Institute of Social Science, University of Tokyo
Discussion Paper Series No. F150

Randomized Strategy Equilibrium andSimultaneous-Move Outcome in
the Action Commitment Gamewith a Small Cost of Leading
Toshihiro Matsumura∗,
Takeshi Murooka†,
Akira Ogawa‡Discussion Paper Series No. F-150
August 18, 2009

Abstract
We introduce a small cost of leading (small gain from waiting) in the two-player action
commitment game formulated by Hamilton and Slutsky (1990). We investigate a quantity
competition model with linear demand and constant marginal costs. We ﬁnd that there exists a unique randomized strategy equilibrium as long as the leading cost is positive and not
too large. The randomized strategy equilibrium converges to the simultaneous-move equilibrium (Cournot equilibrium) as the cost of leading approaches zero. We also investigate
a price competition model in diﬀerential product markets. We show that there exist randomized strategy equilibria, and any of them converge to the simultaneous-move equilibrium(Bertrand equilibrium).
∗

Institute of Social Science, University of Tokyo
Department of Economics, University of California, Berkeley‡
Corresponding author: Akira Ogawa, College of Liberal Arts, International Christian University, 3-102, Osawa, Mitaka-shi, Tokyo 181-8585, Japan. Phone: (81)-422-33-3169. Fax: (81)-422-34-6982. E-mail:ogawaa@icu.ac.jp†

1

JEL classification numbers: L13, C72
Key words: endogenous timing, gain for waiting, mixed strategy equilibria

21

Introduction

In their pioneering work, Hamilton and Slutsky (1990) formulated two important models of theendogenous timing game: observable delay game and action commitment game. Both models
have been intensively used as frameworks for analyzing the endogenous role in many subsequent
works.
In the action commitment game, Hamilton and Slutsky (1990) showed that three pure strategy equilibria exist: two sequential-move outcomes (either player 1 or player 2 becomes the Stackelberg leader) and one simultaneous-move outcome (both players act in the ﬁrst period such as
in the Cournot model and the Bertrand model). They also emphasized that the simultaneousmove outcome is less plausible because the two sequential-move equilibria are the only purestrategy equilibria in undominated strategies and the simultaneous-move outcome is supported
by weakly dominated strategies.
Further, many subsequent works have shown that the simultaneous-move outcome is vulnerable. Albæk (1992), Mailath (1993), Normann (1997), and Hirokawa and Sasaki (2000)
investigated incomplete information games wherein waiting players obtain additional information on demand or cost. In other words, leading players lose some informational gain. Theyshow that while a slight informational gain from waiting eliminates the simultaneous-move equilibrium where both players act in the ﬁrst period, it does not eliminate the two sequential-moveoutcomes. Robson (1990) and Matsumura (1997, 1999) showed that a slight inventory costyields the same result in complete information games. These results show that the set of equilibrium outcomes is not lower hemi-continuous with respect to the cost of leading and that thesimultaneous-move outcome is an equilibrium only when the cost is strictly zero. This may
indicate that the simultaneous-move outcome (such as Bertrand and Cournot) is less plausible
in the endogenous timing game than the sequential-move outcomes (such as Stackelberg).
However, recent studies on laboratory experiments have suggested the opposite. Huck etal. (2002) and Fonseca et al. (2005) examined quantity competition setting and reported
that the Stackelberg outcomes are rarely observed, while the Cournot outcome often appears.

3In this paper, we bridge the gap between the contrasting theoretical and empirical results byinvestigating a randomized strategy equilibrium in the action commitment game.
In this paper, we introduce a small cost of leading into the action commitment game withquantity competition. In the action commitment game with small leading costs, there are twopure strategy equilibria (either player 1 or player 2 is the leader). Naturally, we expect that
there is a randomized strategy equilibrium, too. In fact, we ﬁnd that a randomized strategy
equilibrium does exist and it converges to the simultaneous-move outcome when the cost ofleading converges to zero. Our result indicates that the simultaneous-move outcome is not as
vulnerable as the current literature insists. We can regard the simultaneous-move equilibrium
as a degenerate randomized strategy equilibrium, which is the unique symmetric equilibrium.
We also show that our main result does not depend on the assumptions of strategic substitutes
and/or the ﬁrst-mover advantage, which typically appear in quantity competition models. We
investigate a price competition model in diﬀerentiated product markets. We ﬁnd that the model
has randomized strategy equilibria and any of them converge to the Bertrand equilibrium under
the standard assumptions in the literature on Bertrand competition.

