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Marco Dall'Aglio

Publications and source records attributed to Marco Dall'Aglio.

10 recordsLinked to original sources
arXiv2025

Coopetitive Index: a measure of cooperation and competition in coalition formation

We extend the coopetition index introduced by Aleandri and Dall'Aglio (2025) for simple games to the broader class of monotone transferable utility (TU) games and to all non-empty coalitions, including singletons. The new formulation allows us to define an absolute coopetition index with a universal range in [-1,1], facilitating meaningful comparisons across coalitions. We study several notable instances of the index, including the Banzhaf, Uniform Shapley, and Shapley-Owen coopetition indices, and we derive explicit formulas that connect coopetition to classical semivalues. Finally, we provide axiomatic characterizations of the Uniform Shapley and Shaple--Owen versions, showing that each is uniquely determined by linearity, symmetry over pure bargaining games, external null player neutrality, and a contraction axiom reflecting its internal distribution. These results position the coopetition index as a versatile tool for quantifying the cooperative and competitive tendencies of coalitions in TU-games.

cs.GT↗
arXiv2025

A Coopetition Index for Coalitions in Simple Games

In 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 two classes of coopetition indices -- one relative and one absolute, the latter ranging from -1 to 1, to measure agents' preferences for cooperation (when positive) or competition (when negative) with the remaining players. These indices, together with a generalized group value, provide a comprehensive picture of the relevance and the cohesion of groups. We discuss the relationship with similar group indices and provide proper coopetition Banzhaf and Shapley-Owen types of indices. By applying our indices 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. An electoral application with real world data is also considered.

cs.GT↗
arXiv2024

With a little help from my friends: essentiality vs opportunity in group criticality

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.

econ.TH↗
arXiv2022

Fair Division of Goods in the Shadow of Market Values

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.

cs.GT↗
arXiv2017

The Shapley Value in the Knaster Gain Game

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.

math.OC↗
arXiv2016

Characterizing and Finding the Pareto Optimal Equitable Allocation of Homogeneous Divisible Goods Among Three Players

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).

math.OC↗
arXiv2013

On Bankruptcy Game Theoretic Interval Rules

Interval bankruptcy problems arise in situations where an estate has to be liquidated among a fixed number of creditors and uncertainty about the amounts of the claims is modeled by intervals. We extend in the interval setting the classical results by Curiel, Maschler and Tijs (1987) that characterize division rules which correspond to solutions of the cooperative bankruptcy game. Finally, we analyze the difficulties with incorporating the uncertainty about the estate.

q-fin.GN↗
arXiv2012

Bayesian Posteriors Without Bayes' Theorem

The classical Bayesian posterior arises naturally as the unique solution of several different optimization problems, without the necessity of interpreting data as conditional probabilities and then using Bayes' Theorem. For example, the classical Bayesian posterior is the unique posterior that minimizes the loss of Shannon information in combining the prior and the likelihood distributions. These results, direct corollaries of recent results about conflations of probability distributions, reinforce the use of Bayesian posteriors, and may help partially reconcile some of the differences between classical and Bayesian statistics.

math.ST↗