It takes two to tango: Equilibria in a model of sales

Tilburg University

It takes two to tango

Baye, M.R.; Kovenock, D.; de Vries, C.G.

Publication date:

1993

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Baye, M. R., Kovenock, D., & de Vries, C. G. (1993). It takes two to tango: Equilibria in a model of sales.

(Reprint Series). CentER for Economic Research.

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It Takes Two to Tango:

Equilibria in a Model of Sales

by

Michael R. Baye,

Dan Kovenock

and Casper G. de Vries

Reprinted from

Games and Economic Behavior

Vol. 4, No. 4, 1992

~P~~~ '

Reprint Series

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It Takes Two to Tango:

Equilibria in a Model of Sales

by

Michael R. Baye,

Dan Kovenock

and Casper G. de Vries

Reprinted from

Games and Economic Behavior

Vol. 4, No. 4, 1992

GAMES AND ECONOMIC BEHAVIOR 4~ 493-510 (1992)

It Takes Two to Tango: Equilibria in a

Model of Sales~`

IVIICHAEL R. BAYE

Department of Economics, 613 Kern Graduate Building, The Pennsyluania State

Universiry, Uniuersiry Park, Pennsyluanèa 16802

DAN ICOVENOCK

Department of Economics, Purdue Uniuersity, West Lajayette, Indiana 47907

AND

CASPER G. DE VRIES

Center for Economic Studies, Katholieke Uniuersiteit Leuven, 3000 Leuven, Belgium

Received October 8, 1990

We show that the Varian model of sales with more than two firms has two types of equilibria: a unique symmetric equilibrium, and a continuum of asymmetric equilibria. ln contrast, the 2-firm game has a unique equilibrium that is symmetric. For the n-firm case the asymmetric equilibria imply mixed strategies that can be ranked by first-order stochastic dominance. This enables one to rule out asymmet-ric equilibria on economic grounds by constructing a metagame in which both firms and consumers are players. The unique subgame perfect equilibrium of this metagame is symmetric. Journal ojEconomic Literature Classification Number: ~22. O 1992 Academic Press.lnc.

' We have benefitted from helpful conversations wíth Raymond Deneckere, Bill Neilson,

and Ton Vorst. An earlier version of this paper was presented at the Fourth Congress of the

European Economic Association, Augsburg, September, 1989. All three authors are grateful

to the CentER for Economic Research at Tilburg and to Erasmus University Rotterdam

for support. Kovenock gratefully acknowledges financial support through an Ameritech

Foundation Summer Faculty Research Grant.

493

0899-8256192 á5.00

494 BAYE, KOVENOCK, AND DE VRIES I. INTRODUCTION

It is now well known that there are conditions under which mixed

strategy equilibria exist in n-person games where players have payoff

functions that are neither quasiconcave nor continuous. For the most part,

analyses of these equilibria have focused either on general conditions for

existence, as in Dasgupta and Maskin (1986) and Simon (1987), or on

the derivation of particular symmetric equilibrium mixed strategies, as in

Varian (1980). To date, a complete characterization of equilibria in this

class of n-person games has not been addressed.'

Our initial goal in starting the present line of research was to investigate

uniqueness of equilibrium in a wide class of games of this type. It turns

out, however, that the question appears to be complex and model specific.

Thus, in this paper, we focus exclusively on Varian's (1980) seminal model

of sales. While this may seem particularly restrictive, many economic

problems, such as the "all-pay auction game,"Z have a similar struciure.

The principal result of this paper is that, when there are more than two

firms in the Varian model of sales, there exist a coniinuum of ásymmetric

equilibria and a unique symmetric equilibrium. This contrasts sharply with

the 2-firm game in which the unique equilibrium is the symmetric one.

Despité the multiplicity of equilibria, in all equilibria at least two agents

randomize continuously over the union of the supports of the equilibrium

price distributions, just as in the 2-firm game. We call this the two-to-tango

property.

While the set of equilibria in the Varian model of sales is large, we show

that all the asymmetric equilibria imply mixed strategies that can be ranked

by first-order stochastic dominance. This ordering enables us to construct

a metagame in which both firms and consumers are players and in which

asymmetric equilibria are ruled out as subgame perfect equilibria.

Intu-itively, the asymmetric strategies are not consistent with price dispersion

~ For particular two-person games, such as the War of Attrition (Hendricks, Weiss and

Wilson, 1988), capacity-constrained price setting games (Osborne and Pitchik, 1986a), and

price setting with loyal consumers (Narasimhan, 1988), uniqueness has been thoroughly

examined. Osborne and Pitchik (1986b) also examine the question of uniqueness in the

3-firm "pure" location model of Hotelling (1929). They show that with a uniform distribution

of consumers, in addition to a symmetric mixed strategy equilibrium (sce Shaked, 1982),

there is a unique (up to symmetry) asymmetric equilibrium within the class of equilibria in

which at least one firm uses a pure strategy.

EQUILIBRIA IN A MODEL OF SALES

495

because they imply that distributions of prices charged by some firms

stochastically dominate those charged by other firms.3

II. TrtE MoDEL

Following Varian (1980), consider a market where n? 2 firms produce

a homogeneous product with an identical technology exhibiting weakly

declining average cost.' The cost curve of each firm is denoted c(q), where

q is quantity produced. We assume ihat there are a large number of

consumers, each of whom will purchase one unit of the good if faced with

a price less than or equal to a reservation value r, and none of the good if

faced with a price greater than r. There are two types of consumers:

informed and uninformed. Informed consumers purchase a unit of the

good from the store charging the lowest price, as long as this price is below

the reservation value. Each uninformed consumer is aware of the price at

one firm only, and purchases from that firm if the price is no greater than

r. We assume that the same number, U, of uninformed consumers shop

at each store, and that rU - c(U) ) O.S The total number of informed

consumers is 1.

Firms are assumed to set prices simultaneously. Let (p,, ..., p„) be

the vector of prices charged by the firms, and define p-; to be the minimum

price charged by any firm other than i, and m-; to be the number of firms

charging p-;. Then firm i's profit is given by

P;(! t U) - c(I f U)

if

p; c p-; and p; ~ r

P; I ~ t U~ - c I 1 t U~

if

p; - p-; s r

`m -;

`m -;

n,(Pi, . . . ,P,~

-P;U- c( U)

if

P-; c P; ~ r

0

if

p;1 r.

