A Dynamic Model of Leap-Frogging Investments and Bertrand Price Competition Appendix

Appendix 1: Proof of Lemma 1 Lemma 3.1 Suppose η = 0. If c1 > c2 > 0 and K < βc2 1−β , then in the unique mixed strategy equilibrium of the pure Bertrand dynamic investment and pricing game in state (c1 , c2 , 0) we have π1 > π2 . Proof. First, note that the condition K < βc2 1−β in Lemma 3.1 ensures that investment is profitable in the βc2 long term even for firm 1 whose potential pay-off is smaller ( 1−β < βc1 1−β ). In other words, this condition ensures that for both firms’ investment decisions are economically justified. Next, observe that when β = 0 in the (c1 , c2 , 0) end game there is unique pure strategy equilibrium where neither of the companies invests. Thus, we only consider the case β > 0. The value functions of the two firms in the (c1 , c2 , 0) end game when c1 > c2 are    βc2 −K + V1 = π1 × π2 · (−K) + (1 − π2) · 1−β +(1 − π1 ) × (π2 · 0 + (1 − π2 ) · βV1 )    βc1 −K + V2 = π2 × π1 · (c1 − c2 − K) + (1 − π1) · c1 − c2 + 1−β +(1 − π2 ) × (π1 · (c1 − c2 ) + (1 − π1 ) · (c1 − c2 + βV2 )) where the definition of the probability π1 of investment by firm 1 in the mixed strategy equilibrium gives   βc2 π2 · (−K) + (1 − π2) · − K = π2 · 0 + (1 − π2 ) · βV1 1−β and thus the value function itself becomes the weighted sum of equal parts, leading to   βc2 V1 = π2 · (−K) + (1 − π2 ) · − K = π2 · 0 + (1 − π2 ) · βV1 1−β Using the second equality in the last expression, we find V1 = 0, and then using the first equality in the same expression, we find 1 − π2 = K(1−β) βc2 . The definition of the probability π2 of investment by firm 2 in the mixed strategy equilibrium, similarly gives V2 βc1 = π1 · (c1 − c2 − K) + (1 − π1) · c1 − c2 + −K 1−β = π1 · (c1 − c2 ) + (1 − π1 ) · (c1 − c2 + βV2 )  Using the second equality in the last expression, we find V2 = c1 −c2 (1−β·(1−π1 )) ,  and using the it once again we get  βc1 π1 (c1 − c2 − K) + (1 − π1) c1 − c2 + −K = π1 (c1 − c2 ) + (1 − π1 )(c1 − c2 + βV2 ) 1−β   βc1 (1 − π1 ) − K − π1 K = (1 − π1 )βV2 1−β c1 K − = V2 1 − β β · (1 − π1 )  Combining the two expressions for the value function V2 , we get the following equation c1 − c2 1 − β · (1 − π1 ) c1 K − 1 − β β · (1 − π1 ) = Multiplying by 1 − β and incerting the expression for 1 − π2 , we have c1 − c2 β 1 + 1−β π1 2 c1 − 1−π 1−π1 c2 c1 − c2 c1 − = c1 − = 1 β 1 + 1−β π1 61 1 − π2 c2 6 c1 − c2 1 − π1 1 − π2 > 1 1 − π1 π1 > π2 The inequalities are due to the fact that 0 6 π1 6 1, is strict unless π1 = π2 = 0, which implies K = π1 > π2 . 