Rationalizable Strategies and Perfect Equilibria in ...

Rationalizable Strategies and Perfect Equilibria in Common Value Auctions with Variable Supply Damian Damianov∗ and Johannes Becker University of Heidelberg, Alfred Weber Institute, Heidelberg, Germany † February 2004

Abstract Auctions with variable supply are multi-unit auctions, in which the seller determines the supply quantity as a function of the bidding. These auctions are used on various markets ranging from Treasury bills and IPOs to rare wine and art. In this work we compare in a common value model the mixed strategy subgame perfect equilibria and the rationalizable strategies of the uniform price and the discriminatory auction. This allows us to establish a revenue and efficiency ranking of both auctions. The uniform price auction is found to be ex-ante more profitable for the seller and to allocate goods more efficiently.

Key Words: sealed bid multi-unit auctions, variable supply auctions, discriminatory and uniform price auctions, subgame perfect equilibria, rationalizable strategies. JEL Classification: D44.

∗

Correspondence to: Damian Damianov, Alfred Weber Institute, Grabengasse 14, D-69117 Heidelberg,

Germany † Email addresses: [email protected]; [email protected]

1

Introduction

The theory of multiunit auctions traditionally assumes that the number of units put up for sale is fixed prior to the auction. Although this assumption fits some applications, common auction practices suggest that sellers often determine the supply quantity after observing the bids. These forms of organizing trade, known as variable supply auctions1 , are pertinent to various markets.

1.1

Examples

An important example comes from the financial markets, where the right of the seller to adjust supply after the bidding is often established by law. The treasury auction formats in several countries share this institutional feature. Nyborg, Rydqvist and Sundaresan (2002) report that the Swedish Treasury reserves the right to withdraw securities from the auction after bids have been submitted. Heller and Lengwiler (2001) point to the same feature in the auction for Swiss Treasury Bonds and add that usually significantly less bonds than the initially announced maximum volume are sold in the auction. Umlauf (1993) explains that the Mexican Treasury has the right to cancel a part of or even the entire quantity of each weekly (uniform price) auction after the bidding. Variable supply auctions govern the trade on IPO markets as well. McAdams (2000) comments that the underwriters of IPOs in the United States are granted the Green Shoe option to increase the number of shares by up to 15 %. Busaba (2002) explores the value of a firm’s option to withdraw an IPO during the bookbuilding process and argues that this option strengthens the firm’s bargaining position. Conditioning the supply quantity on the received bids is practiced also by sellers on markets for wine, art, jewelry, furniture2 and other goods. This is done less explicitly, usually by keeping a secret reserve price3 , which is made public after the bidding4 . Bids 1 2

Back and Zender (2001) use the term endogenous supply auctions. Ashenfelter (1989, pp. 24–25) has collected auction data from Sotheby’s, Christie’s, Phillips and

other auction houses in the early eighties. He reports that sellers often retain part of the supply quantity, since bids are too low and do not meet the (hidden) reserve price of the seller. “Retain” rates (as a result of secret reserve prices) range for wine between 5% and 10% and lie for Impressionist paintings at about 30%. 3 Meanwhile the use of hidden reserve prices is commonly used by sellers on internet auction sites (see Lucking-Reiley 2000, p. 244). 4 The question of whether the seller solicits the secret reserve price prior to the bidding or afterwards is important, as the two scenarios define different auction games. Lengwiler (1999) studies a model in which the reserve price is set after the bidding as a function of the bids. In our model the seller chooses (also after the bidding) quantities rather than stop-out prices, but the formulation is equivalent to that of

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below the reserve price are not served and auctioneers say then that the retained objects are “bought in”. This means that they will be put up for sale at a later auction, sold elsewhere or taken off the market. It seems that variable supply auctions are especially attractive on anonymous markets with high uncertainty about demand, since the seller need not commit to supply quantity ex ante. They can be used when an adjustment of supply is technically easy, of low cost, or if the trade of unsold units can be postponed without substantial losses.

1.2

Theoretical models

Although the literature has well documented the importance of these auctions in practice, theoretical studies of the effects of the variable supply feature on bidding behavior appeared only recently. Back and Zender (2001) and McAdams (2000) analyze a stylized complete information model of the uniform price Treasury auction. In their setting the seller fixes a supply quantity, which can potentially be reduced after the bidding, and sets a reserve price of zero. The bidders have common knowledge of the asset value and compete for shares by submitting left-continuous bid functions. The option to reduce supply is found to eliminate some low-price equilibria5 of the uniform price auction. Lengwiler (1999) studies a model in which the bidders are incompletely informed about the (constant) marginal costs of the seller. The buyers in his model announce quantities to the auctioneer on a discrete price grid consisting only of two prices – high and low. The seller then decides whether to serve the demand on the high or on the low price so as to maximize profit. Lengwiler showed that the uniform price and the discriminatory auctions have perfect equilibria. Since the computation of equilibria in such a setting is rather difficult, both standard auction forms could not be compared in terms of revenue for the seller or efficiency. He claimed, however, that both formats allocate inefficiently and found out that the inefficiency might not decrease with increased number of bidders. The setting we consider here is similar to that of Lengwiler. We allow the bidders to announce a price only for a single unit, which could be any real nonnegative number. In this setting we find that the uniform price auction is both more profitable for the seller Lengwiler, since in our framework the quantity choice uniquely defines a stop-out price. Vincent (1995) analyzes the scenario in which the reserve price is fixed prior to the bidding in a single-unit common value auction setting. He demonstrates that keeping the reserve price secret may increase the revenue of the seller. 5 Low-price equilibria of the uniform price multi unit auction with fixed supply are first discussed by Wilson (1979). Back and Zender (1993) bring that issue in the context of the ongoing discussion on how Treasury bills auctions should be organized. They argue that the uniform price auction has the potential of yielding very low revenues for the Treasury.

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and more efficient.

1.3

Nash equilibria and rationalizable strategies

In this work we provide some new insights about rational bidding behavior in common value auctions with the variable supply feature. We use the term rational bidding intentionally, since for the two-bidder case we do not limit our study to equilibrium behavior only. Rather, we extend our concept of rational bidding to the set of rationalizable 6 strategies. Our main contribution consists in comparing the sets of symmetric mixed strategies Nash equilibria (and rationalizable strategies in the two-bidder case) of the uniform price and the discriminatory auction. We do that by identifying bounds on the supports of these sets. In the setting under consideration these bounds allow us to establish a revenue and efficiency ranking of both auctions. Our approach enables the comparison of the two auction formats without the need to explicitly compute their symmetric equilibria, which is analytically intractable in this complex multi-unit auction setting. Additionally, the bounds we provide on the rationalizable strategy sets in the two-bidder case apply to all Nash equilibria, as the set of Nash equilibria is included in the set of rationalizable strategies (see Bernheim 1984, Pearce 1984). The methodology to construct bounds on the set of rationalizable bids is relatively novel to the theory of auctions. It has been recently employed by Battigalli and Siniscalchi (2003), who study interim rationalizable bids in symmetric first-price single-unit auctions with interdependent values and affiliated signals. Cho (2003) also analyzed rationalizable strategies in a single-unit first price auction with many bidders.7 This methodology has advantages reaching beyond the scope of these papers. In complex models, where the characterization of Nash equilibria proves to be intractable or 6

The notion of rationalizability as a criterion for rational strategic choice was introduced independently

by Bernheim (1984) and Pearce (1984). They argued that in a simultaneous game without pre-play communication one cannot expect that players will be able to fully predict their opponents’ behavior and therefore can be in doubt whether in such a game Nash equilibria will be played at all. They proposed that a player’s choice should only be rational given some conjectures (or beliefs) about other players’ behavior. These beliefs need to be rational(izable) in a precisely defined sense but need not necessarily coincide with the actually played (pure or mixed) strategies as the Nash equilibrium concept requires. The rationalizability notion thus allows for more flexibility in the beliefs the players hold. Generally players can have many rationalizable strategies and in simultaneous games the set of Nash equilibria is only a subset of the set of rationalizable strategies (see Bernheim 1984). 7 He extended Wilson’s (1977) result that in single-unit auctions with common value element the equilibrium price converges to the highest valuation among bidders as the number of bidder increases.

