Verifying Payo¤ Security in the Mixed Extension of Discontinuous Games Blake A. Allison and Jason J. Lepore;y

April 29, 2014

Abstract We introduce the concept of disjoint payo¤ matching which can be used to show that the mixed extension of a compact game is payo¤ secure. By putting minor structure on the discontinuities, we need only check payo¤s at each strategy rather than in neighborhoods of each strategy pro…le, placing minimal restriction on the payo¤s at points of discontinuity. The results are used to verify existence of equilibrium in a general model of Bertrand-Edgeworth oligopoly. Keywords: Discontinuous games, Nash equilibrium, Disjoint payo¤ matching, Existence.

Allison : Department of Economics, University of California, Irvine; Lepore : Department of Economics, Orfalea College of Business, California Polytechnic State University, San Luis Obispo. y We would like to thank Adib Bagh and Aric Shafran for their helpful comments. We would also like to thank the editor Xavier Vives and the two excellent referees for helping us improve the article.

1

2

1

Introduction

We provide a new su¢ cient condition for payo¤ security that generalizes the set of games in which existence of mixed strategy equilibrium can be readily veri…ed. Reny (1999) introduced the class of better reply secure games and showed existence of pure strategy Nash equilibrium in such games.1 In addition, Reny provided two conditions that together are su¢ cient for a game to be better reply secure: payo¤ security and reciprocal upper semicontinuity.2 The results apply to both the normal form of a game and its mixed extension. In the context of mixed strategies, the su¢ cient conditions of Reny are di¢ cult to verify since they are based on the mixed extension of a game.3 Our new su¢ cient condition is relatively straightforward to verify for a large class of games in which other su¢ cient conditions have not been readily applicable. We introduce the concept of disjoint payo¤ matching, which imposes minor structure on the discontinuities of the game instead of solely on the payo¤s. A game satis…es disjoint payo¤ matching if, given any strategy of a player, that player possesses a sequence of deviations that are at least as good in the limit and whose discontinuity sets are su¢ ciently disjoint. Su¢ ciently disjoint in this context means that there is no strategy pro…le by the other players that constitutes a discontinuity for a subsequence of these deviations.4 The advantage that disjoint payo¤ matching has over other su¢ cient conditions for existence of mixed strategy equilibrium is that it is easily veri…able, owing to the fact that other conditions appeal to arbitrary probability measures or neighborhoods of strategies. Disjoint payo¤ matching can replace payo¤ security of the mixed extension of the game in Reny’s or Bagh and Jofre’s (2006) theorems that additionally require (weak) reciprocal upper semicontinuity to guarantee better reply security and thus existence of equilibrium in mixed strategies.5 Most closely related to the concept of disjoint payo¤ matching is a su¢ cient condition introduced in Bagh (2010). He introduces the notion of variational convergence of …nite approximations of games. A result of this analysis is a su¢ cient condition for existence that involves computation of limits of mixed strategies of the …nite approximations. To alleviate the di¢ culty of this computation, he establishes a stronger su¢ cient condition on the set of all mixed strategies of the game. Disjoint payo¤ matching places less restriction on the 1

A game is better reply secure if for every nonequilibrium strategy pro…le x and every limiting payo¤ vector u at x , there is a player i that has a strategy that gives payo¤ strictly higher than ui even when other players deviate slightly from x . 2 A game is payo¤ secure if at any strategy pro…le x, each player has a strategy that earns a payo¤ close to that of x against slight deviations from x by the other players. Roughly speaking, a game is reciprocally upper semicontinuous, if whenever some player’s payo¤ jumps down, some other player’s payo¤ jumps up. 3 Recent papers by Tian (2010), Nessah and Tian (2010), and McLennan et al. (2011), have worked to generalize the work of Reny (1999) and made great pushes toward a better understanding of Nash equilibria. 4 We require that the limit superior of the discontinuity sets of the deviations be empty. 5 Bagh and Jofre (2006) show that reciprocal upper semicontinuity can be replaced with the weaker concept of weak reciprocal upper semicontinuity.

