Preemption games under L evy uncertainty Svetlana Boyarchenko and - - PowerPoint PPT Presentation

Outline

Preemption games under L´ evy uncertainty

Svetlana Boyarchenko and Sergei Levendorski˘ i

University of Texas at Austin; University of Leicester

March 9, 2012

Boyarchenko and Levendorski˘ i (UT ) Preemption games 03/09/12 1 / 49

Outline

Outline

1

Introduction

2

Deterministic environment

3

Stochastic environment

4

Preemption under L´ evy uncertainty

5

Subgame perfect equilibria

6

Conclusion

Boyarchenko and Levendorski˘ i (UT ) Preemption games 03/09/12 1 / 49

Introduction

Motivation

Stopping time games are a special case of stochastic games Stopping time games have important applications in economics and finance Main part of the research on stopping games is focused on existence

• f the value of the game and optimal stopping strategies

Deriving optimal stopping strategies and pricing game options is not less important, especially in economics and finance Preemption games are a subset of stopping time games In preemption games, there is a region called the preemption zone, where each of the players prefers to be the leader (be the first to move) In the preemption zone, it may happen that it is optimal to move for

• ne player, but not for both, hence there is a coordination problem

Boyarchenko and Levendorski˘ i (UT ) Preemption games 03/09/12 2 / 49

Introduction

Aims of the paper

For a stochastic preemption game with two players characterize subgame perfect equilibria study effects of jumps on equilibria study effects of players’ asymmetry For simplicity, consider one-factor jump-diffusion models (L´ evy models) Technical challenges In entry-exit problems with strategic interactions under non-Gaussian uncertainty and other stopping games, non-monotone payoffs are the rule rather than exception. In our model, the value function of the low cost player is not only non-monotone, but, for some equilibria, it is also discontinuous.

Boyarchenko and Levendorski˘ i (UT ) Preemption games 03/09/12 3 / 49

Introduction

Our paper builds on the following

Existing models

• 1. Fudenberg and Tirole (1985) – preemption games in continuous time,

deterministic environment, symmetric firms

• 2. Dixit and Pindyck (1994) – preemption games in continuous time,

Gaussian uncertainty, symmetric firms

• 3. Thijssen et al. (2006) – preemption games in continuous time,

imperfect information, symmetric firms

• 4. Pawlina and Kort (2006) – preemption games in continuous time,

Gaussian uncertainty, asymmetric firms (sunk cost asymmetry)

• 5. Thijssen et al. (2002) – preemption games in continuous time,

non-Gaussian uncertainty, symmetric firms

Boyarchenko and Levendorski˘ i (UT ) Preemption games 03/09/12 4 / 49

Deterministic environment

Deterministic environment

Two firms consider a single investment opportunity to enhance their profit flow time t ≥ 0 is continuous; r > 0 – discount rate; πninj – instantaneous profit of firm i ∈ {1, 2}, where, for k ∈ {i, j}, nk =

• if firm k has not invested

1 if firm k has invested . Assume π10 > π00 ∨ ∨ π11 > π01 . ci(t) - the present value of the cost of investment at time t of firm i;

• ci(t)ert′ < 0;
• ci(t)ert′′ > 0 ∀ t.

Additional assumptions guarantee that investment is not optimal at t = 0; investment happens in finite time.

Boyarchenko and Levendorski˘ i (UT ) Preemption games 03/09/12 5 / 49

Deterministic environment

Let Ti be the time of investment of firm i

Value functions at time t = 0; Ti < Tj V i(Ti, Tj) = Ti π00e−rtdt + Tj

Ti

π10e−rtdt − ci(Ti) + ∞

Tj

π11e−rtdt V j(Ti, Tj) = Ti π00e−rtdt + Tj

Ti

π01e−rtdt + ∞

Tj

π11e−rtdt − cj(Tj) Value functions at time t = 0; Ti = Tj = Ts Pi(Ts) = Ts π00e−rtdt + ∞

Ts

π11e−rtdt − ci(Ts). If the firms precommit to investment times then optimal values Ti < Tj < Ts can be easily found from the FOC.

