Games and Economic Behavior 28, 105–129 (1999)
Article ID game.1998.0687, available online at http://www.idealibrary.com on
Endogenous Stackelberg Leadership*
Eric van Damme
CentER, Tilburg University, P.O. Box 90153, 5000 LE Tilburg, The Netherlands
Department of Economics, Universitat Pompeu Fabra, Ramon Trias Fargas 25-27,
08005 Barcelona, Spain
Received May 8, 1997
We consider a linear quantity setting duopoly game and analyze which of the
players will commit when both players have the possibility to do so. To that end,
we study a two-stage game in which each player can either commit to a quantity
in stage 1 or wait till stage 2. We show that committing is more risky for the high
cost firm and that, consequently, risk dominance considerations, as in Harsanyi and
Selten (1988), allowthe conclusion that only the lowcost firmwill choose to commit.
Hence, the low cost firm will emerge as the endogenous Stackelberg leader. Journal
of Economic Literature Classification Numbers: C72, D43.
© 1999 Academic Press
Ever since Von Stackelberg wrote his Marktform und Gleichgewicht in
1934, it hasbeen well known that in many duopoly situationsa firmisbetter
off when it acts as a leader than when it acts as a follower. Since each firm
will strive to obtain the most favorable position for itself, the question arises
as to which of the two duopolists will gain victory and obtain this leadership
position. Von Stackelberg concluded that in general it is not possible to
answer this question theoretically (Von Stackelberg, 1934, pp. 18–20). In
this paper, we consider the special case of a linear quantity setting duopoly
game and show that in this case the role assignment may follow from risk
* Hurkens gratefully acknowledges financial support fromEC Grant ERBCHGCT 93-0462,
CIRIT, Generalitat de Catalunya, Grant 1997SGR 00138, and DGES Grant PB96-0302.
Copyright © 1999 by Academic Press
All rights of reproduction in any form reserved.
106van damme and hurkens
considerations. Specifically, we demonstrate that committing islessrisky for
a low cost firm so that such a firm will emerge as the Stackelberg leader.
Our work isinspired by an idea of ThomasSchelling. Of course, Schelling
is most well known for his general demonstration of the value of commit-
ment, i.e., that committing is beneficial for a player who is the only one
able to make a commitment. Schelling realized that, as a consequence, all
players in the game will attempt to commit themselves and that a coordina-
tion problemmight arise: committing isbeneficial only if the opponent does
not commit; it might be (very) costly if the opponent also commits himself.
This in turn implies that a player might decide not to commit himself since
he fears that the opponent might commit as well and since the costs as-
sociated with the resulting “Stackelberg war” might be too high (Schelling,
1960, p. 39). Hence, there is a fundamental trade-off between flexibility and
commitment. Schelling pointed out this trade-off, but he did not provide a
formal analysis of it, and he did not solve the game. Our aim in this paper
is to provide a full solution for the linear two-person duopoly game.
We consider a quantity setting duopoly game with linear demand and
constant marginal cost. One firm is more efficient, i.e., has lower marginal
cost, than the other. The formal model used to analyze the trade-off be-
tween commitment and flexibility is the two-stage action commitment game
from Hamilton and Slutsky (1990). The rules are as follows. Each duopolist
hasto move (i.e., to choose a quantity) in one of two periods; choicesare si-
multaneous, but if one player chooses to move early while the other moves
late, the latter is informed about the first-mover’s choice before making his
decision. Hence, moving early is profitable if one is the only player to do
so, but it is costly if the other commits as well. This timing game has several
equilibria; in particular, each of the Stackelberg outcomes of the underly-
ing duopoly game is an equilibrium. As Hamilton and Slutsky pointed out,
these are the only pure undominated equilibria of the game. We select the
solution of the game by using the risk-dominance concept from Harsanyi
and Selten (1988). This concept allows one to quantify the risks involved
with the two candidate solutions and, hence, it enables to resolve the trade-
offs. Risk considerations show that committing is less risky for the firm that
has the lower marginal cost. This safer equilibrium in which the low cost
firm moves first is the neutral focal point and, adopting the risk dominance
concept, the players will coordinate on it.
Some intuition for this result might be obtained by looking at the 2× 2
game in which each player isrestricted to use one of two strategies: either to
commit himself to hisStackelberg leader quantity or to wait until the second
period and then best respond to the quantity chosen by the opponent, with
players choosing their Cournot quantities in the second period if neither
player moved in the first period. (See Fig. 1 in Section 3 for the payoff
matrix.) Both Stackelberg outcomes appear as strict equilibria in this game
endogenous stackelberg leadership107
and it is well known that risk dominance allows a simple characterization
for such 2 × 2 games: the equilibrium with the highest (Nash) product of
the deviation losses is the risk dominant one (Harsanyi and Selten, 1988,
Lemma 5.4.4). In Section 3 we show that the equilibrium where the low
cost firm commits is risk dominant in this reduced game. The intuition is
that, if player 1 has higher marginal costs, then his reaction curve is below
the reaction curve of player 2, so that his Stackelberg and Nash quantities
are closer together, which implies that he can gain less from committing
himself than player 2 can. On the other hand, player 1 incurs greater losses
than player 2 does if both players commit themselves. As a consequence,
player 1 is in a weaker bargaining position to push for his most favored
outcome and he will lose the battle.
If the risk dominance relation between our two candidate solutions could
alwaysbe decided on the basisof the 2×2 game spanned by them, then our
problem could be solved by straightforward computation. Unfortunately,
the problem posed in this paper is not that simple to solve and the above
mentioned characterization of risk dominance isof limited use for the prob-
lem addressed. In our “action commitment” game, a player has infinitely
many strategies available; the choice is not simply between committing to
the Stackelberg leader quantity and waiting. Furthermore, it is known that,
in general, the reduced 2 × 2 game spanned by the two equilibrium can-
didates may capture the overall risk situation rather badly. Consequently,
to find the solution of the game, there is no recourse but to apply risk
dominance to the overall game. Now risk dominance is defined by means
of the tracing procedure and the fact that this procedure is rather com-
plex and difficult to handle forces us to restrict ourselves to the linear case.
