Existence of equilibrium in a discriminatory price auction

mathematical social sciences ELSEVIER

Mathematical Social Sciences 30 (1995) 285-292

Existence of equilibrium in a discriminatory price auction Flavio M. M e n e z e s " ' * , Paulo K. M o n t e i r o b aDepartment of Economics, Faculty of Economics and Commerce, Australian National University, Canberra, A C T 0200, Australia ~IMPA- lnstituto de Matematica Pura e Aplicada, Estrada Dona Castorina, 110, Rio de Janeiro, RJ. 22460, Brazil Received December 1994; revised June 1995

Abstract

In this paper, we examine a specific discriminatory price auction of Y divisible objects. Players have demand schedules for the objects and are allowed to bid any amount in the interval [0, Y]. This game describes the main features of Treasury bill auctions. We characterize the set of Nash equilibria of this game. In particular, Nash equilibria may not exist unless bidders face demand functions satisfying special restrictions. This result raises questions about using the theory of auctions of single objects to predict the outcome of multi-object auctions.

Keywords: Discriminatory price auction; Treasury bill auction; Existence of equilibrium

I. Introduction

The debate about the optimal procedure to sell Treasury bills has been reopened. ~ This debate, which has its roots in Friedman's proposal to use uniform price auctions (see, for example, Goldstein, 1962), has concentrated on the advantages of the uniform versus the discriminatory price auction. "~ * Corresponding author. See. for example Stevens and Dumitru (1992) and the Joint Report on the Government Securities Market (1992). In fact, the Federal Reserve has been experimenting with the uniform price auction since September 1992. 2 In the first procedure, bids (which include both price and quantity) are ordered by price, from the highest to the lowest. The auctioneer accepts quantities up to the amount it is selling, but all winners pay the price equivalent to the highest losing bid. In the second procedure, bids are ordered similarly, but each agent pays the price he/she bid. (}165-4896/95/$09.50 (~) 1995 - Elsevier Science B.V. All rights reserved S S D I 0165-4896(95)00796-2

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F.M. Menezes, P.K. Monteiro / Mathematical Social Sciences 30 (1995) 285-292

In this debate, theorists and practitioners usually take for granted that the theory of common value auctions, with symmetric bidders who have some (uncertain) value for a single object, can be generalized to the case of asymmetric bidders with demand functions for a divisible object. For example, Chari and Weber (1992) use standard auction theory, where bidders are allowed to buy only one unit (i.e. they consider the case of an indivisible commodity), to argue that the uniform price auction yields higher prices and a lower probability of collusion, and inhibits the acquisition of too much information. They conjecture that, if bidders have demand schedules rather than a value for the objects, then their arguments would still be valid. The difficulty in obtaining general existence results for discriminatory auctions is well known. Therefore, our motivation is to provide an existence result on a specific, yet natural, bidding game. In this paper, we examine the existence of equilibrium in a type of discriminatory auction that describes the main features of Treasury bill auctions. In particular, we show that the existence of equilibrium may be a problem even when we consider a simple model of Treasury bill auctions where players face demand functions for a divisible object. This result raises questions about using the theory of auctions of a single object to predict the outcome of Treasury bill auctions. This paper is organized as follows. Section 2 describes the auction game. In Section 3, we show that the discontinuity and the lack of quasi-concavity of the payoff functions in this game restrict us from using the Dasgupta and Maskin (1986) existence result. Further, we characterize completely the set of pure strategy Nash equilibria for the auction game. In particular, for a pure strategy Nash equilibrium to exist, demand functions must satisfy some very restrictive assumptions. Finally, Section 4 summarizes our results.

2. Model and definitions

In this paper we examine the outcome of a discriminatory auction of a divisible commodity. The auction rules are as follows. The auctioneer announces a minimum price p0 and the available quantity Y. Without loss of generality, we set p 0 = 0 for the remainder of the paper. Two players, denoted by 1 and 2, participate in such an auction.3 Their bids have two components: the amount of Treasury bills and the price. In the real auction, bidders submit an integer number of bills. However, the number of bills is usually very large, so we simplify our analysis, by allowing bidders to bid any number on the interval [0, Y]. The auctioneer orders bids according to the price, starting from the highest, until the total amount Y is sold. Here our assumption about the number of There is no loss of generality here. Our results may be extended for more than two players.

