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3.2. Open loop Nash games 67 3.2 Open loop Nash games
In order to define the Nash equilibrium concept, we introduce a cost- -functional for each player, which are for sake of simplicity of purely quadratic type
(C)
R11 <H 0, R22 <H 0 i.e. negative definite
Q1 ≤H 0, Q2 ≤H 0 i.e. negative semidefinite.
Ji(u1,u2)=
∞ 2 xTQix+
uTj Rijuj dt,
i=1,2, (3.2)
0
j=1

where Qi ∈ Rn×n and Rij ∈ Rmj×mj, are supposed to be symmetric for i, j = 1, 2.
In gametheoretic terminology, an “open loop” strategy means that the
players have to choose their strategies u1 and u2 prior to the game start and
that their only information on the state of the game is the initial state x0. It
is reasonable to assume that A is stable since then any pair of controls (u1,u2)
with u ∈ L2,mi is admissible. Here, L2,mi denotes the space of square Lebesgue i++
integrable functions on the positive real line taking values in Rmi .
We are inspired by Example 3.1 in [DoJoLoSo], where the problem of fish harvesting rates is studied gametheoretically. On the one hand each player aims to maximize his cost functional, on the other hand the harvesting rate cannot be negative. Hence, this example constitutes a (1–dimensional) positive system, where each player seeks to maximize his utility function. With this example in
mind we define the equilibrium concept in a Nash game.
Definition 3.2.1. The pair (u∗1,u∗2) is called a (two player) Nash (equilibrium) strategy if
J1(u∗1, u∗2) ≥ J1(u1, u∗2) and J2(u∗1, u∗2) ≥ J2(u∗1, u2)
for all admissible strategies u1, u2.
For open loop Nash games it is possible to find existence results for Nash strategies, which are usually given in a ’convex setting’. As we introduced Nash equilibria by maximizing the utility function, we recall relevant results in a ’concave setting’, which is
In contrast to the elementwise ordering used so far, here we use the ordering on symmetric matrices defined by its eigenvalues. Here Rii <H 0 means for example that R11 = R1T1 and all eigenvalues are negative. In [En] and [BaOl] it is shown under condition (C) that a sufficient condition for the existence of