2

Quantity Competition

We formulate a duopoly model with quantity competition. Firms produce perfectly substitutablecommodities. Let yi denote the output of ﬁrm i (i = 1, 2). Let each ﬁrm’s marginal production
cost be c. The market demand is given by P (Y ) = a − Y (price as a function of quantity), whereY = yi + yj is the total output of the duopolists. We assume that a > c.
We consider the action commitment game of Hamilton and Slutsky (1990) where ﬁrms ﬁrst
choose the timing of their actions. The game is a complete information game. There are two
possible time stages for production choice and each ﬁrm chooses its output in only one of thetwo stages. In the ﬁrst stage, ﬁrm i independently chooses its output yi , or waits to produce
until the next stage. At the beginning of the second stage, each ﬁrm observes the rival’s output
produced in the ﬁrst stage. In the second stage, only the ﬁrm i that has been waiting until the4second stage produces its output yi .
The payoﬀ of ﬁrm i (i, j = 1, 2, i = j) is given by Ui = (P (Y ) − c)yi − ε if ﬁrm i produces inthe ﬁrst period and Ui = (P (Y ) − c)yi if it produces in the second period, where ε indicates thecost of leading. We assume that ε ∈ [0, (a − c)2 /72).1

2.1

Fixed Timing Games

Before discussing the endogenous timing game formulated in the previous section, we investigate
the ﬁxed timing games (simultaneous-move and sequential-move games).
2.1.1

Cournot game

Consider the simultaneous-move game where ﬁrm i (i = 1, 2) independently chooses yi . The
ﬁrst-order condition isa − 2yi − yj − c = 0.

(1)

Let Ri (yj ) be the reaction function of ﬁrm i in the Cournot game. It is given by1
Ri (yj ) = (a − yj − c).
2(2)

Let the superscript ‘C’ denote the equilibrium outcome in the Cournot game. We have1
y1C = y2C = (a − c).
3
2.1.2

(3)

Stackelberg game

Consider the sequential-move game where ﬁrm i chooses yi ﬁrst and then ﬁrm j chooses yj
after observing yi (i, j = 1, 2, i = j). Firm j chooses yj = Rj (yi ) and ﬁrm i maximizes P (yi +Rj (yi ))yi − cyi − ε. Let the superscript ‘L’ and ‘F’ denote the equilibrium outcome of the leader
and the follower in the Stackelberg game, respectively. We have
11yiL = (a − c), yjF = (a − c).
2
4
1

(4)

If ε = (a − c)2 /72, the three pure strategy equilibria again appear. One is a Cournot equilibrium whereinboth firms choose the second stage and the other two are Stackelberg equilibria. If ε > (a − c)2 /72, the uniqueequilibrium exists and it is a Cournot equilibrium.

5

2.2

Results

We investigate the action commitment game. First, we present two results on the pure strategy
equilibria in the general context of the action commitment game.
Result 1 (Hamilton and Slutsky 1990) Suppose that ε is zero. There are three pure strategy
equilibria. One is the Cournot equilibrium where both ﬁrms choose to produce in the ﬁrst stage.
The other two are Stackelberg equilibria where one ﬁrm chooses to produce in the ﬁrst stage andthe other chooses to produce in the second stage.
Result 2 (Matsumura 1999) Suppose that ε is positive and suﬃciently small. There are twopure strategy equilibria and both are Stackelberg equilibria.
We now present our result. We discuss the randomized strategy equilibrium.
Proposition 1 Suppose that ε ∈ (0, (a − c)2 /72). (i) There exists a unique randomized strategy
equilibrium and it is symmetric where each ﬁrm chooses the ﬁrst stage with probability q E (ε) ∈(0, 1). (ii) Let q E (ε) ∈ (0, 1) be the probability with which each ﬁrm chooses to produce in theﬁrst stage in the unique randomized strategy equilibrium. Then q E (ε) is continuous and strictly
decreasing with limε→0 q E (ε) = 1 and limε→(a−c)2 /72 q E (ε) = 0.
Proof See Appendix.
Three equilibria exist regardless of whether ε is zero or positive (as long as ε < (a − c)2 /72).
When ε is zero, only pure strategy equilibria exist.2 When ε is strictly positive and suﬃciently
small, two pure strategy equilibria and one randomized strategy equilibrium exist. Although
the set of pure strategy equilibria is discontinuous at ε = 0, this discontinuity disappears if
we consider randomized strategy equilibrium along with pure strategy equilibrium. We canregard the Cournot equilibrium when ε = 0 as a degenerate randomized strategy equilibrium.
Our result indicates that the Cournot equilibrium is not so vulnerable if we pay attention to
randomized strategy equilibrium. We believe that as is suggested in the empirical works such as
2

Hamilton and Slutsky (1990) discussed that in the case with linear demand and equal constant marginal

costs, no equilibrium, symmetric or asymmetric, with mixing over the timing of actions exists.