Boyarchenko and Levendorski˘ i (UT ) Preemption games 03/09/12 6 / 49

Deterministic environment

Value functions: t–the current state

V i

lead(t) – the value at entry of firm i if this firm is the leader

V i

f (t) – the value of firm i if this firm is the follower preempted at t

Pi(t) – the value at entry of firm i if the firms enter simultaneously V i

lead(t) =

• V i(t, Tj), if t ≤ Tj,

Pi(t), if t ≤ Tj. V i

f (t) =

• V i(t, Ti), if t ≤ Ti,

Pi(t), if t ≤ Ti.

Boyarchenko and Levendorski˘ i (UT ) Preemption games 03/09/12 7 / 49

Deterministic environment

Value functions – symmetric firms

Vi

lead(t)

Vi

f(t)

Pi(t) Tp Ts t Ti Tj

Boyarchenko and Levendorski˘ i (UT ) Preemption games 03/09/12 8 / 49

Deterministic environment

Value functions – asymmetric firms, c2(t) = kc1(t), k > 1

t V1

lead(t)

V1

f (t)

V2

lead(t)

V2

f (t)

t1 t2 t3 t4 Boyarchenko and Levendorski˘ i (UT ) Preemption games 03/09/12 9 / 49

Deterministic environment

Strategies and equilibrium concepts: Fudenberg and Tirole (1985)

A simple strategy

• f player i in the game starting at t is a pair of real valued functions

(Gi, αi) : [t, ∞) × [t, ∞) → [0, 1] × [0, 1], satisfying certain properties. For any t, Gi(t) is the cumulative probability that i invested by time t; αi(t) is the intensity with which the player moves right before Gi(t) jumps. Nash equilibrium is defined in a standard way. If simple strategies satisfy a certain intertemporal consistency condition, they are called closed-loop. A perfect equilibrium is a Nash equilibrium in closed-loop strategies.

Boyarchenko and Levendorski˘ i (UT ) Preemption games 03/09/12 10 / 49

Stochastic environment

Uncertainty

(Ω, F, (Ft)0≤t<∞, P) is a filtered probability space X = {Xt}t≥0 is a L´ evy process on R Can be defined by the generating triplet (σ2, µ, F(dx)), where σ2 is the variance of the Brownian motion (BM) component, µ ∈ R, and F(dx) is the L´ evy density. For (α, β) ⊂ R \ 0, F((α, β))dt is the probability of a jump from 0 into (α, β) during time interval dt. Alternatively, a L´ evy process can be uniquely defined by L´ evy exponent: E

• eβXt

= etΨ(β). If X is the BM, Ψ(β) = σ2 2 β2 + µβ.

Boyarchenko and Levendorski˘ i (UT ) Preemption games 03/09/12 11 / 49

Stochastic environment

L´ evy-Khintchine formula and example: DEJD

The L´ evy-Khintchine formula expresses the L´ evy exponent in terms of generating triplet. If the jump part is a finite variation process, then the L´ evy-Khintchine formula can be written in the form Ψ(β) = σ2 2 β2 + bβ +

• R\0

(eβy − 1)F(dy), (1) Double-exponential jump-diffusion (DEJD) model: σ > 0, F(dy) = c+λ+e−λ+y1(0,+∞)(y)dy + c−(−λ−)e−λ−y1(−∞,0)(y)dy, (2) where λ− < 0 < λ+ and c± ≥ 0 If c+ = 0, then there are no positive jumps (process is spectrally negative). Substituting (2) into (1): Ψ(β) = σ2 2 β2 + bβ + c+β λ+ − β + c−β λ− − β . (3)

Boyarchenko and Levendorski˘ i (UT ) Preemption games 03/09/12 12 / 49

Stochastic environment

Main objects

(i) τ +

h , the first entrance time of X into the semi-infinite interval [h, +∞);

(ii) the normalized expected present value (EPV) operator (Eqf )(x) = Ex +∞ qe−qtf (Xt)dt

• (iii) the supremum and infimum processes X t = sup0≤s≤t Xs and

X t = inf0≤s≤t Xs; and the EPV operators under supremum and infimum processes (E+

q f )(x) = Ex

+∞ qe−qtf (X t)dt

• ,

(E−

q f )(x) = Ex

+∞ qe−qtf (X t)dt

• ;

(iv) the notation κ±

q (β) =

• E±

q eβx

|x=0; (v) the operator version of the Wiener-Hopf factorization formula which states that Eq = E+

q E− q = E− q E+ q ,

as operators in spaces of measurable semi-bounded functions

Boyarchenko and Levendorski˘ i (UT ) Preemption games 03/09/12 13 / 49

Stochastic environment

Example

Brownian motion E+

q u(x) = β+

+∞ e−β+yu(x + y)dy, E−

q u(x) = −β− −∞

e−β−yu(x + y)dy, where β− < 0 < β+ are the roots of q − σ2

2 β2 − µβ = 0.