Thus, if a firm does not set the lowest price, it services only the U

uninformed consumers who shop at the store. If a firm sets the lowest

price, it services all of the informed consumers plus the U uninformed

~ An important by-product of our analysis of this metagame is that we genera(ize the model

of Narasimhan (1988) from 2 órtns to n firms.

~ To focus on essentials, we ignore the entry decision and view n as fixed. Varian assumes

strictly declining average cost in order to pin down the equilibrium number of firms when

there is free entry.

496 BAYE, KOVENOCK, AND DE VRIES

consumers who shop at the store. In the event several firms tie in charging

the lowest price, they share the informed consumers equally.

Varian shows in his Proposition 2 that there is no equilibrium where all

firms charge the same price. In a sequence of propositions, he derives the

symmetric mixed-strategy Nash equilibrium to the game. In the following

section we expand the scope of analysis to determine whether there exist

asymmetric mixed-strategy equilibria.

III. THE FULL S ET OF EQUILIBRIA

ln the normal form of the above game, firm i's strategy is p; E[0, x) and

its payoff function is II;(p~, ..., p„), i- 1, ..., n. The complete set of

Nash equilibria will be derived in a series of lemmas. In what follows we

define

rU f c(I f U) - c(U)

p-

IfU

'

(D.1)

i.e., the price at which a firm selling to both its uninformed consumers and

the informed consumers obtains the same profit that it would obtain by

charging the reservation price r and selling only to its uninformed

consum-ers. A price below p is strictly dominated by setting r.

Let s; and s; denote the lower and upper bounds of firm i's equilibrium

price distribution G;. When s; - s;, firm i adopts a pure strategy; otherwise

it employs a mixed strategy. Let a; denote the size of a mass point in i's

distribution.

One equilibrium of this game, a symmetric equilibrium, has been

ana-lyzed by Varian (1980).6 Our main result, summarized in the following

theorem, is that there is a continuum of asymmetric equilibria as well.

THEORÉi~t 1.

The Varian model ofsales possesses two types of

equilib-ria. Either all firms use the same continuous mixed strategy with support

[ p, r] , or at least two firms randomize ouer [ p, r], with each otherftrm i

rándomizing over [ p, x;), x; c r, and havingá mass point at r equal to

(1 - G,{x;)).~ When two or more ,firms haue a positive density ouer a

common interual they play the same (continuous) mixed strategy ouer that

interual.

6 After this paper was completed, it was brought to our attention that BagnoG ( 1986) found

a 5nite number of asymmetric equilibria (see Example 2 below).

~ We could have s; ~ p, in which case the interval [ p, z;) is empty and firm i places all

-EQUILIBRIA IN A MODEL OF SALES 497

To prove the theorem we need a sequence of lemmas.

LEMMAI.

Hir?s;?S;?p10.

Proof. By setting p; r each firm can guarantee itself at least rU

-c(U). This rules out prices greater than r, at which firms earn zero. For

prices less than p, II; c p(1 t U) - c(I t U) - rU - c(U). Hence, a

firm will never pnce below p, as more could be earned by charging r.

~

LEMMA 2.

If 31, j s.t. s; ~ s~ and a;(s~) - 0 then s~ - r. If s; c s; limyt,

G~ ( p) - G~ (s;). !f in addition a;(s;) - 0 then limp t, G~{p) - Iimo t 5. G~{ p).

Proof. Ih (s~, G-~) - s~ U- c(U) c rU - c(U) for s~ c r. Since the

same holds for II;(p, G-~) for p ~ s; and p- s; if a;(s;) - 0, the claim

follows.

~

LEMMA 3.

!f s, - . - . - s,„ G s,,,t ~ , . . . , s„ for n ? m ? 2 then

3 i ~ m such that a;(s;) - 0.

Proof. Suppose not. Then any i ~ m has an incentive to undercut s; by

small e 7 0.

~

LEMMA 4.

If s~ -..~- sm e Sm}~, ..., s„ for n ? m? 2 then

s;-rtl;.

Proof. Immediate from Lemmas 2 and 3.

~

LEMMA S.

There exists no ftrm i such that s; c s~ dj ~ i.

Proof. Suppose such a firm did exist. If a;(s;) - 0, from Lemma 2

limpt, G~(p) - limots; G~(p), b'j ~ i, which implies that II;(s;, G-;) c

llmP t, II;(p, G-;). If the claim held and a;(s;) ~ 0 then dj ~ i a~(s;) - 0,

which implies that limP t, G;(p) - limo t s; G;( p), leading to a similar

contra-díction.

~

LEMMA 6.

s; - r b~i.

Proof. Immediate from Lemmas 4 and 5.

~

Let II ~` represent the equilibrium profit of firm i- 1, ..., n. Then we

have:

LEMMA 7.

I~I;` - II~ tÍi~j.

Proof. Without loss of generality suppose II;` c II; . With s~ being the

lower bound ofj's support, II' c II; - II~L~, G-;) s limD tS~ II;(p;, G-;),

a contradiction.

~

LE~tMA 8.

II! - rU - c(U) bi.

498 BAYE, KOVENOCK, AND DE VRIES

II; - rU - c( U) from Lemmas 3 and 6, and with firms earning equal

profit from Lemma 7, II;` - rU - c(U) di.

r.

LEMMA 9. ~l

, j such that s; - sj - p.

Proof. Suppose not. Let s; be the second lowest s. Then the lowest s

firm can set a price p slightly below s; and earn TI; - p(1 f U) - c(1 f

U) ~ II ~`.

r.

The previous nine lemmas establish that s; - r di; there exist two i's,

say i- i, Z, such that s i - s, - p ; and II ` - rU - c( U) di. We now

proceed to pin down the equilibrium distributions. Let W(p) - p( U f 1)

- c(U -t- I), L(p) - pU - c(U),

n ~

A; - ~ (1 - Gj),

and

A;j - ~ (1 - Gk).

j-~ k~l

j,~; kfj,i

LEMMA 10.