3 discontinuity sets along with more restriction on the payo¤s than does Bagh’s condition, facilitating greater ease of veri…cation. Other su¢ cient conditions in the literature that have attempted to alleviate the burden of computations of payo¤s in the mixed extensions of games are uniform payo¤ security due to Monteiro and Page (2007) and uniform diagonal security due to Prokopovych and Yannelis (2012). Uniform payo¤ security, a condition on the set of pure strategies, is a su¢ cient condition for payo¤ security of the mixed extension of a compact game. Uniform diagonal security is similarly a condition on the set of pure strategies which under certain conditions is a generalization of uniform payo¤ security, but with the advantage of being a su¢ cient condition for existence of equilibrium rather than just payo¤ security of the mixed extension.6 The rest of the paper is formatted as follows. In Section 2, we introduce the necessary preliminaries. Section 3 de…nes disjoint payo¤ matching and proves that it implies payo¤ security for the mixed extension of the game. In Section 4, we use our results to show existence of equilibrium for a Bertrand-Edgeworth price setting oligopoly and we use an example from Sion and Wolfe (1957) to demonstrate how an equilibrium may not exist when disjoint payo¤ matching does not hold.

2

Preliminaries

An N -player compact game is a 2N -tuple G = (Xi ; ui )N i=1 , where the strategy space of each player i is a compact Hausdor¤ space Xi and the payo¤ of each player ui : X1 ::: XN 7! R is bounded and measurable. The mixed extension of the game is G = (Mi ; Ui )N i=1 where the strategy space of each player i is Mi , the set of regular probability , which is R measures on XiQ compact and convex, and the payo¤ function of player i is Ui = ui d , 2 M = N i=1 Mi . Note that, as de…ned, G and G are also the graphs of the games. The closure of the graph is denoted clG. Finally, the frontier of G, denoted FrG, is de…ned to be the elements of the closure that are not in the graph, that is FrG =clG r G. The closure and frontier of the mixed extension are de…ned analogously. Our condition and proofs will make reference to the sets of discontinuities of each player. Speci…cally, we reference the points at which a player’s payo¤ is discontinuous in the other players’ strategies. These are given by the discontinuity map Di : Xi 7! P (X i ), where P (X i ) is the power set of X i , de…ned for all xi 2 Xi as Di (xi ) = fx 6

i

2X

i

: ui is discontinuous in x

i

at (xi ; x i )g :

Prokopovych and Yannelis (2012) also adapt the concept of hospitality from Duggan (2007) to the domain of nonzero sum games. This condition involves the veri…cation of deviations to a speci…c subset of the set of mixed strategies.

4

3

Disjoint payo¤ matching and payo¤ security

We now introduce disjoint payo¤ matching. The condition has two parts: the …rst is that any player can deviate from any strategy and remain almost as well o¤, while the second imposes that the discontinuity sets of each deviation have limited intersection. The name “disjoint payo¤ matching” for our condition follows from the existence of a sequence of strategies which “match” the payo¤ of the original strategy and for which the discontinuity sets are su¢ ciently disjoint. De…nition 1 The game G satis…es disjoint payo¤ matching if for all xi 2 Xi , there exists Xi such that the following holds: a sequence of deviations xki (i) lim inf k ui xki ; x

i

ui (xi ; x i ) for all x

i

2 X i;

(ii) lim supk Di xki = ;.7 Remark 1 It follows that one need only check this condition for xi such that Di (xi ) is nonempty. That is, if ui (xi ; x i ) is continuous in x i at xi , then the conditions of disjoint payo¤ matching are trivially satis…ed by the constant sequence xki = xi . The second condition is clearly satis…ed when Di xki \ Di xli = ; for all k 6= l. This stronger empty intersection condition holds for the prominent examples in the literature. Unlike security concepts, there is no reference here to neighborhoods of the opponents’ strategies. Further, the payo¤s at the discontinuity points of the deviations are irrelevant since they are completely avoided in the limit, making the condition easy to verify and unrestrictive. It is worth noting that games in which discontinuities do not satisfy part (ii) of DPM often share best responses with a game that does satisfy DPM. That is, if ui (x) is player i’s utility function in the game of interest and does not satisfy DPM, then there is often some strategically equivalent game for which player i’s utility is of the form vi (x) = ui (x) + f (x i ), where vi satis…es condition (ii) of DPM. We need one more de…nition before we prove the main result. The following concept was introduced by Reny (1999). De…nition 2 The game G satis…es payo¤ security if for all " > 0 and all x 2 X, there exists for each player i a deviation x0i 2 Xi and a neighborhood N (x i ) of x i such that ui (x0i ; z) ui (x) " for all z 2 N (x i ). 7

Given a sequence of sets En , the limit superior is lim supn En = all points x 2 X such that x 2 En for in…nitely many n.

