endogenous stackelberg equilibria with identical firms

GAMES AND ECONOMIC BEHAVIOR 12, 81–94 (1996)ARTICLE NO. 0006

Endogenous Stackelberg Equilibria with Identical Firms*

DEBASHIS PAL

Department of Economics, University of Cincinnati, Cincinnati, Ohio 45221

Received January 26, 1993

A symmetric duopoly with two production periods is considered. In the firstperiod firms simultaneously choose initial production levels. These outputs becomecommon knowledge and then additional second period outputs are chosen simulta-neously. The market clears after the second period. If production costs fall slightlyover time, a symmetric equilibrium in pure strategies does not exist. There are twoStackelberg leader–follower outcomes; in each outcome one firm produces only inthe first period and the other produces its best response in the second period. Thispaper allows the firms to use mixed strategies and analyzes a symmetric mixedstrategy equilibrium. Journal of Economic Literature Classification Numbers: D43,L13. 1996 Academic Press, Inc.

1. INTRODUCTION

In analyzing quantity setting duopoly games, the two most widely usedmodels are Cournot and Stackelberg. These models differ in the timing ofthe firms’ moves. In Cournot, firms move simultaneously while in Stackel-berg they move sequentially. Traditionally this feature has been treated asexogenously given. Recent work recognizes that whether duopolists movesequentially or simultaneously should instead be determined endogenouslyfrom the model. Gal-Or (1985) and Dowrick (1986) study whether eachfirm prefers to move first or to move second. In particular, they show thatfor quantity competition with substitute goods a leader earns more profitthan a follower. Neither Gal-Or nor Dowrick determine conditions thatlead to Cournot as opposed to Stackelberg outcomes emerging endoge-nously from the model. Hamilton and Slutsky (1990) address this question

* I am indebted to two anonymous reviewers for their helpful comments. I am also gratefulto Simon Anderson, Robert Drago, John Gross, Barnali Gupta, Jonathan Hamilton, JohnHeywood, and Steven Slutsky.

810899-8256/96 $12.00

Copyright 1996 by Academic Press, Inc.All rights of reproduction in any form reserved.

82 DEBASHIS PAL

in an extended duopoly game by allowing the firms to choose the time ofplay in addition to choosing a specific action. If the firms choose actionsat different times, then the firm choosing a later time observes the actionchosen by the firm playing first, giving rise to a sequential outcome. If thefirms choose actions in the same period, a simultaneous play occurs. If afirm can play early only by selecting an action, multiple equilibria exist:both simultaneous and sequential outcomes can emerge.1 Saloner (1987)extends the usual quantity setting duopoly game by allowing two productionperiods before the market clears. In the first period the firms simultaneouslychoose initial production levels. These outputs become common knowledgeand then the firms choose additional second period outputs simultaneously.The market clears only once, after the second period. If production costsdo not vary across the two periods, Saloner shows that both Cournot andStackelberg outcomes can emerge. In fact, any point on the outer envelopeof the reaction functions between (and including) the firms’ Stackelbergoutputs is sustainable as a subgame perfect Nash equilibrium (SPNE) out-come. Pal (1991) shows that this indeterminacy is eliminated when produc-tion costs vary across periods. If production is cheaper in the first period,then there is a unique SPNE at which both firms produce only in the firstperiod, producing their single-period Cournot–Nash quantities. If produc-tion costs are slightly higher in the first period, then there are two SPNE.At each, one firm behaves as a Stackelberg leader producing only in thefirst period, while the other behaves as a follower and produces its bestresponse in the second period. If costs are significantly higher in the firstperiod, then again there is a unique SPNE, but now the firms produce theirsingle-period Cournot–Nash quantities in the second period. Thus, whencosts fall slightly over time, Stackelberg outcomes are generated endoge-nously; otherwise the Cournot outcome is the unique SPNE outcome.

Although these studies have made significant progress toward endoge-nously determining Cournot or Stackelberg outcomes, one conceptual ob-jection remains. To illustrate, consider two identical firms and a case whenthe costs of production fall slightly over time so that the Stackelberg out-comes are the only SPNE outcomes (for this result, see Robson, 1990a;and Pal, 1991).2 That is, there are two SPNE, at each of which one firmbehaves as a leader while the other behaves as a follower. Thus, althoughthe firms are identical, a symmetric equilibrium does not exist. Now, the

1 Anderson and Engers (1994) introduce endogenous order of entry in a Stackelberg modelwith n $ 2 number of firms. Firms compete over entry times to ensure a specific position inthe sequence of moves. If entry costs decrease monotonically over time, profit differencesamong firms disappear.

