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In monotone simple games, larger coalitions typically wield more power, but do all players align their efforts effectively? Consider a voting scenario where a coalition forms, but needs more voters to pass a bill. The cohesion of the new group of voters hinges on whether all the new members can proficiently collaborate with the existing players to ensure the bill's passage or if subgroups form that pursue an independent alternative, thus generating antagonism among the new voters. This research introduces the coopetition index, ranging from -1 to 1, to measure agents' preferences for cooperation (near 1) or competition (near -1) with the remaining players. We also introduce the Banzhaf and Shapley-Owen coopetition indices, addressing limitations of previous indices. By applying our index to the apex game and symmetric majority games, we observe that cooperation and competition frequently balance each other out, leading to null values for the Shapley-Owen and Banzhaf coopetition indices. To distinguish balanced scenarios from those involving powerless coalitions, we define a decisiveness index that measures the extent of coalition involvement in negotiations., Comment: 24 pages, 4 tables

We define a notion of the criticality of a player for simple monotone games based on cooperation with other players, either to form a winning coalition or to break a winning one, with an essential role for all the players involved. We compare it with the notion of differential criticality given by Beisbart that measures power as the opportunity left by other players. We prove that our proposal satisfies an extension of the strong monotonicity introduced by Young, assigns no power to null players and does not reward free riders, and can easily be computed from the minimal winning and blocking coalitions. An application to the Italian elections is presented. Our analysis shows that the measures of group criticality defined so far cannot weigh essential players while only remaining an opportunity measure. We propose a group opportunity test to reconcile the two views., Comment: 28 pages, 5 tables, 5 figures. First presented at the 12th Conference on Economic Design, Padova, June 9, 2022 and at EURO 2022, Espoo, July 6, 2022

Inheritances, divorces or liquidations of companies require common assets to be divided among the entitled parties. Legal methods usually consider the market value of goods, while fair division theory takes into account the parties' preferences expressed as utilities. I combine the two practices to define a procedure that optimally allocates divisible goods with market values to people with easily elicited preferences. Imposing exact equality in the bundles' monetary values may produce unacceptable solutions. I drop the tight requirement and suggest a procedure in which the differences in the monetary values are explained in terms of satisfaction per monetary share as perceived by the agents. A robustness study shows the consequences of misspecification in the model parameters., Comment: 49 pages, 9 figures, 14 tables

In Briata, Dall'Aglio and Fragnelli (2012), the authors introduce a cooperative game with transferable utility for allocating the gain of a collusion among completely risk-averse agents involved in the fair division procedure introduced by Knaster (1946). In this paper we analyze the Shapley value (Shapley, 1953) of the game and propose its use as a measure of the players' attitude towards collusion. Furthermore, we relate the sign of the Shapley value with the ranking order of the players' evaluation, and show that some players in a given ranking will always deter collusion. Finally, we characterize the coalitions that maximize the gain from collusion, and suggest an ad-hoc coalition formation mechanism., Comment: 20 pages, 1 table. This new version corrects some mistakes and improves the layout

We consider the division of a finite number of homogeneous divisible items among three players. Under the assumption that each player assigns a positive value to every item, we characterize the optimal allocations and we develop two exact algorithms for its search. Both the characterization and the algorithm are based on the tight relationship two geometric objects of fair division: the Individual Pieces Set (IPS) and the Radon-Nykodim Set (RNS)., Comment: 28 pages, 10 figures

We consider upper and lower bounds for maxmin allocations of a completely divisible good in both competitive and cooperative strategic contexts. We then derive a subgradient algorithm to compute the exact value up to any fixed degree of precision., Comment: 20 pages, 3 figures. This third version improves the overll presentation; Optimization and Control (math.OC), Computer Science and Game Theory (cs.GT), Probability (math.PR)