There are no point masses on the half open interval

[P~ r).

-Proof Suppose one of the cumulative distribution functions, say G;,

has a mass point at p;. Since b~p E[p, r), (1 - G;)A;j 7 0, (1 - G;)A~ has

a downward jump at p;, Vj ~ i. This follows directly from the monotonicity

of the c.d.f.'s. For p; 1 p this implies that it is worthwhile for j to transfer

all mass from an s-neighborhood above p; to some S-neighborhood below

p;. At p; - p it pays forj to transfer mass from an e-neighborhood above

p; to r. Thus, there would be an s-neighborhood above p; in which no other

firm j would put mass. But then it cannot be an equilibrium strategy for

player i to put mass at p;.

~

Lemma 11 is a generalization of Varian's Proposition 4.

LEMMA. I1.

The integrand

B~{P;) - W(Pi)At(Pr) } L(Pi)(1 - A,{Pr))

(D.2)

is constant and equal to rU - c(U) at the points of increase of G; in the

half open inlerval [ p, r) for all i.

Proof. By Lemma 10 there are no point masses in the interval. Thus,

B,{p;) is the expected profit of firm i from setting p; E [ p, r). Ifp; is a point

of increase of G; then firm i must make its equilibrium profit at p;. t.

LEMMA 12.

Suppose p is a point of increase of G; and G~ in [ p, r).

EQUILIBRIA IN A MODEL OF SALES 499

Proof. B,{p) - B~(p) - rU - c(U). From ( D.2) we have

W(P)A~(P)(1 - G~(P)) f L(P)(1 - A~(P)(1 - G~(P))) - rU - c(U).

This implies that

A~(P)(1 - G~(P)) -

rU - c(U) - L(p) -

,q~r(P)(1 - G(P)).

W(P) - L(P)

Division by AU(p) - A~;(p) 1 0 gives G~(p) - G;(p).

t.

LEMMA 13.

For euery i and every point of increase p of G; in [ p, r)

there is at least one G~ j~ i such that G~ is increasing at p.

-Proof. Because B;(p) is constant in a half-open neighborhood about

p by Lemma 11, dB;(p) - 0. Suppose contrary to the hypothesis that

dA;(p) - 0. Totally differentiating B;(p) gives

A;dWf (1 -A;)dL-O.

However, both dW and dL are positive and A;(p) E(0, 1]. Hence for dB;(p)

to be zero dA; is necessarily negative. By the monotonicity of the G;'s at

least one has to increase.

t.

LEMMA 14.

If G; is strictly increasing on some open subset (x, y),

p G x C y C r, then G; is strictly increasing on the whole interual [ p, y).

Proof. Without loss of generality, suppose, to the contrary, that G;

were constant on (z, x), p ~ z c x. Then from Lemma 10, G;(z) - G;(z).

It is evident that there ezlsts an e~ 0 such that on the interval (x - e, x)

there exist at least two firms, say ! and m, with strictly increasing c.d.f.'s

over the interval (otherwise mass would be moved up to x by some firm).

Thus, for every p E(x - e, x), B~(p) - B,,,(p) - rU - c( U). Furthermore,

since there are no mass points in the interval [ p, r), B~(x) Bm(x)

-B,{z) - rU - c( U) which, from arguments similar to those used in proving

Lemma 12, implies that G;(x) - Gm(z) - G;(x) c 1. But with

B,{z) - B~(x) - B;(p) b~p E(x - e, x)

it must be that B,{p) ~ B;(p) dp E (x - e, z), since such values of p do

not lie in i's support. This implies that A,(p) ? A;(p), and hence that

1-G,(p) ? 1- G;(p). This is a contradiction to the fact that G;(x) - Gi(x),

j00 BAYE, KOVENOCK, AND DE VRIES

LEMMA l5.

(It Takes Two to Tango). At least two firms randomize

continuously on [ p, r].

Proof. Three cases are possible at r: (i) all firms allocate positive mass

at r, (ii) all firms have G,{x;) - 1 at some x; c r, or (iii) there is at least one

firm i that has a positive left-derivative of G; at r. Cases (i) and (ii) are

easily ruled out by previous lemmas. Lemmas 12, 13, and 14 then imply

that there are at least two firms that randomize continuously over [ p,

r].

~

LEMMA 16.

Once G; is constant on a subset (x, y), p ~ x c y c r, it

is constant on (x, r) and has a mass point at r.

-Proof. The first part is a direct implication of Lemma 14. The second

part follows from Lemma 6.

~

The above lemmas together establish our Theorem 1.

Note that in the case where n - 2, Lemmas 12 and 15 and Theorem 1

imply that the equilibrium of Varian's model of sales is unique and

symmet-ric. This illustrates an important property that appears to have implications

for other games with discontinuous payoffs. The 2-person game may have

a unique equilibrium but the n-person game does not. In 2-person games,

in order to make one player indifferent between all pure strategies in its

support, the other player's strategy is uniquely determined. I"n n-player

games this is generally not true.e

Exact expressions for the equilibrium distributions may be obtained

recursively over the interval [ p, r], conditional on the points at which

firms stop randomizing continuóusly and move remaining mass to r. These

expressions are provided in Appendix A. Here we give some instructive

examples.

EXAMPLE 1.

Symmetric Equilibrium (Varian, 1980). From the proof

of Lemma 12, for n? 2 the symmetric strategies are

(r - p)U

vc~-p

1- G-

_{[Ip - c(I f U) f c( U)}

_~

In this case, all firms randomize continuously on the interval [p, r], and

use the same strategy.

-ExAMPLE 2.

Pure and Mixed Strategies (Bagnoli, 1986). Completely

EQUILIBRIA IN A MODEL OF SALES SO1

asymmetric strategies arise when k? 2 firms randomize over [ p, r] and

n- k firms load all mass at r. The respective strategies are

-(r - p)U

uck-i~

1 G;

-Ip - c(I t U) t c( U)]

for i- 1, ..., k

lforpcr

1-G'-Ofor

_{P -}

r

forj-kfl,...,n.

ExAMPLE 3.

