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)

Page 6

Proof. The only variable parameter in the auction that correlates the utility of

different bidders is the the clock value α and the only way individual bidders

affect that variable is through their activity points. If a bidder such as i keeps her

activity points fixed and changes her bid vector, the outcome of the auction will

not change. However, among all the bid vectors that generate the same amount

of activity point, the one with the lowest wi·biproduces the highest utility for

bidder i.

Based on Lemma 1 to describe a best strategy for a bidder i we only need

to specify the activity points ai that she should bid and then Lemma 1 tells

us what condition the corresponding bid vector should satisfy. The next lemma

describes how we can efficiently compute bi(ai) for any given ai.

Lemma 2. For any given ai ∈ [0,r · ¯ qi] we can compute bi(ai) by using the

following procedure.

Without loss of generality, assume securities are sorted in decreasing order

of

wi,j≥

bid vector and increase each qi,j up to ¯ qi,j starting at j = 1 until the generated

activity point reaches ai. The following is a more formal definition of bi(a):

rj

wi,jso that

rj

rj+1

wi,j+1. To find the bid vector, we start from an initial zero

bi(a) = (¯ qi,1,··· , ¯ qi,y−1,bi,y,0,··· ,0)(3.6)

such that:

rybi,y+

y−1

?

j=1

rj¯ qi,j= a

(3.7)

Proof. The proof is by contradiction. Suppose bidder i is submitting a bid vector

biwhich is not according to the mentioned schema but minimizes wi·bisubject

to bi∈ [0,¯ qi] and r · b = ai. So there should be two different securities j and

k such that

wi,j>

that she can decrease bi,kby some ? > 0 and increase bi,jby ?rk

change in her activity points is −?rk+?rk

wi·biby ?wi,k−?rk

about w · bibeing the minimum.

Intuitively, Lemma 2 is saying that a strategic bidder should never sell any

shares of a security j unless for any other security k for which

has already sold all of her shares of security k.

rj

rk

wi,kand in her bid vector bi,j< ¯ qi,jand bi,k> 0. We argue

rj. Note that the

rjrjwhich is 0. However that decreases

rjwi,jwhich is always positive and contradicts our assumption

rk

wi,k>

rj

wi,jshe

Definition 1. We can define a cost function ci(a) : [0,wi· ¯ qi] → R for each

bidder i which only depends on her activity points:

ci(a) = wi· bi(a)0 ≤ a ≤ r · ¯ qi

(3.8)

Page 7

Intuitively, for bidder i, ci(a) is the minimum cost of generating ’a’ activity

points.

At this point, we can define the bid vectors and all the equations only in

terms of ai. Bidders only need to specify their activity point ai. We denote the

final activity points of bidder i when the auction concluded by a∗

total activity point by A∗. The utility of each bidder i can now be written as

the following:

iand the final

ui= vi(α∗a∗

i) − ci(a∗

i) (3.9)

Also, the auction concludes at the highest clock α∗such that:

n

?

i=1

α∗A∗= M

(3.10)

Next, we define the Nash-equilibrium. Before that, notice we can write the

utility of each bidder i as ui(a,A) which is a function of her own bid and the

total aggregate bid. Formally:

ui(a,A) = vi(a

AM) − ci(a) (3.11)

Now we are ready to describe the Nash-equilibrium. Suppose a∗

the activity points at which the auction has concluded. We say the outcome of

the auction is stable or is a Nash-equilibrium if for every bidder i, a∗

response to a∗

are sufficient. Assume that ¯ aidenotes the maximum possible activity points that

bidder i can generate (i.e., ¯ ai= r · ¯ qi). The first order and boundary conditions

of the Nash-equilibrium are the following:

1,··· ,a∗

nare

iis a best

−i. For a Nash equilibrium, the first order and boundary conditions

∀i ∈ N :

d

da∗

iui(a∗

or

iui(a∗

or

iui(a∗

n

?

i,A∗) = 0and0 < a∗

i< ¯ ai

d

da∗

i,A∗) ≤ 0and

a∗

i= 0

d

da∗

i,A∗) ≥ 0 and

a∗

i= ¯ ai

(3.12)

A∗=

i=1

a∗

i

(3.13)

Note that, to use the first order conditions, we need ui(a,A) to be a con-

tinuous and differentiable function in its domain. We can however relax the

differentiability requirement and allow ui(a,A) to have different left and right

derivatives at a finite number of points. In that case, if assume that ρ−

iis the

Page 8

left derivative of

condition, we can replace

proofs simple, we do not use this general form but we will refer to it later when

we explain how to compute the equilibrium.