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possible only in special cases, one might still be able to identify some general properties of rationalizable strategy sets. These properties are valid not only for the correct, selffulfilling beliefs required by the equilibrium notion of rationality, but also for all other not necessarily correct, but sophisticated or rational(izable) beliefs. This additional generality might be important, if one tests theoretical predictions like those presented here with experiments in which the subjects play simultaneously and are not allowed to communicate.

1.4

Organisation of the paper and results

In the following section we present a common value multi-unit auction model. Subsection 2.1 contains the preliminaries of the model. In subsection 2.2 we introduce a general notation for the pure and mixed strategies and for the payoffs of the players in the trade mechanisms considered. Section 3 contains definitions of equilibrium and rationalizability. In section 4 we analyze rational bidding in the uniform price and discriminatory auctions. We prove that both auctions have a (symmetric) mixed strategy Nash equilibrium and the supports of the equilibrium sets of the two auctions are disjoint (theorem 5). The equilibrium bids in the uniform price auction are with probability one higher than those in the discriminatory auction. This result applies for the rationalizable strategy sets in the two-bidder case (theorem 3). It also has immediate implications for the seller’s ranking of both auctions in terms of revenue and (under one additional assumption) for the exante efficiency of these trade mechanisms (see subsection 4.4). The obtained results are illustrated with a numerical example in subsection 4.5. Section 5 concludes.

2

The model

2.1

Preliminaries

We consider a bidding game between n ≥ 2 buyers and a monopolistic seller. The monopolist sells off multiple units of a common value asset via a variable supply auction. Each buyer i ∈ {1, 2, . . . , n} is risk neutral and submits a price bid for a single unit. The following assumptions further specify the setting of the model. Assumptions (A1) No proprietary information8 . 8

The assumption and the term “no proprietary information” were introduced into the auction literature

by Wilson (1979) and Bernheim and Whinston (1986). This simple information structure is often assumed in multi-unit auction models (see, e. g. Back and Zender 2001, Kremer and Nyborg 2002). The assumption

5

The buyers are identically informed about the common value of the good. It is given by a random variable with an expected value9 of v > 0. The seller is uninformed about v, which explains the use of an auction as a trade mechanism. (A2) Private information about seller’s marginal costs. The monopolist “produces” the good with constant, but privately observed marginal production costs10 c. c is a random variable with support [0, c], where c ≥ v. The distribution function of that random variable is denoted by F (c) and the density function by f (c). The latter is taken to be continuous, strictly positive in the interval [0, c] and differentiable in the interval (0, c). (A3) Monotone hazard rate. Further it is assumed that the distribution function is log-concave, i.e. (ln F (c))0 =

f (c) F (c)

is a monotonically decreasing function11 . (A4) Bid constraints. Bidders are allowed to submit a price bid from the interval M = [0, m], where m > v is an arbitrarily large, but finite number12 . is definitely a restriction, since one cannot discuss the effects on bidding behavior any more, which arise from the interaction of privately informed bidders. Such effects, which are well known from the singleunit common value auction literature, are the winner’s curse and the linkage principle (see Wilson 1977, Milgrom 1981, Milgrom and Weber 1982). Since these effects are related to the private information of the bidders, they are now excluded by assumption. The assumption allows, however, to study some unexplored effects related particularly to multi-unit auction environments and to the endogenous supply feature. 9 One can think for example of an IPO or Treasury bill auction. These assets are traded on a secondary market after the auction. The pure common value v is then the expected price on the resale market. Of course on such markets the buyers do not demand a single unit and also can submit multiple bids at multiple prices. 10 The same assumption is used in Lengwiler (1999). 11 This property of the distribution, called “monotone hazard rate” is a standard assumption in auction theory. It guarantees in single-unit first-price auction models that bidders with higher valuations submit higher bids. It is satisfied by most common distributions: uniform, normal, logistic, chi-squared, exponential and Laplace. See Bagnoli and Bergstrom (1989) for a more complete list and for results allowing the identification of distributions with monotone hazard rates. 12 The bidders are not able to pay infinitely large bid prices.

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2.2

Auction games

After receiving the bids the seller decides on supply quantity (so as to maximize profit), a scenario that we model as a two-stage game. The payoffs of the players depend on the bids, the supply quantity and the payment rule of the auction. We will first introduce some general notation for the payoffs of the bidders in order to provide standard definitions of equilibrium and rationalizability for an arbitrary variable supply auction game Γ. Then we will specify these payoffs separately for the uniform price and the discriminatory auction and will analyze rational bidding in both auction formats. 2.2.1

Pure strategies

Each bidder i submits a price bid xi to the auctioneer, indicating the (highest) price he is willing to pay for a unit. The vector of submitted bids is denoted by x and the bid vector of all bidders except bidder i by x−i . Let us consider an arbitrary trade mechanism Γ. Since the seller can condition the supply on the received bids, her strategy is a mapping from the set of bid vectors and possible values of the private information c into supply quantity: φΓ : M n × [0, c] → {0, 1, 2, .., n}.

(2.1)

Assume that after observing the bids x and the marginal costs c the seller supplies the quantity q. We denote her profit by rSΓ (x, q, c) and the payoff (or the net consumer surplus) of bidder i by riΓ (x, q). If the seller supplies according to the strategy φΓ , the expected payoff of bidder i is

Z RiΓ (x; φΓ )

c

= 0

¡ ¢ riΓ (x; φΓ x, c) · f (c) dc

(2.2)

and the (ex ante) expected profit of the seller is Z RSΓ (x; φΓ ) 2.2.2

c

= 0

¡ ¢ rSΓ x; φΓ (x, c) · f (c) dc.

(2.3)

Mixed strategies

A mixed strategy of bidder i, σi is a probability distribution over the set of pure strategies M . The set of mixed strategies of bidder i, Σi ≡ Σ is the set of probability distributions defined on (M, B), where B is the Borel σ-algebra on M . A mixed strategy profile of all bidders is denoted by σ and a mixed strategy profile of all fellow bidders of bidder i by σ −i . The payoff of bidder i in the collapsed game is defined as Z Γ Ri (σ; φΓ ) = RiΓ (x; φΓ )dσ(x).

7

(2.4)

The ex ante profit of the seller is defined as Z Γ RS (σ; φΓ ) = RSΓ (x; φΓ )dσ(x).

3

(2.5)

Definitions

Definition 1 (subgame perfect equilibrium). The mixed strategy profile σ ∗ and the supply function of the seller φ∗Γ constitute a subgame perfect equilibrium (short equilibrium) of the auction Γ, if the following conditions (SS) and (MS) hold. An equilibrium in which the bidders play pure strategies is called a pure strategy equilibrium. Second stage For every vector of declared bids13 x and every value of the marginal costs c, the auctioneer sets the supply quantity so as to maximize her profit: φ∗Γ (x, c) ∈ arg max rSΓ (x, q, c).

