In this paper we examine a number of different definitions of strategic stability and the relations among them. In particular, we show that the stability requirement given by Hillas (1990) is weaker than the requirements involved in the various definitions of stability in Mertens' reformulation of stability (Mertens 1989, 1991). To this end, we introduce a new definition of stability and show that it is equivalent to (a variant of) the definition given by Hillas (1990). We also use the equivalence of our new definition with the definition of Hillas to provide correct proofs of some of the results that were originally claimed (and incorrectly "proved") in Hillas (1990).
Vermeulen, A. J., Jansen, M. J. M., Potters, J. A. M., & Hillas, J. (2001). On the relation between some definitions of strategic stability. Mathematics of Operations Research, 26(26), 611-635. https://doi.org/10.1287/moor.26.3.611.10585