6

Huck et al. (2002) and Fonseca et al. (2005), the Cournot equilibrium is plausible in endogenous
timing contexts.

3

Price Competition

In the previous section we discussed quantity competition and showed that the randomized
strategy equilibrium converges to the Cournot equilibrium. In this section, we examine pricecompetition in diﬀerentiated product markets and show that the equilibrium also converges tothe Bertrand equilibrium.

3.1

Fixed Timing Games

Before discussing the endogenous timing game, we investigate the ﬁxed timing game (simultaneousmove and sequential-move games). We assume that the following ﬁxed timing games have aunique pure strategy equilibrium and that is stable. We also assume the interior solution.
3.1.1

Bertrand game

Consider the simultaneous-move game where ﬁrm i (i = 1, 2) independently chooses pi . The
payoﬀ of ﬁrm i is given by Vi (pi , pj ) (i = 1, 2, i = j). We assume that Vi (pi , pj ) is strictly concave,
continuously diﬀerentiable and increasing in pj . We also assume that |∂ 2 Vi (pi , pj )/∂ 2 pi | >|∂ 2 Vi (pi , pj )/∂pi ∂pj |. This is a so called stability condition.3 These assumptions are standard
in the ﬁeld.
The ﬁrst-order condition is∂Vi (pi , pj )
= 0.
∂pi
B
Let (pBi , pj ) be the Bertrand equilibrium prices.

3.1.2

Stackelberg game

Consider the sequential-move game where ﬁrm i chooses pi ﬁrst and then player j chooses pj
after observing pi (i, j = 1, 2, i = j). Let Rj (pi ) be the reaction function of ﬁrm j. We assume3

See, among others, Vives (1985).

7

that Rj (pi ) is continuously diﬀerentiable and upward-slopping, i.e., Rj > 0. Firm i maximizes
Vi (pi , Rj (pi )) − εi with respect to pi . The ﬁrst-order condition is∂Vi (pi , pj ) ∂Vi (pi , pj ) dRj (pi )
+
= 0.
∂pi
∂pj
dpi
We assume that Vi (pi , Rj (pi )) is strictly concave. Let superscript ‘L’ and ‘F’ denote the equiF
librium outcome of the leader and the follower in the Stackelberg game. Let (pLi , pj ) be theFequilibrium actions in the sequential-move game. We assume that pL
i > pi . This always holds

under the assumptions made above if two ﬁrms are symmetric. Needless to say, the symmetryof the ﬁrms is not a necessary condition.

3.2

Results

We discuss the equilibria of the action commitment game. As in quantity competition, three purestrategy equilibria exist when ε = 0, and two pure strategy equilibria (Stackelberg equilibria)
exist when ε is positive and small.
F
We now present our results on randomized strategy equilibria. Deﬁne ε̄i := Vi (pL
i , pj ) −B
Vi (pBi , pj ). We can show that ε̄i is strictly positive.

Proposition 2 Suppose that 0 < ε1 < ε̄1 and 0 < ε2 < ε̄2 . (i) For any ε1 , ε2 , a randomized
strategy equilibrium exists. (ii) Take any sequence of randomized strategy equilibria where ﬁrm
i’s cost of leading is εni and ﬁrm i chooses the ﬁrst stage with probability qin . If limn→∞ εni = 0,then limn→∞ qjn = 1. If limn→∞ εni = ε̄i , then limn→∞ qjn = 0.
Proof. See Appendix.
Under this general setting, the uniqueness of the randomized strategy equilibrium is notguaranteed. However, any randomized strategy equilibria converge to the Bertrand equilibrium as the cost of leading approaches zero. Thus, our main result still holds under strategic
complements.