Double-exponential jump-diffusion model: E+

q u(x) =

• j=1,2

a+

j β+ j

+∞ e−β+

j yu(x + y)dy,

E−

q u(x) =

• j=1,2

a−

j (−β− j ) −∞

e−β−

j yu(x + y)dy,

where β−

2 < λ− < β− 1 < 0 < β+ 1 < λ+ < β+ 2 are the roots of q − Ψ(β) = 0,

and a±

j > 0 are constants.

Boyarchenko and Levendorski˘ i (UT ) Preemption games 03/09/12 14 / 49

Stochastic environment

Standing assumptions

Assumption 1. No-bubble condition E ∞ e−qteXtdt

• < ∞

⇔ q − Ψ(1) > 0 Assumption 2. X is a L´ evy process satisfying (ACP)-property, with non-trivial supremum and infimum processes. Assumption 3. Monotonicity of density of negative jumps and complete monotonicity of density of positive jumps (Bernstein condition) (ACP)-property: For any f ∈ L∞(R), Eqf is continuous. A sufficient condition is: for some t > 0, the transition measure Pt is absolutely continuous.

Boyarchenko and Levendorski˘ i (UT ) Preemption games 03/09/12 15 / 49

Stochastic environment

Economic environment

Two firms consider an irreversible investment project q > 0 – discount rate I1 = I - sunk cost of firm 1 I2 = kI - sunk cost of firm 2; k ≥ 1 f 1

i (Xt) = eXtD(1) − qIi - profit flow if firm i is a single producer

f 2

i (Xt) = eXtD(2) − qIi - profit flow if both firms produce;

D(2) < D(1) Value functions: x–the current state V i

lead(x) – the value at entry of firm i if this firm is the leader

V i

f (x) – the value of firm i if this firm is the follower

Pi(x) – the value at entry of firm i if the firms enter simultaneously

Boyarchenko and Levendorski˘ i (UT ) Preemption games 03/09/12 16 / 49

Stochastic environment

Simple strategies

We use the definitions from Thijssen et al. (2002, 2006) Definition 1 A simple strategy for player i ∈ {1, 2} in the subgame starting at t0 ∈ [0, ∞) is given by a pair of real valued functions (G t0

i , αt0 i ) : [t0, ∞) × Ω → [0, 1] × [0, 1] such that

(i) G t0

i (·, ω) is non-decreasing, right continuous with left limits;

(ii) the process G t0

i (t, ·) is adapted to the filtration (Ft)t0≤t<∞;

(iii) αt0

i (·, ω) is right continuous with left limits.

G t0

i (t, ω) – the probability that firm i has invested no later than at t

αt0

i (t, ω) – the “intensity” of atoms in the interval (t, t + dt).

Boyarchenko and Levendorski˘ i (UT ) Preemption games 03/09/12 17 / 49

Stochastic environment

Role of “intensities”

Consider a subgame starting at t such that Xt = x, and V j

lead(x) > V j f (x),

j = 1, 2. Payoff matrix during [t, t + ∆t] is Firm 2 Firm 1 Invest Wait Invest (P1(x), P2(x))

• V 1

lead(x), V 2 f (x)

• Wait
• V 1

f (x), V 2 lead(x)

• repeat game

Player j plays “Invest” with probability αt

j (t, ω) and “Wait” with

probability 1 − αt

j (t, ω).

Boyarchenko and Levendorski˘ i (UT ) Preemption games 03/09/12 18 / 49

Stochastic environment

Nash equilibrium

Let αt0

i (t−, ω) = lims↑t αt0 i (s, ω) and G t0 i (t−, ω) = lims↑t G t0 i (s, ω).

Definition 2 A pair of simple strategies (G t0

i , αt0 i ) (i = 1, 2) for the subgame starting at

t0 ∈ [0, ∞) is α-consistent, if for all ω ∈ Ω and i, j = 1, 2, i = j, αt0

i (t, ω)

− αt0

i (t−, ω) = 0 ⇒ G t0 i (t, ω) − G t0 i (t−, ω) =

=

• 1 − G t0

i (t−, ω)

• αt0

i (t, ω)

αt0

i (t, ω) + αt0 j (t, ω) − αt0 i (t, ω)αt0 j (t, ω).

