- Email: [email protected]
Equilibrium exit in stochastically declining industries
GAMES
AND
ECONOMIC
BEHAVIOR
1,
40-59 (1989)
Equilibrium Exit in Stochastically Declining Industries CHARLES H. FINEANDLODE Sloan School of Management, Cambridge,
LI
Massachusetts Institute of Technology, Massachusetts 02139
We study a complete information model of exit for an industry in stochastic decline. We characterize the entire set of pure strategy subgame-perfect equilibria for the exit game. In general, there are multiple subgame-perfect equilibria of our exit game, in contrast to several papers in the literature. Relaxing an assumption in those models (arguably in the direction of more realism) will give multiple equilibria in those models also. We provide (fairly restrictive) necessary and sufficient conditions for there to be a unique subgame-perfect equilibrium. o 1989 Academic
Press, Inc.
1. INTRODUCTION The decline of products and industries is a natural by-product of the growth and evolution of industrial economies. Economic decline puts pressure on firms’ profits and on their ability to remain in business. Most firms eventually depart if the decline is severe. However, those who manage to stay in after a shakeout can often earn significant profits. In this paper, we analyze firms’ decisions to exit an industry that is suffering from economic decline. In contrast to the literature on entry, the study of exit decisions by firms in oligopolistic industries is not extensive. Ghemawat and Nalebuff (1985) analyze a continuous-time, perfect information model for an asymmetric duopoly in a deterministically declining industry, where the firms differ by their capacities and related fixed costs per period. Unless the high-fixed-cost (large) firm has significantly lower operating costs than the low-fixed-cost (small) firm, the unique subgameperfect Nash equilibrium in their model is for the firm with the larger capacity and fixed costs to exit at the first time that its duopoly profits are nonpositive. The smaller firm then remains until its monopoly profits 40 0899-8256/89$3.00 Copyright 6 1989 by Academic Press, Inc. All rights of reproduction in any form reserved.
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become negative. (Fudenberg et al. (1983) use a similar approach to derive equilibrium strategies in the context of a patent race.) Fudenberg and Tirole (1986) also analyze an asymmetric duopoly exit game in a continuous-time model with deterministic demand paths. However, they assume that each firm does not know the fixed operating cost per period of its rival. Thus their model is a war of attrition where each firm continually revises downward its estimate of the other’s costs, as long as the contest for the market lasts. With an assumption that each firm assesses positive probability that its rival will never find it optimal to exit, the model yields a unique subgame-perfect Nash equilibrium where the higher cost firm exits the market before the firm with lower costs if cost distributions are symmetric. We also model an industry that begins as a duopoly. Like Ghemawat and Nalebuff (1985) and Section 5 of Fudenberg and Tirole (1986), we assume that the industry declines over time so that the profits of the two firms shrink until one of them exits, leaving a monopoly for the other. In contrast to the results of Ghemawat and Nalebuff and Fudenberg and Tirole (each of whom obtains a unique subgame-perfect equilibrium in a continuous-time, deterministic demand model), there may be multiple exit-time equilibria in our stochastic, discrete-time framework. As a consequence of the multiplicity of the equilibria, the stronger firm, which has a stage-payoff advantage, may not always exit last. The multiplicity of equilibria in our model is a consequence of the fact that the industry demand process can jump from a point where both firms are viable as duopolists to a point where neither firm is viable as a duopolist but each is viable as a monopolist. At such a point in time, exit by either could constitute an equilibrium in stopping times. Both Ghemawat and Nalebuff and Fudenberg and Tirole avoid this problem by assuming that demand falls continuously (and deterministically) over time. However, their models would also exhibit multiple equilibria as ours does if their demand process were allowed to have jumps. Thus, an assumption in the direction of more realism (a demand process with discrete jumps) may destroy the equilibrium uniqueness in each of their models. (We will elaborate more on the issue in Section 3.) Jumps in the demand process drive the multiple equilibria result, independent of whether the demand process is stochastic or deterministic. Our model uses a stochastic demand process because we think it is descriptively more accurate of industrial decline and because the random events that influence demand help motivate why the process should have jumps. We characterize the generic form of any equilibrium in our model. These equilibria are obtained by solving a time-indexed sequence of fixedpoint problems. Among the equilibria, two are particularly interesting because they provide the upper and lower bounds for any equilibrium exit
42
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LI
times and because each player prefers one of the two equilibria to all other equilibria. In particular, each firm favors the equilibrium which makes his active period the longest. We provide (fairly restrictive) necessary and sufficient conditions for the “natural” equilibrium (in which the stronger firm always outlasts its rival) to be the unique subgame-perfect equilibrium. The conditions require either that there is a significant asymmetry in payoffs between the strong and the weak firm or that the sizes of the demand process jumps are small. To model the exit game, we adopt a natural multiperson extension of the optimal stopping-time concept (see, e.g., Shiryayev, 1978) which is the concept of stopping-time equilibrium, a Nash equilibrium in stoppingtime strategies. The game-theoretic extension of optimal stopping theory was initiated by Dynkin (1969) and followed by Chaput (1974) in an analysis of a class of two-person zero-sum stopping games. The recent modifications and extensions of Dynkin’s work include Mamer (1986) and Ohtsubo (1986, 1987). Ohtsubo (1987) obtains value relations that characterize equilibria in non-zero-sum stopping games and provides an existence theorem for the Markov case, of which our model is an example. Our contribution to this literature is to look at the semiextensive form (instead of the strategic form) of stopping games, characterize explicitly the complete set of subgame-perfect equilibria, and provide a uniqueness theorem for one class of stopping games. In general, any stochastic dynamic game in which each player’s strategy is a single dichotomous decision at each stage can be formulated as a stopping-time problem. Further, in such games, the notion of equilibrium stopping time can be applied. The stopping-time equilibrium concept of Ghemawat and Nalebuff or Fudenberg and Tirole is analogous to ours, except that in their models the single-firm optimal stopping problem is of little intrinsic interest because their models have deterministic demand paths. One characteristic of the firms’ optimal policies in our model is that one or both firms may earn negative profits in some period(s), but remain in the market. This can occur for either of two reasons. First, industry demand can suffer a large stochastic negative shock, such that demand will increase in the next period, in expectation. In such cases, firms will absorb the loss in that period, knowing that demand is likely to recover. This result accords with observed practice. Most industries experience stochastic downturns that do not lead to mass exit from the industry. A firm may also choose to absorb losses and remain in the market in the expectation that its rival will exit the industry first, leaving a more profitable, monopolistic industry for the remaining firm. Although the former effect is unique to our model, the latter effect is also present in the Fudenberg-Tirole formulation.
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For the single-firm model, we obtain comparative static results that we illustrate in an example with a linear demand function and linear costs. For the single-firm example, the optimal exit time is decreasing in the firm’s unit costs, the slope of the demand curve, and the per unit fixed cost of being in the market. The optimal exit time increases in the firm’s discount factor. In a two-firm example with Cournot competition, we calculate two equilibria to illustrate the multiplicity. The remainder of the paper is organized as follows: In Section 2, the single-firm exit problem is formulated as a stopping-time problem and several basic results are established. In Section 3 the main results regarding equilibrium exit behavior in a duopoly are proved and discussed. Concluding remarks follow in Section 4.
2.
THE SINGLE-FIRM
MODEL
In this section, we formulate the dynamic programming model for the single-firm problem and provide some comparative statics on the optimal stopping rule. We assume that at time zero, a single firm is in the market earning nonnegative profits. At any later point in time, if the firm concludes, after observing demand, that remaining in the market is no longer profitable, it may exit the market. Reentry is assumed to be prohibitively costly. We let industry demand in period t be indexed by X,. Firm profits are denoted by a reward function r(X,) where n(e) is an increasing, realvalued function. Industry demand {X, ; t = 1,2, . . .} is a Markov process defined by a family of transition functions, p(x, t; A, s), x E SR,A E '%A('%), and s > t, s, t E SIT+ = (0, 1, 2, . . .}, where a(%) is the collection of Bore1 sets on a real line. For each period t > 0 and x E an, denote the probability measure of X, conditional on X,-i = x by P:(X,) = P,{X,jX,-i = x}. To model stochastic decline of the industry, we assume that for all t 2 0, P: dominates Pf+l in the sense of first-order stochastic dominance (Hadar and Russell, 1969). Formally, for any t 2 0, x, y E 3, sZZ, dPf(.z) 5: J,+ dPf+,(z). We denote this ordering by P: 2 Pf+l. At each time t, the firm observes {X, ; s 5 t} and then decides whether to remain in the market and get n(X,) in period t or to exit and get zero payoffs thereafter. We assume that for each t, the likelihood of having a high future state is increasing in the current state X,. Formally, for x L x’, p:+i stochastically dominates pfi:-;1 for all t 1 0. That is, the better the industry is currently doing, the better is its future prospects. A firm’s strategy is a random time T: Cl* N+, where R is the space of sample paths of {X, ; t B 0). We require T to be the stopping time. That is, T:IR-,X+issuchthat{Tst}~4,foreverytEX+,where$,r(~(X,; s 5 t), the u-field generated by {X,; s 5 t). Let 6 be the firm’s discount
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factor. The firm’s problem is to choose a stopping time T so as to maximize the expected discounted reward
(2.1)
(By convention, we assume that this term is identically zero if T = 0.) This isa standard optimal stopping problem (see, e.g., Shiryayev, 1978). Define t* = inf{t: n(X,) < 0, a.s.}, the first time that stage payoffs are negative almost surely. Note that the stochastic dominance assumption implies that rr(X,) < 0, a.s. for s 2 t*. We assume that 0 < t* < ~0; i.e., the industry becomes nonprofitable to the firm in finite time. Proposition 2.1 below characterizes the optimal stopping times for this problem. The proposition is preceded by a technical lemma that orders the expectations of the positive parts of monotone ordered functions of random variables that are ordered by stochastic dominance. LEMMA 2.1. Suppose that fi and fi are increasing functions and h(z) >fi(z), for z E CR.And suppose that Z, and Zz are random variables with probability measures ~1 and ~2, respectively. Moreover, ~1 stochastically dominates ~2. Then E[fi(Z,)+] 2 E[fz(Zz)+], where
EUXZ;>+l= EUX-5)1co,c&X&)>l, l,(*) is an indicator
function
for
i = 1, 2,
set S, and strict inequality
holds if
~~~~,,cdfd&))l> 0. Proof.
Appendix
A.
n
PROPOSITION 2.1. (i) The optimal stopping time for the problem described in (2.1) can be characterized by T = inf{ t 2 0: X, E B,}, where B, = {x: u,(x) < O}, u,(x) = 77(x) + 6 w+dx), and w+dx> = E[ul+dXl+J+lXl = x] with w,(a) = 0 for t L t*. (ii) For t 2 0 andx E CR,w,(x) 2 w,+ I(x) and u,(x) 2 u,+,(x). Let y, = inf{x E CR: u,(x) 2 O}. Then yr is increasing in t, B, = {x E ‘3: x < y,}, and B, C Br+, for t z 0. Proof. Part (i) follows from the optimality principle of dynamic programming (cf. Shiryayev, 1978, Chap. 2). To prove (ii), note that wt = 0, for t 2 t*. By induction, if w[+,(.) is increasing, then u,(e) = r(.) + 6w,+,(*) is increasing since rr is increasing. Hence wI(.) = E[u,(X,)+I*] is increasing by Lemma 2.1. To show that wr and uI are monotone as functions of t, note that w,*-,(x)
= E[v(X,*-,)+~X,~-~
= x] 2 0 = w,.(x).