8

4Concluding Remarks

In their pioneering work on endogenous timing, Hamilton and Slutsky (1990) showed that threepure strategy equilibria exist: one simultaneous-move equilibrium (Cournot or Bertrand) andtwo sequential-move equilibria (Stackelberg). Many works pointed out that the Cournot or
Bertrand equilibrium is not robust since it does not constitute a pure strategy equilibrium if asmall gain from waiting (such as avoiding inventory costs or additional informational gain) is
introduced. On the contrary, empirical works using laboratory experiments showed that in thequantity competition, Cournot behavior often appears while Stackelberg behavior is rarely seen.
We bridge the gap between the two contrasting results by investigating randomized strategy
equilibrium. We ﬁnd that there exists a randomized strategy equilibrium and it converges to
the Cournot or the Bertrand equilibrium as the gain from waiting approaches zero. Thus, weshow that the pure strategy simultaneous-move equilibrium in the action commitment game isa degenerate randomized strategy equilibrium.
If we restrict our attention to pure strategy equilibria, we get that the set of equilibriumoutcomes is not lower hemi-continuous with respect to the gain from waiting, and that theCournot or Bertrand outcome is an equilibrium only when the gain from waiting is exactly zero.
However, if we extend our attention to randomized strategy equilibria, the set of equilibria iscontinuous. The equilibrium outcome and equilibrium welfare which are very close to those of
the Cournot or the Bertrand equilibrium appear in the randomized strategy equilibrium when
the gain from waiting is suﬃciently small. Thus, the Cournot or the Bertrand equilibrium may
be as robust as the other two Stackelberg equilibria; this result is consistent with the results ofHuck et al. (2002) and Fonseca et al. (2005).
Although we show the result under fairly general conditions in strategic complement cases,
we failed to show it under general conditions in strategic substitute cases. This remains for
future research.

9

Appendix
Proof of Proposition 1(i) Suppose that ﬁrm i chooses to produce in the second stage. At
the beginning of the second stage, ﬁrm i observes whether or not ﬁrm j has produced in theﬁrst stage. When ﬁrm j chooses to produce in the ﬁrst stage, ﬁrm i chooses yi = Ri (yj ) =(a − c − yj )/2. When ﬁrm j chooses to produce in the second stage, both ﬁrms face Cournot
competition. As a result, ﬁrms choose y1 = y2 = (a − c)/3.
Suppose that ﬁrm i chooses to produce in the ﬁrst stage. Let superscript ‘1’ implies the ﬁrst
period production. For example, yi1 is the output of ﬁrm i when it chooses to produce in theﬁrst stage. Let qj be the probability that ﬁrm j chooses to produce in the ﬁrst stage. Firm ichooses yi1 so as to maximize qj (a − yi1 − yj1 − c)yi1 + (1 − qj )(a − yi1 − Rj (yi1 ) − c)yi1 − ε. The
ﬁrst-order condition isqj (a − c − yj1 − 2yi1 ) + (1 − qj )((a − c)/2 − yi1 ) = 0.
The best reply of ﬁrm i when it chooses to produce in the ﬁrst stage is
1(1 + qj )(a − c)11
− qj yj .
yi =1 + qj2(5)(6)

In the randomized strategy equilibrium, the expected payoﬀ of ﬁrm i when it chooses to
produce in the ﬁrst stage must be equal to its expected payoﬀ when it chooses to produce inand the second stage. Thus,
qj (a − yi1 − yj1 − c)yi1 + (1 − qj )(a − yi1 − Rj (yi1 ) − c)yi1 − ε= qj (a − yj1 − Ri (yj1 ) − c)Ri (yj1 ) + (1 − qj )(a − yiC − yjC − c)yiC .

(7)

Substituting i = 1, j = 2 and i = 2, j = 1 into (6) and (7), we have four equations. Solving
these four equations with respect to y11 , y21 , q1 , and q2 , we have the equilibrium outcome. From
the two equations in (6), we haveyi1 =1 + qi(a − c) (i, j = 1, 2, i = j).
2(1 + qi + qj )(8)

Substituting (8) into (7), we have
(1 − q2 )(2 + 4q1 − 5q2 + 4q1 q2 + 2q12 − 7q22 )(a − c)2= ε.
144(1 + q1 + q2 )210(9)

(1 − q1 )(2 + 4q2 − 5q1 + 4q2 q1 + 2q22 − 7q12 )(a − c)2= ε.
144(1 + q2 + q1 )2(10)

Subtracting (10) from (9), we obtain(q1 − q2 )(11 + 4q1 + 4q2 − 5q1 q2 − 7q12 − 7q22 )(a − c)2= 0.
144(1 + q1 + q2 )2(11)