Intermediate Asymmetric Strategies. The finál example

is a situation where two or more firms randomize over [ p, r], and other

6rms randomize over proper subsets [ p, x~], p s x~ c r.In the case of

three firms with strategies H, G, and F, an example of the Nash equilibrium

strategies is

where

1 H(x)

-(r - p) U

in

Ip - c(I f U) f c(U)]

for p E[ p, x]

(r - p)U

(1 - H(x))-i

Ip - c(I t U) t c( U)]

for p E [x, r]

( l- H(x)

for p E[z, r)

jl0

forp-r,

(r - x) U

~n

Ix - c(I f U) f c( U)] ~

Note that H, G, and F have a kink at x, but not a jump.

To conclude, there are an uncountable infinity of payoff-equivalent

equilíbrium mixed strategies.

IV. ORDERING THE ASYMMETRIC STRATEGIES

S~~ BAYE, KOVENOCK, AND DE VRIES

a"reasonable" equilibrium selection by constructing a metagame in which

ít is the unique subgame perfect Nash equilibrium.

DEFINITION 1.

Let F and G be two cumulative distribution functions.

F is said to strictly first-order stochastically dominate G if f~x dF ~

f`, dG for all t, wiih strict inequality holding for some t.

TxeoREM 2.

If, in a Nash equilibrium, firm i has a larger mass point

at r t{tan firm j, then the distribution of prices charged by firm i strictly

first-order stochastically dominates the distribution of prices charged by

firm j. If two firms load the same mass at r, then their price distributions

are stochasticaliy equiualent (i.e., G;(p) - G~(p) Hp).

Proof. Let (Gi, ..., G„) be Nash equilibrium mixed sirategies, and

suppose that firm i loads more mass at r than some firmj. Then by Theorem

1, associated with firms i and j are x;'s and x~'s with x; c x~ such that firm

i randomizes continuously on [ p, x;) and loads remaining mass at r, while

firm j randomizes continuously on [ p, x~) and loads any remaining mass

at r. If x; - p, firm i loads all mass at rand hence the proposition is trivially

proved.

-Hence, suppose p c x; c x~. Then G; and G~ are both strictly

increas-ing for p E [ p, x;), ánd Lemma 12 reveals that G; - G~ on [ p, x;). Hence

G;(p) G~(p~} for p E[ p, x;]; G,(p) c G~(p) for p E(x;, r)ánd G;(r)

-G;(r). That is, G;(p) strictly first-order stochastically dominates G;(p).

If the two firms load the same mass at r, then Theorem 1 implies that

the firms randomize on the same interval, say [ p, x), and load remaining

mass at r. Lemma 12 thus implies that the firms í~ave identical distribution

funciions, so that their strategies are stochastically equivalent.

~

The basic idea behind Theorem 2 is depicted in Fig. 1 for the case when

n- 3. Here, firms 2 and 3 randomize continuously on the interval [p, rJ,

while firm 1 randomizes continuously on the interval [ p, xi) and lóads

mass at r(see Example 3). On the interval [ p, x,) all three firms are equally

likely to charge low prices. On the interval [xi, r], firms 2 and 3 are equally

likely to charge low prices, but firms 2 and 3 charge lower prices than firm

1 with probability one, since all of its mass in the interval is at r. Hence,

G~(p) ~ Gz(p) - G3(p) for all p, with strict inequality holding for p E

(x~, r).

V. RECONSIDERING THE SYMMETRIC EQUILIBRIUM

EQU[LIBRIA IN A MODEL OF SALES SU3

1

Ftc. 1. Stochastic dominance. The figure is drawn on the presumption that U ~ I. This

guarantees G" ~ 0 for n~ 2, while G" ~ 0 necessarily for n- 2.

In a market exhibiting temporal price dispersion, we would see each store

varying its price over time. At any moment, a cross section of the market would

exhibit price dispersion; but because of the intentional Huctuations in price,

consumers cannot learn from experience about stores that consistently have

low prices, and hence price dispersion may be expected to persis[. (p. 65l)

While these arguments are intuitively plausible, the Varian game is a

one-shot game,9 and hence the appeal to learning over time as a disciplining

device is really outside of the model. The idea behind learning is to allow

the consumers to infer the strategies used by the firms. In games with

complete information, players know who their opponents are. Hence by

considering Nash equilibria in a game of complete information, one

circum-vents the "learning" story; one simply asks if, given the strategies of the

other players, any player has an incentive to deviate. Nash theory says

nothing about how agents learn the strategies of opponents, although it

does make a nice story. For this reason we offer an extensive form of the

Varian game that is consistent with the spirit of Varian's arguments.

Suppose the n firms and M- n U uninformed consumers move

simulta-neously, the firms choosing their equilibrium price distributions and the

uninformed consumers each deciding the identity of the firm to which they

will go to make a purchase decision. After the firms and uninformed

consumers have moved (the firms having set prices and the uninformed

S~4 BAYE, KOVENOCK, AND DE VRIES

having chosen to which firm to go), the informed consumers decide

simul-taneously from which, if any, firm to purchase. For simplicity, we assume

that c(q) is zero.'o

We now establish

TttEOREM 3.

The unique subgame perfect eqt~ilibrium in the extensiue

form sales game is the symmetric equilibrium."

The proof proceeds by several lemmata. Before ihese are stated, some

remarks are in order. Note that the only proper subgames of the game

start at the first node along each of the paths at which an informed

con-sumer must make a decision where to shop. Subgame perfection requires

that all informed consumers buy from one of the firms setting the lowest

price. If the uninformed consumers allocate themselves equally across

firms and the firms do not play the symmetric equilibrium then ihey must

play one of the asymmetric equilibria. By Theorem 2 there will exist some

firm, say firm 1, whose distribution stochastically dominates some other

firm's distribution. This implies that it is not a best response for uninformed

consumers to shop at the first firm,~Z since, on average, they will pay

lower prices by shopping elsewhere. Suppose, then, that the uninformed

consumers do not allocate themselves equally across firms. Let U; be the

number of uninformed consumers allocating themselves to firm i. We deal

first with the case where Ut c UZ c U3 ~... ~ U,,. Degenerate cases

where one of the strict inequalities adjacent to UZ is weak require a

separ-ate analysis. This is carried out in Appendix B.