We further expand the first order and boundary conditions. Notice that

d

da∗

d

da∗

da∗

first order and boundary conditions can be rephrased as:

d

da∗

iui(a∗

i,A∗) and ρ+

d

da∗

iis its right derivative, then in the first

i,A∗) = 0 with ρ−

iui(a∗

i≤ 0 ≤ ρ+

i. To keep the

iui(a∗

iA∗= 1, we can write

i,A∗) =

∂

∂aui(a∗

i,A∗) +

d

∂

∂Aui(a∗

i,A∗) =

i,A∗)

∂aui(a∗

d

da∗

iA∗. Because we always have

i,A∗) +

iui(a∗

∂∂

∂Aui(a∗

i,A∗). So the

∀i ∈ N :

∂

∂aui(a∗

i,A∗) +

∂

∂Aui(a∗

i,A∗) = 0 and0 ≤ a∗

i≤ ¯ ai

or

∂

∂aui(a∗

i,A∗) +

∂

∂Aui(a∗

i,A∗) ≤ 0and

a∗

i= 0

or

∂

∂aui(a∗

n

?

i,A∗) +

∂

∂Aui(a∗

i,A∗) ≥ 0 and

a∗

i= ¯ ai

(3.14)

A∗=

i=1

a∗

i

(3.15)

Next, we state the main theorem of this section which gives a sufficient condi-

tion for the existence of a Nash-equilibrium and provides a method for computing

it as well as a bidding strategy.

Theorem 1. Consider the Auction. 1, in which as explained before, each bid-

der’s utility is given by ui= vi(αr·bi)−ci·biif the auction stops at the current

clock α. Assuming that the valuation functions viare continuous, differentiable2

and concave, there exists a unique Nash-equilibrium that satisfies the first order

and boundary conditions of (3.14). Furthermore, there are bid functions gi(α),

such that for every i if bidder i bids b∗

of the auction coincides with the unique Nash-equilibrium. gi(α) is given bellow

(v?

i(ai) where ai= gi(α), then the outcome

iand c?

iare the derivatives of viand ci):

gi(α) = argmina∈[0,¯ ai]

????v?

i(αa)M − αa

M

α − c?

i(a)

????

(3.16)

Furthermore, the gi(α) can be computed efficiently using binary search on ’a’

(the parameter of the argmin) because the expression inside the absolute value is

a decreasing function of a.

Note that the requirement of vi functions being concave is quite natural.

It simply means that the derivative of vi should be decreasing which can be

interpreted as the marginal value of the first dollar received being more than the

marginal value of the last dollar.

2we may relax this to allow vi to have different left and right derivatives at a finite

number of points

Page 9

It is worth mentioning that the bid function gi(α), as described in (3.16)

is not necessarily an increasing function of α. In other words, as the clock goes

down, gi(α) may increase at some points which means bidder i is actually offering

more for sale although the prices are going down. This phenomenon is in fact

quite common when bidder i has liquidity needs as we will explain in section 5 .

We defer the proof of Theorem 1 to section 5. Instead of proving Theorem 1

directly, we prove a more general theorem in the next section. Later, in section 5,

we show that Theorem 1 is a special case of that.

4 Summation Games

In this section, we describe a general class of games which we will refer to as sum-

mation games. Later, we show that the reverse auction explained in the previous

section and some well known problems like the Courant-Nash equilibrium of an

oligopoly game [7] can be expressed in this model. Next, we define a Summation

Game:

Definition 2 (Summation Game). There are n players N = {1,··· ,n}.

Each player can choose a number aifrom the interval [0,¯ ai] where ¯ aiis a con-

stant. The utility of each bidder depends only on her own number as well as the

sum of all the numbers. In other words, assuming that A =?n

We next show that if the utility functions ui(a,A) meet a certain require-

ment, the summation game has a unique Nash-equilibrium that can be computed

efficiently. Before that, we define the following notation

i=1ai, the utility

of each bidder i is given by ui(ai,A).

