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An Analysis for Troubled Assets Reverse Auction

Saeed Alaei (University of Maryland-College Park)Azarakhsh Malekian (University of Maryland-College Park)

Presented by: Vahab Mirrokni (Google Research)

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Structure of the Talk

• Describing Troubled Assets Reverse Auction (Ausubel & Cramton 2008)

• Our Contribution: Computing the bidding strategies and the

equilibrium Generalization to summation games Other applications of summation games

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Economic Crisis• In a normal economy, banks

sell their illiquid assets when they need more liquidity.

• In the crisis, the need for liquidity goes up

• Financial companies try to turn their illiquid assets to liquidity

• Supply of illiquid assets in the market increases, so market clearing prices go down

• Companies lose as the result of the prices of their assets going down.

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Rescue Plan

• Government intervenes to buy illiquid assets assuming that in the future they will reach their original price

• Government should decide: Which securities to buy?

How many from each At what price?

The current market prices are not indicative

• Objective: Quick and effective solution to buy the assets

at a reasonable price

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A Two-Part Reverse Auction (Ausubel, Cramton 2008)

First (Individual Auctions): • Simultaneous descending security by security auctions

are run on securities with enough competitions• Prices from the auctioned securities are used by the

government to estimate reference prices for all the securities Reference prices for securities are computed

Second (Pooled Auction): • Pooled auction is run for the remaining securities

Reference prices from the first part are used in this auction The objective is that bidders with greatest need for liquidity are

most likely to win

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Pooled Reverse Auction• Government declares at the beginning:

Total amount government wants to spend Reference prices for each security (prices are in cents

per each dollar of the face value)• It is a clock auction

Clock value acts as a scaling factor on reference prices.

for example: 3 securities Reference prices:(90¢, 60¢, 80¢) Clock: 80% Current prices: (72¢, 48¢, 64¢)

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Pooled Reverse Auction (AC2008)• Government declares the total budget (i.e., M)• Government announces the reference prices :{r1,…rm}

• ri is the reference price for security i in cents per dollar (i.e., for each dollar of the face value the government pays ri

• The auction uses a single clock “” initially set to its maximum (e.g.100%)

• At each round:• The current prices of the securities are computed by multiplying the

reference prices by the current value of the clock.• Security holders bid the quantities that they would like to sell at the

current prices (the quantities are specified in terms of the dollars of face value)

• Auction concludes:• At the first round in which the total value of the bids at the current

price is less than or equal to government’s budget

Private Valuations

S1 S2 S3

A 91¢ --- ---

B --- 51¢ ---

C --- --- 33¢

Holdings

S1 S2 S3

A $30B --- ---

B --- $35B ---

C --- --- $40B

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(0,0,$24B)(0,$20B,0)(0,0,0)99%

Pooled Reverse Auction Example

(0,0,$40B)(0,$20B,0)($30B,0,0)100%

Bids

CBA

(0,0,$25B)(0,$21B,0)(0,0,0)95%

(0,0,$16B)(0,$22B,0)(0,0,0)90%

GovernmentWe want to buy $15B worth of

security

Reference Prices

S1 S2 S3

92¢ 50¢ 33¢Bidder A has no liquidity need but bidder B has $10B liquidity need and bidder C has $8B liquidity need so they stay in.

Both Bidder B and C increase their bid to demand the same liquidity that they did

before. Bidder B still increases his bid to demand the same liquidity, but bidder C gives up and reduces his bid. The auction ends.

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Our Contributions

• We show how to compute the Nash Equilibrium (NE) and bidding strategy for the Pooled reverse auction

• We show the NE is unique and the auction converges to the NE We reduce pooled reverse auction to a

summation game We show how to compute Nash Equilibrium

for summation game

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Model for the Pooled Auction• m Security

{r1,…rm} : government reference price (announced by government)

• n bidders (qi,1,..,qi,m) : quantity of shares that bidder “i” holds from

each security (wi,1,…, wi,m) : bidder “i”’s valuations of each security vi(l) : valuation of bidder i for receiving a liquidity of l, it

should be a concave function.• The total budget of the government is M

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How to Analyze the Equilibrium?