Let p; - U;rl(I f U;), i- I, ..., n. By assumption p, ~ p, c p3

~''' s p,,. It is easily shown that Lemmas 1 through 6 fróm Section III

hold for this case, where in Lemma 1 we insert p; in place of p in both the

statement and proof, and the proof of Lemma~ is altered in an obvious

fashion. We replace the remaining lemmas of Section III with the lemmas

that follow. Henceforth, let s denote the lower bound of the union of the

supports of the firms' equilibrium price distributions.

LEMMA Í~. S ? pZ.

Proof. Firm i would never put mass below p; since setting price equal

~o We rule out for now mixed strategies on the part of uninformed consumers, although

this does not aft'ect the nature of the outcome because firms care only about the expected

number of uninformed consumers that they serve.

~~ If c(q) were to exhibit sufficiently inereasing returns to scale, i.e., if c(I t M)I(I f M)

G ~ while c(1)II ~ r, there would also be asymmetric equilibria whereby one firm monopolizes

the market. Our assumption that c(q) exhibits constant returns to scale rules out this type

of equilibrium.

EQUILIBRIA !N A MODEL OF SALES SOS

to r strictly dominates such a strategy. Firm 1 clearly has no incentive to

put mass in the interval [ p,, p2).

~

LEMMA 8'.

All firms other than firm 1 must place a mass point at r.

Proof. By Lemma 6, s; - r b~i. Since s? p2 ~ p,, firm 1 must have

equilibrium profit II i of at least (I t U,) pz ~ rU,.~T'hus, firm 1 cannot

have a mass point at r since, by Lemma 3, some firm must put no mass at

r, in which case firm 1 would be undercut at r with certainty and earn rU,

there. Since II i ~ rU,, in every neighborhood below r firm 1 must undercut

every other firm with positive probability. Thus, every firm but firm 1 must

put a mass point at r.

~

LEMMA 9'.

~di ~ 1 II;` - rU;.

Proof. Immediate from Lemmas 3 and 8'.

~

LEMMA 10'.

s- pZ and si - s2 - p~.

Proof. From Lemma 7' s? p:. Suppose s ~ p,. By undercutting s by

an arbitrarily small amount firm~ could earn arblirarily close to (1 f U~)s

) rUZ - II;`, a contradiction. Thus, s- pZ. The second part of the claim

is straightforward.

~

-LEMMA 11'.

There are no point masses on the half open interual

[P~~ r).

Proof. Similar to the proof of Lemma 10, inserting pZ for p, and noting

that if firm 1 has a mass point at p2, firm 2 will move mass up to r, while

if firm 2 has a mass point at p~ firm 1 will move mass slightly below p,.

~

LEMMA 12'.

B; (P;) -(1 t U;)p;A;(p;) t U;p;(1 - A,{p;)) is constant

and equal to II;` at the points of increase of G; in [ p,, r) for al i. B,{p;) ~

II~` if p; is not a point of increase in [p2, r).

-Proof.

Similar to Lemma 11.

~

LEMMA 13'.

tlp E( pZ, r) 3ii , iZ such that tls 7 0 G; (p f e)

-Gr (P - e) 1 0, i- i, , iz.

Proof. Immediate.

~

LEMMA 14'.

s; - r b~i J 2.

Proof. Without loss of generality assume s~ - min;zj s;. Suppose s, ~

r. Then there exists an initial interval of increase [s3, s3 t s) in which

B~(p) - II3 - U3r -(I f U3)pA3(p) -4- U3p(1 - A3(p)). Thus

U3 - A3(P) r p p1

t1p E[s3, s3 t E).

SQ6 BAYE, KOVENOCK, AND DE VRIES

S

UZ - A,(s3)

-3

1.

--

r-s3

Since for s3 c r: A~(s3) - II;tZ (I - G;(s3)) ~ II;~3 (1 - G~{s~)) - A3(s~)

we have a contradiction to the fact that UZ c U3. Thus, s~ - r.

~

We have thus shown that if U, c UZ c U3 ~'~ ' ~ U„ then firms 1 and

2 will continuously randomize over [ p2, r), with firm 2 having a mass point

at r and all other firms setting price équal to r with probability one. This

cannot comprise a subgame perfect equilibrium because the uninformed

consumers shopping at firms 2 through n would prefer to defect to firm 1,

given the strategies played. While the degenerate cases where UZ U;

-...- Um, m s n, will generally lead to multiple equilibria, the stochastic

dominance rankings apply, and as long as all the firms are not symmetric

some uninformed consumers would want to defect. This case is covered

in Appendix B.

VI. CONCLUSION

This paper has derived the complete set of equilibria in Varian's (1980)

model of sales. In addition to the well known symmetric equilibrium,

Theorem 1 reveals the existence of a continuum of asymmetric Nash

equilibrium mixed strategies. While the set of equilibria is thus very large,

Theorem 2 revealed that the set of equilibrium mixed strategies can be

ranked by first-order stochasiic dominance. We then constructed an

exten-sive form of the model which, along with the stochastic dominance

rank-ing, yields Varian's symmetric equilibrium as the unique subgame perfect

Nash equilibrium.

The basic technology used to characterize the complete set of equilibria

to the sales game may also be used to characterize the full set of equilibria

in other games with discontinuous payoffs. For example, results similar

to those in section III and IV reveal a continuum of equilibrium in the all

pay auction (see Moulin, 1986, or Weber, 1985). A detailed examination

of the implications of the asymmetric equilibria in the all pay auction for

lobbying and patent races is contained in Baye, Kovenock, and De Vries

(1990).

APPENDIX A

EQUILIBRIA IN A MODEL OF SALES S07

firms m f 1, . .., n randomizing over [ p, x;), xn ~ xn-, ~.. . s xn,,, c

r, then

-(a) dP E [ P, x~)

(r - p)U

v(n-n

1 G'{P)

-Ip - c(1 f U) f c( U)]

(b) For j- n, n- 1, ..., m f 2 and bp E[x~, x~-,)

(r - p)U

iui-')

1

1 G;(P)

-J

L~ (I

- Gk(xk))

J

-Ip - c(1 t U) } c( U)

k-;

i- 1,..., j- 1

1 - G,{p) - 1 - G;(x;)

(C) dP E[xm t I, r)

i -j,. . .,n.

r

(r - p) U

~ urm - n r

n

1- G,{p) - llp - c(1 f U) f c(U)

Lk-~ i(1 - Gk(xk))

1 - G,{p) - 1 - G;(x;)

(d) 1 - G;(r) - 0

- ii~m- ii

i- 1,...,m

i- m f 1, . .., n

bi.