Definition 3. For each player i, assuming that ui(a,A) is her utility function,

define her characteristic function hi(x,T) as the following:

hi(x,T) =

∂

∂aui(xT,T) +

∂

∂Aui(xT,T) (4.1)

Theorem 2. If all the characteristic functions hi(x,T)3are strictly decreasing

functions in both x and T, then the game has a unique Nash-equilibrium4and

in that equilibrium, the bid of each player i is ai= xi(A)A where xiis defined

as the following:

xi(T) = argminx∈[0,min(1,¯ ai

T)]|hi(x,T)|

(4.2)

3Note that we allow hi(x,T) to be discontinuous at a finite number of points (e.g. a

step function).

4if we relax the requirement of hi’s being strictly decreasing to just being non-

increasing then there is a continuum of Nash-equilibria in which there is one Nash-

equilibrium that is strictly preferred by some players and is just as good as other

Nash equilibria for other players.

Page 10

Furthermore, because hi(x,T) is decreasing in both x and T, xi(T) is also

decreasing in T and the equilibrium can be computed efficiently using two nested

binary searches or using an auction-like mechanism with an ascending clock T

in which the each bidder i submits ai= xi(T)T and the clock T keeps going up

as long as?

Proof. First, it is easy to see that the first order and boundary conditions that

are necessary and sufficient for the Nash-equilibrium are exactly those of (3.14)

which we wrote for the Nash-equilibrium of Auction. 1. We can rewrite those

conditions in terms of hi(x,T) =

the following. Again, note that the following is just a restatement of (3.14) in

which x∗

iai> T.

∂

∂aui(xT,T)+

∂

∂Aui(xT,T) for each player i as

i=

a∗

A∗ and T∗= A∗and a∗

i

i’s are the equilibrium bids:

∀i ∈ N :

hi(x∗

i,T∗) = 0

or

hi(x∗

or

hi(x∗

and0 ≤ x∗

i≤ min(1,¯ ai

T∗)

i,T∗) ≤ 0 and

x∗

i= 0

i,T∗) ≥ 0and

x∗

i= min(1,¯ ai

T∗)

(4.3)

n

?

i=1

x∗

i= 1 (4.4)

Based on our assumption that each hi(x,T) is a decreasing function of both

x and T, it is easy to see that the above 3 conditions can be written in a compact

form as the following single condition:

∀i ∈ N :

x∗

i= argminx∗

n

?

i∈[0,min(1,

¯ ai

T∗)]|hi(x∗

i,T∗)|

(4.5)

i=1

x∗

i= 1(4.6)

To get an intuition of why (4.5) is equivalent to (4.3). Suppose for a given T∗,

we want to find x∗

Since hi(x,T∗) is decreasing in x, to minimize |hi(x,T∗)|, if hi(x,T∗) is nega-

tive we should decrease x until either hi(x,T∗) becomes 0; or x reaches 0 and

hi(x,T∗) is still positive. Otherwise, if hi(x,T∗) is positive we would do the op-

posite. Note that xi(T) as defined in (4.2) returns the value of x that minimizes

|hi(x,T)|. Based on what we just explained, it is easy to see that for any given

T, we can actually do a binary search to find the x that minimizes |hi(x,T)| and

therefore we can efficiently compute xi(T) using a binary search even for fairly

complex hi. Next we show an important property of xi(T).

ithat satisfies (4.5). Take any arbitrary x ∈ [0,min(1,¯ ai

T∗)].

Lemma 1. xi(T) is is a strictly decreasing5function of T.

5When hi(x,T) functions are non-increasing in x and T instead if strictly decreasing

then xi(T) may also be non-increasing instead of strictly decreasing

Page 11

Proof. We only give a sketch of the proof. Take any given T, we know that

xi(T) gives the x that makes hi(x,T) as close to 0 as possible. If we increase T

by ∆T > 0, that may only decrease hi(x,T) by some ? > 0, if hi(x,T) was 0,

now it becomes negative so to counter that and bring hi(x,T) close to 0 we have

to decrease x by some ∆x > 0. On the other hand, if hi(x,T) was positive6it

means we are already in the case of x = min(1,¯ ai

to decrease if¯ ai

T) which may actually cause x

T< 1 because¯ ai

Tdecreases as T increases.

Finally, to find the values of ai’s at the Nash-equilibrium we can use the

following algorithm:

Algorithm 2

– Start with T = 0 (or a sufficiently small positive T).

