N-Player Preemption Games
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1 N-Player Preemption Games Rossella Argenziano Essex Philipp Schmidt-Dengler LSE October 2007 Argenziano, Schmidt-Dengler (Essex, LSE) N-Player Preemption Games Leicester October / 42
2 Timing Games Many settings where timing is key strategic variable: Entry/Exit Product introduction Technology adoption Simple timing games: Each player s only choice is when to stop War of attrition Preemption game Argenziano, Schmidt-Dengler (Essex, LSE) N-Player Preemption Games Leicester October / 42
3 N-Player Preemption Games Extensive literature on war of attrition Preemption game literature focuses on 2-player games Wish to understand role of competition on: timing of entry/product introduction welfare distribution of payo s Argenziano, Schmidt-Dengler (Essex, LSE) N-Player Preemption Games Leicester October / 42
4 Related Literature Unobservable Actions: Reinganum (1981a, 1981b) Park and Smith (2005) Observable Actions: Fudenberg and Tirole (1985) Riordan (1992) Hoppe (2002) Hoppe and Lehmann-Grube (2005) Argenziano, Schmidt-Dengler (Essex, LSE) N-Player Preemption Games Leicester October / 42
5 Outline Timing game of investment with decreasing cost The game with pre-commitment The game without pre-commitment: clusters and di usion Comparative statics Extensions Welfare Cournot example Argenziano, Schmidt-Dengler (Essex, LSE) N-Player Preemption Games Leicester October / 42
6 Model (1) N identical players ( rms) In nite horizon Time is continuous (discrete with in nitely ne grid) t = 0 : New investment opportunity becomes available Investment is irreversible and observable At any t, each active rm chooses whether to Invest or Wait Argenziano, Schmidt-Dengler (Essex, LSE) N-Player Preemption Games Leicester October / 42
7 Model (2) Before investment: zero pro ts (new market/product) After investment: ow pro ts π(m) m = number of investors Present value of the investment cost: c(t) Let T j be the j-th investment time. Firm j 0 s payo is: V j (T 1, T 2,..., T j,..., T N ) = N Z T m+1 π(m) e rt dt c(t j ) m=j T m Argenziano, Schmidt-Dengler (Essex, LSE) N-Player Preemption Games Leicester October / 42
8 Assumptions (1) A1 (i) π (m) > 0 8m (ii) π (m) is decreasing in m (positive and decreasing returns to investment) A2 (i) 8t, (c (t) e rt ) 0 < 0 (ii) (c (t) e rt ) 00 > 0 (cost decreasing and convex) Argenziano, Schmidt-Dengler (Essex, LSE) N-Player Preemption Games Leicester October / 42
9 Assumptions (2) A3 (i) π(1) r c (0) < 0 (no investment at t = 0) A4 (i) 9τ such that c (τ) e r τ < π(n) r (ii)lim t! c 0 (t)e rt 2 ( π (N), 0] (eventually all rms invest) Argenziano, Schmidt-Dengler (Essex, LSE) N-Player Preemption Games Leicester October / 42
10 Game with Pre-Commitment Suppose rms pre-commit to their investment time (or face in nite information lags) Firms maximize payo as if they followed a pre- xed order of investment (Reinganum (1981)) Investment times are given by F.O.C. : T j π(j)e rt j c 0 T = 0 decreases in π(j) and does not depend on N Equilibrium payo s decrease in the order of investment j Argenziano, Schmidt-Dengler (Essex, LSE) N-Player Preemption Games Leicester October / 42
11 Game without Pre-Commitment Information lags are typically very short Payo s decreasing in the order of investment create opportunities for preemption A game with pre-commitment cannot capture preemption. Require subgame perfection Argenziano, Schmidt-Dengler (Essex, LSE) N-Player Preemption Games Leicester October / 42
12 Strategies in Continuous Time (1) How to interpret strategies in continuous time? What is the outcome of a given strategy pro le? Hoppe and Lehmann-Grube (2005) adopt Simon and Stinchcombe (1989) and interpret continuous time as discrete but with a grid that is in nitely ne Argenziano, Schmidt-Dengler (Essex, LSE) N-Player Preemption Games Leicester October / 42
13 Strategies in Continuous Time (2) Restrict play to Pure Strategies such that : when any pro le is played on an increasingly ne sequence of discrete-time grids, it generates a convergent sequence of outcomes, whose limit is independent of the sequence of grids (*) Take this limit to be the outcome of the strategy pro le Simon and Stinchcombe (1989) characterize su cient conditions for (*) Argenziano, Schmidt-Dengler (Essex, LSE) N-Player Preemption Games Leicester October / 42
14 Strategies in Continuous Time (3) In our game, a strategy maps decision nodes into actions Decision Node: Calendar Time and History of previous investments Simon and Stinchcombe s conditions are satis ed if strategies are: piecewise continuous w.r.t. calendar time not too sensitive to the dates of previous investments (payo irrelevant) Argenziano, Schmidt-Dengler (Essex, LSE) N-Player Preemption Games Leicester October / 42