This appendix deals with degenerate rankings of U,, ..., Un in the

proof of Theorem 3. We first deal with the case where U, c U. U3

-~~~- U,,, c Umt, ~, ..., ~ U„ for some 3 s m s n. It is easily seen

that for this case ihe previously altered versions of Lemmas 1 through 6

hold, as do Lemmas 7' through 9'. Lemmas 11' through 13' also continue

to hold (with an obvious alteration in the labelling of players in the proof

of Lemma I 1'). Lemmas 10' and 14' must be altered (slightly) as follows;

the proofs require only a minor change in the labelling of players.

LEMMA 10".

s- pZ. There exists at least one,firm i, 2 ~ i 5 m, such

that ~ - pZ.

-

SQg BAYE, KOVENOCK, AND DE VRIES

lf n 7 m we are through in our proof that such an allocation of consumers

cannot be part of a subgame perfect Nash equilibrium; firms m t 1, ...,

n place all mass at r while other firms place mass below r, which contradicts

the fact that U„ 7 U; b~j s m. .

Suppose then that n- m, so that U~ c UZ - U3 -...- U,,. The

following versions of Lemmas 12 and 14 hold for firms 2, ..., n.

LEMMA 15".

Suppose p is a point of increase of G; and G; in [ p2, r] ,

i, j E{2, ..., n}. Then G; - G~ at p.

Proof. Same as proof of Lemma 12.

~

LEMMA 16".

If G; , i E{2, ..., n}, is strictly increasing on some open

suóset (x, y), pz c z c y c r, then G; is strictly increasing on the whole

interual [p2, y).

Proof. -Similar to proof of Lemma 14 where one of the firms 1, m

must be an element of {2, ..., n} and this firm is used throughout the

continuation of the proof.

~

Lemma 16", together with Lemmas 10" and 13', imply the following:

LEMMA 17".

At least one of the firms 2, ..., n mccst randomize on

the interual [ p~, r].

We are now in a position to show that the indicated allocation of consumers

cannot be part of a subgame perfect equilibrium. To do this we show that

G~ is strictly first-order stochastically dominated by G;, i E{2, ..., n},

LEMMA 18".

s~ - pZ, andfor euery price pz G p c r in the support of

G~ , G~(p) ~ G; (p), i É {2, . . . , n}.

-Proof. From Lemma 17" ai least one of the firms 2, ..., n has support

[ p,, r]. Without loss of generality, suppose this is firm 2. From Lemmas

3ánd 8', firm 1 does not have a mass point at r, and from Lemma 11' no

firm has a mass point in [ p2, r). Thus, there exists some point p E( p2, r)

ai which G~(p) is increasing. At any such point

-Bi(P) ~ p,(1 f U~)

sincc the right-hand side is whai firm 1 can obtain by charging p2.

Rear-ranging this expression, we obtain

-Ai(P) ~ [PZ(1 f U~) -

PUi]Ipl.

EQUILIBRIA 1N A MODEL OF SALES S~9

Az(P) - (r - P)Uz~pl.

Recalling that pz - rUZ~(I f U~) we may subtract A~ from A, to obtain

Ai(P) - Az(p) ~ íU2 - U,)(p - Qz)Ipl ~ 0,

where the strict right-hand inequality follows from the assumption that

UZ 7 U, and p 1 pZ. Thus, at any point of increase of G, in the interval

( p2, r), A, ~ AZ. This directly implies that G, ) G, for any such point. But

since GZ has support [ pZ, r] and G, has no mass points, this implies that

S i- pz. Furthermore, since for any other firm i E{2, ..., n} and for any

p E[ pZ, r], GZ(p) ? G;(p), we have the claim.

r.

An immediate consequence of Lemma 18" and the fact that G, has no

mass points is that G,(p) ? GZ(p) for every p in [pZ, r], with strict

in-equality on the open interval. This contradicts the ~ct that U, c U~, so

the given allocation of consumers cannot be part of a subgame perfect

equilibrium.13

The remaining cases to be covered, where U, - Uz -...- Um c

Um}, ~ ... s U„ for 2 s m ~ n- 1, require a mixture of the analysis

of the symmetric case and asymmetric case. It can be shown that firms 1

through m may play any m-firm equilibrium of the type outlined in Theorem

l, while firms m t I through n put all mass at r. Since at least two of the

firms among {1 ..., m} put all probability mass below r, this cannot be

a subgame perfect equilibrium allocation of consumers.

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Reprint Series, CentER, Tilburg Universlty, The Netherlands:

No. I G. Marini and F. van der Plceg, Monetary and fiscal policy in an optimising model with capital accumulation and finite lives, The Economic Journal, vol. 98, no. 392, 1988, PP. 772 - 786.

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No. 1S R. Alessie, A. Kapteyn and B. Melenberg, The effects of liquidity constraints on consumption: estimation from household panel data, Eutopean Economic Review, vol. 33, no. 2~3, 1989, pp. S47 - SSS.

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No. 31 F. van der Plceg, International interdependence and policy coordination in economies with real and nominal wage rigidity, Greek Economic Review, vol. 10, no. l, June 1988, pp. 1- 48.

No. 32 E. van Damme, Signaling and forward induction in a market entry context,

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A.P. Barten, Toward a levels version of the Rotterdam and related demand

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Th. van de Klundert, On socioeconomic causes of'wait unemployment', European

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No. 36 RJ.M. Alessie, A. Kapteyn, J.B. van Lochem and T.J. Wansbeek, Individual effects in utility consistent models of demand, in J. Hartog, G. Ridder and J. Theeuwes (eds.), Pane! Data and Labor Maticet Studies, Amsterdam: Elsevier Science Publishers B.V. (North-Holland), 1990, pp. 253 - 278.

