Now we state our solution concept.
Definition 2: Given a market game (U ( ),S ) , (x ,x ) with x =
-------------------- i i i∈I 1 i∈I
У x , x ∈ S ∀i ∈ I is said to be a Nash Equilibrium (N.E.) if Vi ∈ I
i∈I i i i
U (x , X ) ≥ U (x , X -X + X ) Vx ∈ S
i ɪ i i i i i i
Now we state and discuss our main assumptions.
1
Assumption 1: U ( ) ∈ G Vi ∈ I.
--------------------------------------- ɪ
*
Notice that under Assumption 1 (A.l in what follows) if ×ι ∈ int. sɪ the
necessary condition of a N.E. reads as follows:
∂U (x*, x*)
—!—i--= 0 VieI.
∂x
∂U (x*, x*)
i i
----------------- +
∂x
i
Let us define
( x ,x) ∂U ( x ,x)
i i i
------- + --------------
X ∂x
∂U
i
Let N be the set of active agents (i.e. those for which x* ∈ int. sɪ in a
N.E. with n players). N+l is defined accordingly. We will assume that N n N+l
≠ 0, i.e. at least one player is active in N.E. with n and n+1 agents
respectively.
Assumption 2: T (x , x) is strictly decreasing on x and x VieI.
----------------------------------------- i i i
12