15 Strategies in Continuous Time (4) Well-known problem with preemption games: no pure strategy equilibria. ) introduce a randomization device: A5 (i) If n rms invest at the same instant t, then only one rm succeeds each with probability 1 n. (ii) At time t, if a rm is indi erent between being the m th investor at t and the (m + 1)-th investor, it invests at t. Argenziano, Schmidt-Dengler (Essex, LSE) N-Player Preemption Games Leicester October / 42
16 Solving the Model In equilibrium: Intuition (i) No rm invests at t = 0. (ii) All rms invest in nite time, and t N = T N. Investment cost is too high at t = 0, but decreases su ciently. Last active player solves same problem as in game with precommitment. =) Finite horizon game: can solve backwards from T N. Argenziano, Schmidt-Dengler (Essex, LSE) N-Player Preemption Games Leicester October / 42
17 Example: Two Firms (1) Like Fudenberg & Tirole (1985) Once the rst rm has invested, it is optimal to invest at t 2 = T 2 What determines t 1? Suppose no rm has invested at t < T 2. If a rm (Leader) invests at t, the second rm (Follower) invests at T 2. Argenziano, Schmidt-Dengler (Essex, LSE) N-Player Preemption Games Leicester October / 42
18 Example: Two Firms (cont.d) The Leader s payo is: Z T 2 L 1 (t) = π(1) e rs ds t{z } Monopoly payo s + π(2) r e rt 2 {z } Duopoly payo s c(t) {z} cost The Follower s payo is independent of the Leader s investment time F 1 (t) = π(2) e rt 2 c(t2 ) r Argenziano, Schmidt-Dengler (Essex, LSE) N-Player Preemption Games Leicester October / 42
19 Example: Two Firms (cont.d) At t < T2, a rm compares being Leader to being Follower: D 1 (t) = L 1 (t) F 1 (t) Z T 2 = π(1) e rs ds t{z } gain from being leader [c(t) c(t 2 )] {z } cost of being the leader Argenziano, Schmidt-Dengler (Essex, LSE) N-Player Preemption Games Leicester October / 42
20 Equilibrium with Two Firms D 1 (t) D 1 (t) 0 t 1 T * t 2 =T * 1 2 Time Argenziano, Schmidt-Dengler (Essex, LSE) N-Player Preemption Games Leicester October / 42
21 Example: Two Firms (cont.d) First investment time pushed back until there is no advantage in being the Leader rather than the Follower Total Payo s are equalized in equilibrium Argenziano, Schmidt-Dengler (Essex, LSE) N-Player Preemption Games Leicester October / 42
22 Example: Three Firms Once one rm has invested, we know how to solve the subgame with two rms At t < t 2, a rm compares being Leader ( rst investor) to being Follower (second or third): D 1 (t) = L 1 (t) F 1 (t) Z t2 = π(1) e rs ds t{z } gain from being leader [c(t) c(t 2 )] {z } cost of being the leader Argenziano, Schmidt-Dengler (Essex, LSE) N-Player Preemption Games Leicester October / 42
23 Equilibrium with Three Firms (Di usion) D 1 (t) D 2 (t) D j (t) 0 t 1 T 1 * t * 2 T 2 t 3 =T * 3 Time Argenziano, Schmidt-Dengler (Essex, LSE) N-Player Preemption Games Leicester October / 42
24 Equilibrium with Three Firms (Cluster) D 2 (t) D j (t) 0 D 1 (t) t =t * * 1 2 T 1 T t =T * Time Argenziano, Schmidt-Dengler (Essex, LSE) N-Player Preemption Games Leicester October / 42
25 Example: Three Firms (cont d) Payo s are equalized in equilibrium Clusters are possible The same holds for N rms Can continue solving the model backwards with N rms. Argenziano, Schmidt-Dengler (Essex, LSE) N-Player Preemption Games Leicester October / 42
26 Example: Five Firms (Di usion) D 1 (t) D j (t) D 2 (t) 0 D 4 (t) t t 3 t 4 t 5 =T * 1 t 2 Time 5 Argenziano, Schmidt-Dengler (Essex, LSE) N-Player Preemption Games Leicester October / 42
27 Example: Five Firms (Cluster) D 1 (t) D j (t) 0 D 2 (t) D 4 (t) t t 3 =t 4 t 5 =T * 1 t 2 Time 5 Argenziano, Schmidt-Dengler (Essex, LSE) N-Player Preemption Games Leicester October / 42
28 Proposition The preemption game with N players admits a unique SPNE in pure strategies. The equilibrium outcome is unique up to a permutation of players and all players receive the same payo : π(n) e rt N c (T r N) The equilibrium investment times (t 1, t 2..., t N ) can be calculated recursively according to the following algorithm: (i) t N = T N ; (ii) For j = 1...N 1, if T j < t j+1, then t j < t j+1, and t j solves Z tj+1 π(j) e rs ds c(t j ) + c(t j+1 ) = 0, t j if T j t j+1, then t j = t j+1. Argenziano, Schmidt-Dengler (Essex, LSE) N-Player Preemption Games Leicester October / 42
29 Comparative Statics: Equilibrium Payo s Equilibrium payo is: π(n) e rt N c (T r N) An increase in π (m) for m < N does not a ect equilibrium payo s. An increase in π (N) increases equilibrium payo of all players Policy implications Argenziano, Schmidt-Dengler (Essex, LSE) N-Player Preemption Games Leicester October / 42