2 Endogenous Stackelberg outcomes are not limited to this illustration and may arise inother occassions as well. For example, in the context of price competition with differentiatedproducts, sequential choice of actions may emerge endogenously (see Robson, 1990b).

ENDOGENOUS STACKELBERG EQUILIBRIA 83

question is, since there are two SPNE outcomes, which one of them willactually occur? Moreover, since a leader earns more profit than a follower,each firm will prefer that equilibrium at which it is the leader and its rivalis the follower. How will the firms coordinate who will be the leader andwho will be the follower? Traditionally, the answer to this question is toassume exogenous factors which would assign the leader. In a perfectlysymmetric situation, however, it is difficult to specify reasonable exogenousconditions making one particular firm the leader. On the other hand, with-out reasonable exogenous conditions we lack convincing arguments fa-voring a specific SPNE outcome. Relying on mechanisms like tossing coinsto decide which one of the two SPNE will be reached is ad hoc. Since eachfirm prefers to be a leader, there is no reason to expect that the observedoutcome will at all correspond to either one of the two SPNE outcomes.3

One crucial assumption which generates this problem is that, in theprevious works, the firms are limited to pure strategies. An alternativeapproach to predicting the outcome of this model is to expand the firms’strategy sets by allowing them to use mixed strategies and to look for asymmetric mixed strategy equilibrium. The advantage of a symmetric mixedstrategy equilibrium is that ex ante we do not need to appoint a particularfirm as the leader. In equilibrium, both firms play identical strategies withcorresponding probabilities, while ex post the outcomes can be either sym-metric or asymmetric. It is shown that a firm may be a leader/follower bychance and we can determine the probability of such an event. Moreover,a symmetric mixed strategy equilibrium provides identical expected profitsfor both firms and thus eliminates the difference in profits which is observedat each of the two SPNE in pure strategies.4

This paper considers the extended version of the usual quantity-settingduopoly model with two identical firms by allowing two production periodsbefore the market clears and analyzes the symmetric equilibria in mixedstrategies. It is shown ex post, that only three types of outcomes are possible:(i) Both firms produce only in the second period, each producing its singleperiod Cournot–Nash output; (ii) there is a ‘‘Stackelberg-like’’ outcome,where one firm produces only in the first period and the other reacts byproducing its best response in the second period; or finally, (iii) there is a‘‘Stackelberg warfare,’’ where both firms produce only in the first period

3 In general, a similar problem always exists in the presence of multiple equilibria. Usually,different refinements are used to select one equilibrium from the set of equilibria. In thiscase, however, the two SPNE are perfectly symmetric in nature. Therefore, without reasonableexogenous conditions making one particular firm the leader, it is not possible to find aconvincing refinement that eliminates one of the two equilibria.

4 It is certainly desirable to focus on pure strategies and allow asymmetric firms to showwhen leadership will endogenously emerge and who will be the leader within the context ofa reasonable economic model.

84 DEBASHIS PAL

but both produce more than their Cournot–Nash outputs. Thus, the useof mixed strategies shows that Stackelberg warfare can arise as a conse-quence of subgame perfect Nash equilibrium behavior and is compatiblewith mainstream game theoretic analysis.

The remainder of the paper is organized as follows. Section 2 describesthe model. Results and proofs are presented in Section 3, and Section 4concludes the paper.

2. THE MODEL

In the spirit of Saloner (1987) and Pal (1991), consider a symmetricduopoly with two production periods, where the market clears at the endof the second period. In the first period, firms x and y simultaneouslyproduce outputs qx

1 and qy1 , respectively. These outputs become common

knowledge and, in the second production period, the firms simultaneouslyproduce nonnegative outputs qx

2 and qy2 . Firms are allowed to use mixed

strategies during output choices in both periods. That is, firm i (i 5 x, y)may choose a cumulative distribution function (cdf) specifying the probabil-ities with which it chooses its first period outputs. After observing therealized (qx

1 , qy1), it may again choose a cdf specifying the probabilities with

which it chooses its second period outputs. After the second period, priceis determined from the inverse demand function P(q x

1 1 qx2 1 qy

1 1 qy2).

The firms choose outputs to maximize expected profits. Within a period,the firms have the same constant marginal cost of production. However,production costs vary across periods. Let cj be the marginal cost of produc-tion in period j, j 5 1, 2.