Definition 3 A pair of simple strategies (G t0

i , αt0 i ) (i = 1, 2) is a Nash equilibrium for the

subgame starting at t0 ∈ [0, ∞) if the strategies are α-consistent, and each player’s strategy maximizes his/her value function, V i(X0), given the other player’s strategy.

Boyarchenko and Levendorski˘ i (UT ) Preemption games 03/09/12 19 / 49

Stochastic environment

Subgame perfect Nash equilibrium

Definition 4 A closed loop strategy for player i ∈ {1, 2} is a collection of simple strategies {(G t

i (·, ω), αt i (·, ω))}t≥0,ω∈Ω that satisfies the following

intertemporal consistency conditions: (i) ∀ ω ∈ Ω, ∀ 0 ≤ t ≤ u ≤ v < ∞ : v = inf{τ > t| Xτ(ω) = Xv(ω)} ⇒ G t

i (v, ω) = G u i (v, ω);

(ii) ∀ ω ∈ Ω, ∀ 0 ≤ t ≤ u ≤ v < ∞ : v = inf{τ > t| Xτ(ω) = Xv(ω)} ⇒ αt

i (v, ω) = αu i (v, ω).

Boyarchenko and Levendorski˘ i (UT ) Preemption games 03/09/12 20 / 49

Stochastic environment

Subgame perfect Nash equilibrium

Definition 5 A pair of closed loop strategies is a subgame perfect equilibrium if for every t ∈ [0, ∞) and ω ∈ Ω, the corresponding pair of simple strategies (G t

i (·, ω), αt i (·, ω)))i=1,2 is a Nash equilibrium.

From now on, we let ω ∈ Ω be fixed. For notational convenience, we drop ω as an argument in players’ strategies.

Boyarchenko and Levendorski˘ i (UT ) Preemption games 03/09/12 21 / 49

Stochastic environment

Summary of existing results: deterministic environment or Gaussian uncertainty

Symmetric firms There is a unique symmetric equilibrium in the preemption zone Each of the players becomes the leader or the follower with positive probability Probability of simultaneous action (coordination failure) is positive if the game starts in the preemption zone If immediate investment is not optimal, then the game starts and ends at the left boundary of the preemption zone, and the probability of coordination failure is zero Asymmetric firms Asymmetry uniquely determines the roles of the firms Low cost firm moves first, and high cost firm follows Coordination problem is irrelevant Two types of equilibria: sequential and preemptive

Boyarchenko and Levendorski˘ i (UT ) Preemption games 03/09/12 22 / 49

Preemption under L´ evy uncertainty

Our setting

In a stochastic version of Fudenberg and Tirole’s (1985) preemption game, two firms contemplate entering a new market where the demand follows a jump-diffusion process. Firms differ is the sunk costs of entry. In the initial state, entry is

• ptimal to none of the firms.

Boyarchenko and Levendorski˘ i (UT ) Preemption games 03/09/12 23 / 49

Preemption under L´ evy uncertainty

Main results

If there are no upward jumps in the stochastic demand, then the low cost firm is the leader, and the high cost firm is the follower:

if the cost disadvantage is sufficiently large, then a sequential equilibrium is played, where the low cost firm chooses the optimal entry threshold as a monopolist; if the cost disadvantage is sufficiently small, then a preemptive equilibrium

• ccurs, where the low cost firm has to enter earlier than the optimal entry

threshold of the monopolist is reached.

If the demand process admits positive jumps, then

simultaneous entry can happen as an equilibrium, or as a coordination failure with positive probability; sequential equilibrium may disappear; the high cost firm may be the first to enter.

Strategies in subgame perfect equilibria are described in terms of stopping times and value functions. Analytical expressions for the value functions and thresholds that define stopping times are derived.

Boyarchenko and Levendorski˘ i (UT ) Preemption games 03/09/12 24 / 49

Preemption under L´ evy uncertainty

Follower’s problem

Assuming that firm j has entered, firm i solves the optimization problem V i

f (x) = sup τ∈M0

Ex +∞

τ

e−qtf 2

i (Xt)dt

• ,

(4) where f 2

i (x) = D(2)ex − qIi, and M0 is the set of stopping times s.t.

τ < +∞, a.s.