Even in this most simple linear case, the computations are already rather
involved; they become very cumbersome in the more general case. Never-
theless, the main result of this paper is that risk dominance indeed selects
the equilibrium in which the low cost firm leads.
The present paper is part of a small, but growing, literature that aims
at endogenizing the first mover in oligopoly models. Ours is the first pa-
per in which a specific Stackelberg outcome is derived from a model in
which the duopolists are in symmetric positions ex ante and in which only
endogenous (strategic) uncertainty is present. Related papers either put
firms in asymmetric positions to start with, or add exogenous uncertainty
(about production costs or market demand), or admit multiple equilibrium
Hamilton and Slutsky (1990) consider the same game as we do and
they show that the two Stackelberg equilibria are the only pure strategy
equilibria in undominated strategies. Hence, they conclude that a Stack-
elberg outcome will result but they cannot tell which one. Sadanand and
Sadanand (1996) analyze the same model when firms face demand uncer-
108 van damme and hurkens
tainty, which is resolved before production in the second stage. (Also see
Sadanand and Green, 1991.) There isalwaysa symmetric (Cournot) equilib-
rium: both firms move late when uncertainty is large and early when there
is no uncertainty. In addition, both Stackelberg outcomes can be sustained
as equilibria provided that uncertainty is not too large. Hence, to select a
unique Stackelberg outcome it is necessary to assume that uncertainty in-
fluences the duopolists in an asymmetric way. An interesting asymmetric
variant that Sadanand and Sadanand analyze is a large firm versus fringe
model. Since each fringe firm individually is too small to influence out-
put, the unique equilibrium now has the large firm committing itself, while
the small firms remain flexible. Spencer and Brander (1992) study a sim-
ilar duopoly model with demand uncertainty. However, they assume that
a firm who moves early is informed about the time at which the oppo-
nent moves, which simplifies the analysis considerably. For example, when
both firms decide to move early, it follows that they will produce Cournot
quantities. In a symmetric setting, both firms will move early (resp. late)
when uncertainty is low (resp. high), so that in each case a Cournot out-
come results. A Stackelberg outcome may result when firms are in asym-
metric positions: when one firm is much better informed about the ex-
ogenous shock than the other, then the better informed firm may emerge
as the Stackelberg leader. A different type of asymmetry is considered in
Kambhu (1984): one firm is risk neutral and the other is risk averse. In
this case, the risk neutral firm may arise as the Stackelberg leader. Mailath
(1993) puts the firms in asymmetric starting positions. One firm is informed
about demand, while the other facesuncertainty and only the informed firm
has the option to move first. In the unique “intuitive” equilibrium the in-
formed firmindeed actsasa Stackelberg leader, even if it could earn higher
ex ante profits by choosing quantities simultaneously with the uninformed
Saloner (1987) considers a model related to the one discussed here in
which two periods of production are also allowed. Firms simultaneously
choose quantities in the first period; these become common knowledge
and then firms simultaneously decide how much more to produce in the
second period before the market clears. Saloner shows that any outcome
on the outer envelope of the two reaction functions lying in between the
two Stackelberg outcomes can be sustained as a subgame perfect equilib-
rium. Ellingsen (1995) notes that only the two Stackelberg outcomes sur-
vive iterated elimination of (weakly) dominated strategies in this game. Pal
(1991) generalizes Saloner’s analysis by allowing for cost differences across
periods. If production is cheaper in the first period (resp. much cheaper
in the second period), then both firms produce their Cournot quantities in
the first (resp. second) period. In the intermediate case, where costs fall
slightly over time, either of the two Stackelberg outcomes can be sustained
endogenous stackelberg leadership109
as a subgame perfect equilibrium. Hence, none of these papers can make
a selection among the Stackelberg outcomes.
The remainder of this paper is organized as follows. The underlying
duopoly game as well as the action commitment game from Hamilton and
Slutsky (1990) are described in Section 2, where relevant notation is also
introduced. Section 3 describes the specifics of the tracing procedure as
it applies in this context and defines the concept of risk dominance. The
main results are derived in Section 4. Section 5 concludes. Some proofs are
relegated to the Appendix.
2. THE MODEL
The underlying linear quantity-setting duopoly game is as follows. There
are two firms, 1 and 2. Firm i produces quantity qiat a constant marginal
cost ci≥ 0. The market price is linear, p = max?0?a − q1− q2?. Firms
choose quantities simultaneously and the profit of firm i is given by
ui?q1?q2? = ?p − ci?qi. We assume that 3ci− 2cj≤ a ?i?j ∈ ?1?2??i ?∈ j?,
which implies that a Stackelberg follower will not be driven out of the
market. We will restrict ourselves to the case where firm 2 is more efficient
than firm 1, c1> c2. We write ai= a − ci.
The best reply of player j against the quantity qiof player i is unique
and is given by
bj?qi? = max?0??aj− qi?/2??
The unique maximizer of the function qi?→ ui?qi?bj?qi?? is denoted
quantity that j will choose as a Stackelberg follower, qF
Nash equilibrium of the game and denote player i’s payoff in this equilib-
rium by Ni. For later reference we note that
i (firms i’s Stackelberg leader quantity). We also write qF
2? for the unique
j? and Fi= ui?qF
j?. We write ?qN
As is well known,
?i = 1?2?
?i = 1?2??
Li> Ni> Fi
Hence each player has an incentive to commit himself.