Assume that qij [ [0, Q̂] ;i 5 x, y and j 5 1, 2, where Q̂ is sufficiently

large so that an output larger than Q̂ will never be chosen by any firm inany period. Then the game can be formally defined as, G 5 (N, S, P),where N 5 hx, yj is the set of players, S 5 Sx 3 Sy is the strategy spacefor the game, and P is the payoff vector. The strategy space for firmi (i 5 x, y), Si, is defined as Si 5 C X I, where C is the set of cdf definedover [0, Q̂] and I is the set of functions that map [0, Q̂] 3 [0, Q̂] into C.That is, firm i’s strategy is to choose a cdf F i

1(qi1) defined over the first

period outputs qi1 [ [0, Q̂] and a cdf F i

2(qi2 u (qx

1 , qy1)) (as a function of the

realized (qx1 , qy

1) in the first period) defined over the second period outputsqi

2 [ [0, Q̂]. For s [ S, define firm i’s payoff function Pi(s) 5

E(fi(qx1 , qx

2 , qy1 , qy

2)), where fi(qx1 , qx

2 , qy1 , qy

2) 5 qi1[P(qx

1 1 qx2 1 qy

1 1 qy2) 2

c1] 1 qi2[P(qx

1 1 qx2 1 qy

1 1 qy2) 2 c2] and E(fi) is the expected value of fi.

For firm i, define the single-period reaction function5

5 By ‘‘single-period reaction function’’ we mean the reaction function corresponding to astandard Cournot model, which has only a single production period.


Ri(qkuc) 5 argzmax z . (P(z 1 qk) 2 c), i 5/ k 5 x, y

We assume that these reaction functions are ‘‘well behaved.’’6 Let (Nx(cj),Ny(cj)) be the unique single-period Cournot–Nash equilibrium outcomegiven the marginal cost cj , j 5 1, 2.

Let fiL(z, c1 , c2) 5 z · [P(z 1 Rk(zuc2)) 2 c1], i 5/ k 5 x, y, be the profit

of firm i if it produces an output z in the first period and firm k (k 5/ i)produces its best response in the second period. For simplicity, we assumea unique Stackelberg point exists for each firm. Denote these by

Si(c1 , c2) 5 argzmax z · [P(z 1 Rk(zuc2)) 2 c1], i 5/ k 5 x, y,

When firms use only pure strategies, the equilibrium in this model isgiven in the following result, shown in Pal (1991). Let f i*

c be the equilibriumprofit of firm i, if both firms produce only in the second period and fi*

L bethe equilibrium profit of firm i, if firm i produces only in the first periodand the other firm produces only in the second period. For a given c2 , letc*

1 be the critical level of first-period marginal production cost such thatfi*

c 5 fi*L .

PROPOSITION 1. (a) If c1 , c2 , then the unique SPNE outcome of thistwo-period Cournot model is h(qx

1 5 Nx(c1), qx2 5 0), (qy

1 5 N y(c1), qy2 5 0)j.

(b) If c1 . c2 then ' a c*1 . c2 such that

(1) if c1 . c*1 , then the unique SPNE outcome of this two-period

Cournot model is h(qx1 5 0, qx

2 5 Nx(c2)), (q y2 5 0, qy

2 5 N y(c2))j,(2) if c1 , c*

1 , then there are two SPNE outcomes, as given by h(qx1 5

Sx(c1 , c2), qx2 5 0), (qy

1 5 0, qy2 5 Ry(Sx(c1 , c2) u c2))j h(qx

1 5 0, qx2 5 Rx(Sy(c1 ,

c2) u c2)), (q y1 5 Sy(c1 , c2), qy

2 5 0)j, or

(3) if c1 5 c*1, then there are three SPNE outcomes, as given in (1)

and (2).

In this paper we assume that c2 , c1 , c*1 . From Proposition 1, it follows

that the two Stackelberg outcomes are the only SPNE outcomes in purestrategies. In each outcome, one firm behaves as a leader by producingonly in the first period, while the other behaves as a follower and producesits best response in the second period.

6 By ‘‘well behaved’’ we mean 0 . [Ri(qk u cj)/qi] $ 21 and [Ri(qk u cj)/cj] , 0. Thefirst condition ensures the existence of a unique single-period Cournot–Nash equilibrium. Aset of sufficient conditions for Ri functions to be ‘‘well behaved’’ is that P(qi 1 qk) is strictlypositive on some bounded interval (0, Q) on which it is twice continuously differentiable,strictly decreasing, and concave. P(qi 1 qk) 5 0 for qi 1 qk $ Q.