Boyarchenko and Levendorski˘ i (UT ) Preemption games 03/09/12 25 / 49

Preemption under L´ evy uncertainty

Follower’s value and entry threshold

V i

f (x) =

• q−1E+

q 1[hi

f ,+∞)E−

q f 2 i (x),

if x < hi

f ,

q−1(Eqf 2

i )(x),

if x ≥ hi

f ,

where the optimal entry threshold is given by Hi

f = ehi

f =

qIi κ−

q (1)D(2).

Since I2 = kI1, we have h2

f > h1 f .

If X is DEJD, then V i

f (x)

= Ii  1(−∞,hi

f )(x)

• j=1,2

a+

j (q)eβ+

j (x−hi f )

β+

j − 1

+ 1[hi

f ,+∞)(x)

• κ+

q (1)ex−hi

f − 1

• .

Boyarchenko and Levendorski˘ i (UT ) Preemption games 03/09/12 26 / 49

Preemption under L´ evy uncertainty

Simultaneous entry

Pi(x) = q−1Eqf 2

i (x) = D(2)ex

q − Ψ(1) − Ii Pi(x) = V i

f (x), for x ≥ hi f

Pi(x) < V i

f (x), for x < hi f

Being the follower is better than simultaneous investment.

Boyarchenko and Levendorski˘ i (UT ) Preemption games 03/09/12 27 / 49

Preemption under L´ evy uncertainty

Leader’s value at entry

If x ≥ hj

f ,

V i

lead(x) = V i f (x) = Pi(x).

If x < hj

f ,

V i

lead(x)

= Ex τ j

f

e−qtf 1

i (Xt)dt +

∞

τ j

f

e−qtf 2

i (Xt)dt

• =

Ex ∞ e−qtf 1

i (Xt)dt

• − Ex

∞

τ j

f

e−qt(f 1

i (Xt) − f 2 i (Xt))dt

• =

Ex ∞ e−qtf 2

i (Xt)dt

• + Ex

τ j

f

e−qt(f 1

i (Xt) − f 2 i (Xt))dt

• V i

lead(x) > Pi(x) – being the leader is better than simultaneous

investment.

Boyarchenko and Levendorski˘ i (UT ) Preemption games 03/09/12 28 / 49

Preemption under L´ evy uncertainty

Leader’s value at entry, cont’d

If x < hj

f ,

V i

lead(x) = V i m(x) − V i loss(x),

where V i

loss(x) = (D(1) − D(2)) q−1E+ q 1[hj

f ,+∞)E−

q ex.

In the case of DEJD model, V i

loss(x) =

D(1) D(2) − 1

• Ij
• k=1,2

a+

k β+ k eβ+

k (x−hj f )

β+

k − 1

.

Boyarchenko and Levendorski˘ i (UT ) Preemption games 03/09/12 29 / 49

Preemption under L´ evy uncertainty

Simultaneous entry zone (SEZ)

Thus, we have for x < min{hi

f , hj f } = h1 f ,

Pi(x) < min

• V i

lead(x), V i f (x)

• , i = 1, 2,

and for x ≥ max{hi

f , hj f } = h2 f ,

Pi(x) = V i

lead(x) = V i f (x), i = 1, 2.

Hence simultaneous entry can be optimal only if x ≥ h2

f .

We call the semi-infinite interval SEZ= [h2

f , +∞) the simultaneous entry zone.

Let τsez denote the first entrance time into the interval SEZ.

Boyarchenko and Levendorski˘ i (UT ) Preemption games 03/09/12 30 / 49

Preemption under L´ evy uncertainty

Preemption zone (PZ)

For x ∈ PZ, V i

lead(x) > V i f (x), i = 1, 2.

For big k, PZ= ∅. If PZ= ∅, then PZ= (xL(k), xH(k)), where xL, xH solve V 2

lead(x) = V 2 f (x).

τpz – the first entrance time into the interval PZ. Simplifying assumption: the equilibrium of the same type will be played in each point of the preemption zone.