T1

N =1

S1

n=N

En . This is equivalently

5 The de…nition is analogous for the mixed extension of the game. Reny (1999) showed that payo¤ security combined with another condition is su¢ cient to guarantee the existence of a pure strategy Nash equilibrium.8 Payo¤ security is easily veri…ed in the set of pure strategies, but is particularly di¢ cult to verify in the mixed extension of a game. Our main result shows that disjoint payo¤ matching implies that the mixed extension of the game is payo¤ secure. Theorem 1 Let G be a compact game. Suppose that G satis…es DPM, then G is payo¤ secure. The advantage of DPM is that it is straightforward to verify and still fairly general. The condition in the following lemma is easier to use in the proof of our main result, but more di¢ cult to verify directly. Lemma 1 Suppose that the compact game G satis…es disjoint payo¤ matching. Then for 0 all " > 0, xi 2 Xi , and i 2 M i there exists a deviation xi 2 Xi and a compact set 0 K X i r Di (xi ) such that the following holds: (i) ui (x0i ; x i ) > ui (xi ; x i ) (ii)

i

(X

i

" for all x

i

2 K,

r K) < ".9

Proof of Lemma 1. Assume that G satis…es disjoint payo¤ matching and consider any player i, " > 0, and i 2 M i . Take xki to be a defection sequence from the de…nition of DPM. De…ne the collection of sets Ek = x i 2 X i : ui xki ; x i > ui (x) " . Then 10 notice that lim inf k Ek = X i , so Further, lim supk Di xki = ;, so i (lim inf k Ek ) = 1. k = 0. By statement (5) in Section 9 of Halmos (1974), i (lim inf k Ek ) i lim supk Di xi lim inf k i (Ek ) and i lim supk Di xki lim supk i Di xki , and so limk i (Ek ) = k 1 and limk i Di xi = 0. It follows that there exists a k such that i (Ek ) > 1 ("=3) and i Di xki < "=3. Choose such a k and by regularity of i , we may choose a closed (and thus compact) subset K Ek rDi xki such that i (K) > i Ek r Di xki ("=3). It follows that i (X i r K) < ". Now we proceed to the proof of Theorem 1 which is based on showing that the condition in Lemma 1 implies the mixed extension is payo¤ secure. 8

Payo¤ security along with reciprocal upper semicontinuity together imply that a game is better reply secure, which in turn guarantees existence of equilibrium in a compact, quasiconcave game. 9 These conditions are equivalent to a slight weakening of disjoint payo¤ matching: for all players i and all k xi 2 Xi and Xi such that (i) lim inf k ui xki ; x i ui (xi ; x i ) i 2 M i , there exists a sequence xi k -almost everywhere, and (ii) lim sup D x is -measure zero. This de…nition seems less useful due i i k i i to its dependence on an arbitrary probability measure. S1 T1 10 The limit inferior of a sequence of sets En is lim inf n En = N =1 n=N En . This is equivalently the set of points are are in En for all but …nitely many n.

6 Proof of Theorem 1. Let " > 0 and suppose that 2 M. Note that for each player i there exists some strategy xi in the support of i such that R

(1)

ui (xi ; x i ) d

R

i

ui (x) d :

From disjoint payo¤ matching and Lemma 1, there exists a deviation x0i and a set K (") X i r Di (x0i ) such that " for all x 6

ui (x0i ; x i ) > ui (xi ; x i ) and i

where M

(X

i

r K (")) <

i

>

sup jui j. It follows that R

(2)

ui (x0i ; x i ) d

K(")

R

i

2 K (")

" ; 6M

ui (xi ; x i ) d

i

K(")

" : 6

Further, we have that R

X

X

(3)

>

i X

i rK(")

> >

ui (x0i ; x i ) d R

i rK(")

R

ui (xi ; x i ) d

(jui (xi ; x i )j + jui (x0i ; x i )j) d

2 sup jui j (X 2" : 6

i

i

i rK(")

i

r K ("))