110 van damme and hurkens
To investigate which player will dare to commit himself when both play-
ers have the opportunity to do so, we make use of the two-period action
commitment game that was proposed in Hamilton and Slutsky (1990). The
rules are as follows. There are two periods and each player has to choose
a quantity in exactly one of these periods. Within a period, choices are si-
multaneous, but if a player does not choose to move in period 1, then in
period 2 this player is informed about which action his opponent chose in
period 1. This game has proper subgames at t = 2 and our assumptions
imply that all of these have unique equilibria. We will analyze the reduced
game, g2, that results when these subgames are replaced by their equi-
librium values. Formally, the strategy set of player i in g2is ?+∪ ?Wi?,
where Widenotes i’s strategy to wait till period 2, and the payoff function
is given by
ui?qi?qj? = ?ai− qi− qj?qi
ui?qi?Wj? = ?ai− qi− bj?qi??qi
ui?Wi?qj? = ?ai− qj?2/4
ui?Wi?Wj? = ?2ai− aj?2/9
It is easily seen that g2has three Nash equilibria in pure strategies: Ei-
ther each player i commits to his Nash quantity qN
one player i commits to his Stackelberg leader quantity qL
player waits till the second period. One also notices (with Hamilton and
Slutsky, 1990) that the first (Cournot) equilibrium is in weakly dominated
strategies (committing to qN
that only the (Stackelberg) equilibria in which players move in different
periods are viable. Below we will indeed show that the Cournot equilib-
rium is risk dominated by both Stackelberg equilibria (Proposition 1). It
should be noted that besides these pure equilibria, the game g2admits sev-
eral mixed equilibria as well. These mixed equilibria will not be considered
in this paper, the reason being that we want to stick as closely as possible
to the general solution procedure outlined in Harsanyi and Selten (1988),
a procedure that gives precedence to pure equilibria whenever possible.
Although mixed strategy equilibria will not be considered, we stress that
mixed strategies will play an important role in what follows. The reason is
that, in the case at hand, a player will typically be uncertain about whether
the opponent will commit or not, and such uncertainty about the oppo-
nent’s behavior can be expressed by a mixed strategy. Let mjbe a mixed
strategy of player j in the game g2. Because of the linear-quadratic speci-
fication of the game, there are only three “characteristics” of mjthat are
relevant to player i, viz., wjthe probability that player j waits, µjthe av-
erage quantity to which j commits himself given that he commits himself,
i in the first period, or
iand the other
iis dominated by Wiin g2), hence one expects
endogenous stackelberg leadership111
and νj, the variance of thisquantity. Specifically, it easily followsfrom(2.6)–
(2.9) that the expected payoff of player i against a mixed strategy mjwith
characteristics ?wj?µj?νj? is given by
ui?qi?mj? = ?1− wj??ai− qi− µj?qi+ wj?2ai− aj− qi?qi/2
ui?Wi?mj? = ?1− wj???ai− µj?2/4+ νj/4? + wj?2ai− aj?2/9
Note that uncertainty concerning the quantity to which j will commit him-
self makes it more attractive for player i to wait: νjcontributes positively to
(2.11) and it does not play a role in (2.10). On the other hand, increasing
wjor decreasing µjincreases the incentive for player i to commit himself.
3. RISK DOMINANCE AND THE TRACING PROCEDURE
The concept of risk dominance capturesthe intuitive idea that, when play-
ers do not knowwhich of two equilibria should be played, they will measure
the risk involved in playing each of these equilibria and they will coordinate
expectations on the less risky one, i.e., on the risk dominant equilibrium
of the pair. The formal definition of risk dominance involves the bicentric
prior and the tracing procedure. The bicentric prior describes the players’
initial assessment about the situation. The tracing procedure is a process
that, starting from some given prior beliefs of the players, gradually adjusts
the players’ plans and expectations until they are in equilibrium. It mod-
els the thought process of players who, by deductive personal reflection, try
to figure out what to play in the situation where the initial uncertainty is
represented by the given prior. Below we describe the mechanisms of the
tracing procedure as well as how, according to Harsanyi and Selten (1988),
the initial prior should be constructed.
First, however, we recall that risk dominance allows a very simple charac-
terization for 2×2 games with two Nash equilibria: the risk dominant equi-
librium is that one for which the product of the deviation losses is largest.
Consequently, if risk dominance could always be decided on the basis of
the reduced game spanned by the two equilibria under consideration (and
if the resulting relation would be transitive), then the solution could be
found by straightforward computations. Unfortunately, this happy state of
affairs does not prevail in general. The two concepts do not always gener-
ate the same solution and it is well known that the Nash product of the
deviation losses may be a bad description of the underlying risk situation
in general. (See Carlsson and van Damme, 1993, for a simple example.) In
our companion paper (van Damme and Hurkens, 1998) we show that also
in duopoly games the two concepts may yield different solutions. In the
present case, however, the two concepts do generate the same solutions.
112 van damme and hurkens
Since the calculations based on the reduced game are easily performed, we
do these first.
Consider, first of all, the reduced game spanned by the Cournot equilib-
leads. In this 2 × 2 game, W2weakly dominates qN
the deviation losses associated with the Cournot equilibrium is zero and,
in the reduced game, the Stackelberg equilibrium is risk dominant. Exactly
the same argument establishes that the Cournot equilibrium is risk domi-
nated by the Stackelberg equilibrium in which firm 2 leads. Next, consider
the reduced game where each player is restricted to either committing him-
self to his Stackelberg quantity or to wait, which is given in Fig. 1,1where
Li?Niand Fiare as in (2.3) and where Didenotes player i’s payoff in the
case of Stackelberg warfare
2? and by the Stackelberg equilibrium ?qL
1?W2? in which firm 1
2; hence the product of
Di= ?ai− aj??2ai− aj?/4?
At the equilibrium where i leads the product of the deviation losses is
?Li− Ni??Fj− Dj? = a2
Consequently, the product of the deviation losses at ?W1?qL
the similar product at ?qL
a1?2a2− a1? > a2?2a1− a2??
which holds since a1< a2. Hence, the product of the deviation losses is
largest at the equilibrium where the efficient firm 2 leads: risk consider-
ations based on reduced game analysis unambiguously point into the di-
rection of the Stackelberg equilibrium where the low cost firm leads. As
already argued, there is, however, no guarantee that this shortcut indeed
identifies the risk dominant equilibrium of the overall game. The only way
to find out isby fully solving the entire game. Thiswe do in the next section.
In the remainder of this section, we formally define the concepts involved.
2? is larger than
1?W2? if and only if
1This game has also been studied by Dowrick (1986), who concludes “that there is no
obvious solution to this game where firms can choose their roles” (p. 259).
FIG. 1. Reduced version of the quantity commitment game.
endogenous stackelberg leadership113
Let g = ?S1?S2?u1?u2? be a two-person game and let mibe a mixed
strategy of player i in g ?i = 1?2?. The strategy mirepresents the initial
uncertainty of player j about i’s behavior. For t ∈ ?0?1? we define the game
?si?sj? = ?1− t?ui?si?mj? + tui?si?sj??