Boyarchenko and Levendorski˘ i (UT ) Preemption games 03/09/12 31 / 49

Preemption under L´ evy uncertainty

Leader-follower game

Y = ex – current demand shock 0.04 0.08 0.12 −60 −20 20 60 Y (a) Values of firm 1; k=1.5 0.04 0.08 0.12 −60 −20 20 60 Y (b) Values of firm 2; k=1.5 0.04 0.08 −40 40 Y (c) Values of firm 1; k=1.1 0.04 0.08 −60 −20 20 Y (d) Values of firm 2; k=1.1 V1

f

V1

lead

P1 V1

f

V1

lead

P1 V2

f

V2

lead

P2 V2

f

V2

lead

P2

Boyarchenko and Levendorski˘ i (UT ) Preemption games 03/09/12 32 / 49

Preemption under L´ evy uncertainty

Subgame perfect equilibria in the preemption zone

Theorem 6 Consider a subgame starting at t ≥ τpz. Then there are 3 SPE given by the following pairs of closed loop strategies. (1) Gt

1(s) = 1, αt 1(s) = 1 for all s ≥ t;

Gt

2(s) =

     if t ≤ s < τ+

h2 f

, 1 if s ≥ τ+

h2 f

, αt

2(s) =

     if t ≤ s < τ+

h2 f

1 if s ≥ τ+

h2 f

. (2) Gt

2(s) = 1, αt 2(s) = 1 for all s ≥ t;

Gt

1(s) =

     if t ≤ s < τ+

h1 f

1 if s ≥ τ+

h1 f

, αt

1(s) =

     if t ≤ s < τ+

h1 f

1 if s ≥ τ+

h1 f

. (3) Gt

i (s) =

      

αt i (t) αt i (t)+αt j (t)−αt i (t)αt j (t)

if t ≤ s < τ+

hi f

1 if s ≥ τ+

hi f

, αt

i (s) =

      

V j lead(Xt )−V j f (Xt ) V j lead(Xt )−Pj (Xt )

if t ≤ s < τ+

hi f

1 if s ≥ τ+

hi f

. In equilibrium (1), the firms’ expected values are V 1(Xt) = V 1

lead(Xt), V 2(Xt) = V 2 f (Xt). In equilibrium (2), the firms’

expected values are V 1(Xt) = V 1

f (Xt), V 2(Xt) = V 2 lead(Xt). In equilibrium (3), the firms’ expected values are

V 1(Xt) = V 1

f (Xt), V 2(Xt) = V 2 f (Xt).

Boyarchenko and Levendorski˘ i (UT ) Preemption games 03/09/12 33 / 49

Preemption under L´ evy uncertainty

Leader’s problem

Let x < xL such that V i

f (x) > V i lead(x). Suppose first that firm 2 finds it

non-optimal to enter (or precommits not to enter) at any level x < h2

f .

Firm 1 solves V 1(x) = sup

τ∈M0

Ex e−qτV 1

lead(Xτ)

• .

Theorem 7 Let F(dy) be non-decreasing on (−∞, 0); define ehl = qI κ−

q (1)D(1).

(5) Then τ +

hl is the optimal entrance time of firm 1 in the class M0.

Boyarchenko and Levendorski˘ i (UT ) Preemption games 03/09/12 34 / 49

Preemption under L´ evy uncertainty

Leader’s value

V 1(x) = E+

q 1[hl,+∞)

• E+

q

−1 V 1

lead(x)

= E+

q 1[hl,+∞)

• E+

q

−1 V 1

m(x) − V 1 loss(x).

In the case of DEJD model, ehl = qIiβ−

1 β− 2 (λ+ − 1)

λ+(β−

1 − 1)(β− 2 − 1)D(1),

V 1(x) = I

• j=1,2

a+

j eβ+

j (x−hl)

β+

j − 1

− V 1

loss(x),

x ≤ hl

Boyarchenko and Levendorski˘ i (UT ) Preemption games 03/09/12 35 / 49

Preemption under L´ evy uncertainty

Leader’s problem, cont’d

Now consider the case when PZ is not empty and precommitment of firm 2 is impossible. Firm 1 solves V 1(x) = sup

τ≤τ +

xL

Ex e−qτV 1

lead(Xτ)

• .

Theorem 8 Let F(dy) be non-decreasing on (−∞, 0); let k < k∗, and, if the density of positive jumps is non-trivial, then let the equilibria in repeated games at points x ∈ (xL, xH) be of type (1). Then the optimal entrance time of firm 1 is τ +

min{hl,xL} .

