy problems. They contain the problems of sharing water in a trans-boundary river and of allocating expedition rewards in projects. We provide three mechanisms for generalizing rules for bankruptcy problems to rules for sequential bankruptcy problems: the upwards, the downwards, and the two-steps mechanisms. Further, we characterize the upwards constrained equal awards, the upwards constrained equal losses, and the upwards proportional rules on the basis of upwards composition and upwards path independence. Moreover, we compare the three mechanisms based on inheritance of well-established properties for bankruptcy rules to the setting of sequential bankruptcy rules.

We consider the problem of sharing water among agents located along a river, who have quasi-linear preferences over water and money, as introduced by Ambec and Sprumont (2002). Given an efficient distribution of river water, where water can be sent from upstream agents to downstream agents but not the other way around, the question is what should be the monetary transfers that downstream agents have to pay to upstream agents as compensation for the upstream agents to abstain from water consumption. Under the quasi-linear utility functions, an efficient water allocation and a transfer scheme determine a welfare distribution. Under more general benefit functions, van den Brink, Estévez-Fernández, van der Laan and Moes (2014) consider three basic axioms for welfare distribution and, additionally, add an independence axiom with respect to benefit functions of upstream, respectively, downstream agents. Both independence axioms yield a unique welfare distribution. In this paper we investigate the impact of similar independence axioms but with respect to water inflows. Surprisingly, together with the three basic axioms, these do not characterize a welfare distribution. Moreover, independence of upstream inflows turns out to be incompatible with the three basic axioms, while independence of downstream inflows yields mutliple solutions. We weaken one of the basic axioms to get compatibility with upstream independence, and then strengthen it to get uniqueness with downstream independence.

This paper analyzes the 1-nucleolus and, in particular, its relation to the nucleolus. It is seen that, contrary to the nucleolus, the 1-nucleolus can be computed in polynomial time due to a characterization using a combination of standard bankruptcy rules for associated bankruptcy problems. Sufficient conditions on a compromise stable game are derived such that the 1-nucleolus and the nucleolus coincide.

In this paper we consider one-to-many matching problems where the preferences of the agents involved are represented by monetary reward functions. We characterize Pareto optimal matchings by means of contractual exchange stability and matchings of maximum total reward by means of compensational exchange stability. To conclude, we show that in going from an initial matching to a matching of maximum total reward, one can always provide a compensation schedule that will be ex-post stable in the sense that there will be no subset of agents who can all by deviation obtain a higher reward. The proof of this result uses the fact that the core of an associated compensation matching game with constraints is nonempty.

This paper extends the notion of individual minimal rights for a transferable utility game (TU-game) to coalitional minimal rights using minimal balanced families of a specific type, thus defining a corresponding minimal rights game. It is shown that the core of a TU-game coincides with the core of the corresponding minimal rights game. Moreover, the paper introduces the notion of the \(k\)-core cover as an extension of the core cover. The \(k\)-core cover of a TU-game consists of all efficient payoff vectors for which the total joint payoff for any coalition of size at most \(k\) is bounded from above by the value of this coalition in the corresponding dual game, and from below by the value of this coalition in the corresponding minimal rights game. It is shown that the core of a TU-game with player set \(N\) coincides with the largest integer below or equal to \(\frac{|N|}{2}\)-core cover. Furthermore, full characterizations of games for which a \(k\)-core cover is nonempty and for which a \(k\)-core cover coincides with the core are provided.

We consider the problem of sharing water among agents located along a river, who have quasi-linear preferences over water and money. The benefit of consuming an amount of water is given by a continuous, concave benefit function. In this setting, a solution efficiently distributes water over the agents and wastes no money. Since we deal with concave benefit functions, it is not always possible to follow the usual approach and define a cooperative river game. Instead, we directly introduce axioms for solutions on the water allocation problem. Besides three basic axioms, we introduce two independence axioms to characterize the downstream incremental solution, introduced by Ambec and Sprumont (J Econ Theory 107:453–462, 2002), and a new solution, called the UTI incremental solution. Both solutions can be implemented by allocating the water optimally among the agents and monetary transfers between the agents. We also consider the particular case in which every agent has a satiation point, constant marginal benefit equal to one up to its satiation point and marginal benefit of zero thereafter. This boils down to a water claim problem, where each agent only has a nonnegative claim on water, but no benefit function is specified. In this case, both solutions can be implemented without monetary transfers.

This paper extends the notion of individual minimal rights for a transferable utility game (TU-game) to coalitional minimal rights using minimal balanced families of a specific type, thus defining a corresponding minimal rights game. It is shown that the core of a TU-game coincides with the core of the corresponding minimal rights game. Moreover, the paper introduces the notion of the k-core cover as an extension of the core cover. The k-core cover of a TU-game consists of all efficient payoff vectors for which the total joint payoff for any coalition of size at most k is bounded from above by the value of this coalition in the corresponding dual game, and from below by the value of this coalition in the corresponding minimal rights game. It is shown that the core of a TU-game with player set N coincides with the ⌊|N |/2⌋-core cover. Furthermore, full characterizations of games for which a k-core cover is nonempty and for which a k-core cover coincides with the core are provided.

This discussion paper led to an article in 'European Journal of Operational Research' , (2014), 237, 606-616. In this paper, we analyze cost sharing problems arising from a general service by explicitly taking into account the generated revenues. To this cost-revenue sharing problem, we associate a cooperative game with transferable utility, called cost-revenue game. By considering cooperation among the agents using the general service, the value of a coalition is defined as the maximum net profit that the coalition may obtain by means of cooperation. As a result, a coalition may profit from not allowing all its members to get the service that generates the revenues. We focus on the study of the core of cost-revenue games. Under the assumption that cooperation among the members of the grand coalition grants the use of the service under consideration to all its members, it is shown that a cost-revenue game has a non-empty core for any vector of revenues if, and only if, the dual game of the cost game has a large core. Using this result, we investigate minimum cost spanning tree games with revenues. We show that if every connection cost can take only two values (low or high cost), then, the corresponding minimum cost spanning tree game with revenues has a non-empty core. Furthermore, we provide an example of a minimum cost spanning tree game with revenues with an empty core where every connection cost can take only one of three values (low, medium, or high cost).

In this paper we establish a relationship between the core cover of a compromise admissible game and the core of a particular bankruptcy game: the core cover of a compromise admissible game is, indeed, a translation of the set of coalitionally stable allocations captured by an associated bankruptcy game. Moreover, we analyze the combinatorial complexity of the core cover and, consequently, of the core of a compromise stable game.

This paper analyzes situations in which a project consisting of several activities is not realized according to plan. If the project is expedited, a reward arises. Analogously, a penalty arises if the project is delayed. This paper considers the case of arbitrary nondecreasing reward and penalty functions on the total expedition and delay, respectively. Attention is focused on how to divide the total reward (penalty) among the activities: the core of a corresponding cooperative project game determines a set of stable allocations of the total reward (penalty). In the definition of project games, surplus (cost) sharing mechanisms are used to take into account the specific characteristics of the reward (penalty) function at hand. It turns out that project games are related to bankruptcy and taxation games. This relation allows us to establish the nonemptiness of the core of project games.