(SS)

q∈{0,1,2,..,n}

Collapsed game The strategy of every bidder i maximizes his expected payoff given the strategies of the other bidders and optimal supply strategy of the seller: RΓi (σi∗ , σ ∗−i ; φ∗Γ ) ≥ RΓi (σi , σ ∗−i ; φ∗Γ ) , ∀σi ∈ Σi .

(MS)

The condition requires that the bid strategies should constitute a Nash equilibrium in the collapsed game. Definition 2 (rationalizable strategies). Assume that in the second stage of the auction game Γ the seller supplies according to the profit-maximizing strategy φ∗Γ . Let e Γ,0 ≡ Σ and for each i recursively define Σ i ½ Γ,k e = σi ∈ Σ e Γ,(k−1) : ∃σ −i ∈ convΣ e Γ,(k−1) such that Σ i i −i RΓi (σi , σ −i ; φ∗Γ )

≥

RΓi (σi0 , σ −i ; φ∗Γ )

for all

σi0

¾ Γ,(k−1) e ∈ Σi .14 ,15

13

The seller observes a realization x drawn from the probability distribution σ. conv stands for convex hull. The convex hull of a set X is the smallest convex set that contains it. 15 For brevity and ease of access we stick to the definition and the notation of Fudenberg and Tirole 14

(1991, pp. 49 Definition 2.3). Although this definition does not introduce the notion of a belief system as the original definition does (see Bernheim 1984, pp.1013-1014, Definitions 3.1-3.3), it is equivalent to Bernheim’s (1984) definition.

8

The set of rationalizable strategies for player i in the trade mechanism Γ is defined as eΓ = Σ i

∞ \

e Γ,k . Σ i

k=0

In words, the rationalizable (or strategically sophisticated) strategy profiles are (mixed) strategy profiles which survive the serial deletion of strategies not belonging to the best responses of the players.

4 4.1

The uniform price and the discriminatory auctions Payoffs

In both the uniform and the discriminatory auction the seller orders the bids in a descending order and serves them until the supply q is exhausted. In the uniform price auction all winning bidders pay a price equal to the lowest winning bid, which is called the stopout price. In the discriminatory auction the seller acts as a perfectly discriminating monopolist and all winners are charged their own bid prices. Let us introduce some additional notation to describe the payoffs of the players. Take an arbitrary bid vector x. Order the bids in a descending order. For that purpose define the function ϕx : {1, 2, . . . , n} → {1, 2, . . . , n},

where ϕx (j) = k,

if bidder j submitted the k -th highest bid. If two or more bids are equal, then the function ϕ orders them arbitrarily. Further we define τ (x) = (τ1 (x), τ2 (x), . . . , τn (x)), where τk (x) is the k-th highest bid if the bids are ordered in a descending order. The stopout price then is τq (x). The payoff of bidder i in the uniform price auction is  v − τq (x) for ϕx (i) ≤ q, riU P (x; q) = 0 for ϕx (i) > q. The payoff of bidder i in the discriminatory price auction is  v − xi for ϕx (i) ≤ q, DA ri (x; q) = 0 for ϕ (i) > q. x

9

The payoff of the auctioneer in the uniform price and in the discriminatory auction is, respectively, rSU P (x; q) = (τq (x) − c)q, q X DA rS (x; q) = (τj (x) − c).

(4.1) (4.2)

j=1

4.2

Discriminatory auction (DA)

Theorem 1. The set of rationalizable strategies of the discriminatory auction contains only one pure strategy for each bidder: © ª e DA ≡ σ DA | σ DA (z) = 1 , Σ i i i

∀i ∈ {1, 2, . . . , n},

where z is the unique solution of the equation v−z =

F (z) . f (z)

(DA)

v F (z) f (z)

z

n−1 n

·

F (z) ; f (z)

n≥2

z

zc n v−z

Figure 1: z is the unique solution of the equation (DA) and zbn of the equation (U Pn ) (see theorem 5).

Proof. Second stage: The monopolist is serving the bids as long as she finds it profitable to sell additional units (see 4.2). Every bid which exceeds (or is at least not lower than) the marginal costs c is

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served. The optimal supply quantity of the auctioneer takes the form: φ∗DA (x, c) = max{k : τk (x) ≥ c.}16 For the payoff of bidder i we obtain riDA (x; φ∗DA ) =

  v − xi

for xi ≥ c,

0

for xi < c.

Observe that the payoff of each bidder is independent of the bids of the other bidder(s). Collapsed game: The expected consumer surplus of that bidder is thus RiDA (x; φ∗DA ) = (v − xi )F (xi ). From the first order condition one obtains that the maximizer z is the unique solution of the equation (DA). Assumptions (A2) and (A3) guarantee that this equation really has a unique solution17 . For the collapsed game, the bid z is a strongly dominant strategy for each player, which completes the proof. If the strategy of the seller were to set a reservation price above which bids are served, then choosing c would have been a dominant strategy. The observation that in the discriminatory auction the optimal strategies of the bidders are independent of the strategies (or the oligopolistic structure) of the other bidders has been discussed also in Lengwiler (1999) in a setting, in which bidders’ strategies are quantities (announced at two different price levels) rather than prices. This observation makes the analysis of subgame perfect equilibria in the discriminatory auction simpler than that in the uniform price auction. For an analysis of a setting in which the bidders perceive the stopout price as a random variable in preparing their bids see Nautz (1995) and Nautz and Wolfstetter (1997). In their models the bidders submit whole demand functions. 16

In fact the auctioneer is indifferent between selling or not selling units to bidders, who quoted a price

equal to the marginal costs. This detail is not important here, as such an event happens with probability 0, since the distribution F (c) is atomless. 17 Consider the function G(z) = v − z −

F (z) f (z) .

(v) Observe that G(0) = v > 0 and G(v) = − Ff (v) < 0 (see

A2). The continuity of G(z) guarantees that the equation G(z) = 0 has a solution in the interval (0, v) (by the Intermediate Value Theorem). (A3) requires that

F (z) f (z)

is a monotonically increasing function,

therefore G(z) is monotonically decreasing. Thus the equation G(z) = 0 has a unique solution.

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4.3 4.3.1

Uniform price auction (UP) The case of two bidders

In this case the bid vector x consist only of two bids. The payoff of the monopolist is   0 for q = 0,    RSUP (x, q, c) = τ1 − c for q = 1,    2(τ − c) for q = 2. 2

Second stage: The optimal supply strategy of the auctioneer is given by   0 for c > τ1 ,    φ∗UP (x, c) = 1 for τ1 > c > 2τ2 − τ1 ,18    2 for 2τ − τ > c. 2

1

Collapsed game: Now one can characterize the expected payoff of bidder i: RiUP (xi , x−i ; φ∗U P ) =  (v − xi ) · ¡F (xi ) − F (2x−i − xi )¢ + (v − x−i ) · F (2x−i − xi ) for xi ≥ x−i , = (v − x ) · F (2x − x ) for xi < x−i . i i −i

(4.3)

Lemma 1. The expected profit function has the following properties19 : ∂i +RiUP (xi , x−i ; φ∗U P ) > 0

for

∂i RiUP (xi , x−i ; φ∗U P ) > 0 f or

xi = x−i < v, x−i < xi < x−i 2

(4.4) and

xi ∈ [0, zb2 ] ,

(4.5)

where zb2 is the unique solution of the equation v−z =

1 F (z) · . 2 f (z)

Proof. (4.4): For xi = x−i we have ∂i +RiUP (xi , x−i ; φ∗U P ) =

(v − xi ) · (f (xi ) + f (2x−i − xi )) − F (xi ) + F (2x−i − xi ) − (v − x−i ) · f (2x−i − xi )

= 18

(v − xi ) · f (xi ) > 0.