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Again, by induction, if w~+~- w~+~L 0 for 0 5 t I t* - 2, then u,(x) ut+r(x) = 6(w,+r(x) - We+*) 1 0. Furthermore, Lemma 2.1 and the assumption of stochastic dominance imply w,(x) 2 w,+,(x) for any x E 93 since u,(a) is increasing. The facts that $I&can be written as {x: x < yr} with y, = inf{x: U,(X) r 0) and that B, C_B,+, for t 2 0 follow from the monon tonicity of u,(x) in t and x. Note that u,(X,) is just the expected discounted payoff from time t if the firm is in the industry at t, decides to stay for one more period, and employs the optimal stopping time from t + 1 on. Therefore, the optimal policy is to stop the first time that X, falls so low that u,(X,) < 0 or, equivalently, X, < yt. Figure 1 illustrates a sample path of X, the cutoff point y, as a function of t, and the optimal stopping time T. The dashed horizontal line represents the zero stage-payoff line, y” = inf{z: r(z) 2 0). The cutoff points yt always lie below this line. That is, at each time t, the firm is willing to sustain a certain level of current loss in the hope of receiving future positive rewards. The cutoff point, y,, increases in t because total future industry profit potential declines over time. The amount of current period losses (?r(yJ when y, < r-‘(O)) that the firm is willing to sustain (in the hope of recovering those losses with future gains) decreases over time because the total potential future gain decreases over time. PROPOSITION 2.2. Suppose r’(x) e n2(x)for x E 9’i or 6’ r a*. Define for i = 1, 2 and t 5 0: wf , ui, yf , and T’ as in Proposition 2.1. Then w: L w:, ui 2 uf, y: s yf for t L 0, and T’ 2 T* as.
Proof. Let t’ = inf{t: 7ri(XI) < 0, a.s.}, i = 1, 2. Obviously, t’ 2 t* and w: = w: = 0 for t B t’. Suppose that w:+, 2 w:+‘. Then U:(X) = n’(x) + &w:+‘(x) 2 n*(x) + ~w:+~(x) = u:(x) for x E ?I&,and this in turn implies
0
1
t’
FIG. 1. Single-firm stopping problem in the monopoly model.
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that w:(x) 2 w:(x) for x E ‘3. That y: 5 yf is obvious by noting that yt = u;‘(O). Therefore, that B: C Bf or that (B:)’ > (B!)” for t 2 0 implies that {T’ > t) = fhcsst{Xs E (Bf)‘} 2 fl~~~~,{X~ E (B?)“} = {T2 > t} for t 2 0. So T’ 2 T2 a.s. The proof is completed by induction. The results for 6’ 2 ij2 follow from a similar argument. n Proposition 2.2 provides a very convenient tool for dealing with the comparative statics analysis of the optimal stopping time. Looking at the stage payoff is sufficient to obtain comparative statics for the dynamic program. For example, consider a monopolist producing a homogeneous good at a constant cost c and having the option in each period whether to stay in the market which is stochastically declining over time, or leave. In period t, let the random variable X1 be the intercept of the linear inverse demand curve; the constant -b is the slope. The opportunity cost of staying in the market is k. (Alternately, the firm must pay a fixed fee, k, each period in order to participate in the production activity.) In each period t, the maximum profit that the monopolist can obtain is 6G)
=
(1/4b)(X, - c>~ - k,
ifX, 2 c;
-k
otherwise.
Clearly r(e) is an increasing function. Also, 7r decreases as c, 6, or k increases. This fact, together with an application of Proposition 2.2, implies that the optimal exit time T decreases almost surely as c, b, or k increases or 6 decreases. That is, the monopolist optimally exits earlier when the marginal cost of production is higher, a better alternative opportunity exists (a higher k), the price is less sensitive to the quantity change (a higher b), or the firm is less patient (a smaller 8).
3.
EQUILIBRIA
IN THE TWO-FIRM
EXIT
GAME
For i, j = 1,2, denoted by rij(Xt) the stage payoff to firm i in period t when there arej firms in the market and the state is X, . For j = 1,2 and for each firm i, the stage payoff, rij(*), is increasing. Since firms prefer monopoly to sharing the market, we assume that each firm has a higher stage payoff when it is the only firm in the market. That is, 7~&) < nil(z) for z E ‘3 and i = 1,2. At each time t, the active firm(s) observe {Xs ; s 5 t} and then (simultaneously) decide whether to stay in the market and earn rij(Xr) (depending upon the number of firms who stay in) or to quit and get zero payoffs thereafter. Firm i’s strategy is a stopping time Ti, and the payoff at time t to firm i given his rival’s strategy T-i = Tj, j f i, is completely determined by observing the history {Xs; s 5 t}, i.e.,
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Let 6i (0 < 6i I 1) be the discount factor for firm i. DEFINITION 3.1. (Tr , T2) is a Nash equilibrium and T2 are stopping times and for i = 1, 2,
E [z
ahf(Xt,
T-i)lXa]
2 E [g
shf(X,,
in stopping times if T,
T-i)lXo],
a.s.,
for any stopping time 7. We are interested in the subgame-perfect equilibria in the sense of Selten (1975) (see also Kreps and Wilson, 1982). To define a pure strategy subgame-perfect equilibrium in the stopping game setting, we must work on the extensive form of the game and specify afamily of stopping times, one for each history. The only restriction is that they satisfy Bayes’ rule. Because of the simple information requirement in the exit game, this can be done as follows. Let T(t): fi + XT be a stopping time defined on XT {T 5 s} E 8, for every s E N: where X: = {t, t + 1, . . .}, and CI is the space of sample paths of {X,; t I O}. For each time t 5: 0, let T!(t) be the time at which firm i exits given that time t has been reached and firm j 0’ # i) is still in the market, and To(t) be the time at which firm i exits given that time t has been reached and firm j (j # i) has exited before t. Note that T:(t) 2 t and T?(t) L t. Firm i employs exit time T!(t) if its rival is still in the market at t; otherwise firm i exits at T?(t). Thus, firms i’s strategy at time t is T,(t) =
T:(t)
if firm j is in the market at time
T?(t),
otherwise.
t;
Bayes’ rule requires that T!(t) = T;(s) for t 5 s 5 T;(t) and that T?(t) = T?(s) for t 5 s 5 T?(t). DEFINITION 3.2. Let (TI , T2) = ({Tl(t)}tbO, { T2(t)},,o) be a pair of stopping-time families defined as above. ( T1, T2) is a subgame-perfect equilibrium if, for each t 2 0, (T,(t), T2(t)) is a Nash equilibrium. That is, if both firms are in the market in period t, for each t 2 0, i = 1,2, and any stopping time S(t).
T:(t) = T;(s) for t 5 s 5 T;(t), and T?(t) must be optimal when firm i is left in the market alone at t.
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DEFINITION 3.3.
j=
AND
LI
Firm 1 is stronger than firm 2 if nij(z)
> nZj(z) for
1,2andzE%.
Ghemawat and Nalebuff first analyze a model where the firms have identical variable costs but the larger firm has larger fixed costs. In their Section 4, for a specific example, they look at how big the variable cost difference, due to economies of scale, would have to be to negate the fixed-cost advantage of the small firm. Our analysis shows that looking at the stage payoff, which nets out the effects of fixed and variable costs, is sufficient to determine which firm has the advantage. DEFINITION 3.4.
Firm 1 is more patient than firm 2 if 6r > &.
To aid the game-theoretic analysis that follows, we first solve four single-firm problems, two for each firm. That is, by applying the results in the previous section, we derive the optimal stopping time for each firm i as if there were j firms in the market throughout i’s stay in the market. Each single-firm problem is indexed by (i, j) and takes mij as the stage payoff. Following the notation in the previous section, define recursively for i,j= 1,2 Uy(X)
=
T;j(X)
+
6iWFil(X),
w&,(x)
= E [ uji+l(Xt+,)+lXt = x],
WY = 0 for t 2 tQ, where t$ E inf{t 2 0: n;j(Xt) < 0, a.s.}, and Tij(s) = inf{t 2 s: X, < yji}, where y? = inf{z E CR: uij(z) 2 0). By Proposition 2.1, we know that e(t) = T,,(t) for i = 1, 2 and for all t. Therefore, the real problem left to determine a subgame-perfect equilibrium is to specify
TI.
By Proposition 2.2, we also have that for i = 1, 2, and for t B 0, wf!+i 2 w:‘,i, yf’ I y:‘, and T,,(t) L Tdt) a.s., and that forj = 1, 2, and for t 2 0, U >- wt+1, 3 Yt‘j <- Yt‘j >and Tv(t) z Tzj(t) a.s. if firm 1 is stronger or more wt+1 patient than firm 2. Intuitively, one may think that since Tii and Ti2 are firm i’s most optimistic and most pessimistic decision rules, Tjl and T,-Jmay impose upper and lower bounds for firm i’s equilibrium stopping time. The following proposition proves that this intuition is correct. Since the equilibrium exit times can be nontrivial to calculate (see below) and are not, in general, unique, it is useful to have these bounds on the exit times. PROPOSITION 3.1. TisT;l,u.s.,fori=
Suppose (TI , T2) is a Nash equilibrium.
Then T,-2I
1,2.
Proof. Suppose Ti > Til with positive probability. Let T s Ti A Ti,. Then, given T-i, the difference between the equilibrium payoff to i and the expected payoff to i if T is employed is
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49
-E ["A2m' 6f(ni,(Xr) - ~dX~))IXo] f=TAT-,
The first term is nonpositive since ~T;I > ~TTT~Z and T A T-i 5 T; A T-;. The second term is negative since, by Proposition 2.1, E [cg;:, S:ni,(X,)lS,,] < 0 if Ti > Ti, with positive probability. In addition, also by Proposition 2.1, E [&,E’ Sf’rrii(X,)]%r,] 2 0 which implies that the third term is nonpositive. This contradicts that T, is the best response to T-;. Therefore, T; CCTi, a.s. for i = 1, 2. n The other part of the proposition is proved similarly. We now turn to an asymmetric situation in which firm 1 is stronger or more patient than firm 2; that is, rlj > n~zjand 61 = 62, or 61 > 82 and rij = n2j. By Proposition 3.1, one natural equilibrium we can easily infer is (T,, , T22).Since firm l’s equilibrium strategies are bounded below by T12 (LT~~), its best response to T22is T,, ; and since firm 2’s equilibrium strategies are bounded above by T2,(IT,,), its best response to T,, is T22. Furthermore, this equilibrium is subgame perfect: given the history up to time t 2 0, (T,,(t), T22(f))is an equilibrium for the subgame beginning at time t. However, as will be shown, this is only one of the many possible subgame-perfect equilibria for this game. Therefore, the assumed asymmetry cannot assure that the stronger or more patient firm always outlasts its rival. Our principal result, characterizing all possible pure-strategy subgameperfect equilibria, is Proposition 3.2. Reference to Fig. 2 aids understanding of the multiplicity of equilibria and the form of the equilibrium exit sets characterized in Proposition 3.2. In region R, , both firms are profitable as duopolists and neither has any incentive to exit. As industry demand declines, X, may fall into region R2, where the strong firm (firm 1) is profitable as a duopolist but the weak firm (firm 2) is not. For X, in this region (approximately), the weak firm will exit. We add the modifier
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0
T 22
T,*
t;2
t;2
T*,
t;,
T,,
=t;,
FIG. 2. Four single-firm stopping problems in the duopoly model.
