A reformulation of von Neumann–Morgenstern stability: mm-stability

The notion of a stable set (introduced by von Neumann and Morgenstern, 1944) is an important tool in the field of Decision Theory. However, stable sets may fail to exist. Other stability notions have been introduced in the literature in order to solve the non-existence problem. We propose a new notion, that we call mm-stability,   and compare it with previous proposals. Moreover, we analyze some properties (existence, uniqueness, unions and intersections, …) of the different notions of a stable set. Finally, we use the Shapley–Scarf market model with indivisible goods in order to show that the non-empty core is an mm-stable set, and does not fulfill, in general, the other stability notions.

► We introduce the notion of m-stability as an alternative to the vNM stable set. ► In a finite context, m-stable sets always exist. ► We compare m-stability with other stability notions. ► The core of a market with indivisible goods is an m-stable set.