86 DEBASHIS PAL

3. MIXED STRATEGY EQUILIBRIUM

The game discussed above is symmetric, so by Dasgupta and Maskin(1986) it must have at least one symmetric equilibrium. As shown in Proposi-tion 1, for c2 , c1 , c*

1 , there is no symmetric equilibrium in pure strategies.Hence, there must be at least one symmetric equilibrium in mixed strategies.

The following lemma is used to prove our results. Saloner (1987) provedthis lemma for c1 5 c2 . The result, however, remains valid for c1 5/ c2 andis used in Pal (1991).

LEMMA 1. Given the first-period outputs (qx1 , qy

1), the second-periodunique equilibrium outputs (in pure strategies) for firm i (i 5 x, y) areas follows:

qi2 5 0 if qi

1 $ Ri(qk1 u c2), qk

1 $ Rk(qi1 u c2)

5 Ni(c2) 2 qi1 if qi

1 # Ni(c2), qk1 # Nk(c2)

5 0 if qi1 $ Ni(c2), qk

1 # Rk(qi u c2)

5 Ri(qk1 u c2) 2 qi

1 if qi1 # Ni(c2), qk

1 $ Nk(c2),

and qi1 # Ri(qk

1 u c2)

This lemma says if (qx1 , qy

1) lies on or outside the outer envelope of thesecond-period reaction functions [area ‘‘D’’ in Fig. 1], then neither firmproduces in the second period. If both firms produce less than the second-period Cournot outputs in the first period [area ‘‘A’’ in Fig. 1], then eachproduces up to its second-period Cournot level. If one exceeds its second-period Cournot output and the other does not [areas ‘‘B’’ and ‘‘C’’ in Fig.1], the latter produces in the second period its best response to the former’sfirst-period output and the former does not produce. Given a pair of first-period outputs (qx

1 , qy1), Fig. 1 shows the second-period equilibrium outputs

(with c1 , c2). Firm x does not produce in the second period, whereas firmy’s second-period output equals Ry(qx

1 u c2) 2 qy1 .

From Lemma 1 it follows that, for each level of (qx1 , qy

1), a unique purestrategy equilibrium solution follows for the second-period outputs. Also,it can be shown that mixed strategy equilibria can be ruled out duringsecond-period output choice.7 Therefore, the firms mix only over the levels

7 If (qx1 , qy

1) lies on or outside the inner envelope of the reaction functions Rx(qy u c2) andRy(qx u c2), then qx

2 5 0 is a dominant strategy for firm x or qy2 5 0 is a dominant strategy for

firm y, or both qx2 5 0 and qy

2 5 0 are dominant strategies for firms x and y, respectively.Together with ‘‘well behaved’’ single-period reaction functions, this rules out mixed strategyequilibrium. If (qx

1 , qy1) lies inside the inner envelope, then the second-stage game is (strict)

dominance solvable and mixed-strategy equilibria can be ruled out.


FIG. 1. First-period and second-period reaction functions.

of first-period output and in the second period; after the realization of(qx

1 , qy1) a unique nonrandom (qx

2 , qy2) follows accordingly.

Recall that Fi1(qi

1) is the cumulative distribution function (cdf) specifyingthe probability with which firm i chooses qi

1 in the first period. In choosingFi

1(qi1), firm i takes Fk

1(qk1) as given (i 5/ k 5 x, y). Let Fi

1(qi1) 5 Fk

1(qk1) 5

F*1(q1), i 5/ k 5 x, y, be the cdf corresponding to a symmetric mixed

strategy equilibrium.The next three propositions characterize F*

1(q1).

PROPOSITION 2. F*1(N(c2)) 5 F*

1(0) where N(c2) 5 Nx(c2) 5 N y(c2).8 Thatis, the firms never mix over outputs q1 [ (0, N(c2)].

Proof. It is sufficient to show that for any qk1 the payoff for firm i from

choosing qi1 5 0 is always higher than choosing qi

1 [ (0, Ni(c2)].If qk

1 . Nk(c2) then qk2 5 0 for any qi

1 . (This follows from Lemma 1.)Then, since c2 , c1 , it is best for firm i to choose qi

1 5 0 and produce itsbest response in the second period.

If qk1 [ [0, Nk(c2)] and qi

1 [ (0, Ni(c2)], then from the second period’sequilibrium strategies (Lemma 1) we have qi

2 5 Ni(c2) 2 qi1 and qk

2 5Nk(c2) 2 qk

1 . Clearly firm i can do better by choosing qi1 5 0, because c2

, c1 . n

PROPOSITION 3. F*1(0) . 0; that is, the firms choose q1 5 0 with posi-

tive probabilities.