30 When do we expect clusters? For su ciently small impact of extra investors on the individual payo : If [π (m) π (m + 1)] is small, T m is close to T m+1 Preemption pushes back the (m + 1) to t m+1 < Tm+1 th investment time For small π (m) π (m + 1), t m+1 < T m < T m+1 and there is a cluster Argenziano, Schmidt-Dengler (Essex, LSE) N-Player Preemption Games Leicester October / 42
31 Comparative Statics: Equil. Investment Times (1) Suppose t m 1 < t m < t m+1. t m π (m) t m 1 π (m) < 0 > 0 When π(m) increases: m-th investment is accelerated (Preemption incentive becomes stronger) (m 1)-th investment is delayed (Preemption incentive is weakened) Argenziano, Schmidt-Dengler (Essex, LSE) N-Player Preemption Games Leicester October / 42
32 Comparative Statics: Equil. Investment Times (2) Suppose t m 1 = t m < t m+1. t m π (m) t m 1 π (m) < 0 < 0 In the presence of a cluster, an increase in π(m) accelerates both m-th and (m investment 1)-th Argenziano, Schmidt-Dengler (Essex, LSE) N-Player Preemption Games Leicester October / 42
33 Comparative Statics: Equil. Investment Times (3) Let π(m) increase. What happens to the (m j)-th investment time? Result: If for a given vector [π (1),.., π (m),.., π (N)] in equilibrium t m j < t m j+1... < t m, then (i) t m j π m (ii) t m j π m > 0 if j is odd < 0 if j is even. =) Alternating e ects Argenziano, Schmidt-Dengler (Essex, LSE) N-Player Preemption Games Leicester October / 42
34 Adding one more player The number of players N determines the last adoption time and the equilibrium payo Adding an (N + 1)-th player: π(n) e rt N c (T r N) Delays the last investment time Reduces the equilibrium payo of all players Has alternating e ects on the rst N investment times Argenziano, Schmidt-Dengler (Essex, LSE) N-Player Preemption Games Leicester October / 42
35 Endogenous Number of Investors (1) So far: total number of players (N) given exogenously All N rms invest eventually What if there is a large number M >> N of potential investors? Argenziano, Schmidt-Dengler (Essex, LSE) N-Player Preemption Games Leicester October / 42
36 Endogenous Number of Investors (cont.d) M potential investors, but π (j) e rt c (t) < 0 8t, 8j N + 1 r 9τ such that π (N) e r τ c (τ) 0 r Only N < M rms invest in equilibrium Usual preemption arguments: All rms make zero pro ts. Argenziano, Schmidt-Dengler (Essex, LSE) N-Player Preemption Games Leicester October / 42
37 Endogenous Number of Investors (cont.d) Last investment occurs at t N < T N, where t N solves π (N) e rt N c (t N ) = 0 r There may be a cluster at t N, because we may have that t N < T N 1 Argenziano, Schmidt-Dengler (Essex, LSE) N-Player Preemption Games Leicester October / 42
38 Welfare Analysis Preemption accelerates investment w.r.t. to precommitment game. Negative e ect on industry pro ts (business stealing e ect) Positive e ect on consumer surplus Total e ect: need to specify details of post-entry competition. Argenziano, Schmidt-Dengler (Essex, LSE) N-Player Preemption Games Leicester October / 42
39 Example: Entry with Cournot Competition Suppose entry cost is exponential c(t) = c e (α+r)t (1) and post-entry competition is Cournot with linear demand We solve the model and nd two results Argenziano, Schmidt-Dengler (Essex, LSE) N-Player Preemption Games Leicester October / 42
40 Proposition If ow pro ts π(m) arise from Cournot competition among the investors, with a linear demand function and constant marginal cost, the equilibrium entry times t m are smaller than the socially optimal investment times for m 2. If in addition, the cost of investment declines exponentially, the rst equilibrium entry time t 1 is larger than the socially optimal investment time if r α > gr α Argenziano, Schmidt-Dengler (Essex, LSE) N-Player Preemption Games Leicester October / 42
41 Proposition Let t j (N) denote the j-th equilibrium investment time in a game with N potential investors. If the cost of investment declines exponentially and ow pro ts π(m) arise from Cournot competition among the investors, with a linear demand function and constant marginal cost, then t 1 (N) > t 1 (2). Argenziano, Schmidt-Dengler (Essex, LSE) N-Player Preemption Games Leicester October / 42
42 Conclusions Preemption games with a nite number of players Clusters of investments are possible in the absence of complementarities Extensions: asymmetric players Argenziano, Schmidt-Dengler (Essex, LSE) N-Player Preemption Games Leicester October / 42
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