The equalities occur with probability 0 and are therefore omitted. is the partial derivative from above with respect to xi .

19 + UP ∂i Ri (xi , x−i ; φ∗U P )

12

(U P2 )

(4.5): Follows from the inequalities ∂i RiUP (xi , x−i ; φ∗U P ) = (v − xi ) · f (2xi − x−i ) · 2 − F (2xi − x−i ) h F (2xi − x−i ) i = 2f (2xi − x−i ) (v − xi ) − 2f (2xi − x−i ) h F (xi ) i > 2f (2xi − x−i ) (v − xi ) − 2f (xi ) h F (b z2 ) i > 2f (2xi − x−i ) (v − zb2 ) − = 0. 2f (b z2 ) The last two inequalities apply because

F f

is monotonically increasing function (see (A3)).

Remark 1. zb2 > z. Follows directly from the fact that z solves the equation (DA), zb2 solves the equation (U P2 ) and assumption (A3) (see also figure 1). Corollary 1. The (pure strategy) best response correspondence of bidder i, x∗i (x−i ) has the following properties: x∗i (x−i ) 6= x−i ,

(4.6)

x∗i (0) = z > 0,

(4.7)

x∗i (v) < v.

(4.8)

Proof. (4.6) follows from (4.4). (4.7) and (4.8) follow from (4.3). (4.6) implies that the uniform price auction has no symmetric subgame perfect equilibrium in pure strategies. (4.7) and (4.8) further imply that the best response correspondence is not continuous, which points to the generic difficulty for the existence of pure strategy equilibrium. Indeed, if the best response were continuous, it should cross the 45◦ line, which does not happen here because of (4.4). In subsection 4.5 we calculate the best responses for a uniformly distributed marginal costs example. See figure 6 for an illustration of the best response correspondences for that numerical example. The next theorem provides an existence result. Theorem 2 (equilibrium existence). The uniform price auction has a mixed strategy equilibrium. Proof. The existence is guaranteed by Glicksberg’s (1952) theorem, since the expected payoff functions RiUP (xi , x−i ; φ∗U P ) are continuous in (xi , x−i ) and the support of the bids M is a convex and compact set. 13

Theorem 3 (rationalizable strategies). The set of rationalizable strategies of the uniform price auction contains only (mixed) strategies with support in [b z2 , v]: e Ui P ⊆ {σiU P | σiU P ([b Σ z2 , v]) = 1},

i = 1, 2.

This theorem is proven in the following three lemmas. In lemma 2 and lemma 4 we proceed by contradiction to verify that mixed strategies placing positive probability in the intervals (v, m] and [0, zb2 ) are not rationalizable. Lemma 3 is used for the proof of lemma 4. Lemma 2. Rationalizable strategies do not place a positive probability on the interval (v, m]: e Ui P , σiU P ((v, m]) > 0 ⇒ σiU P ∈ /Σ

i = 1, 2.

Proof. For each σ ∈ Σ, define σ ˆ ∈ Σ by ¡ ¢ ¡ ¢ σ ˆ (B) = σ B ∩ [0; v] + σ (v; m] · 1v∈B

for B ∈ B,

which means, a bidder with strategy σ ˆ bids v whenever a bidder with strategy σ would submit a bid from the interval (v; m]. We first remark that σ ˆ always weakly dominates σ, as bids above v lead to a strictly negative outcome when served. So a strategy σi of ¡ ¢ player i with σi (v; m] > 0 will never be a best answer to a strategy σ−i of player −i, if under the strategy combination (σi , σ−i ) player i has to pay more than v with strictly positive probability. Using this fact we will now show by induction that, with the notation of definition 2, for k = 1, 2, . . . e UP,k , σi ∈ /Σ i

i = 1, 2,

if σi

³¡

max{v, 2−k m}; m

¤´

> 0.

(4.9)

We start with k = 1. As the other bid is never greater than m, bids from the interval ¡ ¤ max{v, m2 }; m are served when the costs of the seller are below v, which will happen with ¡¡ ¤¢ a strictly positive probability. So, by the introductory remark, if σi max{v, m2 }; m > 0, σî will be strictly better than σi , regardless of what −i does. Now assume that equation (4.9) holds for k − 1. Bids above max{v, 2−k m} are served when the other bidder does not submit a bid above max{v, 2−(k−1) m} and the costs are below v, which by induction happens with strictly positive probability if the other bidder ¡¡ ¤¢ e UP,k−1 . So for each strategy σi with σi max{v, 2−k m}; m > 0 plays a strategy from Σ −i

e UP,k−1 , which proves the strategy σbi will be a strictly better answer to any element of Σ −i (4.9) for k.

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Lemma 3. There exists δ > 0 such that ∂i RiUP (xi , x−i ; φ∗U P ) > 0 for x−i < xi < min{x−i + δ, zb2 }. ¯ ª © z2 Proof. On the set K := (yi , y−i ) ¯ 0 ≤ y−i ≤ zb2 , 0 ≤ yi ≤ v−b we define a function 2 g : K → R by g(yi , y−i ) = ∂i RiUP (yi + y−i , y−i ; φ∗UP ) where for yi = 0 we mean the derivative from above. Notice that we simply wrote the partial derivative as a function of y−i = x−i and the difference yi = xi − x−i . g is ¡ ¢ continuous with g(0, y−i ) > 0 for every y−i ∈ [0; zˆ], so the set H := g −1 (0; ∞) of points where the partial derivative is strictly positive is open20 in K with {0} × [0, zb2 ] ⊆ H. z2 ] Therefore, as [0, z] is compact, there exists21 a neighborhood [0, δ], δ > 0, of 0 in [0, v−b 2

with [0; δ] × [0, zb2 ] ⊆ H. For an illustration of the lemma see the figure below. xi 45°

δ 0

zb2

x−i

Figure 2: The payoff of bidder i increases in the direction of the arrows (see lemma 3).

Lemma 4. Rationalizable strategies do not place a positive probability on the interval (0, zb2 ]: eUP , /Σ σiU P ([0, zb2 )) > 0 ⇒ σiU P ∈ i

i = 1, 2.

The proof of this statement is somewhat technical and is relegated therefore to Appendix A. Here we will provide only the basic idea. We divide the interval [0, zb2 ) into small intervals of length δ, where

zb2 δ

= N is an integer number. We denote the intervals

Ik = [(k − 1) · δ, k · δ) 20

for k = 1, 2, . . . N ; I0 = ∅,

Here we used the fact that pre-images of open sets under continuous mappings are open, see e. g.

Königsberger (2002), p. 16. 21 This follows from the so called “tube lemma”, see e. g. Königsberger (2002), p. 32.