1 − π2 c2 1 − π1 βc2 1−β β 1−β > 0, c1 − c2 > 0, c2 > 0. The final inequality thus leading to a contradiction. We conclude then that Appendix 2: Recursions for Simultaneous and Alternating Move Games Bellman equations for the simultaneous move investment game In this game, the two firms are able to invest in every period, but make their investment decisions simultaneously at the start of each period without knowledge of the opponent’s investment decision in that period. However the firms do observe their opponents’ decisions made in the previous period. In terms of our notation, in this case the state variable m denoting the order of moves takes the value m = 0 in every period, corresponding to simultaneous moves. To reduce notational burden we omit the m state variable in this case since it is always equal to 0. The value functions for firms 1 and 2 in the state (c1 , c2 , c) corresponding to deciding to invest or not to invest, respectively, are v11 (c1 , c2 , c), v10 (c1 , c2 , c), and v21 (c1 , c2 , c) and v20 (c1 , c2 , c), where the superscript denotes the firm (1 or 2) and the subscript denotes the investment decision (1 = invest, 0 = don’t invest) except for the investment probabilities where subscripts are used to denote the investment probabilities for firms 1 and 2, respectively, P1 and P2 . Recall that r j (c1 , c2 ) denotes the single eriod profits earned by firm j when the firms’ costs of production are (c1 , c2 ).  Z c 1 1 v0 (c1 , c2 , c) = r (c1 , c2 ) + β P2 (c1 , c2 , c) φ(v11 (c1 , c, c′ ), v10 (c1 , c, c′ ))π(dc′ |c) 0  Z c 1 ′ 1 ′ ′ + (1 − P2 (c1 , c2 , c)) φ(v1 (c1 , c2 , c ), v0 (c1 , c2 , c ))π(dc |c) 0  Z c 1 1 v1 (c1 , c2 , c) = r (c1 , c2 ) − K(c) + β P2 (c1 , c2 , c) φ(v11 (c, c, c′ ), v10 (c, c, c′ ))π(dc′ |c) 0  Z c 1 ′ 1 ′ ′ +(1 − P2 (c1 , c2 , c)) φ(v1 (c, c2 , c ), v0 (c, c2 , c ))π(dc |c) 0  Z c 2 2 v0 (c1 , c2 , c) = r (c1 , c2 ) + β P1 (c1 , c2 , c) φ(v21 (c, c2 , c′ ), v20 (c, c2 , c′ ))π(dc′ |c) 0  Z c 2 ′ 2 ′ ′ +(1 − P1 (c1 , c2 , c)) φ(v1 (c1 , c2 , c ), v0 (c1 , c2 , c ))π(dc |c) 0  Z c 2 2 v1 (c1 , c2 , c) = r (c1 , c2 ) − K(c) + β P1 (c1 , c2 , c) φ(v21 (c, c, c′ ), v20 (c, c, c′ ))π(dc′ |c) 0  Z c 2 ′ 2 ′ ′ +(1 − P1 (c1 , c2 , c)) φ(v1 (c1 , c, c ), v0 (c1 , c, c ))π(dc |c) 0 where P1 (c1 , c2 , c) = exp{v11 (c1 , c2 , c)/η} exp{v11 (c1 , c2 , c)/η} + exp{v10 (c1 , c2 , c)/η} P2 (c1 , c2 , c) = exp{v21 (c1 , c2 , c)/η} exp{v21 (c1 , c2 , c)/η} + exp{v20 (c1 , c2 , c)/η} Bellman equations for the firm 1 investment leadership game In this game both firms are allowed to invest in every period, but there is a slightly different structure of the information flow compared to the simultaneous move investment case. Here we assume that firm 1 is always the first to make a decision about whether or not to invest in any period, and then immediately after firm 1 makes its decision, firm 2 makes its decision, but firm 2 can condition its investment decision on its observation of whether firm 1 decided to invest or not. Since firm 1 is always the leader, to simplify notation we do not carry along the m state variable in the Bellman equations below. There are six functional equations to be solved in this case, with two of them specifying the value functions for firm 1 corresponding to the values of investing or not investing, respectively, v11 (c1 , c2 , c) and v10 (c1 , c2 , c), and four value functions for firm 2 corresponding to the values of investing or not investing, respectively, under the two scenarios where firm 1 either invests or doesn’t invest. We use a double subscript to indicate firm 2’s investment decision and firm 1’s investment decision (with firm 2’s subscript coming first then firm 1’s subscript next). Thus v201 (c1 , c2 , c) denotes the value to firm 2 of not investing given that it observes that firm 1 did invest. These value functions will imply two different investment probability functions for firm 2 which we denote by double subscripts as well. P20 (c1 , c2 , c) denotes the probability that firm 2 will invest given that it observes that firm 1 did not invest, and P21 (c1 , c2 , c) denotes the probability that firm 2 will invest given that it observes that firm 1 did invest. So we have P1 (c1 , c2 , c) = exp{v11 (c1 , c2 , c)/η} exp{v11 (c1 , c2 , c)/η} + exp{v10 (c1 , c2 , c, 1)/η} P20 (c1 , c2 , c) = exp{v210 (c1 , c2 , c)/η} exp{v210 (c1 , c2 , c)/η} + exp{v200 (c1 , c2 , c)/η} P21 (c1 , c2 , c) = exp{v211 (c1 , c2 , c)/η} . exp{v211 (c1 , c2 , c)/η} + exp{v201 (c1 , c2 , c)/η} where  Z c = r (c1 , c2 ) + β P20 (c1 , c2 , c) φ(v10 (c1 , c, c′ ), v11 (c1 , c, c′ ))π(dc′c ) 0  Z c 1 ′ 1 ′ ′ +(1 − P20 (c1 , c2 , c)) φ(v0 (c1 , c2 , c ), v1 (c1 , c2 , c ))π(dc |c) 0   Z c 1 1 1 ′ 1 ′ ′ v1 (c1 , c2 , c) = r (c1 , c2 ) − K(c) + β P21 (c1 , c2 , c) φ(v0 (c, c, c ), v1 (c, c, c ))π(dc |c) 0  Z c 1 ′ 1 ′ ′ +(1 − P21 (c1 , c2 , c)) φ(v0 (c, c2 , c ), v1 (c, c2 , c ))π(dc |c) , v10 (c1 , c2 , c) 1 0 and = r (c1 , c2 ) + β Z c P1 (c1 , c2 , c′ )φ(v201 (c1 , c2 , c′ ), v211 (c1 , c2 , c′ ))π(dc′ |c)  Z c ′ 2 ′ 2 ′ ′ + (1 − P1 (c1 , c2 , c ))φ(v00 (c1 , c2 , c ), v10 (c1 , c2 , c ))π(dc |c) 0 Z c 2 2 v10 (c1 , c2 , c) = r (c1 , c2 ) − K(c) + β P1 (c1 , c, c′ )φ(v201 (c1 , c, c′ ), v211 (c1 , c, c′ ))π(dc′ |c) 0  Z c ′ 2 ′ 2 ′ ′ + (1 − P1 (c1 , c, c ))φ(v00 (c1 , c, c ), v10 (c1 , c, c ))π(dc |c) 0 Z c 2 2 v01 (c1 , c2 , c) = r (c1 , c2 ) + β P1 (c, c2 , c′ )φ(v201 (c, c2 , c′ ), v211 (c, c2 , c′ ))π(dc′ |c) 0  Z c + (1 − P1 (c, c2 , c′ ))φ(v200 (c, c2 , c′ ), v210 (c, c2 , c′ ))π(dc′ |c) 0 Z c 2 2 v11 (c1 , c2 , c) = r (c1 , c2 ) − K(c) + β P1 (c, c, c′ )φ(v201 (c, c, c′ ), v211 (c, c, c′ ))π(dc′ |c) 0  Z c + (1 − P1 (c, c, c′ ))φ(v200 (c, c, c′ ), v210 (c, c, c′ ))π(dc′ |c) . v200 (c1 , c2 , c) 2 0 0 Bellman equations for the deterministically alternating move investment game In this game, at most one of the firms can invest in any period, and the state variable m that signifies which of the firms is eligible to invest in any period alternates deterministically between the values 1 and 2. Thus, when m = 1 only firm 1 can decide whether to invest, and it makes its decision at the start of the period. Then next period m = 2 and firm 2 can decide whether to invest. The value functions for firms 1 and 2 in the state (c1 , c2 , c, m) corresponding to deciding