Stability of networks under horizon-K farsightedness

Abstract

We introduce the concept of a horizon-K farsighted set to study the influence of the degree of farsightedness on network stability. The concept generalizes existing concepts where all players are either fully myopic or fully farsighted. A set of networks G K is a horizon-K farsighted set if three conditions are satisfied. First, external deviations should be horizon-K deterred. Second, from any network outside of G K there is a sequence of farsighted improving paths of length smaller than or equal to K leading to some network in G K . Third, there is no proper subset of G K satisfying the first two conditions. We show that a horizon-K farsighted set always exists and that the horizon-1 farsighted set G 1 is always unique. For generic allocation rules, the set G 1 always contains a horizon-K farsighted set for any K . We provide easy to verify conditions for a set of networks to be a horizon-K farsighted set, and we consider the efficiency of networks in horizon-K farsighted sets. We discuss the effects of players with different horizons in an example of criminal networks.

The Myopic Stable Set for Social Environments

“…Herings, Mauleon, and Vannetelbosch (2009) defined the pairwise myopically stable sets for network problems using pairwise stability as defined in Section 2 of Jackson and Wolinsky (1996). Since an element of V simply represents a closed cycle as defined in Definition 3.8, the following result follows from Theorem 3.9.…”

“…It differs from the ones in the literature that focus on farsightedness (see among others, Chwe (1994), Herings, Mauleon, andVannetelbosch (2009, 2018), Page, Wooders, and Kamat (2005), Page and Wooders (2009), Ray and Vohra (2015)), notions which strengthen farsightedness by using state-dependent expectations (Greenberg (1990), Xue (1998), Dutta and Vohra (2017)), and approaches to farsightedness that are inspired by noncooperative game theory (Herings, Mauleon, and Vannetelbosch (2004), Dutta, Ghosal, and Ray (2005)). It differs from the ones in the literature that focus on farsightedness (see among others, Chwe (1994), Herings, Mauleon, andVannetelbosch (2009, 2018), Page, Wooders, and Kamat (2005), Page and Wooders (2009), Ray and Vohra (2015)), notions which strengthen farsightedness by using state-dependent expectations (Greenberg (1990), Xue (1998), Dutta and Vohra (2017)), and approaches to farsightedness that are inspired by noncooperative game theory (Herings, Mauleon, and Vannetelbosch (2004), Dutta, Ghosal, and Ray (2005)).…”

“…In fact, extending the definition of the vNM stable set by allowing for a finite iteration of one-step dominations, that is, replacing f by f N in Definition 3.14, has also been advocated by several authors; see Harsanyi (1974), van Deemen (1991), Wooders (2009), andHerings, Mauleon, andVannetelbosch (2017). However, the vNM stable set looks at one-step dominations while our notion of asymptotic external stability uses asymptotic dominance, which can be seen as an infinite iteration of one-step dominations.…”

“…As an example that imposes some reasonable structure on the effectivity correspondence, we provide a brief outline of the notion of coalitional sovereignty (Konishi and Ray (2003), Kóczy and Lauwers (2004), Vohra (2014, 2015), Herings, Mauleon, and Vannetelbosch (2017)). As an example that imposes some reasonable structure on the effectivity correspondence, we provide a brief outline of the notion of coalitional sovereignty (Konishi and Ray (2003), Kóczy and Lauwers (2004), Vohra (2014, 2015), Herings, Mauleon, and Vannetelbosch (2017)).…”

“…The MSS extends the idea of level-1 farsighted stability by Herings, Mauleon, andVannetelbosch (2009, 2018) from finite networks to the general class of social environments. The MSS extends the idea of level-1 farsighted stability by Herings, Mauleon, andVannetelbosch (2009, 2018) from finite networks to the general class of social environments.…”