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Extra resources for A Course in Combinatorial Optimization

Example text

1. 86) if there exist continuously twice differentiable functions W i (x) : Rm → R, i ∈ N , satisfying the following set of partial differential equations: rW i (x) − 1 2 n Ω hζ (x) Wxi h xζ (x) = h,ζ=1 max g i [x, φ∗1 (x) , φ∗2 (x) , . . ui . . , φ∗i−1 (x) , ui (x) , φ∗i+1 (x) , . . , φ∗n (x) +Wxi (x) f [x, φ∗1 (x) , φ∗2 (x) , . . . , φ∗i−1 (x) , ui (x) , φ∗i+1 (x) , . . , φ∗n (x) = g i [x, φ∗1 (x) , φ∗2 (x) , . . , φ∗n (x)] +Wxi (x) f [x, φ∗1 (x) , φ∗2 (x) , . . , φ∗n (x)] , for i ∈ N.

2000). Differential games or continuous-time infinite dynamic games study a class of decision problems, under which the evolution of the state is described by a differential equation and the players act throughout a time interval. In particular, in the general n-person differential game, Player i seeks to: T max ui t0 g i [s, x (s) , u1 (s) , u2 (s) , . . 44) for i ∈ N = {1, 2, . . , n} , subject to the deterministic dynamics x˙ (s) = f [s, x (s) , u1 (s) , u2 (s) , . . 45) where x (s) ∈ X ⊂ Rm denotes the state variables of game, and ui ∈ U i is the control of Player i, for i ∈ N .

N (x0 , T − t0 )] that satisfies the conditions: (i) ξi (x0 , T − t0 ) ≥ v ({i} ; x0 , T − t0 ) , for i ∈ N, and (ii) ξj (x0 , T − t0 ) = v (N ; x0 , T − t0 ) j∈N is called an imputation of the game Γv (x0 , T − t0 ). 1 guarantees individual rationality in the sense that each player receives at least the payoff he or she will get if play against the rest of the players. Part (ii) ensures Pareto optimality and hence group rationality. 1. Suppose the function w : 2n × Rm × R1 → R1 is additive in S ∈ 2n , that is for any S, A ∈ 2n , S ∩ A = ∅ we have w (S ∪ A; x0 , T − t0 ) = w (S; x0 , T − t0 ) + w (A; x0 , T − t0 ).

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