1. First we reduce the bidding strategy from multi dimensional to a single dimensional bid

2. We define the summation game problem and show that our problem is a special case of that.

3. We develop the bidding strategies for summation games in general

4. We compute the bidding strategies for the original game

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• Suppose :•auction stops at clock *•Quantity vector submitted by bidder i: •Total money paid to bidder i: •Total quantity of assets sold by i:

•Activity point of bidder i :

First Step:Computing Utility

• Utility of bidder “i” is a non-quasi-linear function:

Valuation function

Reference prices

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First Step: Defining Cost Function• The effect of each bidder on the auction can be captured by its activity

point.• Among all the bids generating the same activity point,

rational bidder would choose the one that would maximize her utility.

• Any rational player for any given ai submits a bid such that

is minimized subject to

• How:• Sell the securities with higher rj/wi,j first• Define Ci(a) as the minimum value of for generating activity point a

then :

The first term depends only on ai

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First Step: Converting to Single Dimension

• Suppose auction stops at * * ai = M so * = M/ ai

A= ai

• We can rewrite the utility function as:

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How to Analyze the Equilibrium?

1. First we reduce the bidding strategy from multi dimensional to a single dimensional bid

2. We reduce the problem to a summation game

3. We develop the bidding strategies for summation games in general

4. We compute the bidding strategies for the original game

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Second Step: Defining the Summation Game

• Summation Game: n player

each player “i” Selects a number ai [0, a’i] Her utility depends on ai

and A where A= ai . In other words, utility can be defined as ui(ai,A)

Not necessarily a symmetric game

• The reverse auction problem is a special instance

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How to Analyze the Equilibrium?

1. First we reduce the bidding strategy from multi dimensional to a single dimensional bid

2. We reduce the problem to a summation game

3. We develop the bidding strategies for summation games in general

4. We compute the bidding strategies for the original game

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Step 3: Computing Nash Equilibrium for Summation Games

• Condition for Nash equilibrium: Assuming nobody is going to change his action, you don’t gain by

changing your action.• In summation game:

Your action: selecting ai

• Nash equilibrium condition: Consider 0

• If u is differentiable and continuous: if then

• Or if , then ui(ai,A) should be decreasing or

• Or If then ui(ai,A) should be increasing or

• We should satisfy this condition for all players at the same time

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Conditions for Nash equilibrium

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Step 3: Computing Nash Equilibrium for Summation Games

• Our main Theorem:• We prove the existence and uniqueness of the NE and an auction based

algorithm for computing the NE under certain conditions.• Define xi = ai/A fraction of aggregate bid made by player i• Define T = A the total aggregate bid• Rewrite the first order condition of ui(a,A) in terms of xi and T and

define hi(x,T) as follows:

• We show that a unique NE exists and it can be computed efficiently if: hi(x,T) is strictly decreasing in both x and T

• For the pooled reverse auction, this condition is satisfied.

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Step 3: Computing Nash equilibrium for Summation Games

• Rewrite Nash conditions in terms of h(x,T)

If ai=0 then xi=0

If ai = ai’ then xi =ai’/T

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Step 3 :Computing Nash Equilibrium for Summation Game

We assume h(x,T) is strictly decreasing in both x and T. To compute Nash equilibrium for summation games, we should find xi for each player and

T for each player such that the equilibrium conditions are satisfied

We show that there is a function zi(T) such that if we set xi= zi(T) then xi and T satisfy the equilibrium condition for bidder i

We show that zi(T) is a decreasing function

Because zi(T) is decreasing, we can easily find T* such that zi(T*) = 1

Final solutionai = zi(T*) T*, A = T*

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Ascending Auction for Summation Games

• It is enough that each bidder computes/submits zi(T)• This suggests:• Ascending Auction for solving summation games when

h(x,T) is strictly decreasing in x and T: Consider T as the clock of the auction When clock is T, i submits bi = zi(T) The clock goes up until bi = 1 Activity rule: Nobody can increase his bid

• Notes: Each person can compute zi(T) locally (i,e. zi(T) can be

computed from ui(a,A) alone)

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How to Analyze the Equilibrium?

