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). 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.
In this paper, we primarily study the equilibrium strategies of the pooled reverse
auction for troubled assets which was described in . 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  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 , 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.
??Dept. of Computer Science, University of Maryland, College Park, MD 20742.
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 , 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
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
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. ). Different aspect of
Cournot equilibrium has been studied (For example, in  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
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:
– 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
– 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:
= p(T) − ci+ p?(T)Tx
= p0(Qmax− T) − ci− p0Tx
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
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.
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