15

xi

45°

v

UP

zb2 IN I2 I1

v

zb2

x−i

Figure 3: The dark colored rectangle illustrates the boundaries of the support of the rationalizable strategies in the uniform price auction. The triangles are ment to represent the serial elimination of mixed strategies placing positive probability in the intervals I1 , I2 , . . . , IN . Such strategies are proven not to be rationalizable.

as shown in figure 3. The gist of the argument consists in showing by an iterative procedure that mixed strategies placing positive probability on I1 , I2 , . . . , IN are not rationalizable. For that purpose we use the properties of the payoff functions proven in lemma 1 and lemma 3. 4.3.2

The n > 2 bidders case

Let x be an arbitrary bid vector and q, q 0 ∈ {0, . . . , n}. Define τ0 (x) := v. The seller weakly prefers to sell q instead of q 0 units if and only if ¡ ¢ ¡ ¢ τq (x) − c · q ≥ τq0 (x) − c · q 0 . Thus the seller will not supply more than q units if and only if  maxq cq (x) = max F cq (x) − F cq (x) , 0 is the probability that exactly q units are sold. Lemma 5. RiUP (x, φ∗UP ) is continuous in x. Proof. Let x be an arbitrary bid vector. We will show that for any sequence of bid vectors x(k) , k = 1, 2, . . . , with x(k) → x we have RiUP (x(k) , φ∗UP ) → RiUP (x, φ∗UP ). Using the (easy to prove) inequality |a0 b0 c0 − abc| ≤ |a0 − a| · b0 c0 + a · |b0 − b| · c0 + ab · |c0 − c|, which holds for arbitrary nonnegative reals a, b, c, a0 , b0 , c0 , we obtain ¯ UP (k) ∗ ¯ ¯Ri (x , φUP ) − RiUP (x, φ∗UP )¯ n X ¯ ¯ ¯τq (x(k) ) − τq (x)¯ · P (q; x(k) ) · 1{ϕ (i)≤q} ≤ x(k) q=0

+ +

n X ¡ q=0 n X

¡

¢ v − τq (x) · |P (q; x(k) ) − P (q; x)| · 1{ϕx(k) (i)≤q} ¢ v − τq (x) · P (q; x) · |1{ϕx(k) (i)≤q} − 1{ϕx (i)≤q} |.

q=0

This inequality can be interpreted as a decomposition of the change in expected revenue of bidder i into a price effect, a quantity effect and an allocation effect. As sums, differences, products, quotients, minimums and maximums of continuous functions are continuous, so + are the functions c− q (·), cq (·), P (q; ·), and therefore

¯ ¯ ¯τq (x(k) ) − τq (x)¯ → 0,

|P (q; x(k) ) − P (q; x)| → 0

for k → ∞, which means price and quantity effect tend to 0. To complete the proof, we will now show that the allocation effect also tends to 0. This effect can be expressed as X¡ ¢ v − τq (x) · P (q; x) · |1{ϕx(k) (i)≤q} − 1{ϕx (i)≤q} |, q∈Lx

17

where Lx = {q | τq (x) > τq+1 (x)} because P (q; x) = 0 for q ∈ / Lx .22 In words, one needs to sum only over the positions in the announced demand curve for which an increase in quantity leads to a decrease in the stopout price. This is so, because if several bids are equal, the seller serves with probability one either none or all of them. Observe now that there exists a k0 , so that for all k ≥ k0 we have: (k)

(k)

xj < x i

(k)

(k)

if xj < xi and xj > xi

if xj > xi for all i, j ∈ {1, . . . , n}.

(4.10)

Then the inequalities ϕx (i) ≤ q

and ϕx(k) (i) ≤ q

are equivalent for q ∈ Lx and k ≥ k0 , which completes the proof. Theorem 4 (existence). The uniform price auction has a symmetric mixed strategy equilibrium. Proof. Let all bidders except bidder i play the mixed strategy σ and define the following correspondence from the set of mixed strategies over the convex and compact set M into itself: BRi (σ) := arg max RUi P (σi , σ −i ; φ∗U P ). σi

As the function

RiUP (x, φ∗U P )

is continuous, Glicksberg’s (1952) fixed point theorem (see

p. 171) and its application to mixed strategy Nash equilibrium points (see p. 173) guarantee that a mixed strategy fixed point σ ∗ ∈ BRi (σ ∗ ) exists23 . Once we have a fixed point, the mixed strategy profile (σ ∗ , σ ∗ , . . . , σ ∗ ) is a Nash equilibrium since the game is symmetric24 . Theorem 5. If σ ∗ is a symmetric equilibrium bid profile of the uniform price auction, then σi∗ ([b zn , v]) = 1,

i ∈ {1, 2 . . . , n}

where zbn is the unique solution of the equation v−z =

n − 1 F (z) · . n f (z)

(U Pn )

For a proof see Appendices B and C. 22 23

+ One observes that c− q = τq and cq ≤ τq . Hence P (q; x) = 0. The best response correspondence BRi is a closed point to convex set mapping in the sense of

Glicksberg. 24 The idea of the proof is adopted from Moulin (1986), pp. 115–116. He proves by similar arguments that symmetric games satisfying the conditions of Kakutani’s (1941) fixed point theorem have symmetric equilibria.

18

4.4

Revenue and efficiency comparison

Theorem 3 states, that in the two-bidder case the rationalizable strategy sets of the uniform price and the discriminatory auction are disjoint. In the uniform price auction all bidders submit with probability one higher bids than in the discriminatory auction. This claim is also true in the case of n > 2 for the symmetric mixed strategy equilibria of both auctions (see theorem 5). We use these findings to establish a ranking of both auction formats in terms of ex ante revenue for the auctioneer and efficiency. 4.4.1

Revenue

Theorem 6. The uniform price auction is ex ante more profitable for the seller than the discriminatory auction for all rationalizable strategies of the bidders in the two-bidder case and all symmetric mixed strategy equilibria for n ≥ 2: ∗ RUS P (σ U P , φ∗U P ) > RDA S (σ DA , φDA )

for all

e DA , σ U P ∈ Σ eUP σ DA ∈ Σ

and

n = 2;

∗ ∗ RUS P (σ ∗U P , φ∗U P ) > RDA S (σ DA , φDA ), n ≥ 2.

(4.11) (4.12)

Proof. We prove (4.12) using theorem 5: RUS P (σ ∗U P , φ∗U P ) ≥ RSU P (b zn , . . . , zbn , φ∗U P ) = RSDA (b zn , . . . , zbn , φ∗DA ) ∗ ∗ > RSDA (z, . . . , z, φ∗DA ) = RDA S (σ DA , φDA ).

The proof of (4.11) is analogous; apply theorem 3. 4.4.2

Efficiency

As an efficiency measure of a mechanism Γ we use the average ex ante probability with which a buyer is served25 : PΓ (σ, φ∗Γ )

1 = n

where

QΓ (x, φ∗Γ )dσ(x),

Z QΓ (x, φ∗Γ )

25

Z

v

= 0

φ∗Γ (x, c) · f (c)dc.

Note that buyers are only served when v ≥ c since they submit bids no higher than v and the seller

does not serve bids below c. An alternative measure of efficiency one could take is the average trading quantity

Z PΓ (σ, φ∗Γ )

=

QΓ (x, φ∗Γ )dσ(x).

19

Theorem 7. The uniform price auction is more efficient than the discriminatory auction for all rationalizable strategies of the bidders in the two-bidder case and all symmetric mixed strategy equilibria for n ≥ 2, if the marginal costs probability distribution of the seller is convex26 (F 00 > 0). Formally PU P (σ U P , φ∗U P ) > PDA (σ DA , φ∗DA ) for all

e DA , σ U P ∈ Σ eUP σ DA ∈ Σ

and

n = 2;

(4.13)

PU P (σ ∗U P , φ∗U P ) > PDA (σ ∗DA , φ∗DA ), n ≥ 2.