F . M . M e n e z e s , P.K. M o n t e i r o / M a t h e m a t i c a l Social Sciences 3 0 (1995) 2 8 5 - 2 9 2

287

players becomes useful. If there is a tie; that is, if both players submit the same price level (but possibly different quantities), there are two possible cases: (i) if the sum of their bids is less than or equal to Y, then each bidder receives the amount he requested; and (ii) if the sum of their bids is greater than Y, then each bidder receives an amount proportional to his bid. Furthermore, if the price submitted by player 1 is the highest, then he receives the amount he bids, and his opponent receives the difference between Y and l's bid (if any). Each bidder i, i = 1, 2, is characterized by a demand function D i ( p ) , which specifies for every price p ~ [0, fi], the desired quantity. We assume that Di(15 ) = 0, i = 1, 2 and ~=1 De(0) > Y. Moreover, we assume that the demand functions are continuous and decreasing. These demand functions are common knowledge. (Of course, the auctioneer does not know the demand functions, otherwise he/she would use a take-it-or-leave-it type of mechanism.) We refer to p* as the price at which the market clears. Consequently, D i ( p * ) denotes bidder i's demand at the market clearing price. Given our assumptions, there is a unique p* > 0 that clears the market. To complete the description of the game, we need to specify strategies and payoff functions. A bidding strategy for player i, bi(D~(p), D / ( p ) ) , j # i, p E [0, f ] , is a mapping from i's information set into player i's set of actions. Hence, a strategy bl(.), for player 1. is a pair ( p , x ) that represents a price p and the quantity x demanded at that price. We denote by y, player 2's quantity, and by q, his/her price. We let ~L: B~ x Bz----~R denote the payoff function of player i, i = 1, 2, where B~ (B2) denotes the set of feasible bids for player 1 (2) (i.e. [0, fi] x [0, Y]). Payoff functions are defined as follows:

7rl( p, x, q, y) =

l

:,;

p>q

Joma
p=q,

min {:c, Y v} -1 (f,, " (D 1 (s)-p)ds,

{

fc:(DfI(s)

7h( p, x, q, y) =

- 1)ds,

f,~m,#T;=,T (D ;'(s) - q) (is, I'min{y " Y - x } / u - 1 /

J,I

\

~L

( u 2 (s) - q) as ,

,

p
,

q>p

,

q=p, q
3. The existence of Nash equilibria

As Dasgupta and Maskin (1986) point out, there are many economic games with existence problems, including models of spatial competition, Hotelling's model of price competition, and models of market-dependent information. These

F.M, Menezes, P.K. Monteiro / Mathematical Social Sciences 30 (1995) 285-292

288

games typically have convex and compact strategy sets, but fail to have continuous and quasi-concave payoff functions. Consequently, it is not possible to apply the classical existence theorems (see, for example, Debreu, 1952; Glicksberg, 1952; and Fan, 1952). However, Dasgupta and Maskin show that, for a slightly modified version of these games, the payoff functions exhibit weaker forms of continuity which (together with the hypothesis of quasi-concavity) imply the existence of equilibrium. Therefore, they argue that the failure of the payoff functions to be quasi-concave is the reason for the non-existence of Nash equilibria. However, the payoff functions in the Treasury bills auction game do not satisfy the continuity assumptions of Dasgupta and Maskin. In particular, ~'i is only weakly upper semi-continuous in bi.4 Moreover, we are unable to use the more general existence result of Baye et al. (1993) because our payoff functions do not satisfy any of the conditions of their Proposition 2. However, we are still able to characterize the set of pure strategy Nash equilibria of the auction game. Theorem 1. The vector (p, x, q, y) is a pure strategy Nash equilibrium if and only if"

(i) p = q, DI(P) + D2(p) = Y , vY DI(P), (max{Y x + v } ) - Dz(p) ' (iii) f f - Y (O?X(s))ds ~ 7r,(p, x, q, y),

(ii)

xg

(max{Y,x +y})

--

(iv) I f -x ( O f ' ( s ) ) d s ~ r 2 ( p , x ' q, Y). Proof. that it bidder If p'

Suppose that (p, x, q, y) satisfy (i), (ii), (iii), and (iv). We want to prove is a Nash equilibrium. Therefore, let us consider another bid (p', x') for 1. >p: x'

x'

7rl(P',x',q,Y)= f (D~I(s)-P') d s ~ f (D~I(s)-P) ds 0

0

DI(P)

<~ f

(D~(s)-p)ds=Tr~(p,x,q,Y) "

0

4 It is e a s y to see t h a t 7ri is d i s c o n t i n u o u s at ( p , x, q, y) if a n d o n l y i f p = q a n d x + y > Y. M o r e o v e r , ~ri is o n l y w e a k l y u p p e r s e m i - c o n t i n u o u s for p > 0. N o t i c e that for this to h o l d we n e e d the fact that x D ~ " " ' f~ ( i ( s ) - p ) d s is i n c r e a s i n g in the i n t e r v a l [0, D ( p ) ] a n d d e c r e a s i n g in the i n t e r v a l [Di(p), ~).