to invest or not to invest, respectively, are v11 (c1 , c2 , c, m), v10 (c1 , c2 , c, m), and v21 (c1 , c2 , c, m) and v20 (c1 , c2 , c, m), where again the superscript denotes the firm (1 or 2) and the subscript denotes the investment decision (1 = invest, 0 = don’t invest) except for the investment probabilities where subscripts are used to denote the investment probabilities for firms 1 and 2, respectively, P1 and P2 , and with the exception that when it isn’t firm j’s turn to move, the subscript now represents its opponent’s investment decision in that period. Clearly, the probabilities of investing are 0 in the periods where it is not the firm’s turn to invest. In periods where it is their turn to invest, the invesment probabilities are given by P1 (c1 , c2 , c, 1) = exp{v11 (c1 , c2 , c, 1)/η} exp{v11 (c1 , c2 , c, 1)/η} + exp{v10 (c1 , c2 , c, 1)/η} P2 (c1 , c2 , c, 2) = exp{v21 (c1 , c2 , c, 2)/η} , exp{v21 (c1 , c2 , c, 2)/η} + exp{v20 (c1 , c2 , c, 2)/η} where = r (c1 , c2 ) + β Z c P2 (c1 , c2 , c′ , 2)v11 (c1 , c2 , c′ , 2)π(dc′ |c)  Z c ′ 1 ′ ′ + (1 − P2 (c1 , c2 , c , 2))v0 (c1 , c2 , c , 2)π(dc |c) 0 Z c 1 1 v1 (c1 , c2 , c, 1) = r (c1 , c2 ) − K(c) + β P2 (c1 , c2 , c′ , 2)v11 (c, c2 , c′ )π(dc′ |c) 0  Z c ′ 1 ′ ′ + (1 − P2 (c1 , c2 , c , 2))v0 (c, c2 , c )π(dc |c) 0 Z c 2 2 v0 (c1 , c2 , c, 2) = r (c1 , c2 ) + β P1 (c1 , c2 , c′ , 1)v21 (c1 , c2 , c′ , 1)π(dc′ |c) 0  Z c + (1 − P1 (c1 , c2 , c′ , 1))v20 (c1 , c2 , c′ , 1)π(dc′ |c) 0 Z c 2 2 v1 (c1 , c2 , c, 2) = r (c1 , c2 ) − K(c) + β P1 (c1 , c, c′ , 1)v21 (c1 , c, c′ , 1)π(dc′ |c) 0  Z c + (1 − P1 (c1 , c, c′ , 1))v20 (c1 , c, c′ , 1))π(dc′ |c) . v10 (c1 , c2 , c, 1) 1 0 0 The Bellman equations for these values for firms 1 and 2 when it is not their turn to invest are given by v11 (c1 , c2 , c, 2) = r1 (c1 , c2 ) + β v10 (c1 , c2 , c, 2) = r (c1 , c2 ) + β 1 v21 (c1 , c2 , c, 1) = r2 (c1 , c2 ) + β v20 (c1 , c2 , c, 1) = r (c1 , c2 ) + β 2 Z c 0 Z c 0 Z c Z0 c 0 φ(v10 (c1 , c, c′ , 1), v11 (c1 , c, c′ , 1))π(dc′ |c) φ(v10 (c1 , c2 , c′ , 1), v11 (c1 , c2 , c′ , 1))π(dc′ |c) φ(v20 (c, c2 , c′ , 2), v11 (c, c2 , c′ , 2))π(dc′ |c) φ(v10 (c1 , c2 , c′ , 2), v11 (c1 , c2 , c′ , 2))π(dc′ |c). Bellman equations for the stochastically alternating move investment game This game is a generalization of the previous case where we allow the ordering of the movers at each stage of the game to evolve stochastically according to a Markov transition probability f (m′ |m). As in the previous case we restrict the variable m, indicating which of the firms is eligible to invest (move) in any given period, to the set {1, 2} so that we do not allow stochastic transitions to simultaneous moves, m = 0. As in the deterministically alternating move game, we assume that at most one of the firms can invest in any period. Thus the deterministically alternating move case is a special case of the stochastically alternating move case when f (m′ |m) = 1 when m′ 6= m and 