Hence, for t = 1, this game gt?mcoincides with the original game g, while
for t = 0 we have a trivial game in which each player’s payoff depends only
on his own action and his own prior beliefs. Write ?mfor the graph of the
equilibrium correspondence, i.e.
?m=??t?s?: t ∈ ?0?1??s is an equilibrium of gt?m??
It can be shown that, if g is a generic finite game, then, for almost any
prior m, this graph ?mcontains a unique distinguished curve that connects
the unique equilibrium s0?mof g0?mwith an equilibrium s1?mof g1?m. (See
Schanuel et al., 1991, for details.) The equilibrium s1?mis called the linear
trace of m. If players’ initial beliefs are given by m and if players’ reason-
ing process corresponds to that as modeled by the tracing procedure, then
players’ expectations will converge on the equilibrium s1?mof g.
In this paper we will apply the tracing procedure to the infinite game g2
that was described in the previous section. To our knowledge, ours is the
first application of these ideas to a game with a continuum of strategies.
For such games, no generalizations of the Schanuel et al. (1991) results
have been established yet, but as we will see in the following sections, there
indeed exists a unique distinguished curve in the special case analyzed here.
Hence, the nonfiniteness of the game g2will create no special problems.
It remains to specify the players’ initial beliefs when they are uncertain
about which of two equilibria of g, s or s?, should be played. Harsanyi
and Selten (1988) argue as follows. Player j, being Bayesian, will assign a
subjective probability zjto i playing siand he will assign the complementary
a best response against the strategy zjsi+ z?
Assume that j chooses all best responses with equal probability and denote
the resulting strategy of j with bj?zj?. Player i does not know the beliefs zj
of player j and, applying the principle of insufficient reason, he considers zj
to be uniformly distributed on ?0?1?. Writing Zjfor a uniformly distributed
random variable on ?0?1?, player i will therefore believe that he is facing
the mixed strategy
2? in which the payoff functions are given by
j= 1− zjto i playing s?
i. With these beliefs, player j will play
ithat he expects i to play.
and this mixed strategy mjof player j is player i’s prior belief about j’s be-
havior in the situation at hand. Similarly, mi= bi?Zi?, where Z1and Z2
114 van damme and hurkens
are independent, is the prior belief of player j, and the mixed strategy pair
m = ?m1?m2? is called the bicentric prior associated with the pair ?s?s??.
Given this bicentric prior m, we say that s risk dominates s?if s1?m= s,
where s1?mis the linear trace of m. In case the outcome of the tracing
procedure is an equilibrium different from s or s?, then neither of the equi-
libria risk dominates the other. Such a situation will, however, not occur
in our two-stage action commitment game, provided that the costs of the
firms are different.
4. COMMITMENT AND RISK DOMINANCE
In this section, we prove our main results. Let g2be the endogenous
commitment game from Section 2. Write Sifor the pure equilibrium in
which player i commits to his Stackelberg leader quantity in period 1,
mits to his Cournot quantity in period 1, C = ?qN
Stackelberg equilibria risk dominate the Cournot equilibrium and that S2
risk dominates S1when c2< c1. The first result is quite intuitive: Commit-
ting to qN
strategy is risky. The proof of this result is correspondingly easy.
i?Wj?, and write C for the equilibrium in which each player com-
2?. We show that both
iis a weakly dominated strategy and playing a weakly dominated
Cournot equilibrium C ?i = 1?2?.
Without lossof generality, we just prove that S1risk dominatesC.
We first compute the bicentric prior that isrelevant for thisrisk comparison,
starting with the prior beliefs of player 1.
Let player 2 believe that 1 plays z2S11+ ?1− z2?C1= z2qL
Obviously, if z2∈ ?0?1?, then the best response of player 2isto wait. Hence,
the prior belief of player 1 is that player 2 will wait with probability 1,
Next, let player 1 believe that 2 plays z1S12+ ?1− z1?C2= z1W2+ ?1−
irrespective of the value of z1. When z1> 0? committing to a quantity that
is (slightly) above qN
is to commit to a certain quantity q1?z1?. The reader easily verifies that
q1?z1? increases with z1and that q1?1? = qL
prior belief of player 2, then for the characteristics ?w1?µ1?ν1? of m1we
have w1= 0, µ1> qN
Now, let us turn to the tracing procedure. The starting point corresponds
to the best replies against the prior. Obviously, the unique best response
against m2is for player 1 to commit to qL
response against m1is to wait. Hence, the unique equilibrium at t = 0
In g2, the Stackelberg equilibrium Sirisk dominates the
1+ ?1− z2?qN
2. Obviously, waiting yields player 1 the Nash payoff N1as in (2.3),
1yields a strictly higher payoff; hence the best response
1. Consequently, if m1is the
1, ν1> 0.
1, while player 2’s unique best
endogenous stackelberg leadership 115
is S1. Since S1is an equilibrium of the original game, it is an equilibrium
for any t ∈ ?0?1?. Consequently, the distinguished curve in the graph ?mis
the curve ??t?S1?: t ∈ ?0?1?? and S1risk dominates C.
We now turn to the risk comparison of the two Stackelberg equilibria.
Again we start by computing the bicentric prior based on S1and S2. Let
player j believe that i commits to qL
with probability 1− z. From (2.2), (2.10) and (2.11) we obtain
= z?3aj− 2ai− 2qj/2? + ?1− z??2aj− ai− qj?qj/2
= z?3aj− 2ai?2/16+ ?1− z??2aj− ai?2/9?
Given z, the optimal commitment quantity qj?z? of player j is given by
qj?z? = ?aj− ai?/2+ aj/2?1+ z??
which results in the optimal commitment payoff equal to
?2aj− ai+ z?aj− ai??2/8?1+ z??
Note that q2?z? > q1?z? for all z ∈ ?0?1?. The reader easily verifies that
committing yields a higher payoff than waiting if and only if z is sufficiently
small. Specifically, committing is better for player j provided that z ≤ zj
iwith probability z and that i waits
i+ ?1− z?Wi?
i+ ?1− z?Wi?
j− ?4aj− 2ai?2?