F.M. Menezes, P.K. Monteiro / MathematicalSocial Sciences30 (1995) 285-292 If

289

p ' = p we have x'Y DI(P)

max{Y,x'+y}

(D (~(s) - p ) ds

7ri(pl, x', q, y) = [/

0

-- ~'i(P, x, q, y ) . Finally, if p' < p: min{t',Y-y}

min{x',Y-y}

(D ['(s)) ds

7r1(p', x', q, y) = 0

0

y-),

<~f

(D~l(s))ds<~Trl(p,x,q, y)

0

by (iii). Therefore, max(p,x,)~5(p',x', q, y) = ~rl(P, x, q, y). Analogously, we verify that maXtq, y,) ~2(P, x, q', y ' ) = ~r2(p, x, q, y). This ends the proof of the 'if' part. Now suppose that ( p , x , q, y) is a Nash equilibrium. I f p < q we have Y

Y

7r2(p,x, q, y ) : f (Dff'(s)-q)ds<~ f (D~',s)_ 0

p+2 q ) d s

0

D2((p+q)/2)

f

(

- P + q ' ] d s = T r ~ ( p , x , p + q D (P+q']'~

0

The first inequality above is strict if y > 0 and the last one is strict if y = 0. This contradicts the Nash equilibrium definition. Therefore, p ~>q. Analogously, we also prove that q ~ p and, therefore, p = q. Take p' > p and x' = D l ( p ' ) : xY max(Y,x+y}

f

(D ?t(s) - p ) ds = ~rl(p,x, q, y) >~rr,(p',x', q, y)

0

DI(P')

(D ~l(s) - p') ds . 1)

In the limit as p ' - - ) p we have xY

maxlY.x+v}

DI(P)

(D ll*(s) - P) ds . 0

0

F.M. Menezes, P.K. Monteiro / Mathematical Social Sciences 30 (1995) 285-292

290

Therefore, xY/(max{Y,x + y}) = D1(p) and, analogously, yY/(max{Y,x + y}) = D2(p). This proves (ii) and that p > 0. From Y-V

D, l(s) ds= ~r~(O, Y, q, y)<~ ~r,(p,x, q, y), 0

we have Y - y <~xY/(max{Y,x + y}), so x + y >1 Y. This also proves (iii), and that Y = xY/(x + y) + yY/(x + y) = D~(p) + Dz(p); hence, we have (i). [] The intuition behind the possible lack of existence of a Nash equilibrium is quite straightforward. Suppose bidder 2 bids (p*, Dz(P*)). Player 1 will not submit a bid with the price component higher than p*. This family of strategies is strongly dominated by bidding (p*, Dl(p*)). However, by bidding (0, D~(p*)), player 1 guarantees an award of D~(p*) at the minimum price. This is his/her best response. However, player 2's best response to that is now to submit a bid of (t~, D2(6)) , for any 6 > 0 sufficiently small. By doing so, player 2 guarantees to receive his/her full demand at this price. Player l's strategic response is then to submit a slightly higher price. This process continues indefinitely until one of the players reaches p*, and then it starts again. Back and Zender (1993), assuming flat demand functions in a common value environment, show that the discriminatory-price auction has an equilibrium. Given these preferences, Theorem 1 suggests that there may be other equilibria as well. The following corollaries follow directly from Theorem 1 and they will not be proven.

Corollary 1. If the demand functions are identical, then (p, Y, p, Y) is a Nash equilibrium if 2 D l ( p ) = y.5 Corollary 2. Suppose p is such that Dl(p) + D2(p) = Y. Then, there is a pure strategy Nash equilibrium if and only if Y( 1 - D I ( P ) / D 2 ( P ) )

D2(P)

f

(Dzl(s)) ds<~ f

0

0

(DzI(S)-P) ds,

ifDl(p)<~D2(p),

and the analogous inequality in case D z ( p ) ~ Dl(p). Thus, it follows that a sufficient condition for a Nash equilibrium to exist is that bidders have identical demands, i.e. D(p) = Dl(p) = D2(p). When demands are Notice that the equilibrium described in Corollary 1 is not perfect. In general, any Nash equilibrium in pure strategies where at least one of the players bids more than his demand at the price component of his/her bid cannot be perfect.