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Th. van de Klundert, Wage differentials and employment in a two-sector model

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F. van der Ploeg and C. Withagen, Pollution control and the ramsey problem,

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No.56 H.A. Keuzenkamp and F. van der P(oeg, Savings, investment, government finance, and the current account: The Dutch experience, in G. Alogoskoufis, L. Papademos and R. Portes (eds.), External Constrainu on Macroeconomic Policy:

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No. 58 M.F.J. Steel and J.-F. Richard, Bayesian multivariate exogeneity analysis - an application to a UK money demand equation, Joumaf oj Econometrics, vol. 49, no. 1~2, 1991, pp. 239 - 274.

No. 59 Th. Nijman and F. Palm, Generalized least squares estimation of linear models ~ containing rational future expectations, Inrernationa! Economic Review, vol. 32,

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No. 62 W. Gilth and E. van Damme, Gorby games - a game theoretic analysis of disarmament campaigns and the defense efficiency - hypothesis -, in R. Avenhaus, H. Karkar and M. Rudnianski ( eds.), Defenre Decision Making -Analytica! Suppon and Crisis Management, Berlin: Springer-Verlag, 1991, pp. 215 - 240.

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Y. Dai, G. van der Laan, AJJ. Talman and Y. Yamamoto, A simplicial

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-The Economics of EMU, Commission of the European Communities, special

edition no. 1, 1991, pp. 136 - 164.

No. 69 H. Keuzenkamp, A precursor to Muth: Tinbergen's 1932 model of rational expectations, The Economèc Joumal, vol. 101, no. 408, 1991, pp. 1245 - 1253. No. 70 L. Zou, The target-incentive system vs. the price-incentive system under adverse

selection and the ratchet effect, Joumal ojPub[ic Economiu, vol. 46, no. 1, 1991, pp. 51 - 89.

No.71 E. Bomhoff, Between price reform and privatization: Eastern Europe in transition, Finanzmarkt und Porrfolio Management, voL S, no. 3, 1991, pp. 241 -251.

No. 72 E. Bomhoff, Stability of velocity in the major industrial countries: a Kalman Fdter approach, lnternational Monerary Fund Staff Papers, vol. 38, no. 3, 1991, pp. 626 - 642.

No. 73 E. Bomhoff, Currency convertibility: when and how? A contribution to the Bulgarian debate, Kredit und Kapital, vol. 24, no. 3, 1991, pp. 412 - 431. No.74 H. Keuzenkamp and F. van der Ploeg, Perceived constraints for Dutch

unemployment policy, in C. de Neubourg ( ed.), The An of Full Employmenr

-Unemployment Policy in Open Economies, Contributions to Economic Analysis

203, Amsterdam: Elsevier Science Publishers B.V. (North-Holland), 1991, pp. 7 - 37.

No. 7S H. Peters and E. van Damme, Characterizing the Nash and Raiffa bargaining solutions by disagreement point axions, Mathematicr of Operations Research, vol. 16, no. 3, 1991, pp. 447 - 461.

No. 76

P.J. Deschamps, On the estimated variances of regression coefficients in

misspecified error components models, Economerric Theory, vol. 7, no. 3, 1991,

pp. 369 - 384.

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No. 77 A. de Zeeuw, Note on 'Nash and Stackelberg solutions in a differential game model of capicalism', Journal of Economic Dynamics and Control, voL 16, no. 1,

1992, pp. 139 - 145.

No. 78 J.R. Magnus, On the fundamental bordered matrix of linear estimation, in F. van der Ploeg (ed.), Advanced Lectures in Quantitarive Economics, London-Orlando: Academic Press Ltd., 1990, pp. S83 - 604.

No. 79 F. van der Ploeg and A. de Zeeuw, A differential game of international pollution eontrol, Systems and Conrrol Letrers, vol. 17, no. 6, 1991, pp. 409 - 414. No. 80 Th. Nijman and M. Verbeek, The optimal choice of controls and

No. 81 M. Verbeek and Th. Nijman, Can cohort data be treated as genuine panel data?,

Empirical Economics, vol. 17, no. 1, 1992, pp. 9- 23.

No. 82 E. van Damme and W. Guth, Equilibrium selection in the Spence signaling game, in R. Selten (ed.), Game Equilibrium ModeLr !I - Methods, MoraLr, wul Markets, Berlin: Springer-Verlag, 1991, pp. 263 - 288.

No. 83 R.P. Gilles and P.H.M. Ruys, Characterization of economic agents in arbitrary communication s[ructures, Nieuw Archief voor Wiskunde, vol. 8, no. 3, 1990, pp. 325 - 345.

No. 84 A. de Zeeuw and F. van der Ploeg, Difference games and po6cy evaluation: a conceptual framework, Oxford Economic Papers, vol. 43, no. 4, 1991, pp. G12 -636.

No. 85 E. van Damme, Fair division under asymmetric information, in R. Selten (ed.),

Rational lnteraction - Essays in Honor of John C. Harsam~i. Berlin~Heidelberg:

Springer-Verlag, 1992, pp. 121 - t44.

No. 86 F. de Jong, A. Kemna and T. Kloek, A contribution to event study methodology with an application to the Dutch stock market, Journal oj Banking and Finance, vol. 16, no. 1, 1992, pp. 11 - 36.

No. 87 A.P. Barten, The estimation of mized demand systems, in R. Bewley and T. Van Hoa (eds.), Contributions to Con.rumer Demand and Econometrics, Essays in Honow of Henri Theil, Basingstoke: The Macmillan Press Ltd., 1992, pp. 31 - 57. No. 88 T. Wansbeek and A. Kapteyn, Simple estimators for dynamic panel data models with errors in variables, in R. Bewley and T. Van Hoa (eds.), Contributions to Consumer Demand and Econometrics, Essays in Honow of Henri Tlteil, Basingstoke: The Macmillan Press Ltd., 1992, pp. 238 - 251.

No. 89 S. Chib, J. Osiewalski and M. Steel, Posterior inference on the degrees of freedom parameter in multivariate-t regression models, Economics Letters, vol. 37, no. 4, 1991, pp. 391 - 397.