Note that 0 < z1< z2, so that both players initially commit with positive
probability, it being more likely that player 2 commits. Hence, denoting the
best response of player j against zqL
Consequently, writing mjfor the prior of player i (mjbeing given by (3.5))
and writing ?wj?µj?νj? for the characteristics of this prior, we have
wj= 1− zj?
µj= ?aj− ai?/2+ ajln?1+ zj?/2zj?
i+ ?1− z?Wiby bj?z?, we have
if z > zj?
if z < zj?
j/4?1+ zj? − a2
116 van damme and hurkens
Straightforward computations now show that
µ1< µ2? and
These inequalities already give some intuition for why committing is more
risky for player 1: he attaches a smaller probability to the opponent waiting,
he expects the opponent to commit to a larger quantity on average, and
he is more uncertain about the quantity to which the opponent commits
himself. All three aspects contribute positively to making waiting a more
In the next Lemma we show that actually waiting is a dominant strategy
for firm1 at the start of the tracing procedure whenever the cost differential
is sufficiently large. Write
for the relative cost advantage of player i (αj> 1 if and only if ci< cj).
Note that zjdepends on ai?ajonly through αj
18− ?4− 2αj?2
and that zjis a decreasing function of αj.
Lemma 1. Write m0
α2is sufficiently small, then u1?q1?m0
this holds if z2≥ 1/2.
2for the prior strategy of player 2 as given by (4.7). If
2? < u1?W1?m0
2? for all q1. In particular,
j? = zj?ai− qi− µj?qi+ ?1− zj??2ai− aj− qi?qi/2
=?ai− aj/2+ zj?aj/2− µj??qi− ?1+ zj?q2
Hence, the optimal commitment quantity against the prior is
i=ai− aj/2+ zj?aj/2− µj?
i is weakly dominated by Wifor player
i in g2; hence such a quantity yields strictly less than Wiagainst any non-
degenerate mixed strategy of player j. Consequently, the result follows if
We know that any quantity qj≤ qN
1. Now, the inequality q∗
2ai− aj+ zj?2aj− ai? ≤ 3ajln?1+ zj?
iis equivalent to
endogenous stackelberg leadership 117
2αj− 1+ zj?2− αj? ≤ 3ln?1+ zj??
A straightforward computation shows that this inequality is satisfied when
zj= 1/2. (In that case αj= 2 −?3/2.) In the relevant parameter range
is larger than the derivative of the RHS of (4.13), hence, the result follows.
?zj≤ 1?αj≥ 2/3?, the derivative of the LHS of (4.13) (with respect to αj)
In the next Lemma we show that, in contrast to the previous result, the
most efficient firm’s best response to the prior is always to commit.
player i can get by committing himself
1?W2? < maxq2u2?m0
Substituting (4.12) into (4.11) yields the optimal payoff that
1for the prior strategy of player 1 as given by (4.7).
j? =?ai− aj/2+ zj?aj/2− µj??2
On the other hand, waiting yields
j? = zj
??ai− µj?2/4+ νj/4?+ ?1− zj??2ai− aj?2/9
so that by rearranging we obtain
1? − u2?m0
??α?z? =1− z
?2− 8α + 8α2+ z?−7+ 10α − α2? + 18?1− α?ln?z + 1??
and where α1and z1are as in (4.9) and (4.10). Note that z1is a function
of α1, so that ? (as appearing in (4.14)) can be viewed as a function of α1
only. A direct computation shows that ??1? > 0; hence player 2 prefers to
commit when the costs are equal. In the Appendix we show that
?α≥ 0??z≤ 0? and zα≤ 0(4.15)
from which it follows that committing becomes more attractive for player 2
when his cost advantage increases. Consequently, firm 2 finds it optimal to
commit against the prior for all parameter constellations.
118 van damme and hurkens
The Lemmas 1 and 2 imply that the Stackelberg equilibrium with firm 2
as leader is the (unique) equilibrium at the start of the tracing procedure
when z2≥ 1/2. Hence, it is an equilibrium of gt?m0for any value of t and,
therefore we have Corollary 1.
z2≥ 1/2), then the Stackelberg equilibrium in which the efficient firm leads
risk dominates the other Stackelberg equilibrium.
If the difference in costs is sufficiently large (specifically, if
In the remainder of this section, we will confine attention to the case
where the cost difference is small enough so that also for the inefficient
firm 1 the best response to the prior involves a commitment. So from now
on z2< 1/2.2The next Lemma shows that it cannot be true that both firms
keep on committing themselves to the end of the tracing procedure: at least
one of the firms has to switch. The Lemma thereafter will then show that it
is the weakest firm that switches first, which implies that the outcome will
always be leadership of the strong firm.
at “time” t if the players priors are as in (4.7). Then there exists i ∈ ?1?2? and
t < 1 such that st
Let stbe the equilibrium on the path of the tracing procedure
each point reached by the tracing path. Writing qt
mitment quantity of player i at time t, it is easily seen that q1
i = 1?2, since the payoff functions at t = 1 coincide with those of the orig-
inal game. Furthermore, qt
is strictly dominated by waiting. Write ut
t when the prior is given by (4.7) and let
be the gain that player i realizes by committing himself. Clearly, gi?1? = 0.
Furthermore, by the envelope theorem
Assume not, so that each player finds it optimal to commit at
ifor the optimal com-
ifor t < 1 since any quantity less than qN
ifor the payoff function at “time”
gi?t? = ut
For t = 1, the first term in this expression is equal to
2Thisbound isnot sharp. It can be shown that committing isoptimal for firm1 if α1> 1?081
(or z2< 7/20).
endogenous stackelberg leadership 119
Furthermore, the partial derivative with respect to qjis equal to
so that g?
assumption that it is optimal to commit for each player for any value of
t < 1.
i+ ?ai− qN
j?/2 = 0
i?1? > 0 and gi?t? < 0 for some t < 1. But this contradicts our
Our strategy for proving that it is the weakest firm that switches first is
to show that this firm will switch first even when the more efficient firm is
more “pessimistic.” Specifically, we will show that even when the efficient
firm believes that the other commits with the same probability as it itself
does, the inefficient firm will switch before. Specifically, write mjfor the
prior strategy of player j as given by (4.6) and write ¯ mjfor the strategy
defined similarly, but with z1replaced by z2. Let m = ?m1?m2? and ¯ m =
? ¯ m1? ¯ m2?. Hence, player 2 is more pessimistic in ¯ m, while player 1’s prior
beliefs are the same in m and ¯ m. (Recall from (4.5) that z1< z2.) Assume
that each player finds it optimal to commit at t = 0 when the prior is m.
