Invariance under Replication and Juxtaposition
William Thomson, Terje Lensberg · Cambridge University Press eBooks · 1989
Introduction In all of the previous chapters, we considered variations in the number of agents without imposing any restriction on the preferences of the new agents in relation to the preferences of the agents originally present. In contrast, we formulate here various notions of “similarity” between old agents and new agents, and we examine the behavior of solutions in these special cases. The chapter starts with notions of “replication,” and we ask whether and in what sense solutions can be said to be invariant under such replications. Then, we consider related notions of “juxtaposition” and ask similar questions. This chapter is partly inspired by the literature on replica economies initiated by Edgeworth (1881) and taken up by Debreu and Scarf (1963) and many subsequent writers. The replication of economies turned out to be a powerful tool in the study of large economies, opening up a line of investigation that provided valuable insights into the classical concept of perfect competition. The cores of replicated games in coalitional form were recently analyzed by Wooders (1981) and various coauthors, but to our knowledge, the only author who concerned himself with the replication of unanimity games of the kind examined in this book is Kalai (1977a). Kalai showed that each two-person asymmetric Nash solution is equivalent to the Nash solution under appropriate replications (or limits of such replications). He also noted the invariance of the Kalai–Smorodinsky solution under replications.