Page 1

An Analysis of Troubled Assets Reverse Auction

Saeed Alaei?, Azarakhsh Malekian??

University of Maryland

Abstract. In this paper we study the Nash-equilibrium and equilibrium

bidding strategies of the Pooled Reverse Auction for troubled assets.

The auction was described in (Ausubel & Cramton 2008[1]). We fur-

ther extend our analysis to a more general class of games which we call

Summation Games. We prove the existence and uniqueness of a Nash-

equilibrium in these games when the utility functions satisfy a certain

condition. We also give an efficient way to compute the Nash-equilibrium

of these games. We show that then Nash-equilibrium of these games can

be computed using an ascending auction. The aforementioned reverse

auction can be expressed as a special instance of such a game. We also,

show that even a more general version of the well-known oligopoly game

of Cournot can be expressed in our model and all of the previously men-

tioned results apply to that as well.

1 Introduction

In this paper, we primarily study the equilibrium strategies of the pooled reverse

auction for troubled assets which was described in [1]. The US Treasury is pur-

chasing the troubled assets to infuse liquidity into the market to recover from

the current financial crisis. Reverse auctions in general have been a powerful tool

for injecting liquidity into the market in places where it will be most useful. As

explained in [1] a simple and naive approach for the government could be to run

a single reverse auction for all the holders of toxic assets as follows. The auction-

eer(government) then sets a total budget to be spent. The auctioneer starts at a

price like 100¢ on a dollar. All the holders, bid the quantity of their shares that

they are willing to sell at the current prices. There can be excess supply. The

auctioneer then lowers the price in steps e.g. 95¢, 90¢, etc. and bidders indicate

the quantities that they are willing to sell at each price. At some point (for exam-

ple at 30¢ on a dollar) the total supply offered by all the holders for sale equals

or falls bellow the specified budget of the treasury. At that point the auction

concludes and the auctioneer buys the securities offered at the clearing price. As

explained in [1], this simple approach is flawed as it leads to a severe adverse

selection problem. Note that at the clearing price the securities that are offered

are only the ones that are actually worth less than 30¢ on each dollar of face

?Dept. of Computer Science, University of Maryland, College Park, MD 20742.

saeed@cs.umd.edu.

??Dept. of Computer Science, University of Maryland, College Park, MD 20742.

malekian@cs.umd.edu.

Page 2

value. They could as well worth far bellow 30¢. In other words, the government

would pay most of its budget to buy the worst of the securities.

In [1], the authors propose the following two type of auctions.

– A Security by Security Reverse Auction

– A Pooled Reverse Auction

They are both part of a two phase plan. The first one can be used to extract

private information of holders about the true value of the securities to give an

estimate on how much each security and similar securities are actual worth of.

Later, that information can be used to establish reference prices in the Pooled

Reverse Auction.

In this paper we focus our attention on the second class of auction. In a Pooled

Reverse Auction, different securities are pooled together. The government puts

a reference price on each security and then runs a reverse auction on all of them

together. We explain this auction in more detail in section 2.

In section 3, we study the Nash-equilibrium and the bidding strategies of the

Pooled Reverse Auctions in detail. We then create a more abstract model of it

at the end of section 2. In section 4 we describe a general class of games that

can be used to model the Pooled Reverse Auction as well as other problems. In

section 4, we give some exciting result on these games. We give a condition which

is sufficient for the existence of a Nash-equilibrium. We further explain how the

Nash-equilibrium can be computed efficiently using a an ascending auction-like

mechanism. Later in section 5, we show how we can apply our result of section 4

to Pooled Reverse Auctions. section 6 explains how a more general version of

the Cournot’s oligopoly game can be expressed in our model.

1.1 Related Work

We partition the related works to two main groups. The first group that is closely

related to our model, are computing equilibrium in Cournot and public good

provision games. The second one with similar model but different objective are

the works related to bandwidth sharing problems and the efficiency of computed

equilibria.