“approximately” because the precise boundaries must be calculated by backward recursion, as is done in the proof of Proposition 3.2. The source of the multiplicity of equilibria arises from the possibility that X, could jump directly from region RI to R3. In region R3 neither firm is viable as a duopolist but either is viable as monopolist. Thus, exit by either firm for X, in this region could constitute equilibrium behavior. Again, the exact boundaries of these regions must be calculated by backward recursion. Any equilibrium can be characterized by an “agreement” set A, E Rj such that if X, jumps into region R3 from region RI , then if X, E A,, firm 1 will exit, and otherwise, firm 2 will exit. Each measurable subset A, corresponds to a distinct equilibrium for the exit game. Proposition 3.2 uses backward recursion to define precisely the sets that generate the equilibrium behavior we describe. PROPOSITION3.2. Supposefirm 1 is stronger or more patient thanJirm 2. Then ( TI , T2) is a subgame-perfect equilibrium if and only if for each s L 0, T?(s) = Tir(s), and T:(s) = inf{t L s: X, E &,}for i = 1,2, where the exit sets Bit, i = 1, 2, t 2 0, are defined recursively
BI, = C-03,Y:‘)
U (AS, II Ad
where A, is a Bore1 set contained Aif E {x: STAT +
w:+,(x) = 0, for t 2 tri, fir where
0 5 t 9 tf,,
U A,,
B2t
=
A2r
II
AC,
in the set A,, fl Azt fl [y:‘, CQ), ~W~+I(X)
and
<
O},
i=
1,2,
wl+dx> = mf+dx,+*)+lxt
= xl,
EXIT
uf+*(x>
=
[ni2(X)
+
IN
DECLINING
~Wf+*(X)Ilg~,,+,(X)
51
INDUSTRIES
+
[ril(X)
+
SWfY2(X)]lgj,,+,(X),
j # i. Moreover
w:’ % wf I wit’.
Proof.
Appendix
B.
n
The proposition shows how we can obtain all subgame-perfect equilibria by varying A,, a Bore1 subset contained in A,, fl Azt n [yf’, a), for every t. Note that if X, E Al, fl AZ, fl [y:‘, m), then firm l’s expected reward is nonnegative if firm 2 exits at t (X, E [y:‘, ~0))and is negative if firm 2 stays (X, E Al,); firm 2’s expected reward is nonnegative if firm 1 exits (X, E [y:‘, m)) and is negative if firm 1 stays (X, E AZ,), assuming some equilibrium strategies will be employed from t + 1 on. We recognize that the normal form of this situation is an asymmetric “battle of sexes” in which there are three Nash equilibria: firm 1 stays and firm 2 exits, firm 1 exits and firm 2 stays, and a mixed strategy equilibrium. As we know, the mixed strategy equilibrium is inferior since it gives positive probabilities to inefficient outcomes (both stay and both exit). Also, we do not consider mixed strategies or behavioral strategies in this paper. Therefore, either the exit of firm 1 or the exit of firm 2 is consistent with equilibrium behavior and the multiplicity of the subgame-perfect equilibria arises. In Proposition 3.2, A, consists of the states in which firm 1 exits, and the remaining states in A,, fl AZt fl [y:‘, x) are consistent with the second equilibrium outcome in which firm 2 exits. Also note that for each t, Ait, i = 1,2, are functions of A,+ 1, . . . , A,;, , the sets picked from t + 1 on. Among the many equilibria for this game are two extreme subgameperfect equilibria which give upper and lower bounds on any equilibrium stopping times (T, , T2). The first of these, ( f1 , f*), is obtained by letting A, = a, = 0 for all t. It is not difficult to check that (fi, ?‘J = (Tll, TZ2)a.s. (see the proof of Proposition 3.3). In fact, (T1, , TZ2)is the equilibrium most preferred by firm 1, the stronger firm, because, as will be shown later, it gives firm 1 the latest possible exit time and gives firm 2 the earliest possible exit time. On the other hand, the second extreme equilibrium, (?;I, Tz), gives firm 2, the weaker firm, the latest possible exit time and gives firm 1 the earliest possible exit time. This equilibrium is obtained by setting!, = A, = Ale fl x2, rl [y:‘, 0~) for all t, where Ait = Air(Ar+l, . . . ) A,;,). 3.3. There are two extreme subgame-perfect equilibria, APROPOSITION ,. (T1 , T2) and (Tl , Tz), in which for i = 1, 2 and each s 2 0, pi(s) = Tii(s) and F?(s) = inf{t 2 s: X, E B1,}, where the exit sets Bit are dejined recursively as . Bz, = AZr n A:‘, Bl, = C-w, yf’) U A,,
52
FINE
AND
LI
Pi;,= {x E a: %-j?(X) + SW;+,(x) < O}, wf+,= 0, for t 2 tfj, and Wj+,(x) = E[if+,(X,)‘lX,
= x],
for 0 5 t 5 17, , where
ui+I(x) = [7Ti2(x) + t5W:+Z(X)llBj,,+,(X) + [Til(X)+ 6wf~I(X)ll~,,,+,(X)~ j f i. Suppose ( T1 , T2)- is a subgame-perfect and Tz2 5 T2 5 Tz, a.s. Proof.
Appendix
equilibrium.
Then T, 5 TI 5 TI1
C.
n
Since firm 1 is assumed to be stronger than firm 2, the first extreme case This equilibrium is also simple in structure, i.e., a threshold equilibrium. The following proposition gives a necessary and sufficient condition for ( T1I , Tz2) to be unique. Before proving the proposition, we need a proper definition of uniqueness of a subgame-perfect equilibrium in stopping games. Basically, for two subgame-perfect equilibria to be equal, we require that the Nash strategies of the two subgame-perfect equilibria for any subgame be equal almost surely with respect to the conditional probability in the subgame. (TI1 , T& is, in some sense, the more appealing equilibrium.
DEFINITION 3.5.
areequaliffori=
Two subgame-perfect equilibria (T; , TI) and (T;’ , T$) T:!(t),P:,a.s.foralltrOandforallxEYL
1,2,Tl(t)=
PROPOSITION3.4. Supposejrm 1 is stronger 2. Then, (TI 1, T& is the unique subgame-perfect Pf-,{X,EAI,fl [y:‘,m)} = 0,forxERaand 15 determined recursively as in Proposition 3.2 by Proof.
Appendix
D.
or more patient thanfirm equilibrium if and only ij t 5 tTI - 1, wherra,,are letting A, = 0 for cl11t. n
Note that a,, n [y:‘, 00) 5 [y:,, yt” ). Therefore, the proposition indicates that the stronger firm never exits first if either the payoffs are so asymmetric that n12 2 7r2, (or equivalently, y!* 5 Y?‘) or the process X is sufficiently smooth that it never jumps over the interval [yf*, y:‘). We show below that Proposition 3.4 implies the uniqueness result of Ghemawat and Nalebuff (GN) and why their uniqueness result fails if industry demand suffers a severe negative shock. We mention here that we do not attempt to discuss the relations between discrete-time and continuous-time models in general, but in the context of a deterministic
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53
world, discretizing a continuous-time model is quite straightforward. Note that tz = inf{t I 0: ril(X,) < 0, a.s.} and that r& I t& by the asymmetric assumption. Thus, if t L tli* , both firms will leave and if t& I r < tf, , firm 1 may stay and firm 2 will definitely leave. Consider t = tir - 1. Note that y:’ is determined by mTT2r(yf’)= 0 since M$’ = 0. If ~TT~z(Y?)+ ti~:!~(y:‘) 2 0, thena,, rl [y:‘, ~0) = 0 sincea,, = {x: nj2(x) + 6w::r(x) < 0) for t = t& - 1. The condition for uniqueness is satisfied for period t. In general, there is a unique subgame-perfect equilibrium in which the strong firm (firm 1) leaves last if and only if the losses which firm 1 may take in period t, 7rr2(x), are outweighed by the expected equilibrium profit from period t + 1 on, 6$!+,(x), for all possible states, x, in which firm 2 is viable as a monopoly, for all periods t. To pass on to GN’s result as a special case, we simply discretize their continuous-time model in such a way that the length of each period is very short, and hence the size of profits or losses per period is very small while the expected future monopoly profit (6~::~) remains strictly positive and sizable (the discount factor is close to one). Therefore, the condition for uniqueness holds and we can work backward. This is exactly the essence of GN’s proof of the uniqueness. However, their proof may fail if we drop the continuity assumption in their model. In particular, if at a certain point of time, say t*, the demand drops drastically from a level at which both firms are viable as a monopoly but neither is viable as a duopolist to a level at which neither firm is viable as a monopoly, then for t = t * - 1, the duopoly loss of firm 1 will never be compensated by the future profits independent of the fineness of the time discretization. Proposition 3.4 provides a guideline to uniqueness results in more general settings. The following example is an application of the model to a market setting in which competing firms have not only the production decisions in each period but also the option to exit the market. A Cournot duopoly faces a stochastic linear inverse demand, P, = X, - hQ, where Q is the total output of the duopoly, and {X,, t 2 0} is Markovian and stochastically decreasing. The firms produce a homogeneous good at constant costs ci, i = 1, 2. Firm i’s opportunity cost of staying in the market is ki. Without loss of generality, we assume cl I c2. If both firms are in the market, the equilibrium stage payoffs are
~dXt)
=
(1/96)(X,
- 2c, + c# - k,,
if X, > 2c2 - cl;
W4bNXt
- cd* - kl,
if cl 4 X, < 2c2 - cl ; otherwise,
-kl, (1/9b)(X, ~22(-&)
- 2~2 + c,)* - k2,
ifX, 2 2c2 - cl;
= -h,
otherwise.
54
FINE
AND
LI
If one firm is in the market alone, the monopoly i=
stage profits are, for
1, 2,
(1/4b)(Xt ~ilW,)
=
-
ci)'
- kiy
if X, L
Ci;
otherwise.
-k/y
It is easy to see that ril I 7~~~for i = 1, 2, and if cl I c2 and kl I k2 then firm 1 is stronger, i.e., mlj 2 m2j for j = 1, 2. From the general results obtained earlier, we know that (Tll, Tz2) is an exit equilibrium. The thresholds y:’ and y:’ can be calculated as y:’ =
CI
+
2~Wl
-
if kl 2 i$wj:,;
~l~::lh
otherwise,
Cl 9
22 _ Yr -
2~2 -
CI
+
3vb(kz
-
if k2 z- &wff,;
82wf:,L
otherwise.