(4.14)

Proof. Take an arbitrary bid vector x. The average quantity sold in the uniform price auction can be written as a function of the ordered bids: QU P (x; φ∗U P )

=

QU P (τ (x); φ∗U P )

=

n X

q · P (q; x) =

q=1

where

X

q · P (q; x)

− {q|c+ q >cq }

© ¡ ¢ ¡ − ¢ ª P (q; x) := max F c+ q (x) − F cq (x) , 0 .

The last equality means that one needs to sum only over the elements τq (x) for which − c+ q > cq as otherwise P (q; x) = 0. We write these quantities in an ascending order:

l1 , l2 , . . . , lh and obtain τl1 > τl2 > · · · > τlh . We will show that QU P (τl1 , τl2 , ..., τlh ; φ∗U P ) ≥ QU P (τl2 , τl2 , ..., τlh ; φ∗U P ).

(4.15)

+ − One can observe that c− l1 = cl2 and recall that cl1 is a solution of the equation − l1 · (τl1 − c− l1 ) = l2 · (τl2 − cl1 ),

(4.16)

which means that the two dark rectangles in figure 4 cover equal areas. Then τl1 − c− l2 l1 . − = l1 τl2 − cl1 From the convexity of F follows 26

The efficiency ranking is valid for all probability distributions F , which are log-concave and convex.

The class of these distributions is not narrow. For    0   F (c) = cs    1

example the probability distributions for c ∈ (−∞, 0), for c ∈ [0, 1], for c ∈ (1, ∞),

where s ≥ 1 belong to that class.

20

price

τ l1 τ l2 c− l1

l1

quantity

l2

Figure 4: Announced demand curve in the uniform price auction. The two dark rectangles cover equal areas.

F (τl1 ) − F (c− τl1 − c− l1 ) l1 − ≥ F (τl2 ) − F (cl1 ) τl2 − c− l1 and therefore F (τl1 ) − F (c− l2 l1 ) . − ≥ l1 F (τl2 ) − F (cl1 ) Then QU P (τl1 , τl2 , ...τlh ; φ∗U P ) − QU P (τl2 , τl2 , ...τlh ; φ∗U P ) = £ ¤ £ ¤ = l1 F (τl2 ) − F (τl2 ) − (l2 − l1 ) F (τl2 ) − F (c− l1 ) £ ¤ £ ¤ − = l1 F (τl1 ) − F (c− ) − l F (τ ) − F (c ) ≥ 0. 2 l 2 l1 l1 After applying the above argument (h − 1) times one obtains QU P (τl1 , τl2 , ...τlh ; φ∗U P ) ≥ QU P (τlh , τlh , ...τlh ; φ∗U P ). To prove (4.14) we apply theorem 5 to obtain Z 1 ∗ ∗ PU P (σ U P , φU P ) = QU P (x; φ∗U P )dσ ∗U P (x) n Z 1 ≥ QU P (τh , τh , ...τh ; φ∗U P )dσ ∗U P (x) n Z 1 ≥ QU P (b zn , zbn , ...b zn ; φ∗U P )dσ ∗U P (x) n Z 1 > QU P (z, z, ...z; φ∗U P )dσ ∗U P (x) n Z 1 = QDA (z, z, ...z; φ∗DA )dσ ∗DA (x) = PDA (σ ∗DA , φ∗DA ). n 21

(4.17)

The proof of (4.13) is analogous (apply theorem 3).

4.5

A numerical example

Consider the following two bidder example: v = 1 and the marginal costs of the auctioneer are uniformly distributed: f (c) = 1 for c ∈ [0, 1].

xi

45°

1

UP

zb2 z

DA

0 0

z

zb2

1

x−i

Figure 5: Numerical example: n = 2 and f (c) = 1 for c ∈ [0, 1]. The supports of the rationalizable strategy sets in both auction forms (the dark areas) are disjoint. The bids in the uniform price auction are higher with probability one.

In the discriminatory auction all bidders submit a bid of z =

1 2

with probability one,

which is their only rationalizable strategy, as the bid z solves the equation (DA). All rationalizable strategies in the uniform price auction have support in the interval [zb2 , v] = [2/3, 1] as zb2 solves the equation (U Pn ). The efficiency values of the discriminatory auction and the uniform price auctions are: PDA PU P

Z 1 1 2 1 = 2dc = , 2 0 2 2 > . 3

22

The revenue for the seller in the two auction formats is Z 1 2 1 1 DA RS = 2( − c)dc = ; 2 4 0 Z 2 3 2 4 2( − c)dc = . RUS P > 3 9 0 The payoff of bidder i in the uniform price auction is:   (1 − xi )xi     (1 − xi )¡2xi − 2x−i ¢ + (1 − x−i )(2x−i − xi ) UP ∗ Ri (xi , x−i , φU P ) =   (1 − xi )(2xi − x−i )      0

for xi > 2x−i , for 2x−i ≥ xi ≥ x−i , for

x−i 2

≤ xi < x−i ,

for xi
, 13 3 Z 8 Z 10 13 13 10 9 82 4 2( − c)dc + ( − c)dc = = > . 8 13 13 169 9 0 13

PU P = RUS P

5

Conclusion

The standard pricing techniques, the uniform pricing rule and price discrimination rule, are widely used by monopolists for the simultaneous sale of multiple units. When a monopolist lacks information about demand, these pricing techniques often take the form of an auction, in which the seller first collects bids from prospective customers and then decides on a supply quantity so as to maximize profit. These auction forms, called variable supply multi-unit auctions, are used on various markets ranging from Treasury bills and IPOs to rare wine and art. They differ from the fixed supply multi-unit auctions in the sense that the seller participates in the price-setting process as she controls the supply after the bidding. We model this scenario as a two-stage game and compare these variable supply pricing mechanisms in a common value model without proprietary information. In our model bidders announce bids for one unit and are uncertain about the constant marginal costs of the seller. We find that the uniform price auction generates higher bids than the discriminatory auction. This finding further implies that the uniform price auction is (ex ante) more profitable for the seller and more efficient as it leads to higher average trade volume. The obtained results can be given the following intuition. As has been discussed, in the discriminatory auction the winning probability of each bidder is not affected by the bids of his fellow bidders, as the seller optimally serves every bid above 24

her marginal costs. Since the bidders share the same information, they submit equal bids. Thus the right of the seller to discriminate among the bidders and charge them different prices has no bite. In the uniform price auction on the other hand the probability of winning and the final price depend on all bids. Submitting higher bids in this auction format proves to be profitable as it raises the probability of winning, but not necessarily the price a bidder has to pay. This simple observation is employed to demonstrate that the uniform price auction induces a more competitive environment and leads to higher equilibrium bids with probability one (see theorems 3 and 5). We obtained this result without the need to compute the equilibria precisely. Rather, we exploited the properties of the bidders’ payoff functions and the definitions of the equilibrium and rationalizability concepts.

25

A

Appendix: Proof of lemma 4

Proof. Denote the intervals Jk =

k [

Il = [0, kδ).

l=0

We will iteratively show that e U P,k ⊆ {σi | σi (Jk ) = 0}, Σ i

for k = 1, 2, .., N ; i = 1, 2,

(A.1)

which proves the lemma. Observe that (A.1) trivially holds for k = 0. Assume that it holds for k − 1 < N for player −i. We will show that e U P,k ⊆ {σi | σi (Jk ) = 0}. Σ i

(A.2)

Assume on the contrary e U P,k ∃σi ∈ Σ i

with σi (Ik ) > 0.