Combining (2) and (3) yields (4) De…ne

R

ui (x0i ; x i ) d

i

R > ui (xi ; x i ) d inf ui x0i ; x0

ui (x i ) = sup V 3x

i

x0

i 2V

" : 2

i

i

;

where the supremum is taken over all neighborhoods V of x i . As noted by Reny (1999) in the proof of Theorem R 3.1, ui (x i ) is lower semicontinuous. From Reny’s proof of Proposition 5.1, it follows that ui (x i ) d i is lower semicontinuous in i . This property implies the existence of a neighborhood N 2N i such that for all i , (5)

R

R ui (x i ) d > ui (x i ) d

i

" : 6

7 Since M bounds ui as well as ui , we have that

X

R

(ui (x i )

ui (x0i ; x i )) d

X

i rK(")

>

Further, since ui (x0i ; x i ) is continuous in x on K (").11 Therefore, ui (x i ) d

i

(6)

= >

R

R

ui (x0i ; x i ) d ui (x0i ; x i ) d

i rK(")

at all x

+ X

Using the fact that ui (x0i ; x i ) 2N i , R

i

i

(jui (x i )j + jui (x0i ; x i )j) d

2M (X 2" : 6

=

R

R

i

K ("))

i

2 K ("), then ui (x i ) = ui (x0i ; x i )

i

R

ui (x0i ; x i )) d

(ui (x i )

i

i rK(")

2" : 6

i

ui (x i ) and combining (5) and (6), we have that for all

ui (x0i ; x i ) d >

(7)

>

R R R

ui (x i ) d ui (x i ) d

" 6

i

ui (x0i ; x i ) d

i

Lastly, we combine (1), (4), and (7) and …nd that for all R

i

ui (x0i ; x i ) d

> >

R

R R

2N

ui (x0i ; x i ) d

i

ui (xi ; x i ) d

i

ui d

" : 2 i

,

" 2 "

":

Therefore, the mixed extension G is payo¤ secure.

4

Examples

In the …rst part of this section, we use Theorem 1 to prove existence of mixed strategy equilibrium for a Bertrand-Edgeworth price-setting oligopoly with general speci…cations of costs, residual demand rationing, and tie breaking rules. In the second part of this section, 11

The continuity here is with respect to the topology on X i , not to be confused with the subspace topology on K ("). Otherwise, if the function were only continuous with respect to the subspace topology, it might be that ui > ui on the boundary of K (").

8 an example from Sion and Wolfe (1957) is used to demonstrate that equilibrium may not exist if disjoint payo¤ matching does not hold.

4.1

Bertrand-Edgeworth oligopoly

A Bertrand-Edgeworth (BE ) price-setting oligopoly is a competition between producers of homogenous products where prices are the only strategic variables. We apply disjoint payo¤ matching to a BE oligopoly speci…cation that subsumes much of the large literature and o¤ers a basis to generalize the analysis in these games.12 Existence of equilibrium in such games has been examined by Dixon (1984), Allen and Hellwig (1986), Dasgupta and Maskin (1986a&b), Maskin (1986), Deneckere and Kovenock (1996), and Bagh (2010). With the exception of Allen and Hellwig, which studies a symmetric oligopoly with constant marginal cost, these papers only demonstrate existence for a BE duopoly.13 Most of these results rely upon Dasgupta and Maskin (1986a) to guarantee existence, while Deneckere and Kovenock construct an equilibrium and Bagh (2010) develops and applies the concept of variational convergence to show existence. In addition to extending existence results to an oligopoly setting, our formulation greatly generalizes the set of rationing rules which are permitted.14 Consider a homogeneous product industry with a set of …rms N , with jN j = n. All …rms simultaneously announce prices, then production decisions are made after demand is realized. Each …rm i has a continuous, nondecreasing cost of production ci with ci (0) = 0.15 The market demand F : R 7! R is continuous and nonincreasing in x with F (0) > 0. Further, x. Note that any price assume that there exists a x > 0 such that F (x) = 0 for all x 0 x > x is weakly dominated by x = x, so we may restrict the strategy space to X = [0; x]n . We denote by pi the price of any …rm i and by p the vector of all …rms’prices. Each …rm i has a capacity ki , which serves as an upper bound on the quantity that it can produce. Thus, the production problem faced by the …rm at a price pi is max

z2[0;ki ]

i

(pi ; z) = pi z

ci (z) .

