We consider cooperative TU-games with unpaid players, which are described by a TU-game and two categories of players, paid and unpaid. Unpaid players participate in the cooperative game but are not rewarded for their participation, for instance for legal reasons. The objective is then to determine how the contributions of unpaid players are redistributed among the paid players. To meet this goal, we introduce and characterize axiomatically three values that are inspired by the Shapley value but differ in the way they redistribute the contributions of unpaid players. These values are unified as instances of a more general two-step allocation procedure.

This study proposes a general framework for aggregation problems, in which agents’ signals given in an infinite sequence are aggregated. Notably, no restriction is imposed on the set of signals; hence, this framework is applicable to any type of attributes, including preferences, ballots, pieces of information, or judgments. The result can include the empty set that corresponds to the silence of the aggregation rule. Our main focus is to analyze the process by which the structure of the family of decisive coalitions forms a filter or an ultrafilter. In our framework, a coalition is regarded as decisive if a signal is socially approved as long as every agent in the coalition picks the signal up unanimously. First, we show that the family of decisive coalitions forms a filter under a certain axiom representing a stability condition. Next, in the case where the set of signals is finite, we provide axiomatic characterizations of the class of agreement-based rules associated with filters or ultrafilters: a signal is approved if and only if it is a unanimous agreement of some coalition in a specified filter or ultrafilter. Our axiomatic results demonstrate the robustness and stability of such rules.

This paper studies cooperative games where agents have multiple participation levels, and are embedded in a communication network. We introduce a solution concept that generalizes the Myerson value. It is constructed through a restriction operator that captures the interaction between the network structure and agents’ participation levels, and the way this interaction constrains feasible coalitions. The solution is axiomatically characterized and shown to satisfy a stability property. To demonstrate its applicability, we extend the classical sequencing problem to a setting where agents can exert effort to reduce processing times. Our solution then determines how the resulting cost savings are distributed when agents exert maximal effort and tasks are ordered in a cost-minimizing sequence.

This paper aims to characterize classes or types of values for transferable utility games in partition function form, accounting for externalities where the worth of a coalition depends on the cooperation structure outside it. We follow an axiomatic approach, focusing on values that represent two-stage anticipations. In the first stage, the worth of each coalition is computed as a weighted average of each of its possible worths (depending on the cooperation structure outside the coalition). In the second stage, these anticipated worths for all coalitions are used to derive the value for participating in the game, treating the game as if it were one without externalities. We extend previous work by Macho-Stadler et al. (Journal of Economic Theory, 135, 339–356, 2007; Games and Economic Behavior, 108, 49–64, 2018) beyond Efficiency and Symmetry, providing axiomatic characterizations for values based on both exogenous and endogenous weights. Additionally, we model first-step anticipations as operators from games with externalities to games without externalities, characterizing the family of weighted average operators. We also explore the relationship of the average approach with marginalism, highlighting that marginalist values constructed from the average approach require weights satisfying a consistency condition, termed recursivity, but not necessarily a symmetry condition. Finally, we contrast these results with Grabisch and Funaki’s (2012) coalition formation value, which involves averages over dynamic coalition formation processes but cannot be constructed from the average approach.

A cooperative game with a coalition structure is formed by a TU-game and a partition of the agent set. For this class of games, the Owen value is computed as a two-step procedure where the relevant coalitions are those formed by the union of some elements of the partition and a coalition of another element of the partition. In this paper, we consider a broader class of games where the partition is replaced by a collection of (not necessarily pairwise disjoint) coalitions over the agent set and where, in each element of this collection, cooperation among the agents is restricted. Agents then organize themselves into a profile of feasible coalitions. This class of games can be applied to several situations such as the problem of allocating aircraft landing fees in the presence of airlines and codeshare flights. We begin by defining and axiomatically characterizing the class of flow methods, which are marginal values whose coefficients induce a unit flow on the graph of feasible coalition profiles. We then define Owen-type values constructed from flow methods. We show that these values are flow methods whose flow is decomposable into two flows. Finally, we introduce two axioms from which we characterize the flows that can be decomposed in this way, and hence the flow methods constructed by our Owen-type procedure. The last part of the paper studies some special cases.