– Keep increasing T for as long as?n

i=1xi(T) > 1.

– Stop as soon as?n

To see why the above algorithm works, we use Lemma 1 to argue that the

value of the equilibrium aggregate bid8T∗and all the xi(T∗) values are unique.

Because for any T?> T∗,?n

as an ascending auction-like mechanism in which each player submits the bid

ai= xi(T)T where T is the ascending clock and in which the clock stops once

?

In the next section we finish our analysis of the pooled reverse auction of

1. Later, in section 6, we give example of a well-known problem that can be

expressed in our model and its Nash-equilibrium can be computed using Alg. 2.

i=1xi(T) ≤ 1 and then set each bid ai= xi(T)T7

i=1xi(T?) < 1 and for any T?< T∗,?n

i=1xi(T?) > 1.

Note that Alg. 2 can be implemented either using binary search on T or

iai≤ T.

5 Back to Pooled Reverse Auction

In the previous section we described a more general class of games and in The-

orem 2 we gave sufficient conditions for the existence of a Nash-equilibrium. We

explained when it is unique and how to compute it. In this section we continue

our analysis of Auction. 1. We first give a proof for Theorem 1 be reducing it to

a special case of Theorem 2.

Next, we give a proof for Theorem 1 which is based on a reduction to Theo-

rem 2.

6In case hi is discontinuous at x and T, the proof will be slightly different

7If

< 1 then arbitrarily choose each a∗

[lim?→0+xi(T∗− ?)T∗,xi(T∗)T∗] such that?n

8When hi(x,T) functions are non-increasing instead if strictly decreasing then this

algorithm finds the equilibrium with the smallest aggregate bid T∗.

?n

i=1xi(T∗)

i

from the interval

i=1a∗

i= T∗(It is easy to show that

each player i is indifferent to all a∗

i∈ [lim?→0+xi(T∗− ?)T∗,xi(T∗)T∗]).

Page 12

Proof (Proof of. Theorem 1). To be able to apply Theorem 2, we first need to

show that the utility function of each bidder in Auction. 1 meets the requirement

of Theorem 2. More specifically, we have to show that hi(x,T) =

∂

∂Aui(xT,T) is a decreasing function in both x and T. Remember that in our

model for Auction. 1, we can write the utility of bidder i as ui(a,A) = vi(a

ci(a). First, we show that ci(a) is a convex function.

∂

∂aui(xT,T)+

AM)−

Lemma 1. The cost function ci(x) as defined in (3.8) is always a convex func-

tion and has an non-decreasing first order derivative in [0,r.¯ q] although its

derivative might be discontinuous in at most m points.

Proof. The proof is based on the construction given in the proof of Lemma 2.

Note that based on the definition of bi(a) from (3.6) and (3.7) ci(a) is a piece-

wise linear function consisting of m segments and its derivative is given by the

following:

∂

∂xci(a) =wy

ry

:

y−1

?

j=1

rj¯ qi,j< a <

y

?

j=1

rj¯ qi,j

(5.1)

Since we assumed that securities are sorted such that

that the derivative of each segment of ciis greater than or equal to the previous

segment which means its derivative is non-decreasing and so ciis convex.

rj

wj≥

rj+1

wj+1we can see

Lemma 2. The hi(x,T) functions for bidders in Auction. 1 are decreasing in

both x and T:

hi(x,T) =

∂

∂aui(xT,T) +

i(xM)1 − x

∂

∂Aui(xT,T) (5.2)

= v?

T

+ c?

i(Tx)(5.3)

Proof. We showed in Lemma 2 that ciis a convex function. We also assumed

in Theorem 1 that viis a concave function. It is then easy to verify that (5.3)

is indeed a decreasing function in both x and T. First, because vi is concave

v?