?qj? for the best commitment quantity of player i at t when the
opponent commitsto qjat that time and denote the (unique) pair of mutual
best commitment quantities by ?qt
?qi?qj? − ut?m
equilibrium on the tracing path for such t. Define ?¯ qt
but with m replaced by ¯ m in the above definitions. We now have Lemma 4.
Lemma 4.Let ti= sup?τ ∈ ?0?1?: gt
We only provide a sketch of the proof here and relegate technical
details to the Appendix. The proof consists of comparing the tracing path
the prior is ¯ m; hence he will commit to a lower quantity. This in turn
gives player 2 an incentive to commit to a higher quantity when the prior
is ¯ m. Furthermore, if player 2 is more pessimistic, then he finds committing
himself less attractive: ¯ gt
has, committing ismore attractive for firm2 than for firm1 when both firms
are equally pessimistic: ¯ gt
observations. Formally, then, in the Appendix we establish the following
< ¯ gt
i?qi?qj? = ut?m
j? > 0 ?1 = 1?2? for t sufficiently small and ?qt
2? is the
1? ¯ qt
2? and ¯ gt
j? ≥ 0for all t ∈ ?0?τ??. Then
2? with the tracing path ?¯ qt
1. These inequalities are intuitive: player 2 is more pessimistic if
1? ¯ qt
2?. We first show that ¯ qt
2. Still, since firm 2 has lower cost than firm 1
1< ¯ gt
2. The result follows by combining the above
2? = ¯ gt
≤ ¯ gt
1? ¯ qt
≤ ¯ gt
1? ¯ qt
1? ¯ qt
≤ ¯ gt
1? ¯ qt
1? ¯ qt
120van damme and hurkens
(The first equality holds since player i’s prior is the same in both cases; the
first and fourth inequalities follow from the monotonicity of the quantities;
the second and sixth inequalities follow from the best response properties,
and the fifth inequality follows since player 2 is more pessimistic when the
prior is ¯ m.)
Lemma 4 implies that at t = t1the tracing path reaches the equilibrium
mitting to qt1
t1∈ I) with equilibria of the form ?m1?t??qI
and commits to qI
two commitment quantities are determined by the optimality condition for
player 1 (qI
ence condition for player 1 (committing optimally yields the same payoff as
waiting). The probability of waiting, w?t?, is determined by the optimality
condition for player 2.
Figure 2 illustrates the argumentation: Time t is on the horizontal axis,
firm 2’s commitment quantity on the vertical axis. The figure contains three
curves. Curve qCplots the commitment strategy of firm 2 when firm 1
commits for sure and play is in equilibrium. As we established in Lemma 4,
both firms keep committing from t = 0 to t = t1, therefore the tracing path
follows this curve up to t = t1. Curve qIplots firm 2’s commitment quan-
2?, with player 1 being actually indifferent between waiting and com-
1. The tracing path must now continue along an interval I (with
2?t??, where player 2 commits
2?t? and player 1 uses a mixed strategy: he waits with probability w?t?
1?t? with the complementary probability 1 − w?t?. The
1?t? must be the optimal commitment quantity) and the indiffer-
FIG. 2. The tracing path initially follows qC, then bends backwards along qIand finally ends
along qWat S2.
endogenous stackelberg leadership 121
tity that leaves firm 1 exactly indifferent between committing and waiting.
The tracing path has to continue along this curve from t = t1. (In the Ap-
pendix we establish that the curve necessarily bends backwards.) Curve qW
describes the optimal commitment quantity when firm 1 waits with proba-
bility 1. The tracing path follows this curve from t = t0to t = 1. It follows
that the endpoint of the tracing path is the equilibrium where player 2
leads; hence we have shown Proposition 2.
leads risk dominates the Stackelberg equilibrium in which the efficient firm
The Stackelberg equilibrium in which the low cost firm
By combining the Propositions 1 and 2 we therefore obtain our main
and the inefficient firm follows is the risk dominant equilibrium of the endoge-
nous quantity commitment game.
The Stackelberg equilibrium in which the efficient firm leads
Furthermore, asa corollary we immediately have that the shortcut via the
reduced games, as taken in Section 3, indeed correctly identified the risk
dominant equilibrium of the overall game. Finally, the Stackelberg equilib-
rium that is selected is the one with the highest produced quantity (hence,
the lowest price) and the highest total profits. So, in this case, the selected
equilibrium is the one where both the producer and the consumer surplus
In this paper we have endogenized the timing of the moves in the linear
quantity-setting duopoly game by means of Harsanyi and Selten’s concept
of risk dominance. To our knowledge, this is the first application of the (lin-
ear) tracing procedure to games where the strategy spaces are not finite.3
We have seen that no new conceptual problems are encountered, but that
the computational complexities are quite demanding. Ex post we could ver-
ify that these computations were not necessary: The shortcut by means of
a comparison of the Nash products of the deviation losses yields the same
answer. However, as already said, there is no guarantee for this to happen
in general and in our companion paper (van Damme and Hurkens, 1998)
we show that the two concepts yield different solutions in a price setting
context. In that paper we analyze endogenous price leadership in a linear
3Harsanyi and Selten (1988) and G¨ uth and van Damme (1991) considered discretized ver-
sions of games with infinite strategy sets.
122van damme and hurkens
market for differentiated products. Again, we assume that firms differ in
their marginal costs and we show that the efficient firm is the leader in
the risk dominant equilibrium. In this case, however, that equilibrium has
a smaller Nash product than the Stackelberg equilibrium in which the in-
efficient firm leads. Quite interestingly, if the cost differential is sufficiently
small, the inefficient firm has higher profits than the efficient firm in the
risk dominant equilibrium: It profits from free riding as a follower.
