A New Sufficient Condition for Uniqueness in Continuous Games

Consider the class of games in which each player chooses a strategy from a connected subset of the real line. Many oligopoly models fall into this class. In many of these applications, it would be useful to show that an equilibrium was unique, or at least to have a set of conditions under which uniqueness would hold. In this paper, we first prove a uniqueness theorem that is slightly less restrictive than the contraction mapping theorem for mappings from the subsets of the real line onto itself, and then show how uniqueness in the general game can be shown by proving uniqueness using an iterative sequence of R-to-R mappings. This iterative approach works by considering the equilibrium for an m-player game holding the strategies of all other players fixed, starting with a two-player game. If one can show that the m-player game has a unique equilibrium for all possible values for the remaining players strategies, then one can add one player at a time and consider the R-to-R mapping from that player’s strategy on to the unique equilibrium of the first m players and back onto the (m+1)th player’s strategy. We then show how a general condition for each one of this sequence of mappings to have a unique equilibrium is that the leading principal minors of a matrix derived from the Jacobean matrix of best-response functions be positive, and how this general condition encompasses and generalises some existing uniqueness theorems for particular games