One well known problem that can be considered as an example of our model

is the Cournot’s oligopoly game. It can be described as an oligopoly of firms

producing a homogeneous good. The strategy of firm i is to choose qiwhich is

the quantity it produces. Assuming that the production cost is ciper item, the

utility of firm i is (p(Q) − ci)qi for which Q =?

is a vast amount of literature on Cournot games (e.g. [7]). Different aspect of

Cournot equilibrium has been studied (For example, in [3] Bergstrom and Varian,

studied the effect of taxation on Cournot equilibrium and also showed some

characteristics of the Cournot equilibrium.)

Another set of results, with similar model, but with different criteria are the

works related to bandwidth sharing problem. At a high level, the problem is to

iqi is the total production

and p(Q) is the global price of the good based on the total production. There

Page 3

allocate a fixed amount of an infinitely divisible good among rational competing

users. [8] studies this problem from pricing perspective. Kelly [6], considered

a generalized variant of this problem in the context of routing and charging

(However the equilibrium point of his mechanism was not fully efficient) His

model, for a single resource with fixed supply, is to give each person proportional

to his bid from the resource and charge him his bid. Later, Johari et al in [4],

showed that Kelly’s mechanism is at least 75% efficient at the equilibrium point.

In another work, Johari et al show that, Kelly’s model minimizes efficiency loss

(at the equilibrium point) when price discrimination is not allowed and then they

present a class of mechanisms that has an efficient outcome at the equilibrium

point assuming that price discrimination is allowed ([5]).

2 Model for Pooled Reverse Auction

In this section, we explain the basic model for the reverse auction of pooled

securities. We will use this model throughout the rest of this paper. We start by

explaining our notations:

– There are n bidders N = {1,··· ,n}, and m securities.

– Government has evaluated a reference price of rjfor each security j. Also let

r = (r1,··· ,rm) denote the vector of reference prices for all the m securities.

The reference prices are public information. These prices are in the form of

the ratio of the evaluated price to the face price and are expressed in cents

per dollar. For example rj= 0.75 means every dollar of the face value of the

security is actually worth 75¢.

denote the vector of the quantities of shares that bidder i holds from each

security. The shares are expressed in quantity of the face value.

– Each bidder has a private valuation function vi(l) for receiving a liquidity

amount of l. In a quasi-linear setting, we would assume that vi(l) = l. In

our model, we assume the vicould be an arbitrary function. vican capture

the bonus for acquiring a needed amount of liquidity or can be negative to

account for the cost incurred by the shortage thereof. For example consider

the following:

?

l

We could interpret the above vi as the following. Bidder i has a liquidity

need of Li dollars. She incurs a cost of Li− l dollars if she raises only l

dollars where l < Li. Her value for any liquidity that she receives beyond Li

is just the same as the amount that she receives. The experimental study of

reverse auction for troubled assets in [2] considers two cases for vi. In the

first case, each bidder i has a liquidity need Liand vi(l) = 2l for l ≤ Liand

vi(l) = l+Lifor l > Li. In the second case, bidders don’t have liquidity needs,

so vi(l) = l. In this paper, we consider arbitrary viunder some constraints

as we will see later.

– Each bidder i holds ¯ qi,j shares of security j. Also let ¯ qi = (¯ qi,1,··· , ¯ qi,m)

vi(l) =

l + (l − Li)

l ≤ Li

l ≥ Li

(2.1)

Page 4

– Each bidder i has a private value of wi,j for each dollar of security j. Also

let wi = (wi,1,··· ,wi,m) denote the vector of the valuations of bidder i

for different securities. In reality, we should have assumed a single common

value for each security which is unknown and can only be computed by

aggregating all the private information of all bidders. However, that model

is prohibitively hard to analyze in case of non trivial valuation functions for

liquidity (i.e. when vi(l) is not the identity function). Therefore, we assume

that wiis the private values of bidder i for the securities.