Various occasions require that an individual ranks others in her environment based on how these others rank her; consider a company that seeks an employee who values her back, and a student that works better with a professor who also appreciates her strengths. We introduce a formal framework for the ranking of rankings. A set of objects are weakly ranked by a set of items, and a given object (e.g., the company or the student) must obtain a weak ranking of the items (e.g., all employees or all professors) that depends on their provided rankings of the objects. To conduct an axiomatic analysis of this setting, we propose several normative properties that apply to solutions for the ranking of rankings. Our axioms are inspired by analogous properties in the fields of decision and social choice theory, such as anonymity, monotonicity, and independence. By considering combinations of different axioms, we characterise natural families of solutions, as well as unique solutions therein: lexicographic solutions, and a scoring one.

This paper considers an electoral system in which voters may approve any subset of options. We introduce the class of proportional threshold methods that select the subset of options whose share of approvals in the population meets or exceeds a certain threshold. We provide an axiomatic characterization of these methods using a principle of consistency between populations and profiles of approval voting ballots. A distinctive feature of our approach is to provide a geometric proof of this characterization result.

We consider the cost sharing issue resulting from the maintenance of a hazardous waste transportation network represented by a sink tree. The participating agents are located on the nodes of the network and must transport their waste to the sink through costly network portions. We introduce the Liability rule, which is inspired by the principles applied by the courts to settle cost-allocation disputes in the context of hazardous waste. We provide an axiomatic characterization of this rule. Furthermore, we show that the Liability rule coincides with the Priority Shapley value, a new value on an appropriate domain of multi-choice games arising from hazardous waste transportation problems. Finally, we also axiomatize the Priority Shapley value on the full domain of multi-choice games.

We introduce a new family of values for TU-games with a priority structure, which both contains the Priority value recently introduced by Beal et al. and the Weighted Shapley values (Kalai & Samet). Each value of this family is called a Weighted priority value and is constructed as follows. A strictly positive weight is associated with each agent and the agents are partially ordered according to a binary relation. An agent is a priority agent with respect to a coalition if it is maximal in this coalition with respect to the partial order. A Weighted priority value distributes the dividend of each coalition among the priority agents of this coalition in proportion to their weights. We provide an axiomatic characterization of the family of the Weighted Shapley values without the additivity axiom. To this end, we borrow the Priority agent out axiom from Beal et al., which is used to axiomatize the Priority value. We also reuse, in our domain, the principle of Superweak differential marginality introduced by Casajus to axiomatize the Positively weighted Shapley values. We add a new axiom of Independence of null agent position which indicates that the position of a null agent in the partial order does not affect the payoff of the other agents. Together with Efficiency, the above axioms characterize the Weighted Shapley values. We show that this axiomatic characterization holds on the subdomain where the partial order is structured by levels. This entails an alternative characterization of the Weighted Shapley values. Two alternative characterizations are obtained by replacing our principle of Superweak differential marginality by Additivity and invoking other axioms.

A coalitional ranking problem is described by a weak order on the set of nonempty coalitions of a given agent set. A social ranking is a weak order on the set of agents. We consider social rankings that are consistent with stable/core partitions. A partition is stable if there is no coalition better ranked in the coalitional ranking than the rank of the cell of each of its members in the partition. The core-partition social ranking solution assigns to each coalitional ranking problem the set of social rankings such that there is a core-partition satisfying the following condition: a first agent gets a higher rank than a second agent if and only if the cell to which the first agent belongs is better ranked in the coalitional ranking than the cell to which the second agent belongs in the partition. We provide an axiomatic characterization of the core-partition social ranking and an algorithm to compute the associated social rankings.

A TU game is totally positive if it is a linear combination of unanimity games with nonnegative coefficients. We show that the core on each cone of convex games that contains the set of totally positive games is characterized by the traditional properties Pareto efficiency, additivity (ADD), individual rationality, and the null-player property together with one new property, called unanimity requiring that the solution, when applied to a unanimity game on an arbitrary coalition, allows to distribute the entire available amount of money to each player of this coalition. We also show that the foregoing characterization can be generalized to the domain of balanced games by replacing ADD by “ADD on the set of totally positive games plus super-additivity (SUPA) in general”. Adding converse SUPA allows to characterize the core on arbitrary domains of TU games that contain the set of all totally positive games. Converse SUPA requires a vector to be a member of the solution to a game whenever, when adding a totally positive game such that the sum becomes totally additive, the sum of the vector and each solution element of the totally positive game belongs to the solution of the aggregate game. Unlike in traditional characterizations of the core, our results do not use consistency properties.

A coalitional ranking describes a situation where a finite set of agents can form coalitions that are ranked according to a weak order. A social ranking solution on a domain of coalitional rankings assigns a social ranking, that is a weak order over the agent set, to each coalitional ranking of this domain. We introduce two lexicographic solutions for a variable population domain of coalitional rankings. These solutions are computed from the individual performance of the agents, then, when this performance criterion does not allow to decide between two agents, a collective performance criterion is applied to the coalitions of higher size. We provide parallel axiomatic characterizations of these two solutions.