Although Von Stackelberg (1934) argued that in general it is not possible
to determine theoretically which of the duopolists will become the leader
(“Es is jedoch theoretisch nicht zu entscheiden, welcher der beiden Dy-
opolisten obsiegen wird,” p. 20), he also provides a numerical example for
which he does determine the actual leader. The example is given by
p = 10− Q/100? c1= 2? c2= 1?5? F1= 500? F2= 600?
where Fiis the (unavoidable) fixed costs of firm i. Von Stackelberg argues
that in this case firm 2 (which is the one with the lower marginal cost) will
most likely become the market leader since it makes less losses than firm
1 in the case of Stackelberg warfare: We have qL
−593?75, D2= −487?50. Hence, firm 2 makes less losses during the price
war and, therefore, it can win the war of attrition. Of course, this argument
is entirely different from the one developed in this paper. Von Stackelberg
also remarks that actually this outcome is quite natural and follows from
the model’s assumption that the second firm is a more modern one which
has higher fixed costs, but lower marginal cost.4This last comment is very
intriguing since, if the modern firm would have substantially higher fixed
costs, exactly the same argument would imply that the old-fashioned firm
would become the leader.
Note that we did not provide the solution of the endogenoustiming game
for the case where both firms have the same marginal cost. The reader
might conjecture that in that case the Cournot equilibrium would be se-
lected; however, Lemma 3 shows that that conjecture is wrong. If the out-
come of the tracing procedure at t = 1 would be ?qN
would strictly prefer to wait at t < 1, but clearly ?W1?W2? cannot be an equi-
librium at such t. It follows that, in the symmetric case, the outcome must
be a mixed strategy equilibrium. (It obviously must be a symmetric equilib-
rium as well.) Since mixed equilibria have received almost no attention in
the oligopoly literature, we refrain from providing the explicit solution of
1= 375, qL
2= 450, D1=
2?, then each player
4Von Stackelberg denotesthe first firmby A and the second by B and he writes, “In unserem
Beispiel wird warscheinlich die Unternehmung A der Underlegene sein, weil sie den gr¨ oßeren
Verlust erleidet. Diesentspricht auch der Konstruktion unseresBeispiels, in welchemf¨ ur B ein
modernerer Betrieb (h¨ ohere fixe Kosten, daf¨ ur niedriger proportionaler Satz) angenommen
wurde” (Von Stackelberg, 1934, p. 66).
endogenous stackelberg leadership 123
the symmetric game. Let us note, however, that also in the case where the
costs differ, the endogenous timing game has a variety of mixed strategy
equilibria. We did not take these into consideration since the Harsanyi and
Selten (1988) equilibrium selection theory allows us to neglect them. That
theory gives precedence to pure equilibria whenever these exist and we did
consider all pure equilibria in this paper.
In this paper we only allowed for one point in time where the players can
commit themselves; however, one can easily define the game gtin which
there are ?t −1? periods in which the players can commit themselves. (g1=
g, g2is as in (2.6)–(2.9) and gtis defined by induction for t ≥ 3.) Knowing
the solution of g2, the game gt, with t ≥ 3, can be solved by backward
induction, i.e., by applying the subgame consistency principle fromHarsanyi
and Selten (1988): No matter what the history has been, a subgame gτhas
to be played according to its solution. Adopting this principle, one sees
that in g3waiting is a dominant strategy of player 2: If he waits he can best
respond if the opponent commits, while he is guaranteed his Stackelberg
leader payoff if the other waits as well. Consequently, player 2 will wait and
committing becomes a riskless strategy for player 1. Hence, the solution of
g3is that player 1 will commit itself. In other words, player 1 commits
in order to prevent that player 2 will commit himself. We come to the
conclusion that the predicted outcome is very sensitive to the number of
commitment periods: If t is even, the solution of gtis ?W1?qL
t ≥ 3 is odd, the solution of gtis ?qL
robustness reflects the fact that the discrete time model with t ≥ 3 is not an
appropriate one to model commitment possibilities. In future work we plan
to investigate the issue in continuous time, while possibly also allowing for
commitments to be built up gradually. For earlier work along this direction,
we refer to Spence (1979) and to Fudenberg and Tirole (1983).
2? while, if
1?W2?. In our opinion, this lack of
In this Appendix we complete the proofs of the Lemmas 2 and 4 and
Proof of Lemma 2.
We have to show that the inequalities (4.16) hold.
Hence, we have to show that
?z≤ 0?zα≤ 0?
124van damme and hurkens
It is straightforward to verify that zα≤ 0. It is easily seen that ?α≥ 0 if
and only if
−8+ 16α + z?10− 2α? − 18ln?z + 1? ≥ 0?
Now, ln?1+ z? ≤ z, z ≤1
inequality holds for all ?α?z?-combinations in the relevant domain. Next,
we have that
4+ln?z + 1?
z + 1
z + 1
2, and α ≥ 1, from which it follows that the above
?2− 8α + 8α2+ z?−7+ 10α − α2? + 18?1− α?ln?z + 1??
?−8+ 16α + z?10− 2α? − 18ln?z + 1??
10− 2α −
?−7+ 10α − α2+ 18?1− α?/?z + 1??
?z?α?z? ≤ ?z?1?z? = −1
9+ln?z + 1?
z + 1
where the last inequality follows from z ≤1
This completes the proof of the inequalities (4.16) and, therefore, of
Proof of Lemma 4.
following order: First (1) and (4), next (5), and finally (3). Note that the
inequalities (2) and (6) hold by definition: ¯ qt
mitment quantity against ¯ qt
more, the equality in (4.16) holds since player 1 has the same prior in g as
in ¯ g.
Proof of the inequalities (1) and (4) from (4.19).
strategy of player j defined by
?qj?z? as in (4.3),
We will prove the inequalities from (4.19) in the
2) is the optimal com-
1) in the game ¯ gt(resp. gt). Further-
jfor the mixed
if z ≤ x,
prior is given by mx. It is easily verified that player i’s optimal commitment
quantity in gt?xagainst qjis given by
?1− t??ai− aj/2+ x?aj/2− µx
j= mjand mzi
j= ¯ mj. Write gt?xfor the game at t when the
j??+ t?ai− qj?