1. First we reduce the bidding strategy from multi dimensional to a single dimensional bid

2. We reduce the problem to a summation game

3. We develop the bidding strategies for summation games in general

4. We compute the bidding strategies for the original game

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Step 4: Bidding Strategy for Pooled Auction

• We showed pooled auction is an instance of summation game

• We need to show that h(x,T) is strictly decreasing in both x and T

• Recall that:

We want to show hi(x,T) is decreasing in x and T

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Step 4: Bidding Strategy for Pooled Auction

• vi(l) is concave function• We show that ci(a) is a

convex function:• Recall that we used greedy

method to compute ci(a)• The slope is non decreasing • i.e., d/da ci(a) is none

decreasing

Ci(a)

a

The slope of this part is w3/r3

r1q1 r2q2Non

decreasing

Non increasingStrictly

decreasing in x and T

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Step 4: Computing Bidding Strategy

• In a summation game: Use xi=zi(T) at each clock T Stops when xi = 1

• Or in terms of ai: Submit ai=T.zi(T) Increase until ai =A

• In the Reverse Auction: T = M/ As 0: T

• Bid function: ai = M/. zi(M/ ). Total liquidity received

is ai

Or xi.A = M/A.xi.A=Mxi

Mxi is decreasing

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Bid Function

• In summary the bid function for the pooled reverse auction can be summarized as :

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Other Instances of Summation Games

• There are n firms: The firms are suppliers of a homogenous good At each period they choose quantity q to

produce Producing each unit has cost ci Assuming the total supply is Q= qi

Payment for each item is :P(Q) = p0(Qmax –Q) ui(qi,Q) = (p(Q) – ci)qi

Concave function

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Cournot

• Writing h(x,T) function for Cournot we have:

Decreasing in both x and T

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Summary

• Described Troubled Assets Reverse Auction• We showed:

How to reduce it to summation games How to compute bidding strategy for summation

games How to compute the equilibrium point efficiently Described other instances belonging to summation

game class

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Questions?

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Nash Equilibrium Necessary Condition

a

ui(a,A)

a’i

aa’i

d/da ui(a,A)

ui(a,A)

aa’i

aa’i

d/da ui(a,A)

a

ui(a,A)

a’i

aa’i

d/da ui(a,A)

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Step 3: Computing Nash equilibrium for Summation Games

• Close look at equilibrium point: Suppose T* is fixed

The objective is to find x* to satisfy the constraints.

Start from an arbitrary x h(x,T*)= 0 h(x,T*) > 0

We increase x until h(x,T*) = 0 or x = x’=a’/T

h(x,T*) < 0 We decrease x until h(x,T*) = 0 Or x=0

x

h(x,T*)

x’

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Step 3 :Computing Nash Equilibrium for Summation Game

• Assume zi(T) is the value “x” that will satisfy Nash condition for h(x,T)• It is equivalent to say that:• Function zi(T)

Input: T* Output x* that will minimize |h(x*,T*)|

• We showed: If T* is known zi(T) can be computed efficiently

• We will show• If zi(T) is decreasing We can compute T* efficiently as well

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Step 3 :Computing Nash Equilibrium for Summation Game

We assume h(x,T) is strictly decreasing in both x and T. To compute Nash equilibrium for summation games, we should find xi for each player and

T for each player such that the equilibrium conditions are satisfied

We show that there is a function zi(T) such that if we set xi= zi(T) then xi and T satisfy the equilibrium condition for bidder i

We show that zi(T) is a decreasing function

Because zi(T) is decreasing, we can find T* such that zi(T*) = 1

Final solutionai = zi(T*) T*, A = T*

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Step 3 :Computing Nash equilibrium for Summation Game

• How? Assume zi(T) is decreasing. zi(T) is also decreasing in T We want to compute T such that

zi(T) = 1 • Start from an arbitrary value T

If zi(T) >1 increase T till zi(T) = 1

• We can prove that zi(T) is also decreasing.

T

1

z(T)

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Step 3 :Computing Nash Equilibrium for Summation Game

We assume h(x,T) is strictly decreasing in both x and T. To compute Nash equilibrium for summation games, we should find xi for each player and

T for each player such that the equilibrium conditions are satisfied

We show that there is a function zi(T) such that if we set xi= zi(T) then xi and T satisfy the equilibrium condition for bidder i

We show that zi(T) is a decreasing function

Because zi(T) is decreasing, we can find T* such that zi(T*) = 1

Final solutionai = zi(T*) T*, A = T*