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Single-payoff farsighted stable sets in strategic games with dominant punishment strategies

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Abstract

We investigate farsighted stable sets in a class of strategic games with dominant punishment strategies. In this class of games, each player has a strategy that uniformly minimizes the other players’ payoffs for any given strategies chosen by these other players. We particularly investigate a special class of farsighted stable sets, each of which consists of strategy profiles yielding a single payoff vector. We call such a farsighted stable set as a single-payoff farsighted stable set. We propose a concept called an inclusive set that completely characterizes single-payoff farsighted stable sets in strategic games with dominant punishment strategies. We also show that the set of payoff vectors yielded by single-payoff farsighted stable sets is closely related to the strict \(\alpha \)-core in a strategic game. Furthermore, we apply the results to strategic games where each player has two strategies and strategic games associated with some market models.

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Notes

  1. The farsighted stable set has also been applied to problems that are not formulated in strategic games. Diamantoudi and Xue (2003) considered the hedonic coalition formation problem; Diamantoudi (2005) studied the cartel formation problem; Diamantoudi and Sartzetakis (2015) and Benchekroun and Chaudhuri (2015) studied an international environmental agreement formation problem; Page and Wooders (2009) studied a network formation problem; Kawasaki (2010) and Klaus et al. (2010) studied the exchange economy with indivisible goods under the weak and strong dominance relations, respectively; Klaus et al. (2011) studied the roommate problem; and Mouleon et al. (2011) studied the two-sided matching markets, and which was extended to a more general setting by Roketskiy (2012), among others.

  2. Dutta and Vohra (2017) defined the farsighted stable set with rational expectations for more abstract games, including not only strategic games but also coalitional games and other forms. We can apply their result to strategic games and the farsighted stable set due to Chwe (1994).

  3. For example, Milgrom and Roberts (1990, 1996) considered a subclass of games with strategic complementarities that also has this property.

  4. For NTU coalitional games, Bhattacharya and Brosi (2011) showed the existence of farsighted stable sets but did not argue its characteristics explicitly.

  5. In Ray and Vohra (2015a), the definition of the efficiency of a payoff vector in a (possibly not superadditive) coalitional game (N, V) is slightly different from ours as follows: u is efficient iff \(u\in \bigcap _{S\in \mathcal {P}}V(S)\) for some partition \(\mathcal {P}\) of N and there exist no partition \(\mathcal {P}'\) of N and \(u'\in \bigcap _{S\in \mathcal {P}'}V(S)\) such that \(u'\ge u\) and \(u'\ne u\). This definition is equivalent to that in the present paper for the \(\alpha \)-coalitional games by the generic superadditivity, as mentioned in Remark 1(a).

  6. d’Aspremont et al. (1983) formulated the problem in an abstract model. Later, Kamijo and Muto (2010) reformulated it to a strategic game and investigated the farsighted stable set in the sense of Chwe (1994).

  7. We follow the terminology of Echenique and Oviedo (2006), while a core matching should probably be called a strong core matching, which is more familiar terminology recently.

  8. Our strategic game is very similar to that of Konishi and Ünver (1999). The only difference is that we restrict the strategy sets to individually rational partners.

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Authors and Affiliations

  1. Faculty of Economics, University of Toyama, 3190 Gofuku, Toyama-shi, Toyama, 930-8555, Japan

    Toshiyuki Hirai

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  1. Toshiyuki Hirai
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Correspondence to Toshiyuki Hirai.

Additional information

The author is grateful to an associate editor, two anonymous referees, Parkash Chander, Ryo Kawasaki, Manfred Kerber, Shigeo Muto, Anne van den Nouweland, Tamas Solymosi, and participants at the 5th World Congress of the Game Theory Society for their helpful comments and suggestions. He is grateful for the financial supports by JSPS Grant-in-aid for Young Scientists (B) 26780118. He would like to thank Enago for the English language review.

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Hirai, T. Single-payoff farsighted stable sets in strategic games with dominant punishment strategies. Int J Game Theory 47, 1087–1111 (2018). https://doi.org/10.1007/s00182-017-0597-3

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