An Analysis for Troubled Assets Reverse Auction

1. 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)

2. 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

3. 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.

4. 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

5. 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

6. 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¢)

7. 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

8. Bids  A B C 100% ($30B,0,0) (0,$20B,0) (0,0,$40B) 99% 95% (0,0,0) (0,0,0) (0,$21B,0) (0,$20B,0) (0,0,$24B) (0,0,$25B) 90% (0,0,0) (0,$22B,0) (0,0,$16B) Pooled Reverse Auction Example Government We want to buy $15B worth of security 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.

9. 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

10. 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

11. How to Analyze the Equilibrium? • First we reduce the bidding strategy from multi dimensional to a single dimensional bid • We define the summation game problem and show that our problem is a special case of that. • We develop the bidding strategies for summation games in general • We compute the bidding strategies for the original game

12. First Step:Computing Utility • Utility of bidder “i” is a non-quasi-linear function: • 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 : Reference prices Valuation function

13. 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,jfirst • Define Ci(a) as the minimum value of for generating activity point athen : The first term depends only on ai

14. First Step: Converting to Single Dimension • Suppose auction stops at * • * ai = M so * = M/ ai • A= ai • We can rewrite the utility function as:

15. How to Analyze the Equilibrium? • First we reduce the bidding strategy from multi dimensional to a single dimensional bid • We reduce the problem to a summation game • We develop the bidding strategies for summation games in general • We compute the bidding strategies for the original game

16. 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

17. How to Analyze the Equilibrium? • First we reduce the bidding strategy from multi dimensional to a single dimensional bid • We reduce the problem to a summation game • We develop the bidding strategies for summation games in general • We compute the bidding strategies for the original game

18. 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

19. Conditions for Nash equilibrium

20. 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.

21. 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

22. 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 solution ai = zi(T*) T*, A = T*

23. 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)

24. How to Analyze the Equilibrium? • First we reduce the bidding strategy from multi dimensional to a single dimensional bid • We reduce the problem to a summation game • We develop the bidding strategies for summation games in general • We compute the bidding strategies for the original game

25. 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

26. Step 4: Bidding Strategy for Pooled Auction The slope of this part is w3/r3 • 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) r1q1 r2q2 Non increasing Non decreasing a Strictly decreasing in x and T

27. 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 Step 4: Computing Bidding Strategy

28. Bid Function • In summary the bid function for the pooled reverse auction can be summarized as :

29. 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

30. Cournot • Writing h(x,T) function for Cournot we have: Decreasing in both x and T

31. 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

32. Questions?

33. ui(a,A) ui(a,A) ui(a,A) a a a a’i a’i a’i d/da ui(a,A) d/da ui(a,A) d/da ui(a,A) a a a a’i a’i a’i Nash Equilibrium Necessary Condition

34. 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 h(x,T*) x x’

35. 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

36. 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 solution ai = zi(T*) T*, A = T*

37. Step 3 :Computing Nash equilibrium for Summation Game z(T) • 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. 1 T

38. 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 solution ai = zi(T*) T*, A = T*