Some properties for probabilistic and multinomial (probabilistic) values on cooperative games

Abstract

We investigate the conditions for the coefficients of probabilistic and multinomial values of cooperative games necessary and/or sufficient in order to satisfy some properties, including marginal contributions, balanced contributions, desirability relation and null player exclusion property. Moreover, a similar analysis is conducted for transfer property of probabilistic power indices on the domain of simple games.

Keywords:

• TU cooperative game
• probabilistic value
• multinomial value
• semivalue
• regularity

AMS Subject Classification:

• 91A12

Acknowledgements

The authors wish to thank the anonymous reviewers for their helpful suggestions which have been incorporated to the text.

Notes

No potential conflict of interest was reported by the authors.

1 [26–28] are samples of our work in this line.

2 In [1] this property is called monotonicity, but we prefer to call to it positivity as in [11].

3 Notice that Corollary 4.15 follows from this weighted balanced contributions property but only for players j with .

5 If the argument is similar, with .

6 If the argument is similar, with .

7 It could be directly derived from Equation (3(3) ), but with considerably more effort.

Additional information

Funding

This work was supported by the Catalonia Government (Generalitat de Catalunya) [grant number SGR 2014–435]; the Economy and Competitiveness Spanish Ministry [grant number MTM 2012–34426].