We refer to the solution to this problem as si (pi ).16 We assume that si (pi ) is a continuous 12

The literature on BE games includes: Kreps and Scheinkman (1983), Davidson and Deneckere (1986), Osborne and Pitchick (1986), Deneckere and Kovenock (1992), Allen and Hellwig (1993), Deneckere and Kovenock (1996), Allen et al.(2000), Boccard and Wauthy (2000) and Lepore (2009)). 13 The result of Allen and Hellwig has been used to study BE oligopoly in other settings. Vives (1986) studies the an BE oligopoly as the number of …rms gets large with e¢ cient rationing and constant marginal cost up to capacity. Two recent articles Hirata (2009) and De Francesco and Salvadori (2010) characterize equilibria of a BE triopoly with e¢ cient rationing and constant marginal cost up to capacity. 14 Most of the literature focuses on either e¢ cient or proportional rationing. Maskin (1986) and Bagh (2010) consider a larger class of rationing rules, although many reasonable rules are excluded from their frameworks. 15 It is well known that equilibrium may not exist if ci is discontinuous or ci (0) > 0. 16 The speci…cation of si (pi ) follows from Dixon (1984), Maskin (1986) and Bagh (2010).

9 nondecreasing function. Further, we assume that if q < q 0 < si (pi ) and i (pi ; si (pi )), then 0 17 The quantity si (pi ) i (pi ; q ) > i (pi ; q). This is necessarily true if ci is strictly convex. may be referred to as …rm i’s supply, the maximum quantity that it is willing to produce at any given price. Inherently, si ki , so the supply functions account for the capacity constraints. For any price vector p, order the players so that p1 p2 ::: pn . The demand served by …rm 1 is Q1 = min fF (p1 ) ; s1 (p1 )g. We make minimal assumptions as to which portion of demand is served by …rm i, only that for all j > i there is a continuous function ij (p) which denotes the share of i’s quantity that satiates j’s demand.18 In the event that multiple …rms choose the same price, there are multiple ways to order the players such that prices are nondecreasing. In this case, some tie breaking rule is used to allocate the demand. Speci…cally, serves to give each player some weighted average of the demand they would receive under each possible ordering of the prices. Most commonly in application gives a uniform weight to each possible ordering, however, this is not necessary. Let O be the collection of possible orderings of the prices. For each o 2 O, we let o denote the weight applied to the order o, o (i) the position of player i in the ordering o, and Qoi the quantity served by …rm i as if the ordering under o were a strictly increasing order of prices. That is, o n P o Q (p) ; s (p ) ; Qoi = min F (pi ) i i j
Thus, each …rm i serves the minimum of its capacity, supply, and the demand left by the …rms with lower prices than i. The purpose for this formulation is to allow any possible rationing between tied …rms. When multiple …rms are tied, this allows any order of satiation of supply, be it simultaneous, partially sequentially, or fully sequentially. 17

Notice that for symmetric constant marginal cost c 0, we can restrict the strategy space to X = [c; x]n and the supply functions satisfy our assumptions. 18 A simple way to understand the purpose of ij is to consider the case in which a continuum of consumers have unit demand. In this case, ij speci…es the fraction of consumers served by …rm i that have willingness to pay of at least pj . 19 One way to interpret Qoi is that o represents a strict preference order for consumers, whereby pi at …rm i is strictly preferred to pj at …rm j for all i < j. Thus, the demand Qoi re‡ects the notion that this preference induces consumers to shop at …rm i before …rm i + 1. 20 Both and the functions may depend on the full vector of prices as well as the capacities. We suppress the capacity arguments for clarity. The quantities Qoi and Qi depend on capacities only through the supply functions, , and the functions.