8 Since the firms are identical, Nx(c2) 5 Ny(c2).

88 DEBASHIS PAL

Proof. Suppose F*1(0) 5 0. Now, from Proposition 2, F*

1(N(c2)) 5F*

1(0) 5 0. Therefore, in that case, both firms mix only over outputs qi .N(c2). However, this cannot be an equilibrium outcome because if firm kmixes only over qk

1 . Nk(c2), then firm i (i 5/ k 5 x, y) can do better bychoosing qi

1 5 0 and producing its best response in the second period. n

PROPOSITION 4. F*1(N(c2)) , 1; that is, the firms must mix over outputs

q1 . N(c2).

Proof. Suppose F*1(N(c2)) 5 1. Now, from Proposition 2, F*

1(0) 5F*

1(N(c2)) 5 1. Therefore, both firms must choose q1 5 0 with probability1. However, this cannot be an equilibrium outcome, since if firm k choosesqk

1 5 0 with probability 1 then firm i can do better by choosing its Stackelbergleader output. Since c2 , c1 , c*

1 , the leader output would generate a higherprofit for firm i than qi

1 5 0. n

These results (Lemma 1 and Propositions 2–4) indicate the nature of asymmetric mixed-strategy equilibrium. First, neither firm produces in bothperiods. If a firm produces in the first period then it does not produce inthe second period and vice versa. Second, only three types of outcomesare possible ex post: (i) The firms produce only in the second period, eachproducing its single period Cournot–Nash output; or (ii) we observe a‘‘Stackelberg-like’’ outcome, where one firm acts as a leader by producingonly in the first period and the other acts as a follower by producing itsbest response in the second period; or (iii) we observe an outcome whereboth firms produce only in the first period but both produce more thantheir Cournot–Nash outputs. Since the firms simultaneously produce morethan their Cournot–Nash quantities, we call this outcome a Stackelbergwarfare.

In the case of Stackelberg-like outcomes, sequential moves by the firmsarise as an ex post outcome of a symmetric mixed-strategy equilibrium. Inthis case, however, the quantities produced by the firms do not correspondto the standard Stackelberg leader–follower quantities, where one firm isexogenously specified as a leader. In effect, it is hard to assert whether astandard Stackelberg leader–follower game is being played or whether aspecified outcome of a symmetric mixed-strategy equilibrium is encoun-tered just by observing the timing of the moves.

Dowrick (1986) compares Cournot–Nash (follower–follower), Stackel-berg leadership, and Stackelberg warfare games. He defines Stackelberg war-fare as a situation where both firms produce their respective Stackelberg-leader quantities. Hamilton and Slutsky (1990) point out that Stackelbergwarfare can occur only through error, since this strategy with both firms pro-ducing more than their Cournot–Nash outputs is not sustainable as a SPNEoutcome in pure strategies. In this paper, Stackelberg warfare is defined dif-


ferently to describe a situation where both firms produce more than theirCournot–Nash outputs in the first period. Nonetheless, if we allow only purestrategies then Stackelberg warfare of either type is not sustainable as a SPNEoutcome. However, by allowing mixed strategies, we find that there is a posi-tive probability for Stackelberg warfare to occur when both firms simultane-ously produce more than their Cournot–Nash outputs.

To further explore the nature of a symmetric mixed strategy equilibrium,assume that h2fi

L(z, c1 , c2)/z2j # 0 ;z [ (Ni(c2), Si(c1 , c2)], i 5 x, y, andP(Sx(c1 , c2) 1 S y(c1 , c2)) . 0. The first assumption requires that the Stack-elberg leader’s profit be concave in its own output, and the second assump-tion ensures that the market price is positive if both firms produce theirrespective leader quantities. Note that both these requirements are satisfiedfor a linear demand function. It can then be shown that, in the first period,the firms randomize over only two different levels of outputs, 0 and anoutput between the Cournot–Nash and the Stackelberg leader quantity.(See the Appendix for a proof).

To investigate the uniqueness and the exact nature of a symmetric mixedstrategy equilibrium we use a specific demand function. Proposition 5 showsthat the equilibrium is unique for a linear market demand and characterizesthe equilibrium.