Boyarchenko and Levendorski˘ i (UT ) Preemption games 03/09/12 36 / 49

Preemption under L´ evy uncertainty

Sequential and preemptive equilibria

If hl ≤ xL = xL(k), the low cost firm chooses the optimal entry threshold as if it were a monopolist, firm 2 enters at τ +

h2

f , hence

sequential equilibrium happens. If xL < hl, firm 1 enters at τ +

xL and the value function is

V 1(x) = E+

q 1[xL,+∞)

• E+

q

−1 V 1

lead(x)

= E+

q 1[xL,+∞)

• E+

q

−1 V 1

m(x) − V 1 loss(x).

Firm 2 enters at τ +

h2

f , hence preemptive equilibrium happens. Boyarchenko and Levendorski˘ i (UT ) Preemption games 03/09/12 37 / 49

Preemption under L´ evy uncertainty

Sequential and preemptive equilibria

XL = exL, XH = exH 0.1 20 40 60 (a) Sequential equilibrium 0.06 10 20 (b) Preemptive equilibrium V1

f

V1

lead

V1

precom

V1 V2

f

V2

lead

V1

f

V1

lead

V1 V2

f

V2

lead

XL Hl Hl XL XH

Boyarchenko and Levendorski˘ i (UT ) Preemption games 03/09/12 38 / 49

Preemption under L´ evy uncertainty

Leader’s problem, cont’d

Theorem 9 Let the density of positive jumps be non-trivial, let k < k∗, and let the equilibria in repeated games at points x ∈ (xL, xH) be of types (2) or (3). Then (a) equation (E+

q )−1V (h) = 0 has a unique solution, denote it

hlp = hlp(k); (b) the optimal entrance time of firm 1 is τ +

min{hlp,xL};

(c) for k ∈ (1, k∗), hlp(k) < hl; (d) hlp(k) increases in k on the interval, where hlp(k) < xL(k).

Boyarchenko and Levendorski˘ i (UT ) Preemption games 03/09/12 39 / 49

Preemption under L´ evy uncertainty

Firms’ value functions

If hlp(k) ≤ xL(k), V 1(x) = q−1E+1[hlp,+∞)(E+)−1V (x). If hlp(k) > xL(k), V 1(x) = q−1E+1[xL,+∞)(E+)−1V (x). Since hlp depends on k, and hlp(k) < hl, the sequential equilibrium, where firm 1 chooses the optimal entry threshold of the monopolist, no longer exists. If type (3) equilibrium is played in PZ, then V 2(x) = V 2

f (x).

If type (2) equilibrium is played in PZ, V 2(x) = V 2

f (x) + 1(xL,xH)(V 2 lead − V 2 f )(x).

Boyarchenko and Levendorski˘ i (UT ) Preemption games 03/09/12 40 / 49

Preemption under L´ evy uncertainty

Low cost entry zone (LCZ)

In LCZ, it is optimal to enter for low cost firm, but not for high cost firm. If PZ is empty, LCZ=[hl, h2

f ).

If PZ is not empty, and either demand process is spectrally negative

• r equilibrium of type (1) is played in PZ,

LCZ = [hl, h2

f ) \ (xL, xH) = [hl, xL] ∪ [xH, h2 f ).

If demand process admits positive jumps and type (2) or (3) equilibria are played in PZ, LCZ = [hlp, xL] ∪ [xH, h2

f ).

τlcz – the first entrance time into the interval LCZ.

Boyarchenko and Levendorski˘ i (UT ) Preemption games 03/09/12 41 / 49

Subgame perfect equilibria

Demand shocks with non-trivial positive jump component

Preemption zone is empty LCZ=[hl, h2

f ), where hl is the optimal entry threshold of the

monopolist. If τlcz < τsez, then sequential equilibrium happens: firm 1 enters at τlcz, and firm 2 follows at τsez. If τlcz > τsez, then simultaneous investment happens at τsez.

Boyarchenko and Levendorski˘ i (UT ) Preemption games 03/09/12 42 / 49

Subgame perfect equilibria

SPE when initial shocks are low: demand shocks with non-trivial positive jump component

Preemption zone is non-empty If type (1) equilibrium is played in PZ, then LCZ=[hl, h2

f ) \ (xL, xH).

If τsez < min{τlcz, τpz}, then simultaneous investment happens at τsez. If τlcz < min{τsez, τpz}, then sequential equilibrium happens: firm 1 enters at τlcz, and firm 2 follows at τsez. If τpz < min{τsez, τlcz} then firm 1 enters at τpz, and firm 2 follows at τsez.