10 This very general framework captures the notion that consumers shop …rst at …rms with lower prices. Consider two choice for the functions ij given by eij (p) = 1 and pij de…ned iteratively as ( ) Qi p P ;1 . p ij = min D (pi ) j
The rationing rule under eij is the well known e¢ cient, or parallel rule, whereas the rule under pij is the proportional rationing rule. The pro…t of each …rm i can then be written as ui (p) = pi Qi (p)

ci (Qi (p)) :

We now turn to establishing that this game satis…es DPM. Proposition 1 The BE oligopoly game satis…es disjoint payo¤ matching. Proof of Proposition 1. For any …rm i, ui (0; p i ) = 0 for all p i . Consequently, Di (0) = ;. Let pi > 0. Note that the set of discontinuities Di (pi ) is a subset of points where pj = pi for some i 6= j. Thus, if pi 6= p0i , then Di (pi ) \ Di (p0i ) = ;. It follows that for any sequence pli ! pi with pli < pl+1 < pi for all l, condition (ii) of DPM is satis…ed. Note that i l liml Qi pi ; p i Qi (p) for all p i . Since si is continuous, liml Qi pli ; p i si (pi ). By de…nition, ui is increasing in Qi (p) for Qi (p) si (pi ), and since ui is continuous in Qi (p), it follows that lim pi Qi pli ; p

i

l

= pi lim Qi pli ; p l

pi lim Qi (p) l

i

ci Qi pli ; p

i

ci lim Qi pli ; p l

i

ci (Qi (p)) .

Therefore, the game satis…es DPM. Since this game satis…es DPM, we know from Theorem 1 that the mixed extension is payo¤ secure. Now we establish that the game has a mixed strategy equilibrium by appealing to results from Reny (1999) and Bagh and Jofre (2006). The following de…nitions are necessary for our proof of existence. De…nition 3 The game G is weakly reciprocal upper semicontinuous (WRUSC) if for all (x ; u ) 2FrG, there exists for some player i with a deviation xi 2 Xi such that ui xi ; x i > ui .

11 De…nition 4 The game G is better reply secure if whenever x is not an equilibrium and (x ; u ) is in the closure of the graph of G, there exists for some player i a strategy xi and a neighborhood N (x i ) of x i such that for all x i 2 N (x i ), ui (xi ; x i ) > ui . The de…nitions are analogous for the mixed extension of the game. Reny (1999) showed that a compact game whose mixed extension is better reply secure possesses a Nash equilibrium. He also showed that payo¤ security together with reciprocal upper semicontinuity implies that a game is better reply secure. Bagh and Jofre (2006) showed that the latter condition can be replaced with WRUSC. To prove that the game has a mixed strategy equilibrium we only need to show that the mixed extension of the game is WRUSC. Proposition 2 The BE oligopoly game has a mixed strategy equilibrium. Proof. See Appendix.

4.2

Nonexistence

The following example is constructed by Sion and Wolfe (1957) as an example of a game without equilibrium. There are two players with strategy spaces X1 = X2 = [0; 1]. The game is zero-sum, with 8 1 > > < 0 u1 (x1 ; x2 ) = 1 > > : 1

if x1 > x 2 if x1 = x2 or x1 + 12 = x2 : if x1 < x2 < x1 + 12 if x1 + 12 < x2

It is easy to see that this game does not satisfy disjoint payo¤ matching. The discontinuities occur at ties of the form x1 = x2 and x1 + 1=2 = x2 . For true ties (x1 = x2 ), player 1 bene…ts from deviating to points with x01 > x1 , while at shifted ties (x1 + 1=2 = x2 ), player 1 bene…ts from deviations to points with x01 < x1 or x01 > x1 + 1=2. If we consider x1 = 1=2, then if x2 = 1=2, a improvement requires x1 > 1=2, while if x2 = 1, then an improvement requires x01 < 1=2. This tension where one discontinuity demands deviations upward while another demands deviations downward to improve is what causes DPM to fail. Indeed, given any sequence of deviations from x1 = 1=2, each individual deviation must make player 1 discretely worse o¤ at either x2 = 1=2 or x2 = 1. A lesson in general is that DPM tends to hold whenever players can always improve their payo¤s at all discontinuities by deviations in a single direction, as is the case with the Bertrand-Edgeworth game where …rms can always lower their prices any be at least as well

12 o¤, or in any contests, where players can increase their bids or e¤orts and be at least as well o¤. In the current example, di¤erent discontinuities require con‡icting deviations to improve, and so a single sequence of deviations cannot uniformly improve a player’s positions.

