ON A LESS KNOWN NASH EQUILIBRIUM UNIQUENESS RESULT

Abstract

We generalize a less known Nash equilibrium uniqueness result for games in strategic form. Its power is illustrated by applying it to a Public Goods Game, a Homogeneous Cournot Oligopoly Game and a Formal Transboundary Pollution Game.

Keywords:

• aggregative game
• convex analysis
• game in strategic form
• oligopoly
• public goods
• transboundary pollution
• uniqueness of Nash equilibria

H. F. is also affiliated with Universiteit van Tilburg, The Netherlands and P. v. M. with Universiteit Utrecht, The Netherlands. The authors thank a referee, Luis Corchón and Rein Haagsma for comments. The usual disclaimer holds. Moreover, P. v. M. would like to thank Dave Furth for interesting him in aggregative games and for the notion of “co-strategy”. For the most recent comments on this article readers are invited to consult the home page of P. v. M.

Notes

¹In Tan et al., 1995, it is proved that this result even holds if each strategy set is a compact, convex subset of a Hausdorff topological vector space.

²For instance a closed subset of some ^m .

³In this regard it is worthwhile observing that each convex function defined on an interval of is left and right differentiable in each interior point of its domain. See, if wished, Fact A in the appendix.

⁴We generally use boldfaced notations for Cartesian products of players' objects.

⁵We equip with the usual arithmetical operations.

⁶For a = (a ₁,…, a _n ), b = (b ₁,…, b _n ) ∈  ⁿ we write a ≥ b if a _i  ≥ b _i for all i. We write a > b if a ≥ b and a ≠ b. Let F:Z →  ⁿ be a mapping where Z is a subset or ^m and W ⊆ Z. F is called strictly increasing on W if for all a, b ∈ W, we have: a > b ⇒ F(a) > F(b).

⁷Let F:Z →  ⁿ be a mapping where Z is a subset of ^m and W ⊆ Z. F is called ordered on W if for all a, b∈W one has F(a) ≥ F(b) or F(b) ≥ F(a).

⁸In his variant of Corollary 1 Corchón moreover assumes that each Xⁱ is compact, that Y ⁱ = Y, that each π ⁱ is twice continuously totally differentiable, that t ⁱ is strictly decreasing in y ⁱ and that all Nash equilibria are interior. All these additional assumptions are not necessary for our corollary to hold.

⁹In this article we adopt the following usual notion of total differentiability for a function f:A →  where A is a (not necessarily open) subset of ⁿ :f is totally differentiable if it can be extended to a totally differentiable function on an open subset U of ⁿ containing A.

¹⁰Note that in the case π ⁱ is twice totally differentiable for t ⁱ to be strictly decreasing in x ⁱ it is sufficient that and for t ⁱ to be decreasing in y ⁱ it is sufficient that

¹¹Note that this implies that for p:[0, M] →  one has: p ≥ 0, p is decreasing and p is continuous.

¹²The special proof we present here with Fact E (in the appendix) seems to be new.

¹³Note that we do not assume that .

¹⁴Note that although in the right-hand side of (3) some derivatives may be −∞ or ∞, never undefined operations like ∞ − ∞ will occur.

¹⁵This only concerns the points (x ⁱ , r ⁱ ) with in case (𝒟 ⁱ )₋′(r ⁱ ) =  ∞.

¹⁶It is quite interesting to note that in each of these Nash equilibria n for all i one has Q ⁱ (n) = 1 and that 𝒟 ⁱ is not differentiable in Q ⁱ (n).

¹⁷Facts B and C will not be used in the main text, but are only mentioned for completeness.