Boyarchenko and Levendorski˘ i (UT ) Preemption games 03/09/12 43 / 49

Subgame perfect equilibria

Demand shocks with non-trivial positive jump component

Type (2) equilibrium in PZ LCZ=[hlp, h2

f ) \ (xL, xH), where hlp < hl.

If τsez < min{τlcz, τpz}, then simultaneous investment happens at τsez. If τlcz < min{τsez, τpz}, then firm 1 enters at τlcz, and firm 2 follows at τsez. If τpz < min{τsez, τlcz} then firm 2 enters at τpz, and firm 1 follows at τ +

h1

f . Boyarchenko and Levendorski˘ i (UT ) Preemption games 03/09/12 44 / 49

Subgame perfect equilibria

Demand shocks with non-trivial positive jump component

Type (3) equilibrium in PZ LCZ=[hlp, h2

f ) \ (xL, xH), where hlp < hl.

If τsez < min{τlcz, τpz}, then simultaneous investment happens at τsez. If τlcz < min{τsez, τpz}, then firm 1 enters at τlcz, and firm 2 follows at τsez. If τpz < min{τsez, τlcz} then either firm i enters at τpz with positive probability, and firm j follows at τ +

hj

f

; or firms enter simultaneously at τpz with positive probability.

Boyarchenko and Levendorski˘ i (UT ) Preemption games 03/09/12 45 / 49

Conclusion

Conclusion

In a preemption game under L´ evy uncertainty, we studied effects of jumps in demand and firms’ asymmetry on equilibrium strategies. Without positive jumps, equilibrium strategies and types of equilibria are standard. With positive jumps, equilibrium where low cost firm does not preempt may disappear. Simultaneous entry can happen either as an equilibrium or a coordination failure. High cost firm can become the leader. Assuming that the same type of equilibrium is played at each point of the preemption zone, we characterized strategies in subgame perfect equilibria in terms of stopping times and value functions. The model can be used to calculate the probabilities of potentially

• bservable outcomes of the preemption game as well as the expected

waiting time until such outcomes may be observed, given the current state of the demand.

Boyarchenko and Levendorski˘ i (UT ) Preemption games 03/09/12 46 / 49

Conclusion

Stochastic expressions

For a stopping time τ and functions f , g, satisfying Assumption 1, define Vex(τ; f ; x) = E τ−0 e−qtf (x + Xt)dt

• ,

Ven(τ; f ; x) = E +∞

τ

e−qtf (x + Xt)dt

• ,

Vinst(τ; g; x) = Ex[e−qτg(Xτ)]. For h ∈ R, denote by τ +

h the first entrance time into [h, +∞).

Theorem 10 Vex(τ +

h ; f ; x)

= q−1E+

q 1(−∞,h)E− q f (x),

Ven(τ +

h ; f ; x)

= q−1E+

q 1[h,+∞)E− q f (x),

Vinst(τ +

h ; g; x)

= E+

q 1[h,+∞)(E+ q )−1g(x).

Boyarchenko and Levendorski˘ i (UT ) Preemption games 03/09/12 47 / 49

Conclusion

Optimal entry threshold

Set w(x) = E−

q f (x).

Since E+

q is a positive operator, Ven(τ + h ; f ; x)) is maximized only if

1[h,+∞)w(x) is maximized. Hence optimal h satisfies w(h) = 0.

Boyarchenko and Levendorski˘ i (UT ) Preemption games 03/09/12 48 / 49

Conclusion

Optimal stopping for entry problems

For a stopping time τ and a measurable f , define Ven(τ; f ; x) = q−1E[1τ≤Tqf (x + XTq)] = E +∞

τ

e−qtf (x + Xt)dt

• .

Vwait.en(τ; f ; x) = −q−1E[1Tq<τf (x + XTq)] = −E τ−0 e−qtf (x + Xt)dt

• .

Theorem 11 Let there exist h ∈ R such that E−

q f (x) ≤ 0,

x ≤ h, and E−

q f (x) ≥ 0,

x ≥ h. Then (a) τ +

h maximizes Ven(τ; f ; x) in the class of stopping times of the threshold type.

(b) If, in addition, h−x

−∞

Vwait.en(τ +

h ; f ; x + y)F(dy) ≤ f (x),

x > h, a.e., then τ +

h maximizes Vex(τ; f ; x) in the class of all stopping times. Boyarchenko and Levendorski˘ i (UT ) Preemption games 03/09/12 49 / 49