Top

Theory and Decision

Published in:

01-01-2015

Properties based on relative contributions for cooperative games with transferable utilities

Published in: Theory and Decision | Issue 1/2015

Activate our intelligent search to find suitable subject content or patents.

search-config

AI-assisted search

Off

Abstract

By focusing on players’ relative contributions, we study some properties for values in positive cooperative games with transferable utilities. The well-known properties of symmetry (also known as “equal treatment of equals”) and marginality are based on players’ marginal contributions to coalitions. Both Myerson’s balanced contributions property and its generalization of the balanced cycle contributions property (Kamijo and Kongo Int J of Game Theory 39:563–571, 2010; BCC) are based on players’ marginal contributions to other players. We define relative versions of marginality and BCC by replacing marginal contributions with relative contributions, and examine efficient values satisfying each of the two properties. On the class of positive games, a relative variation of marginality is incompatible with efficiency, and together with efficiency and the invariance property with respect to the payoffs of players under a player deletion, a relative variation of BCC characterizes the proportional value and egalitarian value in a unified manner.

previous article Cooperation and signaling with uncertain social preferences

next article Catastrophe insurance equilibrium with correlated claims

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

über 102.000 Bücher
über 537 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Finance + Banking
Management + Führung
Marketing + Vertrieb
Maschinenbau + Werkstoffe
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 340 Zeitschriften

aus folgenden Fachgebieten:

Bauwesen + Immobilien
Business IT + Informatik
Finance + Banking
Management + Führung
Marketing + Vertrieb
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Note that a value on \(\mathbf {G}\) is defined as \(f: \mathbf {G} \rightarrow \mathbb {R}^n\).

An order \((i_1,i_2,\ldots ,i_{n})\) is a sequence of \(n\) players such that each player appears only once.

Note that the quotient is well-defined by the definition of a value on \(\mathbf {G}_+\).

In Ortmann (2000), this property is called “preserving ratios.”

Note that if we consider the class of one-person games, this does not hold because any value satisfies relativity on the class of one-person games.

In the definitions of proportional and quasi-proportional players on \(\mathbf {G}\), we (implicitly or explicitly) require a condition \(v(k)=0\) (see Kamijo and Kongo 2012). This condition is irrelevant on \(\mathbf {G}_+\), and thus, we omit it.

Tijs and Driessen (1986) also examine ID. They call it the “dummy out property.”

Note that when \(Q(S,v)=v(S)\), a \(Q\)-player is a null player that is a dummy player with its singleton coalition worth being zero; there is no null player in games on \(\mathbf {G}_+\).

Theorem 3 seems to contradict Lemma 1 (ii) in Kamijo and Kongo (2012). However, the domains are different. On \(\mathbf {G}\), the contradiction mentioned in the proof of Theorem 3 does not matter when \(v(i)=v(j)=v(\{i,j\})=0\).

Casajus (2011a) proves that on any convex cone \(\mathcal {C} \subseteq \mathbf {G}\), differential marginality is equivalent to fairness introduced by Brink (2001). Differential marginality also characterizes the Banzhaf value (Banzhaf 1965; Casajus 2011b) and the Owen value (Owen 1977; Casajus 2010).

The null player property requires that a null player obtains zero value. Note that in the class of positive game, there are no null players. However, most characterizations using the null player property hold when we replace null players with dummy players. Therefore, to compare the results on the classes of all games and positive games, we sometimes replace null players with dummy players.

Banzhaf, J. F. III. (1965). Weighted voting doesn’t work: A mathematical analysis. Rutgers Law Review, 19, 317–343.

Casajus, A. (2010). Another characterizations of the Owen value without the additivity axiom. Theory and Decision, 69, 523–536.CrossRef

Casajus, A. (2011a). Differential marginality, van den Brink fairness, and the Shapley value. Theory and Decision, 71, 163–174.CrossRef

Casajus, A. (2011b). Marginality, differential marginality, and the Banzhaf value. Theory and Decision, 71, 365–372.CrossRef

Kamijo, Y., & Kongo, T. 2009. Axiomatizations of the values for TU games using the balanced cycle contributions property. G-COE GLOPEII Working Paper Series No.15. Retrieved from http://www.waseda-pse.jp/gse/jp/gcoe/modules/mydownloads/pdf/GCOE_WP24.pdf. Last accessed 28 Nov 2013.

Kamijo, Y., & Kongo, T. (2010). Axiomatization of the Shapley value using the balanced cycle contributions property. International Journal of Game Theory, 39, 563–571.CrossRef

Kamijo, Y., & Kongo, T. (2012). Whose deletion does not affect your payoff? The difference between the Shapley value, the egalitarian value, the solidarity value, and the Banzhaf value. European Journal of Operational Research, 216, 638–646.CrossRef

Myerson, R. B. (1980). Conference structures and fair allocation rules. International Journal of Game Theory, 9, 169–182.CrossRef

Ortmann, K. M. (2000). The proportional value for positive cooperative games. Mathematical Methods of Operations Research, 51, 235–248.CrossRef

Owen, G. (1977). Values of games with a priori unions. In R. Henn & O. Moeschlin (Eds.), Mathematical economics and game theory. Berlin: Springer.

Shapley, L. S. (1953). A value for \(n\)-person games. In H. Kuhn & A. Tucker (Eds.), Contributions to the theory of games II (pp. 307–317). Princeton: Princeton University Press.

Tijs, S. H., & Driessen, T. S. (1986). Extensions of solution concepts by means of multiplicative \(\epsilon \)-tax game. Mathematical Social Sciences, 12, 9–20.CrossRef

van den Brink, R. (2001). An axiomatization of the Shapley value using a fairness property. International Journal of Game Theory, 30, 309–319.CrossRef

Young, H. (1985). Monotonic solutions of cooperative games. International Journal of Game Theory, 14, 65–72.CrossRef

Title: Properties based on relative contributions for cooperative games with transferable utilities
Publication date: 01-01-2015
Published in: Theory and Decision / Issue 1/2015
Print ISSN: 0040-5833
Electronic ISSN: 1573-7187
DOI: https://doi.org/10.1007/s11238-013-9402-3

Springer Professional

Abstract

Please log in to get access to your license.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Springer Professional "Wirtschaft"

Other articles of this Issue 1/2015

A note on equivalent comparisons of information channels

Catastrophe insurance equilibrium with correlated claims

Collectively ranking candidates via bidding in procedurally fair ways

A Bayesian model of Knightian uncertainty

Gender differences when subjective probabilities affect risky decisions: an analysis from the television game show Cash Cab

An axiomatization of Choquet expected utility with cominimum independence