2c2 - c,,
Also, it is easy to see that y:’ increases as cl, b, kl increase or 6, decreases, while y:’ increases as CZ, b, k2 increase or cl, a2 decrease. To see that there might be other subgame-perfect equilibria, we further assume that X, is uniformly distributed on the interval (c2 + 14A - 7At, c2 + 15A - 7At), where A = c2 - cl > 0, and that k = ki = (llb)A2, i = 1, 2. Note that both firms will exit for sure at t = 2 since r;‘(O) = c2 + A, and no firm will exit at I = 1 since 7G1(0) = c2 + 411.< cz + 78. Direct calculation shows that Wl21= E[T2,(Xt)+l = iiq = E[7r,2(X1)+] =
(S3 - 73)A2 157 -k=-k 12b (lo3 - 93)A2 27b
-k=27k
244
Let 6 = 6i = &$. Then yi’ = cz and Alo = (-m, y:], where y;~c,+2~b(k-6iT;)=c,+
l.llA.
Therefore y:, - y;' = c, - c2 + l.llA
= O.llA > 0.
Suppose the initial state X0 = x, x E [yi’, y;). Then firm 2 can stay in and credibly force firm 1 out right away.
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Note that in this example, we need a drastic downward jump of demand in order to make yf’ below yj . The decreasing rate of X,, 7, is carefully picked such that there exists a 6 to make A. # 0. 4.
CONCLUDING
REMARKS
This paper analyzes the pure-strategy subgame-perfect equilibria of an exit game in a stochastically declining market. With trivial modification, an equivalent analysis applies to the case of increasing payoffs which can be interpreted as the entry games played by firms in a growing industry. We find that these games generally have multiple subgame-perfect equilibria. The models of both Ghemawat and Nalebuff, and Fudenberg and Tirole, have demand processes that decline deterministically and continuously. Each obtain a unique subgame-perfect equilibrium. The Huang and Li (1986) paper shows that a unique subgame-perfect equilibrium also obtains with a continuous stochastic process. In Fine and Li (1987) we analyze a continuous-time two-player exit game with stochastic (Poisson) jumps and demonstrate the multiplicity of equilibria in that setting as well. Thus we conclude (Proposition 3.4.) that an underlying stochastic demand process with sizable jumps can yield multiple exit equilibria if no further assumption is made on the magnitude of the asymmetry of the firms’ payoff structure. APPENDIX
A
Proof of Lemma 2.1. First note thatfi >f2 implies {z E %:fi(z) {z E 3: h(z) > 0). Since pl 2 p2, E[fi(Zl)l,,,ffi,(f,(Z1))1
> 0) >
= li/,(z,>*{ h(z) &l(Z)
2 I (f2(L)>O) h(z) 44(z) ’ l,,,.,>o,f2(d &l(Z) L I (f2(z)>o) f-i(z) &2(z) = E[fi(Z2)1(O,m)(f2(Z2))1. APPENDIX
B
Proof of Proposition 3.2. Backward induction. ~P{~~~(X,)r:O}=0,fori=1,2.Also,w~=wf=wf’=0,fori=1,2. The proposition is trivially true.
For t L tf,, P{X, E BiC,}
56
FINE
AND
Now assume that the proposition
A;,Ct-w,yf2)
LI
is true for s 2 t + 1. Note that and
Ai C [yf’, ~1,
(3.1)
since w:‘+r 5 ~f+~ 5 wf:, for i = 1,2. Suppose both firms are in the market at time t. Given any strategy B2, adopted by firm 2 at time t, firm 1 will be in the market alone from t on if X, E B2t, and he will be involved in a game which gives expected payoff w:+~ if X, E B$, and equilibrium strategies follow from t + 1 on. Remaining active at t, firm 1 expects to get
Firm l’s best response then is to exit if u:(X,) < 0 and stay if u,‘(X,) 2 0. Let B1, = {x: u:(x) < O}. Firm 1 will exit if firm 2 exits and X, < yf’ or if firm 2 stays and X, E A,,. That is BI,
Similarly,
=
(B2r
n
Y;‘))
t--w,
U
CBS,
n
(3.2)
Ad.
given firm l’s strategy B1,, 2’s best response should satisfy B2r
=
WI,
n
Y:‘))
C--w,
U
VT,
n
(3.3)
A2r).
Using fact (3.1), we can rewrite (3.2) and (3.3) to be
B,,= {t-w,y:‘)W, n C-m,Y;‘>)>U Wt n AI,) = C--w,yf’) U (Et n Ad, et = m, n [Y?, 9 u {A;,\(B,, n AS,)) 21 = WI, n [yt , 30)) U AL. Substituting
(3.4)
(3.5)
(3.4) into (3.5), we have BSt
=
CBS,
n
AI,
[it
n
21
, ~1)
U
(3.6)
AS,,
since y:’ L y,” . Thus B$ is part of the equilibrium if and only if B$‘, satisfies (3.6) or BS, is a fixed point of&(.) where.&(B) = (B fl A,, rl [yf’, m)) U A$‘, for any B E %(?A). The only possible solution is of the form B& = A, U A;,, where A, C A,, fl [y:‘, ~0). We let A, c A,, fI Azr n [y:‘, w) simply to make A, and A& two disjoint sets. Similarly, we can prove that BI, is part of the equilibrium if and only if
BI,= WI,n At, n [Y:‘, which gives us the equivalent
~1)
U
64%
characterization
n
Al,)
U
t-w,
Y:‘>>
(3.7)
of a generic equilibrium.
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Finally, u:’ I uf d u:’ since w;‘+t % w:+, 5 wf!+t . Therefore, w:’ for i = 1, 2 and for all t. We conclude the induction.
w:’ 5 wf 5 n
C
APPENDIX
Proof of Proposition 3.3. Let ( f1 , fz) be obtained by letting A, = A, = 0 for all t. That (fr, pz) and (Tr , Tz) are subgame-perfect equilibria is a direct corollary of Proposition 3.2. We shall use the induction argument to show the second assertion as well as fi = Tii a.s. as a by-product. First note that for t 2 tfl , iZ: = w,’ = G,,’ = wI“=()andw22=$~~=~~= iV: = 0. This implies that A,, = (-co, y,“) and a21 = (--co, ;:‘), and A11 n as, = 0. Thus B,, = (-co, y)‘), and &, = (-a, y:‘). Assume Wf+t 5 wj+, I $;+I 5 w::,, and w:t, = *?+I 5 w;+, 5 W:+,. Then
C-00,Y:‘) Y:‘)
C-T
2 &r
2 AI, 2 a,, 2 C-00, Y:‘),
=
1
-42l
AZ,
2
x2,
2
(3.8)
Y:‘),
C--x,
and hence d,, n a$ = 0. Using (3.8), we have El,
Y,“>
=
GL
fl
[YE
=')I
U
t-w,
2
(AI,
n
[Y!‘,
~1)
U
(A%
n
AI,)
2
BI,
U
(A&
n
AI,)
U
(-a,
=
A
2 (&, n A,) u (-T
2
(AI, u
II
C-m,
[Y:‘,
~1)
U
t-m,
Y;‘>
Y,“>
(3.9)
yf’)
of’)
= I&, = (--co, yf’). It can be similarly shown that C--m,
Y:‘>
=
B2t
1
B2t
2
(3.10)
B2r.
Therefore, ~7,’5 uf 5 I;f 5 uf’ and u,22=ti:~~~~U:.Weju~tshowiif~ ut , as an example, 8(X,)
= [~IZW,)
+ ~~f+lw,)llE~,(&)
5
+ ~w:+,(m~B;,GG) +
C97l2bK)
+
[~,l(X)
+
~w::I(x,)Il&w,)
[ill
+
~w~:lw,)lls,,(xt)
= d(X)
-
since W:+, 5 w:+li2B2, 2 Bz,, and B5r C B s,. Hence, we have Gf 5 w) 5 r+j I wi’, and wI - $1: 5 w: I ii?:. We conclude the induction. As a
58
FINE AND LI
result, (3.9) and (3.10) imply that FI 5 TI I TI = TII and T22 = f2 5 T2, a.s.
T2 5 n
D
APPENDIX
Proof of Proposition 3.4. For t 1 tf, - 1 and for i = 1, 2, Ai, = a, = (--co, 9;) = (--cc, y:), and A, = [y:‘,-yj2) since $+I = G’j+, = 0. Then Pf-,{& E [yf’, y:“)} = 0 implies T/i = Ti, P:-a.s., for s 2 t - 1, and W: = 6; and 67: = wf2. Suppose Tii(S) = T/(s), i= 1, 2, PS-a.s.,fors z tandforxE%. Then #+I = G:+, and i?j?+, = w::, imply that Alt = a,, and x2, = (-cc, y:‘). Therefore, Bl,\Bl, = B&V& = A, = Al, f~ [y:‘, tQ), and Tii = Ti, i = I, 2, P:-a.s., for s 2 t - 1 and x E CR, by the condition that Pf-,{X[ E A,, n m)} = 0. The induction is completed. [Y?, To prove the necessary condition, let t = sup{s Z- 0: Pz-,{X, E [yz’, yi’)} > 0). Such a t exists if P:-1(X, E als II [y:‘, 03)) > 0 because AI, f~ ru5’9 ~0)C [yj’, yj’). Since Ti = Tit, P:-a.s., for s z t and x E CRand hence 8$ n BZt = [y:‘, y:‘) and a,, = (-w, y:‘), we have P:-,{T2(t)
> T22(t)} z P;-,{X,
E B& fl l?2t}
= P:-,{x, E A,, n [$I, ~1)> 0.
n
ACKNOWLEDGMENTS The authors gratefully acknowledge comments of John Cox, Drew Fudenberg, Chi-fu Huang, Garth Saloner, Jean Tirole, and several anonymous referees. We also acknowledge fruitful discussions with Erhan Cinlar and Michael Harrison, and financial support from Coopers and Lybrand Inc.
REFERENCES CHAPUT, H. (1974). “Markov Games,” Technical Report No. 33, Department of Operations Research, Stanford University. DYNKIN, E. (1969). “Game Variant of a Problem on Optimal Stopping,” Sovier Math. Dokl. 10,270-274. FINE, C. H., AND LI, L. (1987). “Stopping Time Equilibria in Games with Stochastically Declining Payoffs,” Working Paper, OR 158-87, Operations Research Center, MIT. FUDENBERG, D., GILBERT, R. J., STIGLITZ, J. E., AND TIROLE, J. (1983). “Preemption, Leapfrogging, and Competition in Patent Race,” Europ. Econ. Rev. 24, 3-31. FUDENBERG, D., AND TIROLE, J. (1986). “A Theory of Exit in Duopoly,” Econometrica 56(4), 943-%O. GHEMAWAT, P., AND NALEBUFF, B. (1985). “Exit,” Rand J. &on. 16(2), 184-194.
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HADAR, J., AND RUSSELL, W. R. (1%9). “Roles for Ordering Uncertain Prospects,” Amer. Econ. Rev., March, 25-34. HUANG, C., AND LI, L. (1986). “Continuous Time Stopping Games,” Working Paper No. 17%-86, Sloan School of Management, MIT. KREPS, D., AND WILSON, R. (1982). “Sequential Equilibria,” Econometrica 50, 836-894. MAMER, J. (1986). “Monotone Stopping Games,” Working Paper, Graduate School of Management, UCLA. OHTSUBO, Y. (1986). “Optimal Stopping in Sequential Games with or without a Constraint of Always Terminating,” Math. Oper. Res. 11(4), 591-607. OHTSUBO, Y. (1987). “A Nonzero-Sum Extension of Dynkin’s Stopping Problem,” Math. Oper. Res. 12(2), 277-297. SELTEN, R. (1975). “Reexamination of the Perfectness Concept for Equilibrium Point in Extensive Games,” ht. J. Game Theory 4, 25-55. SHIRYAYEV, A. (1978). Optimal Stopping Rules. New York/Berlin: Springer-Verlag.
