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By Haurie A., Krawczyk J.

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BIMATRIX GAMES 41 game (A, B). t. 13) v 1 1m v 2 1n 0 1 1 IR. 19) (ii) (x, y) is an equilibrium for the bimatrix game. Proof: From the constraints it follows that xT Ay ≤ v 1 and xT By ≤ v 2 for any feasible (x, y, v 1 , v 2 ). Hence the maximum of the program is at most 0. Assume that (x, y) is an equilibrium for the bimatrix game. e. 1) a value 0. 19). 19). We know that an equilibrium exists for a bimatrix game (Nash theorem). 19) with optimal value 0. Hence the optimal program (x∗ , y∗ , v∗1 , v∗2 ) must also give a value 0 to the objective function and thus be such that xT∗ Ay∗ + xT∗ By∗ = v∗1 + v∗2 .

E. t. u∗ ∈ Γ(u∗ , r). Hence a coupled equilibrium exists. 8. One is required to show that the point to set mapping is upper semicontinuous. 2 This existence theorem is very close, in spirit, to the theorem of Nash. e. it does not provide a computational method. However, the definition of a normalised equilibrium introduced by Rosen establishes a link between mathematical programming and concave games with coupled constraints. 4. 39) can be defined by a set of inequalities hk (u) ≥ 0, k = 1, . .

M are given weights. The precise role of this weighting scheme will be explained later. For the moment we could take as well rj ≡ 1. Notice that, even if u and v are in U, the combined vectors (u1 , . . , vj , . . , um ) are element of a larger set in U1 × . . × Um . This function is continuous in u and concave in v for every fixed u. We call it a reaction function since the vector v can be interpreted as composed of the reactions of the different players to the given vector u. This function is helpful as shown in the following result.

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