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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

and

Sjaak Hurkens†

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

1. INTRODUCTION

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.

†E-mail: hurkens@upf.es.

105

0899-8256/99 $30.00

Copyright © 1999 by Academic Press

All rights of reproduction in any form reserved.

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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

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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

outcomes.

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-

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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

firm.

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

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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

by qL

quantity that j will choose as a Stackelberg follower, qF

Li= ui?qL

Nash equilibrium of the game and denote player i’s payoff in this equilib-

rium by Ni. For later reference we note that

(2.1)

i (firms i’s Stackelberg leader quantity). We also write qF

jfor the

i?, and

j= bj?qL

2? for the unique

i?qF

j? and Fi= ui?qF

i?qL

j?. We write ?qN

1?qN

qL

i=2ai− aj

Li=?2ai− aj?2

As is well known,

2

?qN

i=2ai− aj

Ni=?2ai− aj?2

3

?qF

i=3ai− 2aj

Fi=?3ai− 2aj?2

4

?

(2.2)

8

?

9

?

16

?

(2.3)

qL

i> qN

i> qF

i

?i = 1?2?

?i = 1?2??

(2.4)

Li> Ni> Fi

(2.5)

Hence each player has an incentive to commit himself.

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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

(2.6)

(2.7)

(2.8)

(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

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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.

(2.10)

(2.11)

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.

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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-

rium ?qN

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

1?qN

2? and by the Stackelberg equilibrium ?qL

1?W2? in which firm 1

2; hence the product of

Di= ?ai− aj??2ai− aj?/4?

(3.1)

At the equilibrium where i leads the product of the deviation losses is

equal to

?Li− Ni??Fj− Dj? = a2

j?2ai− aj?2/1152?

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

(3.2)

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.

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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

gt?m= ?S1?S2?ut?m

ut?m

i

?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

probability z?

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

1?ut?m

2? in which the payoff functions are given by

(3.3)

(3.4)

j= 1− zjto i playing s?

i. With these beliefs, player j will play

js?

ithat he expects i to play.

mj= bj?Zj?

(3.5)

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

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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,

Si= ?qL

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-

1?qN

2?. We show that both

iis a weakly dominated strategy and playing a weakly dominated

Proposition 1.

Cournot equilibrium C ?i = 1?2?.

Proof.

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,

m2= W2.

Next, let player 1 believe that 2 plays z1S12+ ?1− z1?C2= z1W2+ ?1−

z1?qN

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

1.

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

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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

uj?qj?zqL

= z?3aj− 2ai− 2qj/2? + ?1− z??2aj− ai− qj?qj/2

uj?Wj?zqL

= 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

where

iwith probability z and that i waits

i+ ?1− z?Wi?

(4.1)

i+ ?1− z?Wi?

(4.2)

(4.3)

(4.4)

zj=

?4aj− 2ai?2

18a2

j− ?4aj− 2ai?2?

(4.5)

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

?Wj

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?

νj= a2

i+ ?1− z?Wiby bj?z?, we have

if z > zj?

qj?z?

bj?z? =

if z < zj?

(4.6)

(4.7a)

(4.7b)

j/4?1+ zj? − a2

jln2?1+ zj?/4z2

j?

(4.7c)

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116 van damme and hurkens

Straightforward computations now show that

w1> w2?

µ1< µ2? and

(4.8a)

(4.8b)

ν1< ν2?

(4.8c)

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

attractive strategy.

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

αj=ai

aj

(4.9)

for the relative cost advantage of player i (αj> 1 if and only if ci< cj).

Note that zjdepends on ai?ajonly through αj

zj=

?4− 2αj?2

18− ?4− 2αj?2

(4.10)

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.

Proof.

We have

2for the prior strategy of player 2 as given by (4.7). If

2? < u1?W1?m0

2? for all q1. In particular,

ui?qi?m0

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/2?

(4.11)

q∗

i=ai− aj/2+ zj?aj/2− µj?

1+ zj

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

q∗

?

(4.12)

We know that any quantity qj≤ qN

1≤ qN

1. Now, the inequality q∗

2ai− aj+ zj?2aj− ai? ≤ 3ajln?1+ zj?

i≤ qN

iis equivalent to

Page 13

endogenous stackelberg leadership 117

or

2αj− 1+ zj?2− αj? ≤ 3ln?1+ zj??

(4.13)

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.

Lemma 2.

Then u2?m0

Proof.

player i can get by committing himself

Write m0

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).

1?q2?.

uc

i?m0

j? =?ai− aj/2+ zj?aj/2− µj??2

2?1+ zj?

?

On the other hand, waiting yields

ui?Wi?m0

j? = zj

??ai− µj?2/4+ νj/4?+ ?1− zj??2ai− aj?2/9

so that by rearranging we obtain

uc

2?m0

1? − u2?m0

1?W2? =

a2

1

4?1+ z1???α1?z1?

(4.14)

where

??α?z? =1− z

36

4+ln2?1+ z?

?2− 8α + 8α2+ z?−7+ 10α − α2? + 18?1− α?ln?z + 1??

−z

2

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.

Page 14

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.

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.

Lemma 3.

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

i= Wi.

Proof.

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

?qt

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-

i= qN

i

for

i> qN

ifor t < 1 since any quantity less than qN

ifor the payoff function at “time”

i

gi?t? = ut

ii?qt

j

?− ut

i

?Wi?qt

j

?

(4.16)

g?

i?t? =

∂

∂t

?

ut

i

?qt

i?qt

j

?− ut

i

?Wi?qt

j

??

+

∂

∂qj

?

ut

i

?qt

i?qt

j

?− ut

i

?Wi?qt

j

??∂qt

j

∂t?

For t = 1, the first term in this expression is equal to

ui

j

?Wi?m0

2Thisbound isnot sharp. It can be shown that committing isoptimal for firm1 if α1> 1?081

(or z2< 7/20).

?qN

i?qN

?− ui

?Wi?qN

j

j

?− ui

?qN

j

i?m0

?> 0?

j

?+ ui

?Wi?m0

j

?

= ui

?− ui

?qN

i?m0

Page 15

endogenous stackelberg leadership 119

Furthermore, the partial derivative with respect to qjis equal to

−qN

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.

Write qt?m

i

?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

gt

i

?qi?qj? − ut?m

Then gt

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

t2> t1.

Proof.

We only provide a sketch of the proof here and relegate technical

details to the Appendix. The proof consists of comparing the tracing path

?qt

¯ qt

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

inequalities:

?qt

< ¯ gt

1?qt

2?. Write

i?qi?qj? = ut?m

j? > 0 ?1 = 1?2? for t sufficiently small and ?qt

i

?Wi?qj??

(4.17)

i?qt

i?qt

1?qt

isimilarly,

2? is the

1? ¯ qt

2? and ¯ gt

i?qt

i?qt

j? ≥ 0for all t ∈ ?0?τ??. Then

1?qt

1≥ qt

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< qt

2and

2≤ gt

2. Still, since firm 2 has lower cost than firm 1

1< ¯ gt

2. The result follows by combining the above

gt

1?qt

1?qt

2? = ¯ gt

?3?

11?qt

2

? ?1?

? ?4?

≤ ¯ gt

1

?qt

?qt

1? ¯ qt

2

? ?2?

? ?5?

≤ ¯ gt

1

?qt

?¯ qt

1? ¯ qt

2

? ?6?

?

2

?¯ qt

1? ¯ qt

2

≤ ¯ gt

2

1? ¯ qt

2

≤ gt

2

1? ¯ qt

2

≤ gt

2

?qt

1?qt

2

??

(4.18)