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On effectivity functions of game forms

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  • Boros, Endre
  • Elbassioni, Khaled
  • Gurvich, Vladimir
  • Makino, Kazuhisa

Abstract

To each game form g an effectivity function (EFF) Eg is assigned. An EFF E is called formal (formal-minor) if E=Eg (respectively, E[less-than-or-equals, slant]Eg) for a game form g. (i) An EFF is formal iff it is superadditive and monotone. (ii) An EFF is formal-minor iff it is weakly superadditive. Theorem (ii) looks more sophisticated, yet, it is simpler than Theorem (i) and instrumental in its proof. In addition, (ii) has important applications in social choice, game, and even graph theories. Constructive proofs of (i) were given by Moulin, in 1983, and by Peleg, in 1998. Both constructions are elegant, yet, sets of strategies Xi of players i[set membership, variant]I might be doubly exponential in size of the input EFF E. In this paper, we suggest another construction such that Xi is only linear in the size of E. Also, we extend Theorems (i), (ii) to tight and totally tight game forms.

Suggested Citation

  • Boros, Endre & Elbassioni, Khaled & Gurvich, Vladimir & Makino, Kazuhisa, 2010. "On effectivity functions of game forms," Games and Economic Behavior, Elsevier, vol. 68(2), pages 512-531, March.
    • Handle: RePEc:eee:gamebe:v:68:y:2010:i:2:p:512-531
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    References listed on IDEAS

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    1. Moulin, H. & Peleg, B., 1982. "Cores of effectivity functions and implementation theory," Journal of Mathematical Economics, Elsevier, vol. 10(1), pages 115-145, June.
    2. repec:dau:papers:123456789/13220 is not listed on IDEAS
    3. Boros, Endre & Gurvich, Vladimir, 2000. "Stable effectivity functions and perfect graphs," Mathematical Social Sciences, Elsevier, vol. 39(2), pages 175-194, March.
    4. Abdou, Joseph, 1987. "Stable effectivity functions with an infinity of players and alternatives," Journal of Mathematical Economics, Elsevier, vol. 16(3), pages 291-295, June.
    5. Gurvich, Vladimir, 2008. "War and peace in veto voting," European Journal of Operational Research, Elsevier, vol. 185(1), pages 438-443, February.
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    1. Peters, H.J.M. & Schröder, M.J.W. & Vermeulen, A.J., 2013. "Ex post Nash consistent representation of effectivity functions," Research Memorandum 049, Maastricht University, Graduate School of Business and Economics (GSBE).
    2. Kretz, Claudio, 2021. "Consistent rights on property spaces," Journal of Economic Theory, Elsevier, vol. 197(C).
    3. Hans Peters & Marc Schröder & Dries Vermeulen, 2015. "On existence of ex post Nash consistent representation for effectivity functions," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 45(2), pages 287-307, September.

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