PROPOSITION 5. When P(q) 5 max [0, a 2 bq], a . 0, b . 0, and c2 ,c1 , c*

1 5 (0.03a 1 0.97c2), then there exists a unique symmetric mixedstrategy equilibrium such that the firms randomize in the first period over 0and some q*

1 , q*1 . Ni(c2), with probabilities p and (1 2 p), respectively.

q*1 and p are the solutions of the following two equations:

q*1 5 h(a 2 c1) 2 p[(a 2 c2)/2]j/h(3 2 3p)bj (5.1)

p[(a 2 c2)2/9b] 1 (1 2 p)(a 2 c2 2 bq*1)2/4b 5 q*

1[a 2 c1 2 2bq*1]

2q*1ph[a 2 c2)/2] 2 [(3/2)bq*

1]j with p [ (0, 1). (5.2)

Proof. See Appendix. n

From Proposition 5 and Lemma 1, it follows that the following outcomesoccur with respective probabilities:

h(qx1 5 0, qx

2 5 Nx(c2)), (qy1 5 0, qy

2 5 Ny(c2))j w.p. p2

h(qx1 5 q*

1 , qx2 5 0), (qy

1 5 0, qy2 5 Ry(q*

1 u c2))j w.p. p(1 2 p)

h(qx1 5 0, qx

2 5 Rx(q*1uc2)), (qy

1 5 q*1 , qy

2 5 0)j w.p. p(1 2 p)

h(qx1 5 q*

1 , qx2 5 0), (qy

1 5 q*1 , qy

2 5 0)j w.p. (1 2 p)2.

90 DEBASHIS PAL

TABLE IDETERMINATION OF p AND q*

1, FOR DIFFERENT VALUES OF a AND c1. bAND c2 ARE NORMALIZED TO 1. c1 [ (c2 , (0.03a 1 0.97c2)).

a 5 50 a 5 80 a 5 100 a 5 200

c1 5 1.1 c1 5 1.1 c1 5 1.1 c1 5 1.1p 5 0.43 p 5 0.35 p 5 0.32 p 5 0.24q*

1 5 17.93 q*1 5 28.33 q*

1 5 35.22 q*1 5 69.43

c1 5 1.2 c1 5 1.2 c1 5 1.2 c1 5 1.2p 5 0.56 p 5 0.47 p 5 0.43 p 5 0.32q*

1 5 18.65 q*1 5 29.22 q*

1 5 36.20 q*1 5 70.78

c1 5 1.5 c1 5 1.5 c1 5 1.5 c1 5 1.5p 5 0.76 p 5 0.65 p 5 0.60 p 5 0.47q*

1 5 20.15 q*1 5 31.07 q*

1 5 38.25 q*1 5 73.57

c1 5 2.0 c1 5 2.0 c1 5 2.0 c1 5 2.0p 5 0.92 p 5 0.81 p 5 0.75 p 5 0.60q*

1 5 21.94 q*1 5 33.26 q*

1 5 40.67 q*1 5 76.85

c1 5 2.5 c1 5 2.5 c1 5 2.5p 5 0.90 p 5 0.85 p 5 0.69q*

1 5 35.01 q*1 5 42.59 q*

1 5 79.45

c1 5 3.0 c1 5 3.0 c1 5 3.0p 5 0.97 p 5 0.92 p 5 0.75q*

1 5 36.52 q*1 5 44.26 q*

1 5 81.70

c1 5 3.5 c1 5 3.5p 5 0.97 p 5 0.80q*

1 5 45.75 q*1 5 83.72

c1 5 6.0p 5 0.97q*

1 5 91.89

The first outcome is equivalent to the Cournot outcome with both firmsproducing only in the second period. Outcomes h(qx

1 5 q*1 , qx

2 5 0), (qy1 5

0, qy2 5 Ry(q*

1 u c2))j and h(qx1 5 0, qx

2 5 Rx(q*1 u c2)), (qy

1 5 q*1 , qy

2 5 0)jresemble the Stackelberg outcomes where one firm produces only in thefirst period and the other produces its best response in the second period.The probability that a specific firm becomes the leader is p(1 2 p). Thelast outcome h(qx

1 5 q*1 , qx

2 5 0), (qy1 5 q*

1 , qy2 5 0)j generates Stackelberg

warfare when both firms produce simultaneously in the first period andproduce more than their Cournot–Nash quantities.

