Josep M. Izquierdo

Average Monotonic Cooperative Games

Abstract A subclass of monotonic transferable utility (T.U.) games is studied: average monotonic games. These games are totally balanced. We prove that the core coincides with both the bargaining set a la Davis and Maschler and the bargaining set a la Mas-Colell. To obtain this result a technique based on reduced games is used. Journal of Economic Literature Classification Number: C71

A Simple Procedure to Obtain the Extreme Core Allocations of an Assignment Market

Given an assignment market, we introduce a set of vectors, one for each possible ordering on the player set, which we name the max-payoff vectors. Each one of these vectors is obtained recursively only making use of the assignment matrix. Those max-payoff vectors that are efficient turn out to give the extreme core allocations of the market. When the assignment game has a large core, all the max-payoff vectors are extreme core allocations.

A characterization of convex TU games by means of the Mas-Colell bargaining set (à la Shimomura)

Within the class of zero-monotonic games, we prove that a cooperative game with transferable utility is convex if and only if the core of the game coincides with the Mas-Colell bargaining set (à la Shimomura, in Int J Game Theory 26:283–302, 1997).

Rationing problems with ex-ante conditions

An extension of the standard rationing model is introduced. Agents are not only identified by their respective claims to some amount of a scarce resource, but also by some exogenous ex-ante conditions (initial stock of resource or net worth of agents, for instance), other than claims. Within this framework, we define a generalization of the constrained equal awards rule and provide two different characterizations of this generalized rule. Finally, we use the corresponding dual properties to characterize a generalization of the constrained equal losses rule.

A characterization of convex games by means of bargaining sets

The aim of the paper is to characterize the classical convexity notion for cooperative TU games by means of the Mas-Colell and the Davis–Maschler bargaining sets. A new set of payoff vectors is introduced and analyzed: the max-Weber set. This set is defined as the convex hull of the max-marginal worth vectors. The characterizations of convexity are reached by comparing the classical Weber set, the max-Weber set and a selected bargaining set.

Sequentially compatible payoffs and the core in TU-games

Abstract In the context of cooperative TU-games, we introduce a recursive procedure to distribute the surplus of cooperation when there is an exogenous ordering among the set of players N. In each step of the process, using a given notion of reduced games, an upper and a lower bound for the payoff to the player at issue are required. Sequentially compatible payoffs are defined as those allocation vectors that meet these recursive bounds. For a family of reduction operations, the behavior of this new solution concept is analyzed. For any ordering of N, the core of the game turns out to be the set of sequentially compatible payoffs when the Davis–Maschler reduced games are used. Finally, we study which reduction operations give an advantage to the first player in the ordering.

The core and the steady bargaining set for convex games

Within the class of zero-monotonic and grand coalition superadditive cooperative games with transferable utility, the convexity of a game is characterized by the coincidence of its core and the steady bargaining set. As a consequence it is proved that convexity can also be characterized by the coincidence of the core of a game and the modified Zhou bargaining set à la Shimomura.

The incentive core in co-investment problems

We study resource-monotonicity properties of core allocations in co-investment problems: those where a set of agents pool their endowments of a certain resource or input in order to obtain a joint surplus or output that must be allocated among the agents. We analyze whether agents have incentives to raise their initial contribution (resource-monotonicity). We focus not only on looking for potential incentives to agents who raise their contributions, but also in not harming the payoffs to the rest of agents (strong monotonicity property). A necessary and sufficient condition to fulfill this property is stated and proved. We also provide a subclass of co-investment problems for which any core allocation satisfies the aforementioned strong resource-monotonicity property. Moreover, we introduce the subset of core allocations satisfying this condition, namely the incentive core.

Constrained Multi-Issue Rationing Problems

We study a variant of the multi-issue rationing model, where agents claim for several issues. In this variant, the available amount of resource intended for each issue is constrained to an amount fixed a priori according to exogenous criteria. The aim is to distribute the amount corresponding to each issue taking into account the allocation for the rest of issues (issue-allocation interdependence). We name these problems constrained multi-issue allocation situations (CMIA). In order to tackle the solution to these problems, we first reinterpret some single-issue egalitarian rationing rules as a minimization program based on the idea of finding the feasible allocation as close as possible to a specific reference point. We extend this family of egalitarian rules to the CMIA framework. In particular, we extend the constrained equal awards rule, the constrained equal losses rule and the reverse Talmud rule to the multi-issue rationing setting, which turn out to be particular cases of a family of rules, namely the extended α-egalitarian family. This family is analysed and characterized by using consistency principles (over agents and over issues) and a property based on the Lorenz dominance criterion.

Rationing problems with payoff thresholds

An extension of the standard rationing model is introduced. Agents are not only identi fied by their respective claims over some amount of a scarce resource, but also by some payoff thresholds. These thresholds introduce exogenous differences among agents (full or partial priority, past allocations, past debts, ...) that may influence the final distribution. Within this framework we provide generalizations of the constrained equal awards rule and the constrained equal losses rule. We show that these generalized rules are dual from each other.We characterize the generalization of the equal awards rule by using the properties of consistency, path-independence and compensated exemption. Finally,we use the duality between rules to characterize the generalization of the equal losses solution.