We can show that 11 + 4q1 + 4q2 − 5q1 q2 − 7q12 − 7q22 = 0 if and only if q1 = q2 = 1. Thus, at the
randomized strategy equilibrium, q1 = q2 must hold.
Substituting q1 = q2 = q into (9), we have
(1 − q)2 (2 + q)(a − c)2(1 − q)(2 − q − q 2 )(a − c)2== ε.
144(1 + 2q)2
144(1 + 2q)2

(12)

Let
h(q) :=(1 − q)2 (2 + q)(a − c)2.
144(1 + 2q)2

We haveh(0) =(a − c)2dh(q)(1 − q)(11 + 5q + 2q 2 )(a − c)2> 0, h(1) = 0, and=−< 0 ∀q ∈ [0, 1).
72
dq
144(1 + 2q)3Thus, (12) has a unique solution q E ∈ (0, 1) for all ε ∈ (0, (a − c)2 /72)—a randomized strategy
equilibrium.
Proof of Proposition 1(ii) By the implicit function theorem,
1dq E (ε)= dh(q) < 0dεdq

holds for all q ∈ [0, 1). Thus q E (ε) is decreasing in ε.
q E (ε) is the solution of h(q) = ε. Since h(1) = 0, we have q E (0) = 1. Since h(0) = (a−c)2 /72,
we have q E ((a − c)2 /72) = 0. We can show that q E (ε) is continuous for all ε ∈ [0, (a − c)2 /72),
so q E (ε) → 1 as ε → 0 and q E (ε) → 0 as ε → (a − c)2 /72.
Proof of Proposition 2(i) Suppose that ﬁrm i chooses to name its price in the second stage.
At the beginning of the second stage, ﬁrm i observes whether or not ﬁrm j named its price in theﬁrst stage. When ﬁrm j chooses to name its price in the ﬁrst stage, ﬁrm i chooses pi = Ri (pj ).
11

When ﬁrm j chooses to name its price in the second stage, ﬁrms face Bertrand competition. As
Ba result, ﬁrms choose (pi , pj ) = (pB
i , pj ) in equilibrium.

Suppose that ﬁrm i chooses to name its price in the ﬁrst stage. Let p1i be the price of ﬁrm iwhen it chooses to name its price in the ﬁrst stage. Further, let p2j be the price of ﬁrm j whenﬁrm i chooses p1i in the ﬁrst stage. Let qj be the probability that ﬁrm j chooses to name its pricein the ﬁrst stage. Firm i chooses p1i so as to maximize qj Vi (p1i , p1j ) + (1 − qj )Vi (p1i , Rj (p1i )) − εi .
The ﬁrst-order condition is∂ qj Vi (p1i , p1j ) + (1 − qj )Vi (p1i , Rj (p1i ))=qj∂Vi (p1i , p1j )
∂p1i

∂p1i
+ (1 − qj )∂Vi (p1i , p2j )∂p1i∂Vi (p1i , p2j ) dRj (p1i )
+
·
= 0.
∂p2j
dp1i

(13)

In the randomized strategy equilibrium, the expected payoﬀ of ﬁrm i when it chooses toname its price in the ﬁrst stage must be equal to its expected payoﬀ when it chooses to nameits price in the second stage. Thus,B
qj Vi (p1i , p1j ) + (1 − qj )Vi (p1i , Rj (p1i )) − εi = qj Vi (Ri (p1j ), p1j ) + (1 − qj )Vi (pB
i , pi ).

(14)

Substituting i = 1, j = 2 and i = 2, j = 1 into (13) and (14), we have four equations. Solving
these four equations with respect to p11 , p12 , q1 , and q2 , we have the equilibrium outcome.
Since we assume |∂ 2 Vi (pi , pj )/∂ 2 pi | > |∂ 2 Vi (pi , pj )/∂pi ∂pj |, we can apply the implicit function theorem to the two equations in (13). From the two equations in (13), we have the solution
p1i (qi , qj ), p1j (qi , qj ), which is continuously diﬀerentiable for all qi , qj ∈ [0, 1].
Substituting p1i (qi , qj ), p1j (qi , qj ) into (14), we have
qj [Vi (p1i (qi , qj ), p1j (qi , qj )) − Vi (Ri (p1j (qi , qj )), p1j (qi , qj ))]B+(1 − qj )[Vi (p1i (qi , qj ), Rj (p1i (qi , qj ))) − Vi (pB
i , pi )] = εi .