5

Appendix

Proof of Proposition 2. Since the strategy space X is compact and Hausdor¤, we need only show that the game satis…es WRUSC. We begin by de…ning for each player i ui (p) = lim sup ui (x) x!p

and u = (u1 ; :::; un ), noting that each ui is upper semicontinuous. Since si (p) is continuous and Qi (p) si (p) for all p, then lim supx!p Qi (x) si (p). Further, by assumption, for any Q such that Qi (p) < Q si (p), i (pi ; Q) > i (pi ; Qi (p)). Thus, it follows that ci lim sup Qi (x) .

ui (p) = pi lim sup Qi (x) x!p

x!p

Note that lim Qi (xi ; p i ) = lim sup Qi (x) xi !pi

x!p

Qi (p) . Thus, for any p 2 X with pi > 0; lim ui (xi ; p i ) = ui (p) .

(8)

xi !pi

Let ( ; u ) 2FrG and let

l

! u

R be such that ud l ! u . Note that Z = lim ud l l Z lim sup ud l . l

Since each ui is upper semicontinuous, then as Reny (1999) shows in the proof of Proposition 5.1, Z Z lim sup l

Thus, we have that u

u ( ).

ud

l

ud :

13 De…ne Yi = p 2 X : Qi (p) < Qi (p) and Y = 0, and (ii) (Y ) > 0.

S

i

Yi . We consider two cases: (i)

(Y ) =

(i) In this case, Qi (p) = Qi (p) -almost everywhere for all players i. Thus, as noted above, ui = ui -almost everywhere, so we conclude that u ( ) = u ( ) u . Since ( ;u ) 2 = G, then it must be that ui ( ) > ui for some player i. It follows that i = i satis…es the de…nition of WRUSC. (ii) We begin by showing that ui ( ) > ui for some player i. For any p 2 Y , at least two …rms must charge the same positive price, and at least one such …rm i must have Qi (p) < Qi (p) si (p). Let I be the set of …rms j with pj = pi and J the set of …rms j with pj < pi . Note that P

(9)

j2I

Qj (p)

F (pi )

P

Qj

j2J

ji

(p)

for any choice of . The inequality in (9) must hold with equality else there would be excess demand that …rm i would be able to satiate. Pn Let A (p) lim supx!p A (x). We will show that j=1 uj (p) and A (p) A (p) <

Pn

j=1

uj (p) .

ej (p) Let xm ! p be such that A (xm ) ! A (p), and for each j let Q Pn

j2I

Pn

ej (p) = F (pi ) Q

j2J

ej Q

ji

limm Qj (xm ). Since

(p) ,

ei0 (p) < Qi0 (p). By our assumption, then there still exists at least one …rm i0 such that Q e i0 (pi0 ; Qi0 (p)) < i0 pi0 ; Qi0 (p) , so (10)

Pn

j=1

uj (p)

A (p) =

Pn

j=1

i0

> 0.

j

pi0 ; Qi0 (p)

pi0 ; Qi0 (p)

e

j (pi0 ; Qi0 (p))

e

i0 (pi0 ; Qi0 (p))

The inequality in (10) holds based on the facts that: (i) uj (p) limm uj (xm ) for all players j, ei0 (p)). Therefore and (ii) based on (8), uj (p) = i0 pi0 ; Qi0 (p) and limm uj (xm ) = i0 (pi0 ; Q for all p 2 Y , P A (p) < ni=1 ui (p) .

14 Since A is upper semicontinuous, then Pn

i=1

ui =

Z Z

< The …nal line follows from the fact that

Z

A (p) d

l

A (p) d Pn

i=1

ui (p) .

(Y ) > 0.

Let i be the player with ui ( ) < ui . Consider the deviation functions fm (pi ) =

1

1 m

pi .

We construct a sequence of measures that transfers any mass or density from each pi to 1 fm (pi ). For each m and every measurable set E, de…ne m i (E) = i (fm (E)). By Theorem 39C in Halmos (1974), Z Z Z m lim ui (pi ; p i ) d i d i = ui (fm (pi ) ; p i ) d . m

Since ui is bounded, there is a constant, integrable function which bounds ui , so the Lebesgue dominated convergence theorem states that Z Z lim ui (fm (pi ) ; p i ) d = lim ui (fm (pi ) ; p i ) d . m

m

As noted, for all p, limm Qi (fm (pi ) ; p i ) = lim supx!p Qi (x), so limm ui (fm (pi ) ; p i ) = ui . It follows that Z Z Z m lim ui d i d i = u i d . m

Therefore, for su¢ ciently large m,

m i

satis…es the de…nition of WRUSC.

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