Obtaining explicit expressions for p and q*1 is difficult since the reduced


form of p is a cubic expression. However, from Eqs. (5.1) and (5.2) it canbe seen that, as c1 decreases to c2 , p decreases to 0 and, as c1 increases to(0.03a 1 0.97c2), p increases to 1. Intuitively, as c1 decreases to c2 , earlyproduction becomes relatively less expensive and the firms produce in thefirst period with higher probabilities to take advantage of being leaders.Also, as p decreases to zero, q*

1 converges to the Cournot level. On theother hand, as c1 increases to (0.03a 1 0.97c2), first-period productionbecomes more expensive and the firms assign lower probabilities to earlyproduction. Values for p and q*

1 corresponding to the other values of theparameters can be found by numerical computation. From Eqs. (5.1) and(5.2) it follows that the value of p is independent of b, so that b 5 1 canbe set without loss of generality. We also normalize c2 to be 1.

Table I presents some results for different values of a and c1 . Observethat p increases with c1 and decreases with a. An increase in a shifts thedemand to the right and thus increases the gain from being a leader. As aresult, the firms assign higher probabilities to early production. In Table Iobserve also that for given a, p and q*

1 are directly related, or equivalently,(1 2 p) and q*

1 are inversely related. Intuitively, since the Stackelbergwarfare occurs with probability (1 2 p)2, the firms choose lower q*

1 asStackelberg warfare is more likely.

4. SUMMARY AND CONCLUSION

In the context of quantity-setting duopoly games with identical firms,situations may arise in which the Stackelberg outcomes are the only SPNEoutcomes in pure strategies. At each of these equilibria one firm is a leaderand its rival is a follower.9 Without reasonable exogenous conditions makinga particular firm the leader, we lack a convincing argument favoring aspecific SPNE outcome. Instead of relying on ad-hoc mechanisms to decidewhich one of the two SPNE will be reached, this paper allows the firms to usemixed strategies and characterizes a symmetric mixed-strategy equilibrium.This equilibrium provides identical expected profits for both firms andthus eliminates the difference in expected profits (between a leader and afollower) which is observed at each of the two pure strategy SPNE.

It is shown that sequential moves by the firms may arise as an expostoutcome of a symmetric mixed-strategy equilibrium. Therefore, just from

9 The same problem may arise even with nonidentical firms. To illustrate, suppose firmsx and y have marginal costs cx and cy in period I and ucx and ucy in period II, respectively(0 , u , 1). Then, for some values of u, it is possible that the two Stackelberg outcomes arethe only equilibrium outcomes. In each outcome one firm behaves as a leader by producingonly in period I, and the other behaves as a follower by producing only in period II.

92 DEBASHIS PAL

observing the timing of the moves it is hard to assert whether a standardStackelberg leader–follower game is being played or whether a specificoutcome of a symmetric mixed-strategy equilibrium is encountered. In thiscase, however, the quantities produced by the firms do not correspond tothe standard Stackelberg leader–follower quantities, where one firm isexogenously specified as a leader.

This paper also relates to some recent works (e.g., Banerjee and Cooper,1991) which are skeptical of theories relying heavily on Cournot behavior,since situations may arise when Stackelberg outcomes are the only SPNEin pure strategies. While some recent papers (e.g., Robson, 1990 a; Banerjeeand Cooper, 1991; Pal, 1991) describe situations in which Stackelberg out-comes are the only equilibria even with identical firms, it should be notedthat these studies restrict the firms to only pure strategies. Once the firmsare allowed to use mixed strategies, this paper shows that the probabilityof observing ex-post Cournot behavior must be positive. In fact, dependingon the parameters, this probability can be significantly high (close to one).Therefore, rejecting Cournot behavior on the grounds that Stackelbergoutcomes are the only equilibrium outcomes in pure strategies is clearly pre-mature.

An interesting extension of this work would be to allow for N (N $ 2)production periods before the market clears. First, restrict the firms to purestrategies only. If c1 , c2 , ? ? ? , cN , there is a unique equilibrium wherethe firms simultaneously produce their Cournot–Nash quantities in period1. If c1 . c2 . ? ? ? . cN , then a symmetric equilibrium may arise wherethe firms simultaneously produce their Cournot–Nash quantities in periodN. It is possible, however, to have two asymmetric leader–follower equilib-ria, at each of which one firm produces in period N 2 1 and the otherproduces only in period N. It is interesting to allow the firms to use mixedstrategies in this situation and to find out the likelihood of an ex postCournot outcome. It appears that if cN22 is sufficiently larger than cN21 ,then a symmetric mixed strategy equilibrium exists which is exactly similarto that in the two-period case, with periods N 2 1 and N being periods 1and 2, respectively. As a result, the probability of an ex-post Cournotoutcome is positive. To characterize a symmetric mixed-strategy equilib-rium in general, however, is challenging and remains an issue of future re-search.

