Abstract
Young’s theorem implies that every core concept violates monotonicity. In this paper, we investigate when such a violation of monotonicity by a given core concept is justified. We introduce a new monotonicity property for core concepts. We pose several open questions for this new property. The open questions arise because the most important core concepts (the nucleolus and the per capita nucleolus) do not satisfy the property even in the class of convex games.
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Notes
See Shapley (1953). In this paper we denote by \(Sh(v)\) the Shapley value of a game \((N,v).\)
We define the property for core concepts.
Other definitions may be considered. For example:
A violation of monotonicity by \(\phi \) with respect to games \(v\) and \(w\) is justified if there exists a nonempty set of monotonic transformations of \(v\) , \(A(v)\subseteq \Gamma _{0}\backslash \left\{ v,w\right\} ,\) such that:
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1.
There exist monotonic core concepts in \(A(v)\cup \left\{ v\right\} .\)
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There does not exist monotonic core concept in \(A(v)\cup \left\{ v,w\right\} \).
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1.
To be precise this paper deals with the class of non negative veto balanced games. Non negativity implies that the worth of any coalition is non negative. Therefore, in this section, by \(\Gamma _{VB}\) we refer to the class of non negative veto balanced games.
Not necessarily games with a large core.
Let \(N=\left\{ 1,2,3\right\} \) and consider the following 3-person veto balanced games:
$$\begin{aligned} v(S)&= \left\{ \begin{array}{cc} 4&\text{ if} S\in \left\{ \left\{ 1,2,3\right\} ,\left\{ 1,2\right\} ,\left\{ 1,3\right\} \right\} \\ 0&\text{ otherwise} \end{array} \right. \text{ and} \\ w(S)&= \left\{ \begin{array}{cc} v(S)&\text{ if} S\ne N \\ 6&\text{ if} S=N. \end{array} \right. \end{aligned}$$Any egalitaran concept selects \((4,0,0)\) in game \(v\) and \((2,2,2)\) in game \( w.\)
Zhou (1991) shows that the nucleolus satisfies 1-monotonicity. Our approach is different and we show that any violation of 2-monotonicity is not justified.
In case that \(f_{l2}(y,v)\ge f_{l1}(y,v)\ge 0\) the bilateral transfer is from player \(l\) to player \(1.\)
For example, consider the following case. Let \(N=\left\{ 1,2,3,4\right\} \) be a set of players and consider the following two games:
$$\begin{aligned} v(S)&= \left\{ \begin{array}{cc} 2&\text{ if} \left| S\right| =2 \text{ and} S\notin \left\{ \left\{ 1,2\right\} ,\left\{ 3,4\right\} \right\} \\ 4&\text{ if} S=N \\ 0&\text{ otherwise} \end{array} \right. \text{ and} \\ w(S)&= \left\{ \begin{array}{cc} 2&\text{ if} S=\left\{ 1,2\right\} \\ v(S)&\text{ otherwise.} \end{array} \right. \end{aligned}$$Consider the allocation \((0,0,2,2).\) Only simultaneous transfers from player \(3\) and \(4\) to players \(1\) and \(2\) allow the construction of allocation \( (1,1,1,1).\)
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Acknowledgments
J. Arin acknowledges the support of the Spanish Ministerio de Ciencia e Innovació n under projects SEJ2006-05455 and ECO2009-11213, co-funded by ERDF, and by Basque Government funding for Grupo Consolidado GIC07/146-IT-377-07. We thank an Associated Editor and two anonymous reviewers for their very helpful comments and suggestions. The final version of the paper benefits very much from them.
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Arin, J. Monotonic core solutions: beyond Young’s theorem. Int J Game Theory 42, 325–337 (2013). https://doi.org/10.1007/s00182-013-0368-8
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DOI: https://doi.org/10.1007/s00182-013-0368-8