Non-differentiability of Payoff Functions and Non-uniqueness of Nash Equilibria Pierre von Mouche Abstract—Given non-degenerate intervals X i of R and an increas- ing ordered mapping Φ: X 1 ×···× X N → R N , games in strategic form between N players with the X i as action sets with the following three properties are studied: the set of Nash equilibria E is convex, Φ is constant on E and in each Nash equilibrium at least one payoff function is not partially differentiable w.r.t. its own action. The results are illustrated for a special class of aggregative games that include the formal transboundary pollution games with global transboundary pollution. Keywords—Aggregative game, convex analysis, formal trans- boundary pollution game, non-differentiable payoff functions, unique- ness of Nash equilibria. I. I NTRODUCTION Consider the following game in strategic form Γ 0 between two players taken from Folmer and von Mouche (2004). Each player i has action set X i = [0, 2] and payoff function f i (x 1 ,x 2 ) = ln(x i + 1) −D i (x 1 + x 2 ), where D 1 (Q)=  1 3 Q (Q ∈ [0, 1]) 2 3 Q 1 − 1 3 (Q ∈ [1, 2]) , D 2 (Q)=  1 4 Q (Q ∈ [0, 1]) 4Q − 15 4 (Q ∈ [1, 2]) . A straightforward calculation shows that there are well-defined reaction functions given by R 1 (x 2 )=  1 − x 2 (0 ≤ x 2 ≤ 1/2) 1/2 (1/2 ≤ x 2 ≤ 1) , R 2 (x 1 )=1 − x 1 . This implies {(x, 1 − x) | 1/2 ≤ x ≤ 1} for the set of Nash equilibria. Observe the following three properties of Γ 0 : (1) the sum of actions is constant (i.e. 1) in each Nash equilibrium, (2) in each Nash equilibrium no payoff function is partially differentiable w.r.t. its own action, and (3) the set of Nash equilibria is convex. The aim of this article is to give sufficient conditions for games in strategic form that imply these properties directly. To this end a theory will be presented for a class of games G 1 in strategic form with special attention to the subclass of so-called formal transboundary pollution games with global transboundary pollution, and in particular Γ 0 . An action of a player in such a game has the real-world interpretation of the emission level of a country and the sum of emission levels Present coordinates: Wageningen Universiteit, Sociale Wetenschap- pen, Hollandseweg 1, 6700 EW Wageningen, The Netherlands. Email: pierre.vanmouche@wur.nl. Article presented at the International Conference on Applied Mathematics and Numerical Analysis, Tokyo, Japan, May 27–29, 2009. across the countries is interpreted as a deposition level (see, for instance, Folmer and von Mouche, 2002). A direct result of this theory is that in a formal transboundary pollution game with global transboundary pollution each Nash equilibrium has the same deposition level. II. SETTING AND NOTATIONS Let N be a positive integer, and write N := {1,...,N }. Fix non-degenerate intervals X i (i ∈N ) of R, and with X := X 1 ×···× X N , a mapping Φ=(ϕ 1 ,...,ϕ N ): X → R N , which will be called co-strategy mapping. 1 It is supposed that Φ is increasing and ordered. 2 Write Y i := ϕ i (X)(i ∈N ), Y := Y 1 ×···× Y N . Sufficient for Y i to be an interval is that ϕ i is continuous. Given X 1 ,...,X N and Φ, let G 0 be the class of games in strategic form with N as set of players, and for each player i action set X i . The payoff function of player i will be denoted by f i : X → R. For the moment there are no further restrictions for the payoff functions. In the next definition a subclass G 1 of G 0 will be defined by assuming some specific properties for the payoff functions. For Γ ∈G 0 denote the set of Nash equilibria by E(Γ) 1 I thank D. Furth for suggesting this terminology. 2 Given a positive integer n, the relations ≥, >,  on R n are defined by: x ≥ y : x k ≥ y k (1 ≤ k ≤ n); x > y : x ≥ y and x = y; x  y : x k >y k (1 ≤ k ≤ n). And ≤, <,  denote the dual relations of respectively ≥, >, . Consider a mapping F : Z → R n , where Z ⊆ R m . In this article, F is called – ordered if for all a, b ∈ Z it holds that F (a) ≥ F (b) or that F (a) ≤ F (b);– strictly ordered if for all a, b ∈ Z it holds that F (a)  F (b) or that F (a)= F (b) or that F (a)  F (b);– increasing if for all a, b ∈ Z one has a ≤ b ⇒ F (a) ≤ F (b);– strongly increasing if for all a, b ∈ Z one has a < b ⇒ F (a) <F (b). World Academy of Science, Engineering and Technology 53 2009 731