T

is decreasing in both x and T, ciis convex and c?

non-decreasing which means −c?

them all together, hi(x,T) is a strictly decreasing function of both x and T.

iis non-increasing,

1−x

iis

i(Tx) is non-increasing in both x and T. Putting

Since in Lemma 2 we proved that hi(x,T) is decreasing in both x and T,

we can now apply Theorem 2 and all of the claims of Theorem 1 follow from

Theorem 2. Also note that gi(α) which was defined in (3.16) is actually the same

as xi(T)T, where T =M

α

It is interesting to notice that the auction-like mechanism of Theorem 2 and

Auction. 1 are actually equivalent. In fact, the xi(T) where T = M/α, has a

very natural interpretation in Auction. 1. It specifies the fraction of the budget

of the auctioneer that the bidder i is demanding at the clock α. In fact we may

Page 13

modify the auction to ask the bidders to submit the amount of liquidity that

they are demanding directly at each step of the clock and then the auction stops

when the demand becomes less than or equal to the budget of the auctioneer.

Then, each bidder will be required to sell enough quantity of her shares at the

current prices to pay for the liquidity that she had demanded.

It is easy to see that the liquidity that each bidder demands may only decrease

as the α increases. However, the value of the bid, xi(T)T, may actually increase

because bidder I may want to maintain her demand for the liquidity.

6 Application to Cournot’s Oligopoly

In this section, we show how the well-known problem of Cournot’s Oligopoly can

be expressed in our model of a summation game and all the results of Theorem 2

can therefore be applied:

Definition 4 (Cournot’s Oligopoly).

– There are n firms. The firms are oligopolist suppliers of a homogenous good.

– At each period, each firm chooses a quantity qito supply.

– The total supply Q on the market is the sum of all firms’ supplies:

Q =

?

i

qi

(6.1)

– All firms receive the same price p per unit of the good. The price p on the

market depends on the total supply Q as:

p(Q) = p0(Qmax− Q) (6.2)

– Each firm i incurs a cost ciper unit of good. These costs can be different for

different firms and are private information

– Each firm i’s profit is given by:

ui(qi,Q) = (p(Q) − ci)qi

(6.3)

– After each market period, firms are informed of the total quantity Q and the

market price p(Q) of the previous period.

If we write down the hi(x,T) for each firm i we get:

hi(x,T) =

∂

∂aui(xT,T) +

= p(T) − ci+ p?(T)Tx

= p0(Qmax− T) − ci− p0Tx

∂

∂Aui(xT,T) (6.4)

(6.5)

(6.6)

Notice that clearly the above hi(x,T) is a decreasing function of both x and

T and therefore all of the nice results of Theorem 2 can be applied. Notice in

fact that as long as p(Q) is concave and a decreasing function of Q, hi(x,T) is

still a decreasing function of both x and T and all of the results of Theorem 2

still holds.

Page 14

7 Conclusion

In this paper we studied the Nash-equilibrium and equilibrium bidding strategies

of the troubled assets reverse auction. We further generalized our analysis to

a more general class of games with non quasi-linear utilities. We proved the

existence and uniqueness of a Nash-equilibrium in those games and we also gave

an efficient way to compute the equilibrium of those games. We also showed that

finding the Nash equilibrium can be implemented using an ascending mechanism

so that the participants don’t need to reveal their utility functions. We also,

showed that even a more general version of the well-known problem of Cournot’s

Oligopoly can be expressed in our model and all of the previously mentioned

results apply to that as well.

References

1. L. Ausubel and P. Cramton. A troubled asset reverse auction. In Working paper,

2008.

2. L. M. Ausubel, P. Cramton, E. Filiz-Ozbay, N. Higgins, E. Ozbay, and A. Stock-

ing. Common-value auctions with liquidity needs:an experimental test of a troubled

assets reverse auction. University of maryland, economics working paper series,

Department of Economics, University of Maryland, 2008.

3. T. Bergstrom and H. Varian. Two remarks on cournot equilibria. University of cali-

fornia at santa barbara, economics working paper series, Department of Economics,

UC Santa Barbara, Feb. 1985.

4. R. Johari, S. Mannor, and J. N. Tsitsiklis. Efficiency loss in a network resource

allocation game. Mathematics of Operations Research, 29:407–435, 2004.

5. R. Johari and J. Tsitsiklis. Efficiency of scalar-parameterized mechanisms. Opera-

tion Research, 2008.

6. F. Kelly. Charging and rate control for elastic traffic. European Transactions on

Telecommunications, 8:33–37, 1997.

7. A. Mas-colell, M. D. Whinston, and G. J. R.

8. S. Shenker, D. Clark, D. Estrin, and S. Herzog. Pricing in computer networks:

Reshaping the research agenda. ACM Computer Communication Review, 26:183–

201, 1996.