Rationality, aggregate monotonicity and consistency in cooperative games: some (im)possibility results

Calleja, Pedro; Llerena, Francesc

doi:10.1007/s00355-016-0966-z

Cited by 7 publications

(8 citation statements)

References 35 publications

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“…Note that, for single-valued solutions, our version of RSAM coincides with the corresponding property introduced by Calleja and Llerena (2017). The following theorem characterizes the weighted prenucleoli that satisfy RSAM.…”

Section: Notation Definitions and Preliminariesmentioning

confidence: 79%

“…However, the conditions only remain necessary in the general n-person case. Without assuming symmetry, we show that a weighted (pre)nucleolus satisfies regular SAM as defined by Calleja and Llerena (2017) if and only if the weight system is regular (i.e., associated with a positive payoff vector). As a consequence, a weighted prenucleolus satisfies ESD if and only if it is the per capita prenucleolus.…”

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confidence: 98%

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Monotonicity and Weighted Prenucleoli: A Characterization Without Consistency

2020

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A solution on a set of transferable utility (TU) games satisfies strong aggregate monotonicity (SAM) if every player can improve when the grand coalition becomes richer. It satisfies equal surplus division (ESD) if the solution allows the players to improve equally. We show that the set of weight systems generating weighted prenucleoli that satisfy SAM is open which implies that for weight systems close enough to any regular system the weighted prenucleolus satisfies SAM. We also provide a necessary condition for SAM for symmetrically weighted nucleoli. Moreover, we show that the per capita nucleolus on balanced games is characterized by single-valuedness (SIVA), translation and scale covariance (COV), and equal adjusted surplus division (EASD), a property that is comparable but stronger than ESD. These properties together with ESD characterize the per capita prenucleolus on larger sets of TU games. EASD and ESD can be transformed to independence of (adjusted) proportional shifting and these properties may be generalized for arbitrary weight systems p to I(A)S p . We show that the p-weighted prenucleolus on the set of balanced TU games is characterized by SIVA, COV, and IAS p ; and on larger sets by additionally requiring IS p .

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Section: Notation Definitions and Preliminariesmentioning

confidence: 79%

mentioning

confidence: 98%

Monotonicity and Weighted Prenucleoli: A Characterization Without Consistency

2020

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“…the monotone path f w as defined in (3)). Moreover, Calleja and Llerena (2016) show that it also satisfies PC. To prove uniqueness, suppose there is a single-valued solution σ on Γ satisfying these three properties.…”

Section: Proofmentioning

confidence: 91%

“…Nevertheless, in many applications, and because of external features of the players, the assumption that every player has the same abilities may not be appropriated. The weighted Shapley values (Shapley, 1953a) and the weighted surplus division solutions (Calleja and Llerena, 2016) take care of this aspect by assigning exogenously each player to a strictly positive weight, representing such abilities. A different prominent rule is the prenucleolus (Schmeidler, 1969) that takes specially care of minimizing complaints of coalitions to a particular allocation.…”

Section: Introductionmentioning

confidence: 99%

Path monotonicity, consistency and axiomatizations of some weighted solutions

Calleja

Llerena

2019

Int J Game Theory

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On the domain of cooperative games with transferable utility, we introduce path monotonicity, a property closely related to fairness (van den Brink, 2001). The principle of fairness states that if a game changes by adding another game in which two players are symmetric, then their payoffs change by the same amount. Under efficiency, path monotonicity is a relaxation of fairness that guarantees that when the worth of the grand coalition varies, the players' payoffs change according to some monotone path. In this paper, together with the standard properties of projection consistency (Funaki, 1998) and covariance, we show that path monotonicity characterizes the weighted surplus division solutions. Interestingly, replacing projection consistency by either self consistency (Hart and Mas-Colell, 1989) or max consistency (Davis and Maschler, 1965) we obtain new axiomatic characterizations of the weighted Shapley values and the prenucleolus, respectively. Finally, by the duality approach we provide a new axiomatization of the weighted egalitarian non-separable contribution solutions using complement consistency (Moulin, 1985).

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“…For the formal definitions of these intuitive properties, see Section 2. It is well known (see, e.g., Calleja and Llerena [2]) that the per capita prenucleolus satisfies ESD (and, hence, SAM and AM). However, when considering the class of balanced games, ESD is a rather weak property.…”

Section: Introductionmentioning

confidence: 97%

Monotonicity and Weighted Prenucleoli: A Characterization Without Consistency

2018

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A solution on a set of transferable utility (TU) games satisfies strong aggregate monotonicity (SAM) if every player can improve when the grand coalition becomes richer. It satisfies equal surplus division (ESD) if the solution allows the players to improve equally. We show that the set of weight systems generating weighted prenucleoli that satisfy SAM is open, which implies that for weight systems close enough to any regular system, the weighted prenucleolus satisfies SAM. We also provide a necessary condition for SAM for symmetrically weighted nucleoli. Moreover, we show that the per capita nucleolus on balanced games is characterized by single-valuedness (SIVA), translation covariance (TCOV) and scale covariance (SCOV), and equal adjusted surplus division (EASD), a property that is comparable to but stronger than ESD. These properties together with ESD characterize the per capita prenucleolus on larger sets of TU games. EASD and ESD can be transformed to independence of (adjusted) proportional shifting, and these properties may be generalized for arbitrary weight systems p to I(A)S p . We show that the p-weighted prenucleolus on the set of balanced TU games is characterized by SIVA, TCOV, SCOV, and IAS p and on larger sets by additionally requiring IS p . Funding: P. Calleja and F. Llerena were supported by Spanish Ministerio de Economía y Competitividad [Grants ECO2016-75410-P and ECO2017-86481-P (AEI/FEDER)]. P. Sudhölter was supported by the Department of Applied Economics IV of the University of the Basque Country at Bilbao and the Spanish Ministerio de Economía y Competitividad [Grant ECO2015-66803-P (MINECO/FEDER)].

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Rationality, aggregate monotonicity and consistency in cooperative games: some (im)possibility results

Cited by 7 publications

References 35 publications

Monotonicity and Weighted Prenucleoli: A Characterization Without Consistency

Monotonicity and Weighted Prenucleoli: A Characterization Without Consistency

Path monotonicity, consistency and axiomatizations of some weighted solutions

Monotonicity and Weighted Prenucleoli: A Characterization Without Consistency

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