?1− t??1+ x? + 2t
endogenous stackelberg leadership 125
sion for µx
jis as in (4.7b) but with zjreplaced by x. Substituting that expres-
jand rewriting yields
i?qj? =ai+ ?1− t??xai− aj− ajln?1+ x??/2− tqj
1+ t + x?1− t?
< 0 if x ≤1
1and t < 1?
A straightforward computation shows that the derivative is negative if
and only if t < 1 and
< 2a2+ ?1− t??xa2− a1− a1ln?1+ x?? − 2tq1?
and, as both sides of this inequality are linear in t, it suffices to check
that the inequality holds at both endpoints. Now, at t = 0 the inequality
??1+ t + x?1− t??
a1ln?1+ x? < a2
which holds since a1≤ a2. At t = 1, the inequality simplifies to
q1< a1?1+ x?
and this holds because of our restrictions on the parameters. (Recall that
these restrictions are without loss of generality: player 1 will not commit to
a quantity that is larger than the Stackelberg leader quantity and if z2≥1
then Lemma 1 applies.)
Since z1< z2(cf. (4.8)), Claim 1 implies that player 2’s best response
quantity is lower in gt? ¯ mthan it is in gt?m. Since player 1 has the same best
response correspondence in these two games, it follows that player 2 (resp.
player 1) commits to a lower (resp. higher) quantity in gt? ¯ mthan in gt?m.
(Formally, if t ≤ min?t1?t2?, then the map q2→ qt?m
and cuts the 45◦line at a point lower than the one where the first graph
cuts the diagonal.) Hence, Claim 1 establishes that for t < 1:
1< ¯ qt
2> ¯ qt
The proof of the inequalities (1) and (4) can now be completed by showing
that the gain from committing is decreasing in the opponent’s quantity.
126van damme and hurkens
Because of the linearity of the payoff function in t? it suffices to show that
this holds for t = 1, i.e., for the original game. Now
In the relevant range where both players find it optimal to commit them-
selves ?t ≤ min?t1?t2?? we have qi≥ qN
−qi+ ?ai− qj?/2 ≤ 0?
which completes the inequalities (1) and (4).
Proof of inequality (5).
We have that
The second integrand is clearly nonnegative. The first is nonnegative since
Proof of inequality (3).
The proof involves some straightforward but te-
dious calculations. For simplicity, write x = z2. Because of Lemma 1 and
Corollary 1, we may confine ourselves to the case where x <1
may verify that, up to a positive multiplier, ¯ gt
to ??t?x?, where
??t?x? = ?x + 1??x − 2??−2x − 1? + t?x − 1?2?4x + 1?
+ t2?−3− x − x2+ 4x3− 2x4?/?1+ x?
+ 6?x2− 1+ t?−1+ 2x − 2x2? + t2?x − 1?2ln?1+ x?
ui?qi?qj? = −qi?
ui?Wi?qj? = −?ai− qj?/2?
ifor i = 1?2, and therefore
2?q1?q2? − ¯ gt
?u2?W1?q2? − u2?q1?z??q2??dz
?u2?q1?z??W2? − u2?W1?W2??dz
?u2?W1?q2? − u2?W1?W2??dz
?u2?q1?z??W2? − u2?q1?z??q2
2. This establishes inequality (5).
2. The reader
1? ¯ qt
2? − ¯ gt
1? ¯ qt
2? is equal
+ 3?t − 1??2+ 2x + t?3− 2x??ln2?1+ x??
t = 0 or t = 1. Now direct substitution yields
??0?x? = ?x + 1??x − 2??−2x − 1? + 6?x2− 1?ln?1+ x?
− 6?1+ x?ln2?1+ x??
For x ≤
2, ??t?x? is concave in t so that the minimum is attained in
endogenous stackelberg leadership 127
Using the fact that ln?1+ x? ≤ x, we obtain
??0?x? ≥ ?x + 1??x − 2??−2x − 1? − 6?1− x2?x − 6?1+ x?x2
= ?x + 1??2− 3x − 2x2? > 0?
Another direct substitution gives
?1+ x???1?x? = 9x − 6?1+ x?ln?1+ x?
≥ 3x?1− 2x? > 0?
where we again have used that ln?1+x? ≤ x. Consequently ??t?x? > 0 for
all t and x, which completes the proof of inequality (3).
Proof of Proposition 2.
ment quantity against qjis
?qj? =?1− t??ai− aj/2+ zj?aj/2− µj??+ t?ai− qj?
We first recall that in gt?m0the optimal commit-
1+ t + ?1− t?zj
other. Using (A.1) (applied to qC
1?t? and qC
2?t? be optimal commitment quantities against each
i?t?) we can rewrite it as
i?t??1+ t + ?1− t?zi
?+ t?ai− qj?
1+ t + ?1− t?zj
For t ∈ ?0?1?, let qI
leaves firm 1 indifferent between committing optimally (to qI
ing. We know from previous analysis that firm 1 strictly prefers committing
to waiting when firm 2 commits to qC
gain from committing is decreasing in the opponent’s commitment strat-
egy. Hence, the curve qI
(See Fig. 2.)
The tracing path must continue along the curve qI
We need to establish the direction. On the tracing path it must hold that
w?t? ≥ 0 and committing with the remaining probability to qI
established that the optimal commitment strategy of firm 2 is increasing in
w?t? (keeping firm 1’s quantity fixed). So qI
this is equivalent to
2?t? denote the commitment quantity of firm 2 that
1?t?) and wait-
2?t?, for all t < t1. Moreover, the
2?t? intersects the curve qC
2?t? from above at t = t1.
2?t? for some time.
2?t? is the best reply against firm 1’s strategy of waiting with probability
1?t?. It is easily
2?t? ≥ qt?m0
1?t??. Using (A.2)
?1+ t + ?1− t?z1?qI
≥ −t?a2− qC
1?t?? + qC
2?t??1+ t + ?1− t?z2? + t?a2− qI
128van damme and hurkens
Multiplying both sides by 1+ t + ?1− t?z2and using (A.2) once more, this
is equivalent to
?1+t +?1−t?z2??1+t +?1−t?z1?qI
which, since z2> z1, implies that
2?t? ≥ qC
This implies that the tracing path must bend backwards.
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