# Differential Topology by Koschorke U.

By Koschorke U.

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89) produces n j =1 j =i 1 φj∗ (x) + φi∗ (x) 2 for i ∈ N. 91) yields n j =1 j =i 1 1 φj∗ (x) + φi∗ (x) = n − 2 2 3 ( [c + Vˆxi x 1/2 ]x , n ˆ j 1/2 ])3 j =1 [c + Vx x for i ∈ N. 92) represents a system of equations that is linear in {φ1∗ (x), φ2∗ (x), . . , φn∗ (x)}. 92) yields the game equilibrium strategies φi∗ (x) = x(2n − 1)2 j n [c + Vˆx x 1/2 ]]3 2[ j =1 n j =1 j =i 3 j c + Vˆx x 1/2 − n − 2 c + Vˆxi x 1/2 for i ∈ N. 88), and upon solving it, yields the following. 95) a A. 4. 93). 1 Consider the competitive dynamic advertising game in which there are two firms in a market.

Moreover, V (t0 )i (t, x) is the game equilibrium payoff of agent i at time t ∈ [t0 , T ] with the state being x, that is, T V (t0 )i (t, x) = t g i s, x ∗ (s), φ1∗ s, x ∗ (s) , φ2∗ s, x ∗ (s) , . . , φn∗ s, x ∗ (s) × exp − s r(y) dy ds + q i x ∗ (T ) exp − t0 T r(y) dy . t0 We also call it the value function of agent i in the game. A remark that will be utilized in the subsequent analysis is given below. 2), which starts at time τ for τ ∈ [t0 , T ). Note that the equilibrium feedback strategies are Markovian in the sense that they depend on the current time and current state.

62) yields the condition for a maximum as exp rt (t0 )1 Vx (t, x)(1 − x)1/2 c1 exp rt (t0 )2 φ2∗ (t, x) = Vx (t, x)x 1/2 . 63) exp rt (t0 )1 exp rt (t0 )2 Vx (t, x)(1 − x) − Vx (t, x)x , c1 c2 V (t0 )2 (T , x) = exp(−rT )S2 (1 − x). 63) admits a solution V (t0 )1 (t, x) = exp −r(t) A1 (t)x + B1 (t) , V (t0 )2 (t, x) = exp −r(t) A2 (t)x + B2 (t) , where A(t) and B(t) satisfy [A1 (t)]2 A1 (t)A2 (t) + , A˙ 1 (t) = rA1 (t) − q1 + 2c1 c2 [A1 (t)]2 B˙ 1 (t) = rB1 (t) − , 2c1 B1 (T ) = 0; [A2 (t)]2 A1 (t)A2 (t) A˙ 2 (t) = rA2 (t) − q2 + + , 2c2 c1 [A2 (t)]2 B˙ 2 (t) = rB2 (t) − , 2c2 A1 (T ) = S1 , A2 (T ) = S2 , B2 (T ) = 0.