AND
ECONOMIC
BEHAVIOR
1,
40-59 (1989)
Equilibrium Exit in Stochastically Declining Industries CHARLES H. FINEANDLODE Sloan School of Management, Cambridge,
LI
Massachusetts Institute of Technology, Massachusetts 02139
We study a complete information model of exit for an industry in stochastic decline. We characterize the entire set of pure strategy subgame-perfect equilibria for the exit game. In general, there are multiple subgame-perfect equilibria of our exit game, in contrast to several papers in the literature. Relaxing an assumption in those models (arguably in the direction of more realism) will give multiple equilibria in those models also. We provide (fairly restrictive) necessary and sufficient conditions for there to be a unique subgame-perfect equilibrium. o 1989 Academic
Press, Inc.
1. INTRODUCTION The decline of products and industries is a natural by-product of the growth and evolution of industrial economies. Economic decline puts pressure on firms’ profits and on their ability to remain in business. Most firms eventually depart if the decline is severe. However, those who manage to stay in after a shakeout can often earn significant profits. In this paper, we analyze firms’ decisions to exit an industry that is suffering from economic decline. In contrast to the literature on entry, the study of exit decisions by firms in oligopolistic industries is not extensive. Ghemawat and Nalebuff (1985) analyze a continuous-time, perfect information model for an asymmetric duopoly in a deterministically declining industry, where the firms differ by their capacities and related fixed costs per period. Unless the high-fixed-cost (large) firm has significantly lower operating costs than the low-fixed-cost (small) firm, the unique subgameperfect Nash equilibrium in their model is for the firm with the larger capacity and fixed costs to exit at the first time that its duopoly profits are nonpositive. The smaller firm then remains until its monopoly profits 40 0899-8256/89$3.00 Copyright 6 1989 by Academic Press, Inc. All rights of reproduction in any form reserved.
EXIT
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41
become negative. (Fudenberg et al. (1983) use a similar approach to derive equilibrium strategies in the context of a patent race.) Fudenberg and Tirole (1986) also analyze an asymmetric duopoly exit game in a continuous-time model with deterministic demand paths. However, they assume that each firm does not know the fixed operating cost per period of its rival. Thus their model is a war of attrition where each firm continually revises downward its estimate of the other’s costs, as long as the contest for the market lasts. With an assumption that each firm assesses positive probability that its rival will never find it optimal to exit, the model yields a unique subgame-perfect Nash equilibrium where the higher cost firm exits the market before the firm with lower costs if cost distributions are symmetric. We also model an industry that begins as a duopoly. Like Ghemawat and Nalebuff (1985) and Section 5 of Fudenberg and Tirole (1986), we assume that the industry declines over time so that the profits of the two firms shrink until one of them exits, leaving a monopoly for the other. In contrast to the results of Ghemawat and Nalebuff and Fudenberg and Tirole (each of whom obtains a unique subgame-perfect equilibrium in a continuous-time, deterministic demand model), there may be multiple exit-time equilibria in our stochastic, discrete-time framework. As a consequence of the multiplicity of the equilibria, the stronger firm, which has a stage-payoff advantage, may not always exit last. The multiplicity of equilibria in our model is a consequence of the fact that the industry demand process can jump from a point where both firms are viable as duopolists to a point where neither firm is viable as a duopolist but each is viable as a monopolist. At such a point in time, exit by either could constitute an equilibrium in stopping times. Both Ghemawat and Nalebuff and Fudenberg and Tirole avoid this problem by assuming that demand falls continuously (and deterministically) over time. However, their models would also exhibit multiple equilibria as ours does if their demand process were allowed to have jumps. Thus, an assumption in the direction of more realism (a demand process with discrete jumps) may destroy the equilibrium uniqueness in each of their models. (We will elaborate more on the issue in Section 3.) Jumps in the demand process drive the multiple equilibria result, independent of whether the demand process is stochastic or deterministic. Our model uses a stochastic demand process because we think it is descriptively more accurate of industrial decline and because the random events that influence demand help motivate why the process should have jumps. We characterize the generic form of any equilibrium in our model. These equilibria are obtained by solving a time-indexed sequence of fixedpoint problems. Among the equilibria, two are particularly interesting because they provide the upper and lower bounds for any equilibrium exit
42
FINE
AND
LI
times and because each player prefers one of the two equilibria to all other equilibria. In particular, each firm favors the equilibrium which makes his active period the longest. We provide (fairly restrictive) necessary and sufficient conditions for the “natural” equilibrium (in which the stronger firm always outlasts its rival) to be the unique subgame-perfect equilibrium. The conditions require either that there is a significant asymmetry in payoffs between the strong and the weak firm or that the sizes of the demand process jumps are small. To model the exit game, we adopt a natural multiperson extension of the optimal stopping-time concept (see, e.g., Shiryayev, 1978) which is the concept of stopping-time equilibrium, a Nash equilibrium in stoppingtime strategies. The game-theoretic extension of optimal stopping theory was initiated by Dynkin (1969) and followed by Chaput (1974) in an analysis of a class of two-person zero-sum stopping games. The recent modifications and extensions of Dynkin’s work include Mamer (1986) and Ohtsubo (1986, 1987). Ohtsubo (1987) obtains value relations that characterize equilibria in non-zero-sum stopping games and provides an existence theorem for the Markov case, of which our model is an example. Our contribution to this literature is to look at the semiextensive form (instead of the strategic form) of stopping games, characterize explicitly the complete set of subgame-perfect equilibria, and provide a uniqueness theorem for one class of stopping games. In general, any stochastic dynamic game in which each player’s strategy is a single dichotomous decision at each stage can be formulated as a stopping-time problem. Further, in such games, the notion of equilibrium stopping time can be applied. The stopping-time equilibrium concept of Ghemawat and Nalebuff or Fudenberg and Tirole is analogous to ours, except that in their models the single-firm optimal stopping problem is of little intrinsic interest because their models have deterministic demand paths. One characteristic of the firms’ optimal policies in our model is that one or both firms may earn negative profits in some period(s), but remain in the market. This can occur for either of two reasons. First, industry demand can suffer a large stochastic negative shock, such that demand will increase in the next period, in expectation. In such cases, firms will absorb the loss in that period, knowing that demand is likely to recover. This result accords with observed practice. Most industries experience stochastic downturns that do not lead to mass exit from the industry. A firm may also choose to absorb losses and remain in the market in the expectation that its rival will exit the industry first, leaving a more profitable, monopolistic industry for the remaining firm. Although the former effect is unique to our model, the latter effect is also present in the Fudenberg-Tirole formulation.
EXIT
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43
For the single-firm model, we obtain comparative static results that we illustrate in an example with a linear demand function and linear costs. For the single-firm example, the optimal exit time is decreasing in the firm’s unit costs, the slope of the demand curve, and the per unit fixed cost of being in the market. The optimal exit time increases in the firm’s discount factor. In a two-firm example with Cournot competition, we calculate two equilibria to illustrate the multiplicity. The remainder of the paper is organized as follows: In Section 2, the single-firm exit problem is formulated as a stopping-time problem and several basic results are established. In Section 3 the main results regarding equilibrium exit behavior in a duopoly are proved and discussed. Concluding remarks follow in Section 4.
2.
THE SINGLE-FIRM
MODEL
In this section, we formulate the dynamic programming model for the single-firm problem and provide some comparative statics on the optimal stopping rule. We assume that at time zero, a single firm is in the market earning nonnegative profits. At any later point in time, if the firm concludes, after observing demand, that remaining in the market is no longer profitable, it may exit the market. Reentry is assumed to be prohibitively costly. We let industry demand in period t be indexed by X,. Firm profits are denoted by a reward function r(X,) where n(e) is an increasing, realvalued function. Industry demand {X, ; t = 1,2, . . .} is a Markov process defined by a family of transition functions, p(x, t; A, s), x E SR,A E '%A('%), and s > t, s, t E SIT+ = (0, 1, 2, . . .}, where a(%) is the collection of Bore1 sets on a real line. For each period t > 0 and x E an, denote the probability measure of X, conditional on X,-i = x by P:(X,) = P,{X,jX,-i = x}. To model stochastic decline of the industry, we assume that for all t 2 0, P: dominates Pf+l in the sense of first-order stochastic dominance (Hadar and Russell, 1969). Formally, for any t 2 0, x, y E 3, sZZ, dPf(.z) 5: J,+ dPf+,(z). We denote this ordering by P: 2 Pf+l. At each time t, the firm observes {X, ; s 5 t} and then decides whether to remain in the market and get n(X,) in period t or to exit and get zero payoffs thereafter. We assume that for each t, the likelihood of having a high future state is increasing in the current state X,. Formally, for x L x’, p:+i stochastically dominates pfi:-;1 for all t 1 0. That is, the better the industry is currently doing, the better is its future prospects. A firm’s strategy is a random time T: Cl* N+, where R is the space of sample paths of {X, ; t B 0). We require T to be the stopping time. That is, T:IR-,X+issuchthat{Tst}~4,foreverytEX+,where$,r(~(X,; s 5 t), the u-field generated by {X,; s 5 t). Let 6 be the firm’s discount
44
FINE AND LI
factor. The firm’s problem is to choose a stopping time T so as to maximize the expected discounted reward
(2.1)
(By convention, we assume that this term is identically zero if T = 0.) This isa standard optimal stopping problem (see, e.g., Shiryayev, 1978). Define t* = inf{t: n(X,) < 0, a.s.}, the first time that stage payoffs are negative almost surely. Note that the stochastic dominance assumption implies that rr(X,) < 0, a.s. for s 2 t*. We assume that 0 < t* < ~0; i.e., the industry becomes nonprofitable to the firm in finite time. Proposition 2.1 below characterizes the optimal stopping times for this problem. The proposition is preceded by a technical lemma that orders the expectations of the positive parts of monotone ordered functions of random variables that are ordered by stochastic dominance. LEMMA 2.1. Suppose that fi and fi are increasing functions and h(z) >fi(z), for z E CR.And suppose that Z, and Zz are random variables with probability measures ~1 and ~2, respectively. Moreover, ~1 stochastically dominates ~2. Then E[fi(Z,)+] 2 E[fz(Zz)+], where
EUXZ;>+l= EUX-5)1co,c&X&)>l, l,(*) is an indicator
function
for
i = 1, 2,
set S, and strict inequality
holds if
~~~~,,cdfd&))l> 0. Proof.
Appendix
A.
n
PROPOSITION 2.1. (i) The optimal stopping time for the problem described in (2.1) can be characterized by T = inf{ t 2 0: X, E B,}, where B, = {x: u,(x) < O}, u,(x) = 77(x) + 6 w+dx), and w+dx> = E[ul+dXl+J+lXl = x] with w,(a) = 0 for t L t*. (ii) For t 2 0 andx E CR,w,(x) 2 w,+ I(x) and u,(x) 2 u,+,(x). Let y, = inf{x E CR: u,(x) 2 O}. Then yr is increasing in t, B, = {x E ‘3: x < y,}, and B, C Br+, for t z 0. Proof. Part (i) follows from the optimality principle of dynamic programming (cf. Shiryayev, 1978, Chap. 2). To prove (ii), note that wt = 0, for t 2 t*. By induction, if w[+,(.) is increasing, then u,(e) = r(.) + 6w,+,(*) is increasing since rr is increasing. Hence wI(.) = E[u,(X,)+I*] is increasing by Lemma 2.1. To show that wr and uI are monotone as functions of t, note that w,*-,(x)
= E[v(X,*-,)+~X,~-~
= x] 2 0 = w,.(x).
