By Driessen T.
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Additional resources for A bankruptcy problem and an information trading problem: Applications to k-convex games
4. 4 it does not exist. We include plots of the functional cost in Figure 1, where (1) is the plot under u(l), (2) - under uta), (3) - under (4) - under ujO) and (5) - under Ul. 1 (a) ). 5. Notice that the game cost incurred by the nonlinear controllers goes to negative infinity at a higher rate. Also, the cost incurred by u(1) and ujl) the higher order approximations to the optimal nonlinear and linear controllers - goes to negative infinity at a higher rate than those incurred by uta) and ujO).
Below, BT will denote the transpose of a matrix (or of a column vector) B, and IIBII will denote the sum of absolute values of the components of B. 2 Construction of €-equilibrium Markov strategies Consider a family of infinite horizon stochastic games, all with the same state and action spaces X and A as above, and the same transition probabilities P, parametrized by a vector A E JRn . e. r(A, x, a) is the cost in the MDP A, when at state x and the actions chosen are a. r is given by r(A,x,a) = ATBy(x,a).
We clearly have Ph (x) ~ PI( x ). If there exist some u = (u 1, U 2 ) and 8 such that Pf(x) I + 8>- E(u x 1 ,U 2 )CT Z 1_ > pfII (x) - 8, then u is called 8-saddle point, or 8-equilibrium strategy pair for Q1f (we need not specify Q 1 J or Q 1 II ). If this holds for 8 = 0, then u is called the saddle point or equilibrium strategy for Qf. Remarks: (i) Q1 J is equivalent to the problem: find a policy u 1 E U 1 that achieves SUPu1EUl infu2Eu2 cTZ1, where Zt E 1Rn is given by tE[O,I], Zo = z (4) The same holds for Q1 J1 .