0 otherwise. The value functions below are relevant when the firm index, f , equals the move state variable, m, so each firm has the option to invest in this case. The probabilities each firm invests (in the states where they are allowed to invest, i.e. state m = 1 for firm 1 and state m = 2 for firm 2) are given by P1 (c1 , c2 , c) = P2 (c1 , c2 , c) = exp{v11 (c1 , c2 , c, 1)/η} 1 exp{v1 (c1 , c2 , c, 1)/η} + exp{v10 (c1 , c2 , c, 1)/η} exp{v21 (c1 , c2 , c, 2)/η} . 2 exp{v1 (c1 , c2 , c, 1)/η} + exp{v20 (c1 , c2 , c, 1)/η} These probabilities depend on the four value functions (v10 (c1 , c2 , c, 1), v11 (c1 , c2 , c, 1), v20 (c1 , c2 , c, 2), v21 (c1 , c2 , c, 2)) whose Bellman equations are given below. Z c = r (c1 , c2 ) − K(c) + β f (1|1)φ(v11 (c, c2 , c′ , 1), v10 (c, c2 , c′ , 1)) 0   +[1 − f (1|1)] P2 (c, c2 , c′ )v11 (c, c2 , c′ , 2) + (1 − P2 (c, c2 , c′ ))v10 (c, c2 , c′ , 2)) π(dc′ |c) Z c v10 (c1 , c2 , c, 1) = r1 (c1 , c2 ) + β f (1|1)φ(v11 (c1 , c2 , c′ , 1), v10 (c1 , c2 , c′ , 1)) 0   +[1 − f (1|1)] P2 (c1 , c2 , c′ )v11 (c1 , c2 , c′ , 2) + (1 − P2 (c1 , c2 , c′ ))v10 (c1 , c2 , c′ , 2)) π(dc′ |c) Z c v21 (c1 , c2 , c, 2) = r2 (c1 , c2 ) − K(c) + β f (2|2)φ(v21 (c1 , c, c′ , 2), v20 (c1 , c, c′ , 2)) 0   +[1 − f (2|2)] P1 (c1 , c, c′ )v21 (c1 , c, c′ , 1) + (1 − P2 (c1 , c, c′ ))v20 (c1 , c, c′ , 1)) π(dc′ |c) Z c 2 2 v0 (c1 , c2 , c, 2) = r (c1 , c2 ) + β f (2|2)φ(v21 (c1 , c2 , c′ , 2), v20 (c1 , c2 , c′ , 2)) 0   +[1 − f (2|2)] P2 (c1 , c2 , c′ )v21 (c1 , c2 , c′ , 1) + (1 − P2 (c1 , c2 , c′ ))v20 (c1 , c2 , c′ , 1)) π(dc′ |c). v11 (c1 , c2 , c, 1) 1 Note that the value functions above depend on the values (v10 (c1 , c2 , c, 2), v11 (c1 , c2 , c, 2), v20 (c1 , c2 , c, 1), v21 (c1 , c2 , c, 1)) for the two firms in the states where it is not these firms’ turn to invest. Note that in this case, the subscripts of the value function denote the investment decision of the opponent that does have the turn to invest, and a subscript of 0 denotes the opponent’s decision not to invest and a subscript of denotes the opponent’s decision to invest. v11 (c1 , c2 , c, 2) = r1 (c1 , c2 ) + β Z c 0 f (1|2)φ(v11 (c1 , c, c′ , 1), v10 (c1 , c, c′ , 1))   +[1 − f (1|2)] P2 (c1 , c, c′ )v11 (c1 , c, c′ , 2) + (1 − P2 (c1 , c, c′ ))v10 (c1 , c, c′ , 2) π(dc′ |c) Z c 1 1 v0 (c1 , c2 , c, 2) = r (c1 , c2 ) + β f (1|2)φ(v11 (c1 , c2 , c′ , 1), v10 (c1 , c2 , c′ , 1)) 0   +[1 − f (1|2)] P2 (c1 , c2 , c′ )v11 (c1 , c2 , c′ , 2) + (1 − P2 (c1 , c2 , c′ ))v10 (c1 , c2 , c′ , 2) π(dc′ |c) Z c v21 (c1 , c2 , c, 1) = r2 (c1 , c2 ) + β f (2|1)φ(v21 (c, c2 , c′ , 2), v20 (c, c2 , c′ , 2)) 0   +[1 − f (2|1)] P2 (c, c2 , c′ )v21 (c, c2 , c′ , 1) + (1 − P2 (c, c2 , c′ ))v20 (c, c2 , c′ , 1) π(dc′ |c) Z c 2 2 v0 (c1 , c2 , c, 1) = r (c1 , c2 ) + β f (2|1)φ(v21 (c1 , c2 , c′ , 2), v20 (c1 , c2 , c′ , 2)) 0   +[1 − f (2|1)] P2 (c1 , c2 , c′ )v21 (c1 , c2 , c′ , 1) + (1 − P2 (c1 , c2 , c′ ))v20 (c1 , c2 , c′ , 1) π(dc′ |c).