EXIT IN DECLINING
INDUSTRIES
45
Again, by induction, if w~+~- w~+~L 0 for 0 5 t I t* - 2, then u,(x) ut+r(x) = 6(w,+r(x) - We+*) 1 0. Furthermore, Lemma 2.1 and the assumption of stochastic dominance imply w,(x) 2 w,+,(x) for any x E 93 since u,(a) is increasing. The facts that $I&can be written as {x: x < yr} with y, = inf{x: U,(X) r 0) and that B, C_B,+, for t 2 0 follow from the monon tonicity of u,(x) in t and x. Note that u,(X,) is just the expected discounted payoff from time t if the firm is in the industry at t, decides to stay for one more period, and employs the optimal stopping time from t + 1 on. Therefore, the optimal policy is to stop the first time that X, falls so low that u,(X,) < 0 or, equivalently, X, < yt. Figure 1 illustrates a sample path of X, the cutoff point y, as a function of t, and the optimal stopping time T. The dashed horizontal line represents the zero stage-payoff line, y” = inf{z: r(z) 2 0). The cutoff points yt always lie below this line. That is, at each time t, the firm is willing to sustain a certain level of current loss in the hope of receiving future positive rewards. The cutoff point, y,, increases in t because total future industry profit potential declines over time. The amount of current period losses (?r(yJ when y, < r-‘(O)) that the firm is willing to sustain (in the hope of recovering those losses with future gains) decreases over time because the total potential future gain decreases over time. PROPOSITION 2.2. Suppose r’(x) e n2(x)for x E 9’i or 6’ r a*. Define for i = 1, 2 and t 5 0: wf , ui, yf , and T’ as in Proposition 2.1. Then w: L w:, ui 2 uf, y: s yf for t L 0, and T’ 2 T* as.
Proof. Let t’ = inf{t: 7ri(XI) < 0, a.s.}, i = 1, 2. Obviously, t’ 2 t* and w: = w: = 0 for t B t’. Suppose that w:+, 2 w:+‘. Then U:(X) = n’(x) + &w:+‘(x) 2 n*(x) + ~w:+~(x) = u:(x) for x E ?I&,and this in turn implies
0
1
t’
FIG. 1. Single-firm stopping problem in the monopoly model.
46
FINE AND LI
that w:(x) 2 w:(x) for x E ‘3. That y: 5 yf is obvious by noting that yt = u;‘(O). Therefore, that B: C Bf or that (B:)’ > (B!)” for t 2 0 implies that {T’ > t) = fhcsst{Xs E (Bf)‘} 2 fl~~~~,{X~ E (B?)“} = {T2 > t} for t 2 0. So T’ 2 T2 a.s. The proof is completed by induction. The results for 6’ 2 ij2 follow from a similar argument. n Proposition 2.2 provides a very convenient tool for dealing with the comparative statics analysis of the optimal stopping time. Looking at the stage payoff is sufficient to obtain comparative statics for the dynamic program. For example, consider a monopolist producing a homogeneous good at a constant cost c and having the option in each period whether to stay in the market which is stochastically declining over time, or leave. In period t, let the random variable X1 be the intercept of the linear inverse demand curve; the constant -b is the slope. The opportunity cost of staying in the market is k. (Alternately, the firm must pay a fixed fee, k, each period in order to participate in the production activity.) In each period t, the maximum profit that the monopolist can obtain is 6G)
=
(1/4b)(X, - c>~ - k,
ifX, 2 c;
-k
otherwise.
Clearly r(e) is an increasing function. Also, 7r decreases as c, 6, or k increases. This fact, together with an application of Proposition 2.2, implies that the optimal exit time T decreases almost surely as c, b, or k increases or 6 decreases. That is, the monopolist optimally exits earlier when the marginal cost of production is higher, a better alternative opportunity exists (a higher k), the price is less sensitive to the quantity change (a higher b), or the firm is less patient (a smaller 8).
3.
EQUILIBRIA
IN THE TWO-FIRM
EXIT
GAME
For i, j = 1,2, denoted by rij(Xt) the stage payoff to firm i in period t when there arej firms in the market and the state is X, . For j = 1,2 and for each firm i, the stage payoff, rij(*), is increasing. Since firms prefer monopoly to sharing the market, we assume that each firm has a higher stage payoff when it is the only firm in the market. That is, 7~&) < nil(z) for z E ‘3 and i = 1,2. At each time t, the active firm(s) observe {Xs ; s 5 t} and then (simultaneously) decide whether to stay in the market and earn rij(Xr) (depending upon the number of firms who stay in) or to quit and get zero payoffs thereafter. Firm i’s strategy is a stopping time Ti, and the payoff at time t to firm i given his rival’s strategy T-i = Tj, j f i, is completely determined by observing the history {Xs; s 5 t}, i.e.,
EXIT
IN
DECLINING
47
INDUSTRIES
Let 6i (0 < 6i I 1) be the discount factor for firm i. DEFINITION 3.1. (Tr , T2) is a Nash equilibrium and T2 are stopping times and for i = 1, 2,
E [z
ahf(Xt,
T-i)lXa]
2 E [g
shf(X,,
in stopping times if T,
T-i)lXo],
a.s.,
for any stopping time 7. We are interested in the subgame-perfect equilibria in the sense of Selten (1975) (see also Kreps and Wilson, 1982). To define a pure strategy subgame-perfect equilibrium in the stopping game setting, we must work on the extensive form of the game and specify afamily of stopping times, one for each history. The only restriction is that they satisfy Bayes’ rule. Because of the simple information requirement in the exit game, this can be done as follows. Let T(t): fi + XT be a stopping time defined on XT {T 5 s} E 8, for every s E N: where X: = {t, t + 1, . . .}, and CI is the space of sample paths of {X,; t I O}. For each time t 5: 0, let T!(t) be the time at which firm i exits given that time t has been reached and firm j 0’ # i) is still in the market, and To(t) be the time at which firm i exits given that time t has been reached and firm j (j # i) has exited before t. Note that T:(t) 2 t and T?(t) L t. Firm i employs exit time T!(t) if its rival is still in the market at t; otherwise firm i exits at T?(t). Thus, firms i’s strategy at time t is T,(t) =
T:(t)
if firm j is in the market at time
T?(t),
otherwise.
t;
Bayes’ rule requires that T!(t) = T;(s) for t 5 s 5 T;(t) and that T?(t) = T?(s) for t 5 s 5 T?(t). DEFINITION 3.2. Let (TI , T2) = ({Tl(t)}tbO, { T2(t)},,o) be a pair of stopping-time families defined as above. ( T1, T2) is a subgame-perfect equilibrium if, for each t 2 0, (T,(t), T2(t)) is a Nash equilibrium. That is, if both firms are in the market in period t, for each t 2 0, i = 1,2, and any stopping time S(t).
T:(t) = T;(s) for t 5 s 5 T;(t), and T?(t) must be optimal when firm i is left in the market alone at t.
48
FINE
DEFINITION 3.3.
j=
AND
LI
Firm 1 is stronger than firm 2 if nij(z)
> nZj(z) for
1,2andzE%.
Ghemawat and Nalebuff first analyze a model where the firms have identical variable costs but the larger firm has larger fixed costs. In their Section 4, for a specific example, they look at how big the variable cost difference, due to economies of scale, would have to be to negate the fixed-cost advantage of the small firm. Our analysis shows that looking at the stage payoff, which nets out the effects of fixed and variable costs, is sufficient to determine which firm has the advantage. DEFINITION 3.4.
Firm 1 is more patient than firm 2 if 6r > &.
To aid the game-theoretic analysis that follows, we first solve four single-firm problems, two for each firm. That is, by applying the results in the previous section, we derive the optimal stopping time for each firm i as if there were j firms in the market throughout i’s stay in the market. Each single-firm problem is indexed by (i, j) and takes mij as the stage payoff. Following the notation in the previous section, define recursively for i,j= 1,2 Uy(X)
=
T;j(X)
+
6iWFil(X),
w&,(x)
= E [ uji+l(Xt+,)+lXt = x],
WY = 0 for t 2 tQ, where t$ E inf{t 2 0: n;j(Xt) < 0, a.s.}, and Tij(s) = inf{t 2 s: X, < yji}, where y? = inf{z E CR: uij(z) 2 0). By Proposition 2.1, we know that e(t) = T,,(t) for i = 1, 2 and for all t. Therefore, the real problem left to determine a subgame-perfect equilibrium is to specify
TI.
By Proposition 2.2, we also have that for i = 1, 2, and for t B 0, wf!+i 2 w:‘,i, yf’ I y:‘, and T,,(t) L Tdt) a.s., and that forj = 1, 2, and for t 2 0, U >- wt+1, 3 Yt‘j <- Yt‘j >and Tv(t) z Tzj(t) a.s. if firm 1 is stronger or more wt+1 patient than firm 2. Intuitively, one may think that since Tii and Ti2 are firm i’s most optimistic and most pessimistic decision rules, Tjl and T,-Jmay impose upper and lower bounds for firm i’s equilibrium stopping time. The following proposition proves that this intuition is correct. Since the equilibrium exit times can be nontrivial to calculate (see below) and are not, in general, unique, it is useful to have these bounds on the exit times. PROPOSITION 3.1. TisT;l,u.s.,fori=
Suppose (TI , T2) is a Nash equilibrium.
Then T,-2I
1,2.
Proof. Suppose Ti > Til with positive probability. Let T s Ti A Ti,. Then, given T-i, the difference between the equilibrium payoff to i and the expected payoff to i if T is employed is
EXIT
=
IN
DECLINING
INDUSTRIES
49
-E ["A2m' 6f(ni,(Xr) - ~dX~))IXo] f=TAT-,
The first term is nonpositive since ~T;I > ~TTT~Z and T A T-i 5 T; A T-;. The second term is negative since, by Proposition 2.1, E [cg;:, S:ni,(X,)lS,,] < 0 if Ti > Ti, with positive probability. In addition, also by Proposition 2.1, E [&,E’ Sf’rrii(X,)]%r,] 2 0 which implies that the third term is nonpositive. This contradicts that T, is the best response to T-;. Therefore, T; CCTi, a.s. for i = 1, 2. n The other part of the proposition is proved similarly. We now turn to an asymmetric situation in which firm 1 is stronger or more patient than firm 2; that is, rlj > n~zjand 61 = 62, or 61 > 82 and rij = n2j. By Proposition 3.1, one natural equilibrium we can easily infer is (T,, , T22).Since firm l’s equilibrium strategies are bounded below by T12 (LT~~), its best response to T22is T,, ; and since firm 2’s equilibrium strategies are bounded above by T2,(IT,,), its best response to T,, is T22. Furthermore, this equilibrium is subgame perfect: given the history up to time t 2 0, (T,,(t), T22(f))is an equilibrium for the subgame beginning at time t. However, as will be shown, this is only one of the many possible subgame-perfect equilibria for this game. Therefore, the assumed asymmetry cannot assure that the stronger or more patient firm always outlasts its rival. Our principal result, characterizing all possible pure-strategy subgameperfect equilibria, is Proposition 3.2. Reference to Fig. 2 aids understanding of the multiplicity of equilibria and the form of the equilibrium exit sets characterized in Proposition 3.2. In region R, , both firms are profitable as duopolists and neither has any incentive to exit. As industry demand declines, X, may fall into region R2, where the strong firm (firm 1) is profitable as a duopolist but the weak firm (firm 2) is not. For X, in this region (approximately), the weak firm will exit. We add the modifier
50
FINE AND LI
0
T 22
T,*
t;2
t;2
T*,
t;,
T,,
=t;,
FIG. 2. Four single-firm stopping problems in the duopoly model.
