A Unifying Pair of Cournot–Nash Equilibrium Existence Results

Abstract

Pluralitas non est ponenda sine necessitate. William of OckhamFor games with a measure space of players a tandem pair, consisting of a mixed and a pure Cournot-Nash equilibrium existence result, is presented. Their generality causes them to be completely mutually equivalent. This provides a unifying pair of Cournot-Nash existence results that goes considerably beyond the central result of 11, Theorem 2.1]. The versatility of this pair is demonstrated by the following new applications: (i) uni cation and generalization of the two equilibrium distribution existence results for anonymous games in 44], (ii) generalization of the equilibrium existence result for Bayesian di erential information games in 38], (iii) inclusion of the Bayesian Nash equilibrium existence results in 41, 6] for games with private information in the sense of Harsanyi 33].

Comments on purification in continuum games

“…This result stems from Balder (1984, Theorem 1 ), a result that already contains both the purification theorem of Dvoretzky et al (1950) and Aumann's identity [see also (Balder 1985)]. Employing this purification device, the above two-step proof-scheme was applied in Balder (2002) to obtain Theorem 2.2.1 therein (for the reader's convenience this result is also recalled in the appendix-see Theorem A.5). The generality of this pure equilibrium existence result and its companion mixed equilibrium existence result (this is Theorem 2.1.1 of Balder (2002), which corresponds to Step 1 in the above scheme) is known to be considerable.…”

“…The generality of this pure equilibrium existence result and its companion mixed equilibrium existence result (this is Theorem 2.1.1 of Balder (2002), which corresponds to Step 1 in the above scheme) is known to be considerable. This is not only due to a very general choice of topologies, but also due to the fact that Balder (2002) uses a continuum game model that is in internal-external form. This formal concept stems from Balder (1995).…”

“…While their Theorems 1-2 concern purification per se, their Theorems 3-4 concern the purification of mixed equilibrium profiles as in Step 2 of the above proof-scheme. In the last line of Khan et al (2006) its authors express the hope that their results "may have further application to more general settings", among which they include Balder (2002) and its references. The purpose of this paper is to point out that precisely the contrary is true: when specialized to finite action spaces, certain results in Balder (2002) already imply those of Khan et al (2006).…”

“…Employing this purification device, the above two-step proof-scheme was applied in Balder (2002) to obtain Theorem 2.2.1 therein (for the reader's convenience this result is also recalled in the appendix-see Theorem A.5). The generality of this pure equilibrium existence result and its companion mixed equilibrium existence result (this is Theorem 2.1.1 of Balder (2002), which corresponds to Step 1 in the above scheme) is known to be considerable. This is not only due to a very general choice of topologies, but also due to the fact that Balder (2002) uses a continuum game model that is in internal-external form.…”

“…Games in internal-external form would appear to be a quite natural extension of the concept of games in strategic form and a large variety of games with a measure space of players/types is known to be representable in that form (Balder 1991(Balder , 1995(Balder , 1996(Balder , 1999(Balder , 2002; see also Angeloni and Martins-da-Rocha (2005) for recent discoveries of such representations. Martins-da- Rocha and Topuzu (2006) have recently extended the scope of the internal-external form model even further: by introducing artificial payoffs of a generalized Shafer-Sonnenschein variety (Shafer and Sonnenschein 1975) they have shown that the existence results from Balder (2002) can be extended to continuum games with non-ordered preferences.…”