No. 90

H. Petets and P. Wakker, Independence of irrelevant alternatives and revealed

group preferences, Economenica, vol. 59, no. 6, 1991, pp. 1787 - 1801.

No. 91 G. Alogoskoufis and F. van der Ploeg, On budgetary policies, growth, and ezternal deficits in an interdependent world, lournal oj the Japanese and Intemadona! Economies, vol. 5, no. 4, 1991, pp. 305 - 324.

No. 92

R.P. Gilles, G. Owen and R. van den Brink, Games with permission structures:

The conjunctive approach, Intetnational loumal oj Game Theory, vol. 20, no. 3,

1992, PP. 277 - 293.

No. 93

J.A.M. Potters, IJ. Curiel and S.H. Tijs, Traveling salesman games, Marhematical

Progmmming, vol. 53, no. 2, 1992, pp. 199 - 211.

No. 94

A.P. Jurg, MJ.M. Jansen, J.A.M. Potters and S.H. Tijs, A symmetrization for

fmite two-person games, Zeiuchtift f'w Operations Research - Methods and Modelr

No. 95 A. van den Nouweland, P. Borm and S. Tijs, Allocation rules for hypergraph communication situations, Intemtationalloumal of Game Theory, vol. 20, no. 3,

1992, pP. 255 - 268.

No. 96 E.J. Bomhoff, Monetary reform in Eastern Europe, European Economic Review, vol. 36, no. 2~3, 1992, pp. 454 - 458.

No. 97 F. van der Ploeg and A. de Zeeuw, International aspects of pollution control,

Environmenral and Resource Economics, vol. 2, no. 2, 1992, pp. 117 - 139.

No. 98 P.E.M. Borm and S.H. Tijs, Strategic claim games rnrresponding to an NTU-game, Games and Economic Behavior, vol. 4, no. 1, 1992, pp. 58 - 71.

No. 99 A. van Soest and P. Kooreman, Coherency of the indirect translog demand system with binding nonnegativity constraints, loumal oj Econornetrics, vol. 44, no. 3, 1990, pp. 391 - 400.

No. 100 Th. ten Raa and E.N. Wolff, Secondary products and the measurement of productivity growth, Regional Science and Urban Economics, vol. 21, no. 4, 1991, pp. 581 - 615.

No. 101 P. Kooreman and A. Kapteyn, On the empirical implementation of some game theoretic models of household labor supply, Theloumal ojHuman Resources, vol. 25, no. 4, 1990, PP. 584 - 598.

No. 102 H. Bester, Bertrand equilibrium in a differentiated duopoly, Intemational Economic Review, vol. 33, no. 2, 1992, pp. 433 - 448.

No. 103 J.A.M. Potters and S.H. Tijs, The nucleolus of a matrix game and other nucleoli, Mathematics ojOperotions Research, vol. 17, no. 1, 1992, pp. 164 - 174. No. 104 A. Kapteyn, P. Kooreman and A. van Soest, Quantity rationing and concavity in

a flexible household labor supply model, Review ojEconomics and Statistics, vol. 72, no. 1, 1990, pp. 55 - 62.

No. 105 A. Kapteyn and P. Kooreman, Household labor supply: What kind of data can tell us how many decision makers there are?, European Economic Review, vol. 36, no. 2~3, 1992, pp. 365 - 371.

No. ]06 Th. van de Klundert and S. Smulders, Reconstructing growth theory: A survey, De Economisr, vol. 140, no. 2, 1992, pp. 177 - 203.

No. 107 N. Rankin, Imperfect competition, expectations and the multiple effects of monetary growth, The Economic loumal, vol. 102, no. 413, 1992, pp. 743 - 753. No. 108 J. Greenberg, On the sensitivity of von Neumann and Morgenstern abstract stable sets: The stable and the individual stable bargaining set, International

lourna! of Game Theory, vol. 21, no. 1, 1992, pp. 41 - 55.

No. 110 M. Verbeek and Th. Nijman, Testing for selectivity bias in panel data models,

Intenwtional Economic Review, vol. 33, no. 3. 1992, pp. 681 - 703.

No. 11 ] Th. Nijman and M. Verbeek, Nonresponse in panel data: The impact on estimates of a life c.ycle consumption function, Joarna!ojApplied Econometrics, vol. 7, no. 3, 1992, pp. 243 - 257.

No. ll2 I. Bomze and E. van Damme, A dynamical characterization of evolutionarily stable states, Annals of Operations Research, vol. 37, 1992, pp. 229 - 244. No. l13 P.J. Deschamps, Expectations and intertemporal separability in an empirical

model of consumption and investment under uncertainty, E~npirica! Econon:icr, voL 17, no. 3, 1992, pp. 419 - 450.

No. 114 K. Kamiya and D. Talman, Simplicial algorithm for computing a core element in a balanced game, Jouma! oj the Opemtions Research, vol. 34, no. 2, 1991, pp. 222 - 228.

No. 115 G.W. Imbens, An efficient method of moments estimator for discrete choice models with choicebased sampling, Econometrica, vol. 60, no. 5, 1992, pp. 1187

-1214.

No. 116 P. Borm, On perfectness concepts for bimatrix games, OR Spektrum, vol. 14, no. 1, 1992, pp. 33 - 42.

No. 117 A.P. Jurg, I. Garcia Jurado and P.E.M. Borm, On modifications of the concepts of perfect and proper equilibria, OR Spektrum, vol. 14, no. 2, 1992, pp. 85 - 90. No. 118 P. Borm, H. Keiding, R.P. McLean, S. Oortwijn and S. Tijs, The compromise

value for NTU-games, Intemationa! Joumal of Game Theory, vol. 21, no. 2, 1992, pp. 175 - 189.

No. 119 M. Maschler, JA.M. Potters and S.H. Tijs, The general nucleolus and the reduced game properry, Intemational Joutna! of Came Theory, vol. 21, no. 1,

1992, pp. 85 - 106.

No. 120

K W~rneryd, Communication, correlation and symmetry in bargaining,

Economiu Letrers, vol. 39, no. 3, 1992, pp. 295 - 300.

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