We consider cooperatives games (TU-games) enriched by a system of a priori unions and a communication forest graph which are independent from each other. These two structures reflect the limitations of cooperation possibilities. In this framework, we introduce four Owen-type allocation rules, which are defined by a two-step application of an allocation rule à la Owen (in: Henn R, Moeschlin O (eds) Essays in mathematical economics and game theory, Springer, Berlin, 1977) to TU-games with a priori unions where the TU-game is replaced by Myerson’s (Math Oper Res 2:225–229, 1977) graph-restricted TU-game. The four possibilities arise by applying, at each step, either the Myerson value (Myerson 1977) or the average tree solution (Herings et al. in Games Econ Behav 62:77–92, 2008). Our main result offers comparable axiomatizations of these four allocation rules.

In this study, we propose a new direction of research on the axiomatic analysis of approval voting, which is a common democratic decision method. Its novelty is to examine an infinite population setting, which includes an application to intergenerational problems. In particular, we assume that the set of the population is countably infinite. We provide several extensions of the method of approval voting for this setting. As our main result, axiomatic characterizations of the extensions are offered by revealing a direct link between approval voting and the Borda rule. The characterized methods are natural extensions of the standard approval voting method for the finite-population case and are regarded as minimum requirements for other possible infinite-population extensions, which are reasonably democratic.

In many real world situations, the design of social rankings over agents or items from a given raking over groups or coalitions, to which these agents or items belong to, is of big interest. With this aim, we revise the lexicographic excellence solution and introduce two novel solutions which, moreover, take into account the size of the groups. We present some desirable axioms which are interpreted in this context. Next, a comparable axiomatization of these three solutions is established, revealing the main differences among the two new social rankings and the lexicographic excellence solution. Finally, we apply the three social rankings under study to a real scenario. Specifically, the performance of some football players of Paris Saint-Germain during the UEFA Champions League according to these three rules is analyzed.

We study cooperative games with a priority structure modeled by a poset on the agent set. We introduce the Priority value, which splits the Harsanyi dividend of each coalition among the set of its members over which no other coalition member has priority. This allocation shares many desirable properties with the classical Shapley value: it is efficient, additive and satisfies the null agent axiom. We provide two axiomatic characterizations of the Priority value which invoke both classical axioms and new axioms describing the effects of the priority structure on the payoff allocation. Finally, in the special case where agents are ranked by level, a link between the Priority value, the weighted Shapley values and the Owen-type values can be drawn.

We introduce the game INFLUENCE, a scoring combinatorial game, played on a directed graph where each vertex is either colored black or white. The two players, Black and White play alternately by taking a vertex of their color and all its successors (for Black) or all its predecessors (for White). The score of each player is the number of vertices he has taken. We prove that INFLUENCE is a nonzugzwang game, meaning that no player has interest to pass at any step of the game, and thus belongs to Milnor's universe. We study this game in the particular class of paths where black and white are alternated. We give an almost tight strategy for both players when there is one path. More precisely, we prove that the first player always gets a strictly better score than the second one, but that the difference between the score is bounded by 5. Finally, we exhibit some graphs for which the initial proportion of vertices of the color of a player is as small as possible but where this player can get almost all the vertices.

A coalitional ranking problem is described by a weak order on the set of nonempty coalitions of a given agent set. A social ranking is a weak order on the set of agents. We consider social rankings that are consistent with stable/core partitions. A partition is stable if there is no coalition better ranked in the coalitional ranking than the rank of the cell of each of its members in the partition. The core-partition social ranking solution assigns to each coalitional ranking problem the set of social rankings such that there is a core-partition satisfying the following condition: a first agent gets a higer rank than a second agent if and only if the cell to which the first agent belongs is better ranked in the coalitional ranking than the cell to which the second agent belongs in the partition. We provide an axiomatic characterization of the core-partition social ranking and an algorithm to compute the associated social rankings.

We relax the assumption that the grand coalition must form by imposing the axiom of Cohesive efficiency: the total payoffs that the players can share is equal to the maximal total worth generated by a coalition structure. We determine how the three main axiomatic characterizations of the Shapley value are affected when the classical axiom of Efficiency is replaced by Cohesive efficiency. We introduce and characterize two variants of the Shapley value that are compatible with Cohesive efficiency. We show that our approach can also be applied to the variants of more egalitarian values.