(A.3)

We will now demonstrate that U P,(k−1)

∀σ−i ∈ conv Σ−i

, ∃σî : RUi P (σî , σ−i , φ∗U P ) ≥ RUi P (σi , σ−i , φ∗U P ),

(A.4)

which is a contradiction to the above assumption (A.3), namely that σi is a best response U P,(k−1)

to some mixed strategy from the set conv Σ−i

.

Case 1: σ−i (J2k ) > 0. Consider the strategy σî : σî (B) = σi (B ∩ CIk ) + σi (Ik ) · 1kδ∈B

for B ∈ B,

where CIk is the complement set of Ik (CIk = M \Ik ) and B ∈ B: RUi P (σî , σ−i , ; φ∗U P ) − RUi P (σi , σ−i ; φ∗U P ) Z ³Z Z ´ UP ∗ ≥ Ri (xi , x−i ; φU P )dσî (xi ) − RiUP (xi , x−i ; φ∗U P )dσi (xi ) dσ−i (x−i ) Z ³ Z ´ UP ∗ RiUP (xi , x−i ; φ∗U P )dσi (xi ) dσ−i (x−i ) = Ri (kδ, x−i ; φU P ) · σi (Ik ) − Ik Z Z ³ ´ RiUP (kδ, x−i ; φ∗U P ) − RiUP (xi , x−i ; φ∗ ) dσi (xi )dσ−i (x−i ) = Z Ik Z ³ ´ RiUP (kδ, x−i ; φ∗U P ) − RiUP (xi , x−i ; φ∗U P ) dσi (xi )dσ−i (x−i ) > 0. = CJ(k−1)

(A.5)

(A.6) (A.7)

Ik

(A.7) follows from (A.6) because we assumed that (A.1) holds for (k − 1) < N for player −i. The last inequality holds because: 26

RiUP (xi , x−i ; φ∗U P ) < RiUP (kδ, x−i ; φ∗U P ) for ∀xi ∈ Ik and (k − 1) · δ ≤ x−i < 2kδ; RiUP (xi , x−i ; φ∗U P ) ≤ RiUP (kδ, x−i ; φ∗U P ) for ∀xi ∈ Ik and (k − 1) · δ ≤ x−i (see Lemma 3, Lemma 1), and σ−i (J2k ) > 0 by assumption. Case 2: σ−i (J2k ) = 0. Consider the strategy σî : σî (B) = σi (B ∩ CIk ) + σi (Ik ) · 1 3v ∈B 4

for B ∈ B,

Then RUi P (σî , σ−i ; φ∗U P ) − RUi P (σi , σ−i ; φ∗U P ) = Z Z ³ ´ 3v = RiUP ( , x−i ; φ∗U P ) − RiUP (xi , x−i ; φ∗ ) dσi (xi )dσ−i (x−i ) 4 ZCJ2k ZIk ³ ´ 3v = RiUP ( , x−i ; φ∗U P ) dσi (xi )dσ−i (x−i ) > 0. 4 CJ2k Ik

(A.8)

The inequality (A.8) holds because RiUP ( 3v , x−i ; φ∗U P ) > 0 for x−i ∈ [0, v]. 4

B

Appendix

In this appendix we provide some lemmata that we need for the proof of theorem 5. Lemma 6. The partial derivative of the bidder that submitted the lowest bid is not arbitrary close to 0 if that bidder submitted a bid lower than some z < zbn and is served with positive probability. Formally, for any i the expected profit function has the following property: For any z < zbn there exists ∂ > 0 so that for any x with c+ n (x) > 0 and 0 ≤ xi < min{{z} ∪ {xj | j 6= i}}, the following inequality is satisfied: ∂i RiUP (x; φ∗U P ) ≥ ∂. Proof. Observe that

qτq (x) − nxi 0f· · (v − z − · := ∂(ˆ q) (B.2) n − qˆ n − 1 f (z) h n n F (zbn ) i >f· · (v − zbn ) − · = 0. (B.3) n − qˆ n − 1 f (zbn )

∂i RiUP (x; φ∗U P ) = (v − xi ) · f (

Here we denoted f :=

˜ > 0. inf f (k)

˜ k∈[0,c zn )

The inequalities (B.1),(B.2) and (B.3) are valid because c+ n < xi < z < zbn and

F f

is

monotonically increasing function (see (A3)). We define now min ∂(ˆ q ) := ∂ > 0, qˆ

which completes the proof. Lemma 7. Assume that all bidders except bidder i submit a bid of z˜ ∈ [0, v): xj = z˜, j 6= i, then there exist small ε > 0 and ∂˜ > 0 such that27 for xi ∈ [˜ z , z˜ + ε) ˜ ∂i RiUP (x, φ∗U P ) ≥ ∂. Proof. RiUP (xi , z˜, . . . , z˜; φ∗U P ) = (v − z˜) · F (

³ n˜ z − xi n˜ z − xi ´ ) + (v − xi ) F (xi ) − F ( ) . n−1 n−1

(B.4)

∂i RiUP (xi , z˜, . . . , z˜; φ∗U P ) ³ n˜ z − xi −1 n˜ z − xi 1 ´ )· + (v − xi ) f (xi ) + f ( ) n−1 n−1 n−1 n−1 n˜ z − xi − F (xi ) + F ( ) n−1 = (v − z˜) · f (

27

For xi = z˜ we mean the derivative from above.

28

(B.5) (B.6)

is continuous in xi . As ∂i RiUP (˜ z , z˜, . . . , z˜; φ∗U P ) = (v − z˜) · f (˜ z ) > 0 there exists ε > 0 such that for xi ∈ [˜ z , z˜ + ε) we have ∂i RiUP (xi , z˜, . . . , z˜; φ∗U P ) > 0. Let then ∂˜ :=

inf

∂i RiUP (xi , z˜, . . . , z˜; φ∗U P ) > 0.

xi ∈[˜ z ,˜ z +ε)

Lemma 8. For any x such that z˜ ≤ xj ≤ Proof. c+ n (x)

n n−1

· z˜ for every j, c+ n (x) > 0.

n n˜ z − (n − 1) n−1 · z˜ > = 0. n−n+1

Lemma 9. Let x be such that: xi ∈ [˜ z , z˜ + ε); let there exist a bidder k 6= i who submitted a bid of z˜ and another bidder j 6= i, who submitted a bid of xj > z˜ + (n − 1)2 ε. Then ∂i+ RiUP (x; φ∗U P ) = 0. Proof. Observe that ϕx (i) ≤ (n − 1)28 . We have c+ ϕx (i) (x) ≤

(˜ z + ε) · (n − 1) − xj ; n−2

c− ˜ − (n − 1) · (˜ z + ε). ϕx (i) (x) ≥ n · z It is now easy to verify that − c+ ˜ + (n − 1)2 ε, ϕx (i) (x) < cϕx (i) (x) for xj > z

which completes the proof.

C

Appendix: Proof of theorem 5

Proof. The proof that in equilibrium bidders do not submit with positive probability bids higher than v is straightforward. Consider first the case in which a bidder wins with a bid higher than v and has to pay a stop-out price lower than v. Then if he submitted a bid of v instead he would also have won, as the seller would supply the same quantity and the stop-out price would have remained the same. On the other hand if a bidder wins with a bid higher than v and has to pay a stop-out price higher than v (which would arise with 28

If bidder i submits also a bid of z˜, then we choose ϕx such that bidder i obtains a number lower than

that of bidder k.

