On Team Games: A Computational Structural Analysis

A Controllability Study for a Team Linear Quadratic Game

“…Let the individual contribution of player Pi to the team be JiN(uj, v) = 0.5 a2 Iyi(tf)I2 + 0.5 J (11 ui(t) 12Ri -11v(t)1122R IN) dt (2( where Pi plays strategy ui and E strategy v such that the total team payoff is j E JiN(u, v) (2] Let the payoff of the 1 versus 1 game (Pi,E) be Ji (ui,v) = 0.5 a2fy1(t)II2 + 0.5 f (|ui(t) 12R (28) and, since the control spending of the evader is a positive 9) quantity divided by N in the definition of JiN (29) Tih(ui,e , VNc) s-Ji,o(uis VN) The conclusion is that Ji1 (uaj* v*) > J (U.A * VN) (30) or that pursuer Pi is better off as a team member than playing alone. 0) Therefore, from (18) and (15), a sufficient condition to ensure existence of a solution, or for pursuer Pi to be "efficient" is that (31) holds. Since Me is semipositive definite, another sufficient condition is…”

“…If the matrices defined in (31) are semipositive definite, then multiplying each end by the vector K' I(tf,t) yi(t) and using definition (10) yields the scalar inequality -yT(t) Kl T(tf,t) Me(tf, t) E K 1(tf,t) yj(t)-0 and, as t tf, Me(tf,t) is (from (17) and (6)) proportional to C(tf) = A Ge(tf) Re 1(tf) G (tf) AT.…”