A d-person Differential Game with State Space Constraints by Ramasubramanian S.

By Ramasubramanian S.

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2), 5–54 (2000) 6. : Ergodic control for constrained diffusions: characterization using HJB equations. SIAM J. Control Optim. 43, 1467–1492 (2004/2005) 7. : Stochastic differential games: occupation measure based approach. J. Optim. Theory Appl. 73, 359–385 (1992) 8. : Small BV solutions of hyperbolic noncooperative differential games. SIAM J. Control Optim. 43, 194–215 (2004) 9. : Hamilton-Jacobi equations with state constraints. Trans. S. 318, 643–683 (1990) 10. : Existence and uniqueness for a simple nonzero—sum differential game.

Res. 28, 801–835 (2003) 3. : Optimal Control and Viscosity Solutions of Hamilton-JacobiBellman Equations. Birkhauser, Boston (1997) 4. : Stochastic games for N players. J. Optim. Theory Appl. 105, 543–565 (2000) 5. : Stochastic games with risk sensitive pay offs for N players. Le Matematiche 55(Suppl. 2), 5–54 (2000) 6. : Ergodic control for constrained diffusions: characterization using HJB equations. SIAM J. Control Optim. 43, 1467–1492 (2004/2005) 7. : Stochastic differential games: occupation measure based approach.

Fix yˆ1 (·). Let 0 ≤ y2 (t) < λ2 t for some t. Corresponding to yˆ1 (·), y2 (·) note that z1 (t) ≥ 0 but z2 (t) = −t + y2 (t) + R21 yˆ1 (t) < 0. So with yˆ1 (·) fixed, y2 (·) cannot be feasible unless y2 (t) ≥ λ2 t, ∀t. In an entirely analogous manner with yˆ2 (·) fixed, y1 (·) cannot be feasible unless y1 (t) ≥ λ1 t for all t. Therefore it follows that for any λ1 , λ2 as above (uˆ 1 (·), uˆ 2 (·)) ≡ (λ1 , λ2 ) gives a Nash equilibrium for each t ≥ 0. So even Nash equilibrium serving for all t need not be unique.

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