Next, we briefly explain the reverse auction mechanism for pooled securities

as described in [1].

Auction 1 (Pooled Reverse Auction) Initially, the auctioneer (government)

establishes the reference prices for all the securities. These reference prices are

supposed to be the best estimate of the government about the true value of the

securities. The reference prices are announced publicly.

The auction uses a single descending clock α which specifies the current prices

as a percentage of the reference prices. For example, α = 110% means the current

price of each security is 110% of its reference price. As the clock goes down,

participants update their bids. Bidder i submits a bid bi= (bi,1,··· ,bi,m), where

bi,jis the quantity of shares from security j that bidder i would like to sell at the

current prices. These quantities are specified in terms of dollars of face value.

The auctioneer collects all the bids and computes the activity points for each

bidder i as ai= r · bi(remember r is the vector of reference prices) . In other

words, the activity points of each bidder is her bid quantity for each security

times the reference price of that security summed over all the securities. The

auctioneer also computes the total activity point A =?

as Aα > M. In practice, the clock goes down in discrete steps. At each step

the auctioneer collects all the bids and computes the aggregate activity point.

At the first step that Aα becomes less than or equal to M, the clock stops and

the auction concludes. The auctioneer then buys from each bidder the quantity

of shares specified in her bid. Bidders are paid at the current prices (i.e. the

reference price scaled by the current value of the clock). Assuming that α∗was

the final value of the clock and for each bidder i, b∗

i, the amount of liquidity that bidder i receives is α∗ri· b∗

In the next section we study the equilibrium of the above auction.

iai. Assuming that M

is the total budget of the government, the clock keeps going down for as long

iwas the final bid of bidder

i.

3 The Equilibrium of Pooled Reverse Auction

In this section, we study the Nash-equilibrium of Auction. 1 and propose a

method that can be used to efficiently compute that. We also develop a bidding

strategy that leads to the Nash-equilibrium.

First, we show how to compute the utility of each bidder i. Assume that biis

the bid of bidder i and α is the current value of the clock. Also, as we defined in

Page 5

section 2, vi(l) is the valuation of bidder i for receiving amount l of liquidity and

wi = (wi,1,··· ,wi) is the vector of her valuations for different securities. We

denote by ui, the tentative utility of bidder i which is her utility if the auction

stops at the current value of the clock. uican be computed as the following:

ui= vi(αr · bi) − wi· bi

(3.1)

Before we start with the bidding strategies, we restate some of the definitions

from Auction. 1.

– For a bidder i with current bid bi, we use ai to define her activity point

which is defined as:

ai= r · bi

(3.2)

– The total activity point of all bidders is defined as:

A =

n

?

i=1

ai

(3.3)

– The auction clock, α, keeps going down for as long as αA > M where M is

the total budget of the auctioneer. If we denote the value of the clock when

the auction stops by α∗, then α∗A ≤ M. Note that, to simplify the analysis,

we assume quantities do not need to be integers. We also assume that the

clock changes continuously and bidders update their bids continuously as

well. Respectively, we may assume that when the auction concludes, the

auctioneers budget constraint is met with equality so:

α∗A = M

(3.4)

Next, we show that the best strategy for each player i can be described by

just specifying the activity points that she needs to generate. In other words,

the only thing that bidder i has to decide is how much activity point to generate

and her best bid vector can be specified as a function of that.

Lemma 1. In order to play her best strategy, bidder i only needs to choose her

activity points ai and then among all the bid vectors bi ∈ [0,¯ qi]1such that

r·bi= aiher best strategy is to submit a bid bithat minimizes wi·bi. We will

refer to one such bid vector as bi(ai). Formally:

bi(a) = argminbwi· b : b ∈ [0,¯ qi] ∧ r · b = a

1We use the notation [a,b] to denote all the vectors that are componentwise greater

than or equal to a and less than or equal to b

(3.5)