Mixed strategies in discriminatory divisible‐good auctions

• Date: 01 March 2013
• Author: Andrew B. Philpott,Pär Holmberg,Edward J. Anderson
• DOI: http://doi.org/10.1111/1756-2171.12008
• Published date: 01 March 2013

RAND Journal of Economics

Vol.44, No. 1, Spring 2013

pp. 1–32

Mixed strategies in discriminatory

divisible-good auctions

Edward J. Anderson∗

P¨

ar Holmberg∗∗

and

Andrew B. Philpott∗∗∗

We introduce the concept of an offer distribution function to analyze randomized offer curves

in multiunit procurement auctions. We characterize mixed-strategy Nash equilibria for pay-as-

bid auctions where demand is uncertain and costs are common knowledge, a setting for which

pure-strategy supply function equilibria typically do not exist. We generalize previous results

on mixtures over horizontal offers as in Bertrand-Edgeworth games and also characterize novel

mixtures over partly increasing supply functions. We show that the randomization can cause

considerable production inefficiencies.

1. Introduction

We analyze multiunit auctions where bidders are free to choose a separate price for each

object. We assume that the number of traded objects is large and that each bid consists of a

complete curve of price-quantity pairs. Such auctions are called divisible-good auctions (Back

and Zender, 1993; Wangand Zender, 2002), auctions of shares (Wilson, 1979), or supply function

auctions (Green and Newbery, 1992; Klemperer and Meyer, 1989). Important markets with

this character are treasury auctions, electricity auctions, and auctions of emission permits. We

focus on procurement auctions (reverse auctions), such as electricity markets, where producers

compete to sell their goods. But results are analogous for sales auctions. In particular, we are

interested in circumstances when pure-strategy Nash equilibria (NE) do not exist, and in these

settings we calculate equilibria with randomized offer curves and giveestimates of the production

inefficiencies that they cause.

∗University of Sydney; edward.anderson@sydney.edu.au.

∗∗Research Institute of Industrial Economics; par.holmberg@ifn.se.

∗∗∗University of Auckland; a.philpott@auckland.ac.nz.

Weare grateful to Ralph Bailey, TalatGenc, William Hogan, Ali Hortacsu, Erik Lundin, anonymous referees, and seminar

participants at the Research Institute of Industrial Economics (May,2010) for comments. Holmberg has been financially

supported by The Jan Wallanderand Tom Hedelius Foundation and the Research Program “The Economics of Electricity

Markets.” Anderson and Philpott acknowledge the financial support of the New Zealand Marsden Fund under contract

no. UOA719WIP.

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Divisible-good auctions have two typical mechanisms. In a uniform-price procurement

auction, all sellers are paid the clearing price (the highest accepted offer) for all of their accepted

supply. The alternative, which we analyze in this article, is a pay-as-bid (or discriminatory)

procurement auction, where the auctioneer pays each accepted offer according to its offer price.

A survey by Bartolini and Cottarelli (1997) has found that 39 out of 42 countries used the

discriminatory format in their treasury auctions. On the other hand, most electricity markets use

the uniform-price format. But there are exceptions. The electricity market in Britain switched to

a pay-as-bid format in 2001, and Italy has recently decided to follow suit. A similar move has

been considered in California (Kahn et al., 2001). Some of the power system reserves are also

procured by the system operator using a discriminatory mechanism, for example in Germany

(Swider and Weber, 2007).

It has been shown that pure-strategy NE are normally nonexistent in pay-as-bid electricity

markets with capacity constraints and costs that are common knowledge (Fabra,von der Fehr, and

Harbord, 2006; Genc, 2009; Holmberg, 2009).1There are previous studies of mixed-strategy NE

in such auctions, but they are limited to mixtures where each producer offers its entire capacity

at one price.2They occur when producers are pivotal, that is, competitors do not have enough

capacity to meet maximum demand, and they are essentially Bertrand-Edgeworth NE (Allen and

Hellwig, 1986; Beckmann, 1967; Levitan and Shubik, 1972; Maskin, 1986; Vives, 1986) with

the added complexity that either the auctioneer’s demand (Anwar, 2006; Fabraet al., 2006; Genc,

2009; Son et al., 2004) or the bidders’ costs/valuations are uncertain, as in Back and Zender

(1993).

A mixed-strategy NE in a game with complete information can be interpreted as a pure-

strategy BayesianNE in a game with incomplete information (Harsanyi, 1973). Privately observed

small random variations in an individual firm’s payoff function (e.g., due to private costs) can

effectively serve as a randomizing device (Wilson, 1969). Thus, our derivation of mixed-strategy

NE is consistent, by virtue of this interpretation, with Reny’s (1999) existence results for pure-

strategy Bayesian NE.

In this article, we generalize previous equilibrium studies of discriminatory auctions by

considering general cost functions and general probability distributions for the auctioneer’s

demand. Weare also the first to characterize equilibria with mixtures over increasing offer curves

in such auctions. We start the analysisby deriving optimality conditions for producers who know

their own production costs but face an uncertain residual demand. A first-order condition for

strictly increasing offers has previously been derived by Hortacsu and McAdams (2010). We

give a fuller treatment including second-order conditions and optimality conditions when an

offer’s monotonicity constraint is binding; most divisible-good auctions only allow monotonic

offer curves. The latter condition also applies to situations where offers are constrained to be

horizontal by restrictions on the number of allowed steps in an offer, as in models by Fabra et al.