“approximately” because the precise boundaries must be calculated by backward recursion, as is done in the proof of Proposition 3.2. The source of the multiplicity of equilibria arises from the possibility that X, could jump directly from region RI to R3. In region R3 neither firm is viable as a duopolist but either is viable as monopolist. Thus, exit by either firm for X, in this region could constitute equilibrium behavior. Again, the exact boundaries of these regions must be calculated by backward recursion. Any equilibrium can be characterized by an “agreement” set A, E Rj such that if X, jumps into region R3 from region RI , then if X, E A,, firm 1 will exit, and otherwise, firm 2 will exit. Each measurable subset A, corresponds to a distinct equilibrium for the exit game. Proposition 3.2 uses backward recursion to define precisely the sets that generate the equilibrium behavior we describe. PROPOSITION3.2. Supposefirm 1 is stronger or more patient thanJirm 2. Then ( TI , T2) is a subgame-perfect equilibrium if and only if for each s L 0, T?(s) = Tir(s), and T:(s) = inf{t L s: X, E &,}for i = 1,2, where the exit sets Bit, i = 1, 2, t 2 0, are defined recursively
BI, = C-03,Y:‘)
U (AS, II Ad
where A, is a Bore1 set contained Aif E {x: STAT +
w:+,(x) = 0, for t 2 tri, fir where
0 5 t 9 tf,,
U A,,
B2t
=
A2r
II
AC,
in the set A,, fl Azt fl [y:‘, CQ), ~W~+I(X)
and
<
O},
i=
1,2,
wl+dx> = mf+dx,+*)+lxt
= xl,
EXIT
uf+*(x>
=
[ni2(X)
+
IN
DECLINING
~Wf+*(X)Ilg~,,+,(X)
51
INDUSTRIES
+
[ril(X)
+
SWfY2(X)]lgj,,+,(X),
j # i. Moreover
w:’ % wf I wit’.
Proof.
Appendix
B.
n
The proposition shows how we can obtain all subgame-perfect equilibria by varying A,, a Bore1 subset contained in A,, fl Azt n [yf’, a), for every t. Note that if X, E Al, fl AZ, fl [y:‘, m), then firm l’s expected reward is nonnegative if firm 2 exits at t (X, E [y:‘, ~0))and is negative if firm 2 stays (X, E Al,); firm 2’s expected reward is nonnegative if firm 1 exits (X, E [y:‘, m)) and is negative if firm 1 stays (X, E AZ,), assuming some equilibrium strategies will be employed from t + 1 on. We recognize that the normal form of this situation is an asymmetric “battle of sexes” in which there are three Nash equilibria: firm 1 stays and firm 2 exits, firm 1 exits and firm 2 stays, and a mixed strategy equilibrium. As we know, the mixed strategy equilibrium is inferior since it gives positive probabilities to inefficient outcomes (both stay and both exit). Also, we do not consider mixed strategies or behavioral strategies in this paper. Therefore, either the exit of firm 1 or the exit of firm 2 is consistent with equilibrium behavior and the multiplicity of the subgame-perfect equilibria arises. In Proposition 3.2, A, consists of the states in which firm 1 exits, and the remaining states in A,, fl AZt fl [y:‘, x) are consistent with the second equilibrium outcome in which firm 2 exits. Also note that for each t, Ait, i = 1,2, are functions of A,+ 1, . . . , A,;, , the sets picked from t + 1 on. Among the many equilibria for this game are two extreme subgameperfect equilibria which give upper and lower bounds on any equilibrium stopping times (T, , T2). The first of these, ( f1 , f*), is obtained by letting A, = a, = 0 for all t. It is not difficult to check that (fi, ?‘J = (Tll, TZ2)a.s. (see the proof of Proposition 3.3). In fact, (T1, , TZ2)is the equilibrium most preferred by firm 1, the stronger firm, because, as will be shown later, it gives firm 1 the latest possible exit time and gives firm 2 the earliest possible exit time. On the other hand, the second extreme equilibrium, (?;I, Tz), gives firm 2, the weaker firm, the latest possible exit time and gives firm 1 the earliest possible exit time. This equilibrium is obtained by setting!, = A, = Ale fl x2, rl [y:‘, 0~) for all t, where Ait = Air(Ar+l, . . . ) A,;,). 3.3. There are two extreme subgame-perfect equilibria, APROPOSITION ,. (T1 , T2) and (Tl , Tz), in which for i = 1, 2 and each s 2 0, pi(s) = Tii(s) and F?(s) = inf{t 2 s: X, E B1,}, where the exit sets Bit are dejined recursively as . Bz, = AZr n A:‘, Bl, = C-w, yf’) U A,,
52
FINE
AND
LI
Pi;,= {x E a: %-j?(X) + SW;+,(x) < O}, wf+,= 0, for t 2 tfj, and Wj+,(x) = E[if+,(X,)‘lX,
= x],
for 0 5 t 5 17, , where
ui+I(x) = [7Ti2(x) + t5W:+Z(X)llBj,,+,(X) + [Til(X)+ 6wf~I(X)ll~,,,+,(X)~ j f i. Suppose ( T1 , T2)- is a subgame-perfect and Tz2 5 T2 5 Tz, a.s. Proof.
Appendix
equilibrium.
Then T, 5 TI 5 TI1
C.
n
Since firm 1 is assumed to be stronger than firm 2, the first extreme case This equilibrium is also simple in structure, i.e., a threshold equilibrium. The following proposition gives a necessary and sufficient condition for ( T1I , Tz2) to be unique. Before proving the proposition, we need a proper definition of uniqueness of a subgame-perfect equilibrium in stopping games. Basically, for two subgame-perfect equilibria to be equal, we require that the Nash strategies of the two subgame-perfect equilibria for any subgame be equal almost surely with respect to the conditional probability in the subgame. (TI1 , T& is, in some sense, the more appealing equilibrium.
DEFINITION 3.5.
areequaliffori=
Two subgame-perfect equilibria (T; , TI) and (T;’ , T$) T:!(t),P:,a.s.foralltrOandforallxEYL
1,2,Tl(t)=
PROPOSITION3.4. Supposejrm 1 is stronger 2. Then, (TI 1, T& is the unique subgame-perfect Pf-,{X,EAI,fl [y:‘,m)} = 0,forxERaand 15 determined recursively as in Proposition 3.2 by Proof.
Appendix
D.
or more patient thanfirm equilibrium if and only ij t 5 tTI - 1, wherra,,are letting A, = 0 for cl11t. n
Note that a,, n [y:‘, 00) 5 [y:,, yt” ). Therefore, the proposition indicates that the stronger firm never exits first if either the payoffs are so asymmetric that n12 2 7r2, (or equivalently, y!* 5 Y?‘) or the process X is sufficiently smooth that it never jumps over the interval [yf*, y:‘). We show below that Proposition 3.4 implies the uniqueness result of Ghemawat and Nalebuff (GN) and why their uniqueness result fails if industry demand suffers a severe negative shock. We mention here that we do not attempt to discuss the relations between discrete-time and continuous-time models in general, but in the context of a deterministic
EXIT
IN
DECLINING
INDUSTRIES
53
world, discretizing a continuous-time model is quite straightforward. Note that tz = inf{t I 0: ril(X,) < 0, a.s.} and that r& I t& by the asymmetric assumption. Thus, if t L tli* , both firms will leave and if t& I r < tf, , firm 1 may stay and firm 2 will definitely leave. Consider t = tir - 1. Note that y:’ is determined by mTT2r(yf’)= 0 since M$’ = 0. If ~TT~z(Y?)+ ti~:!~(y:‘) 2 0, thena,, rl [y:‘, ~0) = 0 sincea,, = {x: nj2(x) + 6w::r(x) < 0) for t = t& - 1. The condition for uniqueness is satisfied for period t. In general, there is a unique subgame-perfect equilibrium in which the strong firm (firm 1) leaves last if and only if the losses which firm 1 may take in period t, 7rr2(x), are outweighed by the expected equilibrium profit from period t + 1 on, 6$!+,(x), for all possible states, x, in which firm 2 is viable as a monopoly, for all periods t. To pass on to GN’s result as a special case, we simply discretize their continuous-time model in such a way that the length of each period is very short, and hence the size of profits or losses per period is very small while the expected future monopoly profit (6~::~) remains strictly positive and sizable (the discount factor is close to one). Therefore, the condition for uniqueness holds and we can work backward. This is exactly the essence of GN’s proof of the uniqueness. However, their proof may fail if we drop the continuity assumption in their model. In particular, if at a certain point of time, say t*, the demand drops drastically from a level at which both firms are viable as a monopoly but neither is viable as a duopolist to a level at which neither firm is viable as a monopoly, then for t = t * - 1, the duopoly loss of firm 1 will never be compensated by the future profits independent of the fineness of the time discretization. Proposition 3.4 provides a guideline to uniqueness results in more general settings. The following example is an application of the model to a market setting in which competing firms have not only the production decisions in each period but also the option to exit the market. A Cournot duopoly faces a stochastic linear inverse demand, P, = X, - hQ, where Q is the total output of the duopoly, and {X,, t 2 0} is Markovian and stochastically decreasing. The firms produce a homogeneous good at constant costs ci, i = 1, 2. Firm i’s opportunity cost of staying in the market is ki. Without loss of generality, we assume cl I c2. If both firms are in the market, the equilibrium stage payoffs are
~dXt)
=
(1/96)(X,
- 2c, + c# - k,,
if X, > 2c2 - cl;
W4bNXt
- cd* - kl,
if cl 4 X, < 2c2 - cl ; otherwise,
-kl, (1/9b)(X, ~22(-&)
- 2~2 + c,)* - k2,
ifX, 2 2c2 - cl;
= -h,
otherwise.
54
FINE
AND
LI
If one firm is in the market alone, the monopoly i=
stage profits are, for
1, 2,
(1/4b)(Xt ~ilW,)
=
-
ci)'
- kiy
if X, L
Ci;
otherwise.
-k/y
It is easy to see that ril I 7~~~for i = 1, 2, and if cl I c2 and kl I k2 then firm 1 is stronger, i.e., mlj 2 m2j for j = 1, 2. From the general results obtained earlier, we know that (Tll, Tz2) is an exit equilibrium. The thresholds y:’ and y:’ can be calculated as y:’ =
CI
+
2~Wl
-
if kl 2 i$wj:,;
~l~::lh
otherwise,
Cl 9
22 _ Yr -
2~2 -
CI
+
3vb(kz
-
if k2 z- &wff,;
82wf:,L
otherwise.