29

positive probability in a symmetric equilibrium), then he has a negative payoff. In this case a bid of v would guarantee a payoff of zero. The proof that the bidders do not submit bids lower than zbn with positive probability is somewhat more involved. Assume on the contrary that the lower bound of the support of a symmetric mixed strategy equilibrium is z˜ < zbn . Formally z˜ = max{z|σi∗ ([z, v]) = 1} and z˜ < zbn . Take an arbitrary bidder i and consider a deviation strategy σî , which only shifts the probability mass of the small interval [˜ z , z˜ + ε) to the point z˜ + ε: σî (B) = σi (B ∩ C[˜ z , z˜ + ε)) + σi ([˜ z , z˜ + ε)) · 1{˜z+ε∈B}

for B ∈ B.

We will show that this deviation strategy is more profitable for player i, a contradiction to the equilibrium assumption. One observes first that ∂i+ RiUP (x; φ∗U P ) ≥ −1.

(C.1)

That is the case because a small increase in the own bid leads to a increase in the stop-out price (with some probability), but does not lower the winning chances. Denote h ń−1 n Ω1 = z˜, max{v, · z˜} , n−1 ´ h n−1 n · z˜} , Ω2 = z˜ + ε, max{v, n−1 µ ¶ £ ¢ n−1 n 2 Ω3 = {˜ z } ∪ z˜ + (n − 1) ε, max{v, · z˜} , n−1 h in−1 n 2 Ω4 = z˜ + (n − 1) ε, · z˜ , n−1 Ω5 = {˜ z }n−1 . Case 1. Assume that σi∗ ({˜ z }) = 0. Then RUi P (σî , σ ∗−i ; φ∗U P ) − RUi P (σi∗ , σ ∗−i ; φ∗U P ) Z ³ Z z˜+ε Z z˜+ε ´ UP ∗ UP ∗ ∗ = Ri (xi , x−i ; φU P )dσî (xi ) − Ri (xi , x−i ; φU P )dσi (xi ) dσ ∗−i (x−i ) z˜ z˜ Z ³ Z z˜+ε ³ ´ ´ z + ε, x−i ; φ∗U P ) − RiUP (xi , x−i ; φ∗U P ) dσi∗ (xi ) dσ ∗−i (x−i ) = RiUP (˜ z˜ Z ³ Z z˜+ε ³ ´ ´ = RiUP (˜ z + ε, x−i ; φ∗U P ) − RiUP (xi , x−i ; φ∗U P ) dσi∗ (xi ) dσ ∗−i (x−i ) (C.2) Ω1 /Ω2

Z

³Z

z˜ z˜+ε

+ Ω2

z˜

³

´ ´ RiUP (˜ z + ε, x−i ; φ∗U P ) − RiUP (xi , x−i ; φ∗U P ) dσi∗ (xi ) dσ ∗−i (x−i ). (C.3)

30

For enough small ε > 0 we obtain RUi P (σî , σ ∗−i ; φ∗U P ) − RUi P (σi∗ , σ ∗−i ; φ∗U P ) Z Z z˜+ε ≥ (−1) · (˜ z + ε − xi )dσi∗ (xi )dσ ∗−i (x−i ) Ω1 /Ω2

Z

z˜ z˜+ε

Z

+ Ω2

z˜

= (−1) ·

∂ · (˜ z + ε − xi )dσi∗ (xi )dσ ∗−i (x−i ) Z

σ ∗−i (Ω1 /Ω2 ) Z

z˜+ε z˜

(see C.1, C.2) (see C.3, lemma 6, lemma 8)

(˜ z + ε − xi )dσi∗ (xi )

z˜+ε

+ ∂ · σ ∗−i (Ω2 ) (˜ z + ε − xi )dσi∗ (xi ) z˜ Z z˜+ε ´ ³ = (˜ z + ε − xi )dσi∗ (xi ) · (−1) · σ ∗−i (Ω1 /Ω2 ) + ∂ · σ ∗−i (Ω2 ) > 0

(C.4)

z˜

The last inequality is valid because lim σ ∗−i (Ω1 /Ω2 ) = 0 and

ε→0

lim σ ∗−i (Ω2 ) > 0.

(C.5)

ε→0

Case 2. Assume now that σi∗ ({˜ z }) > 0. Then RUi P (σî , σ ∗−i ; φ∗U P ) − RUi P (σi∗ , σ ∗−i ; φ∗U P ) Z ³ Z z˜+ε ³ ´ ´ = RiUP (˜ z + ε, x−i ; φ∗U P ) − RiUP (xi , x−i ; φ∗U P ) dσi∗ (xi ) dσ ∗−i (x−i ) Ω1 /Ω3

z˜

Z

³Z

z˜+ε

+ Ω3 /(Ω4 ∪Ω5 )

Z

z˜

z˜+ε

+ Z

³Z

z˜ z˜+ε

+ Ω5

³ ´ ´ RiUP (˜ z + ε, x−i ; φ∗U P ) − RiUP (xi , x−i ; φ∗U P ) dσi∗ (xi ) dσ ∗−i (x−i ) (C.7)

³Z Ω4

(C.6)

z˜

³ ´ ´ UP ∗ UP ∗ ∗ Ri (˜ z + ε, x−i ; φU P ) − Ri (xi , x−i ; φU P ) dσi (xi ) dσ ∗−i (x−i )

(C.8)

³ ´ ´ RiUP (˜ z + ε, x−i ; φ∗U P ) − RiUP (xi , x−i ; φ∗U P ) dσi∗ (xi ) dσ ∗−i (x−i ).

(C.9)

For enough small ε > 0 we obtain RUi P (σî , σ ∗−i ; φ∗U P ) − RUi P (σi∗ , σ ∗−i ; φ∗U P ) Z Z z˜+ε ≥ (−1) · (˜ z + ε − xi )dσi∗ (xi )dσ ∗−i (x−i ) Ω1 /Ω3

Z

z˜

Z

z˜+ε

+ Ω3 /(Ω4 ∪Ω5 ) Z Z z˜+ε

Z

Z

z˜ z˜+ε

+ Ω5

z˜

0 · (˜ z + ε − xi )dσi∗ (xi )dσ ∗−i (x−i )

∂ · (˜ z + ε − xi )dσi∗ (xi )dσ ∗−i (x−i )

+ Ω4

z˜

(see C.6, C.1)

∂˜ · εdσi∗ (xi )dσ ∗−i (x−i )

(see C.7, lemma 9)

(see C.8, lemma 6, lemma 8) (see C.9, lemma 7)

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Z

z˜+ε

= z˜

³ ´ ˜ ∗ (Ω5 ) > 0. (C.10) (˜ z + ε − xi )dσi∗ (xi ) (−1) · σ ∗−i (Ω1 /Ω3 ) + ∂ · σ ∗−i (Ω4 ) + ∂εσ −i

The last inequality is valid because: limε→0 σ ∗−i (Ω1 /Ω3 ) = 0; if there exists ε > 0 such that σ ∗−i (Ω4 ) > 0 then limε→0 σ ∗−i (Ω4 ) > 0; if there is no ε > 0 such that σ ∗−i (Ω4 ) > 0 then σ ∗−i (Ω1 /Ω3 ) = 0; σ ∗−i (Ω5 ) = (σi∗ ({˜ z }))n−1 > 0 by assumption.

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