(2006) and Bertrand games.

Weuse our optimality conditions to analyze discriminatory auctions with capacity constraints

when production costs are common knowledge and demand has an additive shock. We show that

the markup times the hazard rate of the demand shock must be nonincreasing, otherwise pure-

strategy supply function equilibria cannot exist. This is a very strong condition, and it is not

satisfied for most probability distributions that one encounters in practice. Thus, we analyze

duopoly markets where pure-strategy equilibria do not existand derive symmetric mixed-strategy

equilibrium offers in such auctions. We consider three types of mixed-strategy equilibria: (i)

mixtures over increasing supply curves, where slope constraints are not binding; (ii) mixtures

over horizontal offers, where the whole output is offered at the same price, as in Bertrand-

Edgeworth NE analyzed in previous studies; and (iii) mixtures with hockey-stick offers, which

1An exception is Holmberg (2009), who finds pure-strategy NE when the hazard rate of the additivenonstrategic

demand shock is decreasing. Corresponding results for treasury auctions are found by Rostek, Weretka,and Pycia, (2010).

2Anwar (2006) shows that multiunit uniform-price auctions sometimes have a mixed-strategy equilibrium with

independent increasing offers.

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are a combination of the other two mixtures; offers are slope constrained for low outputs and

some offers are strictly increasing for high outputs.

The type of equilibria that arise in a duopoly depend on whether producers are pivotal or not.

Apivotal producer is one for which the competitor’s capacity is not sufficient to meet maximum

demand. Thus, the removal of a pivotalproducer from the market would create a supply shortage

with positive probability.We show that symmetric duopoly mixtures over strictly increasing offer

curves only occur in discriminatory auctions with inelastic demand and nonpivotal producers. In

this case there is a continuum of such equilibria.

The slope-constrained mixtures only occur when producers are pivotal, and then the price

cap determines a unique equilibrium among the mixtures that we consider.The horizontal mixture

exists when the price cap is sufficiently high relative to the curvature of the cost curve. When the

price cap becomes sufficiently low, this equilibrium is continuously transformed into a hockey-

stick mixture, so that the top of the mixture has horizontal offers and the bottom of the mixture

has a hockey-stick-shaped mixture.

The slope-constrained equilibria can be intuitively explained as follows. Ex post,afterthe

demand shock has been realized, it is always optimal to offer all accepted offers horizontally in a

pay-as-bid auction, so that the maximum price is obtained for all the quantity supplied. Hence,

unless the demand density is sufficiently decreasing or marginal costs are sufficiently steep

relative to markups (in which case pure-strategy NE can be found), producers have incentives

to offer the very first unit at the same price as some of the units with a higher marginal cost.

Hence, the lowest part of the offer curve becomes horizontal, and producers have incentives to

slightly undercut each other’s lowestoffers down to the marginal cost, as in a Bertrand game. With

constant marginal costs and nonpivotal producers, there is a pure-strategy Bertrand-NE (Wang

and Zender, 2002; Fabra et al., 2006). But similar to a Bertrand-Edgeworth game, there will

be profitable deviations from such an outcome if producers are pivotal (Genc, 2009; Holmberg,

2009), and then the equilibrium must be a mixed one. Increasing marginal costs may become

steep relative to markups for higher outputs, so that the producer will have incentives to increase

the offers of more expensive units. In this case the offer gets a hockey-stick shape.

As firms are symmetric in our model, it is always efficient to clear the market such that

each firm has the same output, as in symmetric pure-strategy NE of uniform-price auctions

(Klemperer and Meyer, 1989). But this rarelyhappens when firms randomize their supply curves.

In our examples, where the two symmetric firms have quadratic costs, the mixed strategies

lead to significant production inefficiencies: between 25% and 100% of the optimal production

cost. For pivotal firms with fixed capacities, the welfare loss becomes higher for higher price

caps. The highest relative welfare loss occurs for horizontal mixtures when firms’ production

capacity is near the maximum demand, so that one of the firms produces the whole demand for

nearly all demand outcomes. This inefficiency is a significant drawback for the discriminatory

format.

The setting that we consider is particularly useful when modelling strategic bidding in a

wholesale electricity market operating as a discriminatory divisible-good auction. Electricity

markets are special in that there are very limited storage possibilities, so supply must equal

consumption at every instant. The system operator runs a real-time auction to match demand and

supply in every period (normally hour or half-hour). Producer’s offer curves to this auction are

submitted before the delivery period starts. The demand is uncertain at this point, mainly because

of unexpected changes in wind generation,3air temperature, and unexpected transmission-line

failures. Production costs in wholesale electricity markets are primarily determined by fuel costs

and plants’ efficiency,which are wellknown and common knowledge. Thus, a standard assumption

of electricity markets is that supplier costs are common knowledge and that the exogenous

3Because (random) generation from wind farms is dispatched at zero price, it can be regarded as subtracting from

demand.

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