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identical, Theorem 1 allows us to characterize the set of all Nash equilibria in pure strategies. Part (i) requires p = q = p * . The remainder of Theorem 1 implies that the set of Nash equilibria can be described as {(p*, c~, p*, a): a E [W, Y]}, where W = rain{w: w > D ( p * ) and 7r1(0, Y - w, p*, w) ~< ~5(P*, w, p*, w) = 7 r ~ ( p * , D ( p * ) , p * , D ( p * ) } . ° For example, if D 1 ( p ) = D 2 ( p ) = l - O . 5 p and Y = 1, then W = 31/2/2. Next, we provide an example of Corollary 2 with a simple parametric form Example 1. Define D i ( p ) = a i / p ' , i = 1, 2. Let 0 < a I < a 2 and e > 1. The inverse demand of player 2 is given by D ~ l ( s ) = a ~ " ' s -x''. Since D~(s)<-D2(s), for any s > 0, there is a Nash equilibrium in pure strategies if and only if Y( I - (st 1 a2)) f

(a2/P ~)

a 21/, s --1,'. ds~<

I1

f

,1/,2 s ta

1,,, - p )

ds,

0

where p is such that (a I + a2)/p" = Y. After integrating both sides and simplifying, we obtain the following restriction on the parameters a~ and a2: al [ 1 > ~ - - ~>

(1),/~,

1)]1/2

1 -

a2

Furthermore, if each player is allowed to submit multiple bids, then we can easily prove that a Nash equilibrium will have only one bid for each bidder and then the theorem on Nash equilibrium existence applies. Next, we provide an example of the non-existence of Nash equilibria. E x a m p l e 2. Define D r ( p ) = a - bp, a > Y, b > 0 , and D2(p) = r D l ( p ) for r > 1. The price p = (a(1 + r) - b Y ) / ( b ( 1 + r)) is such that D ~ ( p ) + Dz(p) = Y. If we use Corollary 2, then we see that, for r large enough, there is no Nash equilibrium.

We were unable to prove whether or not mixed strategies equilibria exist. However, since the continuity problem is binding, we conjecture that having mixed strategies does not solve the existence problem.

4.

Conclusions

In this paper, we presented a simple model of Treasury bill auctions under complete information, where bidders face demand functions for a divisible object At any Nash equilibrium in pure strategies, each player receives half of the awards at the market clearing price.

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F.M. Menezes, P.K. Monteiro / Mathematical Social Sciences 30 (1995) 285-292

r e p r e s e n t i n g the fractions of T r e a s u r y bills. We show that a Nash equilibrium will n o t exist unless bidders face d e m a n d functions that satisfy the conditions of T h e o r e m 1. If d e m a n d s are identical, then the o u t c o m e predicted is such that each player receives half of the awards at the m a r k e t clearing price. T h e non-existence of equilibria for this auction g a m e is a troubling result. It indicates that s o m e of the results of the t h e o r y of auctions of single objects must be t a k e n with caution when examining multi-object auctions. Finally, although a m o r e general a p p r o a c h would allow for incomplete i n f o r m a t i o n , significant complexities arise even in the absence of private inf o r m a t i o n . Further, o u r aim was to provide the simplest possible f r a m e w o r k to m o d e l T r e a s u r y bill auctions. It remains to be shown w h e t h e r or not our existence result is robust to incomplete information.

Acknowledgements W e wish to t h a n k Jfirgen Eichberger, J. Kline, T. K o m p a s , H e r v 6 Moulin and an a n o n y m o u s referee for useful c o m m e n t s . T h e usual disclaimer applies.

References K. Back and J. F. Zender, Auctions of divisible goods: On the rationale for the Treasury experiment, Rev. Finan. Studies 6, no. 4 (1993) 733-764. M. R. Baye, G. Tian and J. Zhou, Characterization of the existence of equilibria in games with discontinuous and non-quasiconcave payoffs, Rev. Econ. Studies 60, no. 4 (1993) 935-948. V.V. Chari and R. J. Weber, How the US Treasury should auction its debt, Quart. Rev., Federal Reserve Bank of Minneapolis (1992) 3-12. P. Dasgupta and E. Maskin, The existence of equilibrium in discontinuous economic games, I: Theory, Rev. Econ. Studies 53 (1986) 1-26. G. Debreu, A social equilibrium existence theory, Proc. Nat. Acad. Sci, 38 (1952) 886-893. K. Fan, Fixed point and minimax theorems in locally convex topological linear spaces, Proc. Nat. Acad. of Sci. 38 (1952) 121-126. I. L. Glicksberg, A further generalization of the Kakutani fixed point theorem with application to Nash equilibrium points, Proc. Amer, Math. Soc. 38, (1952) 170-174. H. Goldstein, The Friedman proposal for auctioning Treasury bills, J. Polit. Econ. 70, no. 4 (1962) 386-392. Joint Report on the Government Securities Market, Washington, DC: Department of the Treasury, Securities and Exchange Commission, and Board of Governors of the Federal Reserve System (1992). E. J. Stevens and D. Dumitru, Auctioning Treasury securities, Econ. Comm., Federal Reserve Bank of Cleveland (1992).