(15)Note that if qj = 0, then p1i (qi , qj ) = pLi and the left-hand side of (15) equals ε̄i . Note alsothat if qj = 1, then p1i (qi , qj ) = Ri (p1j ) and the left-hand side of (15) equals zero. Hence, the lefthand side of (15) is greater (smaller) than the right-hand side of (15) when qj = 0 (1). Thus, we

12can apply the ﬁxed point theorem shown in Garcia and Zangwill (1979). Under these conditions,for any εi , εj there exists (qi∗ , qj∗ ) that comprises a randomized strategy equilibrium.
Proof of Proposition 2(ii).

First, we prove that limn→∞ qjn = 1 if limn→∞ εni = 0. We

prove it by contradiction. Suppose not. Then there exists a sequence of randomized strategy
n ∞equilibrium such that {εni }∞n=1 converges to 0 and {qj }n=1 does not converge to 1.
0
Let qj0 be lim inf n→∞ qjn . Since qjn ≤ 1, qj0 < 1. Since p1i (qi , qj ) = pLi when qj = 0, qj > 0

holds from (15).
n ∞
0
k ∞
Let {qjk }∞k=1 be a subsequence of {qj }n=1 which converges to qj . Note that {εi }k=1 converges

to 0.
By the continuity of p1i , p1j , Rj , and Vi ,lim [qjk (Vi (p1i (qik , qjk ), p1j (qik , qjk )) − Vi (Ri (p1j (qik , qjk )), p1j (qik , qjk )))k→∞B+ (1 − qjk )(Vi (p1i (qik , qjk ), Rj (p1i (qik , qjk ))) − Vi (pB
i , pj ))]=qj0 (Vi (p1i (qi0 , qj0 ), p1j (qi0 , qj0 )) − Vi (Ri (p1j (qi0 , qj0 )), p1j (qi0 , qj0 )))(16)B+ (1 − qj0 )(Vi (p1i (qi0 , qj0 ), Rj (p1i (qi0 , qj0 ))) − Vi (pB
i , pj ))= lim εki = 0k→∞holds from (15). Thus, qj0 ∈ (0, 1) and ε0i = 0 comprise the randomized strategy equilibrium.
L 4 Consider ﬁrm i’s
On the other hand, for any qj ∈ (0, 1) p1i (qi , qj ) must be in (pB
i , pi ).

payoﬀ when i chooses p1i = Ri (p1j (qi0 , qj0 )) in the ﬁrst period:
qj0 Vi Ri (p1j (qi0 , qj0 )), p1j (qi0 , qj0 ) + (1 − qj0 )Vi Ri (p1j (qi0 , qj0 )), Rj (Ri (p1j (qi0 , qj0 ))) .

(17)

1 0 0
B
1 0 0
F
L
Since pB
i < pi (qi , qj ) and Rj is upward slopping, pj < Rj (pi (qi , qj )). Because pj = Rj (pi ) and
1 0 0
F
F
L
1 0 0
L
p1i (qi0 , qj0 ) < pL
i , Rj (pi (qi , qj )) < pj . By the assumption of pi < pi , we have Rj (pi (qi , qj )) < pj .
LSince Vi (pi , Rj (pi )) is strictly concave, Vi (pi , Rj (pi )) increases in pi for pi ∈ (pB
i , pi ). BeL
1 0 0
1 0 0
cause we have shown that Rj (p1i (qi0 , qj0 )) ∈ (pB
i , pi ), Vi (Ri (pj (qi , qj )), Rj (Ri (pj (qi , qj )))) >
B
B B
1 0 0
1 0 0
Vi (pBi , Rj (pi )) = Vi (pi , pj ). Thus, (17) is strictly larger than Vi (pi (qi , qj ), Rj (pi (qi , qj ))) −4

This was suggested in Hamilton and Slutsky (1990), and was shown in Pastine and Pastine (2004).

13

B
1
Vi (pBi , pj ). Since ﬁrm i’s price when it names its price in the ﬁrst stage is pi (qi , qj ), (17) is equal

to or smaller than Vi (p1i (qi0 , qj0 ), p1j (qi0 , qj0 )) − Vi (Ri (p1j (qi0 , qj0 )), p1j (qi0 , qj0 )). Thus, the equality in(16) does not hold—a contradiction.
Next, we prove that limn→∞ qjn = 0 if limn→∞ εni = ε̄i . The ﬁrst term in (15) is non-positive
and the second term in (15) is at largest (1 − qj )ε̄i . Thus qj must be close to zero when εi isclose to ε̄i .

14

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