APPENDIX

We show that if h2fiL(z, c1 , c2)/z2j # 0 ;z [ (Ni(c2), Si(c1 , c2)] and

P(Sx(c1 , c2) 1 Sy(c1 , c2)) . 0, then at a symmetric mixed-strategy equilibriumthe firms randomize (in the first period) over only two different levels of


outputs, 0 and an output between the Cournot–Nash and the Stackelberg-leader quantity.

From Propositions 2–4 and Lemma 1, it follows that at a symmetricmixed-strategy equilibrium the firms randomize over 0 and over outputsgreater than N(c2) [where Nx(c2) 5 Ny(c2) 5 N(c2)].

Let firm k randomize over qk1 5 0 and outputs qk

1 [ (Nk(c2), Sk(c1 , c2)].If qk

1 5 0, then the profit of firm i (i 5/ k), fi(qi1 u qk

1 5 0) 5 fkL(qi

1 , c1 , c2),is concave for all qi

1 [ (Ni(c2), Si(c1 , c2)] (by assumption). Also, for anyqk

1 [ (Nk(c2), Sk(c1 , c2)], f i(qi1 u qk

1 [ (Nk(c2), Sk(c1 , c2)]) is strictly concavefor all qi

1 [ (Ni(c2), Si(c1 , c2)] (since P9 , 0, P0 # 0, and P(Sk(c1 , c2) 1Si(c1 , c2)) . 0). Therefore, for any randomization by firm k (over qk

1 5 0and outputs qk

1 [ (Nk(c2), Sk(c1 , c2)]), the expected profit of firm i, E(fi),is strictly concave for all qi

1 [ (Ni(c2), Si(c1 , c2)]. This implies that firm irandomizes over qi

1 5 0 and a single qi1 [ (Ni(c2), Si(c1 , c2)]. Therefore, at

a symmetric mixed-strategy equilibrium the firms randomize over qi1 5 0

and a single qi1 [ (Ni(c2), Si(c1 , c2)] (i 5 x, y).

Proof of Proposition 5. Note that for a linear demand 2(f kL(z, c1 , c2))/

z2 , 0; that is, the Stackelberg leader’s profit is concave in its own outputand P(Sk(c1 , c2) 1 Si(c1 , c2)) . 0. Thus, from the earlier discussion it followsthat at a symmetric mixed-strategy equilibrium the firms randomize overonly two different levels of outputs, 0 and some output between the Cour-not–Nash and the Stackelberg leader amounts of output.

Let firm k randomize over qk1 5 0 and a single q*

1 [ (Nk(c2), Sk(c1 , c2)]with probabilities p and (1 2 p), respectively. If firm i chooses qi

1 [ (Ni(c2),Si(c1 , c2)], then its expected profit is

Ehfi(qi1 [ (Ni(c2), Si(c1 , c2)])j

5 p (fi(qi1 u qk*

1 5 0) 1 (1 2 p) (f i(qi1 u qk

1 5 q*1)

5 qi1 (a 2 c1 2 bq*

1 2 bqi1) 2 pqi

1((a 2 c2)/2 2 bqi1/2 2 bq*

1).

To get the optimal qi1 5 qi*

1 [ (Ni(c2), Si(c1 , c2)], we solve Ehfi(qi1 [

(Ni(c2), Si(c1 , c2)]j/qi1 5 0 and get

qi*1 5 [h1/(2 2 p)bj] h[a 2 c1 2 bq*

1] 2 [(a 2 c2)/2 2 bq*1]j (A.1)

Now p must satisfy

Ehfi(qi1 [ (Ni(c2), Si(c1 , c2)])j 5 Ehf i(qi

1 5 0)j

⇒ qi*1 (a 2 c1 2 bq*

1 2 bqi*1 ) 2 pqi*

1 ((a 2 c2)/2 2 bqi*1 /2 2 bq*

1)

5 p[(a 2 c2)2/9b] 1 (1 2 p)[(a 2 c2 2 bq*1)2/4b] (A.2)

94 DEBASHIS PAL

Since we are looking for a symmetric mixed strategy equilibrium, thethird condition is

qi*1 5 q*

1 (A.3)

Substituting (A.3) into (A.1) and (A.2), we get Eqs. (5.1) and (5.2). Itcan be checked that there exist unique p [ (0, 1) and q*

1 that satisfy Eqs.(5.1) and (5.2), which proves the uniqueness of the symmetric mixed-strat-egy equilibrium. n

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endogenous stackelberg equilibria with identical firms

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