2c2 - c,,
Also, it is easy to see that y:’ increases as cl, b, kl increase or 6, decreases, while y:’ increases as CZ, b, k2 increase or cl, a2 decrease. To see that there might be other subgame-perfect equilibria, we further assume that X, is uniformly distributed on the interval (c2 + 14A - 7At, c2 + 15A - 7At), where A = c2 - cl > 0, and that k = ki = (llb)A2, i = 1, 2. Note that both firms will exit for sure at t = 2 since r;‘(O) = c2 + A, and no firm will exit at I = 1 since 7G1(0) = c2 + 411.< cz + 78. Direct calculation shows that Wl21= E[T2,(Xt)+l = iiq = E[7r,2(X1)+] =
(S3 - 73)A2 157 -k=-k 12b (lo3 - 93)A2 27b
-k=27k
244
Let 6 = 6i = &$. Then yi’ = cz and Alo = (-m, y:], where y;~c,+2~b(k-6iT;)=c,+
l.llA.
Therefore y:, - y;' = c, - c2 + l.llA
= O.llA > 0.
Suppose the initial state X0 = x, x E [yi’, y;). Then firm 2 can stay in and credibly force firm 1 out right away.
EXIT
IN
DECLINING
55
INDUSTRIES
Note that in this example, we need a drastic downward jump of demand in order to make yf’ below yj . The decreasing rate of X,, 7, is carefully picked such that there exists a 6 to make A. # 0. 4.
CONCLUDING
REMARKS
This paper analyzes the pure-strategy subgame-perfect equilibria of an exit game in a stochastically declining market. With trivial modification, an equivalent analysis applies to the case of increasing payoffs which can be interpreted as the entry games played by firms in a growing industry. We find that these games generally have multiple subgame-perfect equilibria. The models of both Ghemawat and Nalebuff, and Fudenberg and Tirole, have demand processes that decline deterministically and continuously. Each obtain a unique subgame-perfect equilibrium. The Huang and Li (1986) paper shows that a unique subgame-perfect equilibrium also obtains with a continuous stochastic process. In Fine and Li (1987) we analyze a continuous-time two-player exit game with stochastic (Poisson) jumps and demonstrate the multiplicity of equilibria in that setting as well. Thus we conclude (Proposition 3.4.) that an underlying stochastic demand process with sizable jumps can yield multiple exit equilibria if no further assumption is made on the magnitude of the asymmetry of the firms’ payoff structure. APPENDIX
A
Proof of Lemma 2.1. First note thatfi >f2 implies {z E %:fi(z) {z E 3: h(z) > 0). Since pl 2 p2, E[fi(Zl)l,,,ffi,(f,(Z1))1
> 0) >
= li/,(z,>*{ h(z) &l(Z)
2 I (f2(L)>O) h(z) 44(z) ’ l,,,.,>o,f2(d &l(Z) L I (f2(z)>o) f-i(z) &2(z) = E[fi(Z2)1(O,m)(f2(Z2))1. APPENDIX
B
Proof of Proposition 3.2. Backward induction. ~P{~~~(X,)r:O}=0,fori=1,2.Also,w~=wf=wf’=0,fori=1,2. The proposition is trivially true.
For t L tf,, P{X, E BiC,}
56
FINE
AND
Now assume that the proposition
A;,Ct-w,yf2)
LI
is true for s 2 t + 1. Note that and
Ai C [yf’, ~1,
(3.1)
since w:‘+r 5 ~f+~ 5 wf:, for i = 1,2. Suppose both firms are in the market at time t. Given any strategy B2, adopted by firm 2 at time t, firm 1 will be in the market alone from t on if X, E B2t, and he will be involved in a game which gives expected payoff w:+~ if X, E B$, and equilibrium strategies follow from t + 1 on. Remaining active at t, firm 1 expects to get
Firm l’s best response then is to exit if u:(X,) < 0 and stay if u,‘(X,) 2 0. Let B1, = {x: u:(x) < O}. Firm 1 will exit if firm 2 exits and X, < yf’ or if firm 2 stays and X, E A,,. That is BI,
Similarly,
=
(B2r
n
Y;‘))
t--w,
U
CBS,
n
(3.2)
Ad.
given firm l’s strategy B1,, 2’s best response should satisfy B2r
=
WI,
n
Y:‘))
C--w,
U
VT,
n
(3.3)
A2r).
Using fact (3.1), we can rewrite (3.2) and (3.3) to be
B,,= {t-w,y:‘)W, n C-m,Y;‘>)>U Wt n AI,) = C--w,yf’) U (Et n Ad, et = m, n [Y?, 9 u {A;,\(B,, n AS,)) 21 = WI, n [yt , 30)) U AL. Substituting
(3.4)
(3.5)
(3.4) into (3.5), we have BSt
=
CBS,
n
AI,
[it
n
21
, ~1)
U
(3.6)
AS,,
since y:’ L y,” . Thus B$ is part of the equilibrium if and only if B$‘, satisfies (3.6) or BS, is a fixed point of&(.) where.&(B) = (B fl A,, rl [yf’, m)) U A$‘, for any B E %(?A). The only possible solution is of the form B& = A, U A;,, where A, C A,, fl [y:‘, ~0). We let A, c A,, fI Azr n [y:‘, w) simply to make A, and A& two disjoint sets. Similarly, we can prove that BI, is part of the equilibrium if and only if
BI,= WI,n At, n [Y:‘, which gives us the equivalent
~1)
U
64%
characterization
n
Al,)
U
t-w,
Y:‘>>
(3.7)
of a generic equilibrium.
EXIT
IN
DECLINING
57
INDUSTRIES
Finally, u:’ I uf d u:’ since w;‘+t % w:+, 5 wf!+t . Therefore, w:’ for i = 1, 2 and for all t. We conclude the induction.
w:’ 5 wf 5 n
C
APPENDIX
Proof of Proposition 3.3. Let ( f1 , fz) be obtained by letting A, = A, = 0 for all t. That (fr, pz) and (Tr , Tz) are subgame-perfect equilibria is a direct corollary of Proposition 3.2. We shall use the induction argument to show the second assertion as well as fi = Tii a.s. as a by-product. First note that for t 2 tfl , iZ: = w,’ = G,,’ = wI“=()andw22=$~~=~~= iV: = 0. This implies that A,, = (-co, y,“) and a21 = (--co, ;:‘), and A11 n as, = 0. Thus B,, = (-co, y)‘), and &, = (-a, y:‘). Assume Wf+t 5 wj+, I $;+I 5 w::,, and w:t, = *?+I 5 w;+, 5 W:+,. Then
C-00,Y:‘) Y:‘)
C-T
2 &r
2 AI, 2 a,, 2 C-00, Y:‘),
=
1
-42l
AZ,
2
x2,
2
(3.8)
Y:‘),
C--x,
and hence d,, n a$ = 0. Using (3.8), we have El,
Y,“>
=
GL
fl
[YE
=')I
U
t-w,
2
(AI,
n
[Y!‘,
~1)
U
(A%
n
AI,)
2
BI,
U
(A&
n
AI,)
U
(-a,
=
A
2 (&, n A,) u (-T
2
(AI, u
II
C-m,
[Y:‘,
~1)
U
t-m,
Y;‘>
Y,“>
(3.9)
yf’)
of’)
= I&, = (--co, yf’). It can be similarly shown that C--m,
Y:‘>
=
B2t
1
B2t
2
(3.10)
B2r.
Therefore, ~7,’5 uf 5 I;f 5 uf’ and u,22=ti:~~~~U:.Weju~tshowiif~ ut , as an example, 8(X,)
= [~IZW,)
+ ~~f+lw,)llE~,(&)
5
+ ~w:+,(m~B;,GG) +
C97l2bK)
+
[~,l(X)
+
~w::I(x,)Il&w,)
[ill
+
~w~:lw,)lls,,(xt)
= d(X)
-
since W:+, 5 w:+li2B2, 2 Bz,, and B5r C B s,. Hence, we have Gf 5 w) 5 r+j I wi’, and wI - $1: 5 w: I ii?:. We conclude the induction. As a
58
FINE AND LI
result, (3.9) and (3.10) imply that FI 5 TI I TI = TII and T22 = f2 5 T2, a.s.
T2 5 n
D
APPENDIX
Proof of Proposition 3.4. For t 1 tf, - 1 and for i = 1, 2, Ai, = a, = (--co, 9;) = (--cc, y:), and A, = [y:‘,-yj2) since $+I = G’j+, = 0. Then Pf-,{& E [yf’, y:“)} = 0 implies T/i = Ti, P:-a.s., for s 2 t - 1, and W: = 6; and 67: = wf2. Suppose Tii(S) = T/(s), i= 1, 2, PS-a.s.,fors z tandforxE%. Then #+I = G:+, and i?j?+, = w::, imply that Alt = a,, and x2, = (-cc, y:‘). Therefore, Bl,\Bl, = B&V& = A, = Al, f~ [y:‘, tQ), and Tii = Ti, i = I, 2, P:-a.s., for s 2 t - 1 and x E CR, by the condition that Pf-,{X[ E A,, n m)} = 0. The induction is completed. [Y?, To prove the necessary condition, let t = sup{s Z- 0: Pz-,{X, E [yz’, yi’)} > 0). Such a t exists if P:-1(X, E als II [y:‘, 03)) > 0 because AI, f~ ru5’9 ~0)C [yj’, yj’). Since Ti = Tit, P:-a.s., for s z t and x E CRand hence 8$ n BZt = [y:‘, y:‘) and a,, = (-w, y:‘), we have P:-,{T2(t)
> T22(t)} z P;-,{X,
E B& fl l?2t}
= P:-,{x, E A,, n [$I, ~1)> 0.
n
ACKNOWLEDGMENTS The authors gratefully acknowledge comments of John Cox, Drew Fudenberg, Chi-fu Huang, Garth Saloner, Jean Tirole, and several anonymous referees. We also acknowledge fruitful discussions with Erhan Cinlar and Michael Harrison, and financial support from Coopers and Lybrand Inc.
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EXIT IN DECLINING
INDUSTRIES
59
HADAR, J., AND RUSSELL, W. R. (1%9). “Roles for Ordering Uncertain Prospects,” Amer. Econ. Rev., March, 25-34. HUANG, C., AND LI, L. (1986). “Continuous Time Stopping Games,” Working Paper No. 17%-86, Sloan School of Management, MIT. KREPS, D., AND WILSON, R. (1982). “Sequential Equilibria,” Econometrica 50, 836-894. MAMER, J. (1986). “Monotone Stopping Games,” Working Paper, Graduate School of Management, UCLA. OHTSUBO, Y. (1986). “Optimal Stopping in Sequential Games with or without a Constraint of Always Terminating,” Math. Oper. Res. 11(4), 591-607. OHTSUBO, Y. (1987). “A Nonzero-Sum Extension of Dynkin’s Stopping Problem,” Math. Oper. Res. 12(2), 277-297. SELTEN, R. (1975). “Reexamination of the Perfectness Concept for Equilibrium Point in Extensive Games,” ht. J. Game Theory 4, 25-55. SHIRYAYEV, A. (1978). Optimal Stopping Rules. New York/Berlin: Springer-Verlag.