1. 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. 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. 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. 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. 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. 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. 7 Pooled Reverse Auction (AC2008) Government declares the total budget (i.e., M ) Government announces the reference prices :{r 1,…r m } r i is the reference price for security i in cents per dollar (i.e., for each dollar of the face value the government pays r i 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. Private Valuations S1S2S3 A91¢--- B 51¢--- C 33¢ Holdings S1S2S3 A$30B--- B $35B--- C $40B 8 (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% Government We want to buy $15B worth of security Reference Prices S1S2S3 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.

9. 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. 10 Model for the Pooled Auction m Security  {r 1,…r m } : government reference price (announced by government) n bidders  (q i,1,..,q i,m ) : quantity of shares that bidder “i” holds from each security  (w i,1,…, w i,m ) : bidder “i”’s valuations of each security  v i (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. 11 Concave Valuation An example for valuation function v i (l) should be concave Justification: Each bidder values the first dollar she receives, more than the last dollar she receives It can model the penalty for the lack of liquidity For example:

12. 12 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

13. 13 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

14. 14 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 a i submits a bid such that  is minimized  subject to How: Sell the securities with higher r j /w i,j first Define C i (a) as the minimum value of for generating activity point a then : The first term depends only on a i

15. 15 First Step: Converting to Single Dimension Suppose auction stops at  *   *  a i = M so  * = M/  a i  A=  a i We can rewrite the utility function as:

16. 16 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

17. 17 Second Step: Defining the Summation Game Summation Game:  n player  each player “i”  Selects a number a i  [0, a’ i ]   Her utility depends on a i and A where A=  a i. In other words, utility can be defined as u i (a i,A)  Not necessarily a symmetric game The reverse auction problem is a special instance

18. 18 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

19. 19 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 a i Nash equilibrium condition:  Consider   0 If u is differentiable and continuous:  if then Or  if, then u i (a i,A) should be decreasing or Or  If then u i (a i,A) should be increasing or We should satisfy this condition for all players at the same time

20. 20 Conditions for Nash equilibrium

21. 21 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 x i = a i /A fraction of aggregate bid made by player i Define T = A the total aggregate bid Rewrite the first order condition of u i (a,A) in terms of x i and T and define h i (x,T) as follows: We show that a unique NE exists and it can be computed efficiently if:  h i (x,T) is strictly decreasing in both x and T For the pooled reverse auction, this condition is satisfied.

22. 22 Step 3: Computing Nash equilibrium for Summation Games Rewrite Nash conditions in terms of h(x,T) If a i =0 then x i =0 If a i = a i ’ then x i =a i ’/T

23. 23 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 x i for each player and T for each player such that the equilibrium conditions are satisfied We show that there is a function z i (T) such that if we set x i = z i (T) then x i and T satisfy the equilibrium condition for bidder i We show that z i (T) is a decreasing function Because z i (T) is decreasing, we can easily find T* such that  z i (T*) = 1 Final solution a i = z i (T*) T*, A = T*

24. 24 Ascending Auction for Summation Games It is enough that each bidder computes/submits z i (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 b i = z i (T)  The clock goes up until  b i = 1  Activity rule: Nobody can increase his bid Notes:  Each person can compute z i (T) locally (i,e. z i (T) can be computed from u i (a,A) alone)

25. 25 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

26. 26 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 h i (x,T) is decreasing in x and T

27. 27 Step 4: Bidding Strategy for Pooled Auction v i (l) is concave function We show that c i (a) is a convex function: Recall that we used greedy method to compute c i (a) The slope is non decreasing i.e., d/da c i (a) is none decreasing C i (a) a The slope of this part is w 3 /r 3 r1q1r1q1 r2q2r2q2 Non decreasing Non increasing Strictly decreasing in x and T

28. 28 Step 4: Computing Bidding Strategy In a summation game:  Use x i =z i (T) at each clock T  Stops when  x i = 1 Or in terms of a i :  Submit a i =T.z i (T)  Increase until  a i =A In the Reverse Auction:  T = M/   As   0: T  Bid function:  a i = M/ . z i (M/  ).  Total liquidity received is  a i  Or  x i.A = M/A.x i.A=Mx i  Mx i is decreasing

29. 29 Bid Function In summary the bid function for the pooled reverse auction can be summarized as :

30. 30 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 c i  Assuming the total supply is Q=  q i  Payment for each item is :P(Q) = p 0 (Q max –Q)  u i (q i,Q) = (p(Q) – c i )q i Concave function

31. 31 Cournot Writing h(x,T) function for Cournot we have: Decreasing in both x and T

32. 32 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

33. 33 Questions?

34. 34 Nash Equilibrium Necessary Condition a u i (a,A) a’ i a d/da u i (a,A) u i (a,A) a a’ i a d/da u i (a,A) a u i (a,A) a’ i a d/da u i (a,A)

35. 35 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’

36. 36 Step 3 :Computing Nash Equilibrium for Summation Game Assume z i (T) is the value “x” that will satisfy Nash condition for h(x,T) It is equivalent to say that: Function z i (T)  Input: T*  Output x* that will minimize |h(x*,T*)| We showed:  If T* is known z i (T) can be computed efficiently We will show If z i (T) is decreasing  We can compute T* efficiently as well

37. 37 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 x i for each player and T for each player such that the equilibrium conditions are satisfied We show that there is a function z i (T) such that if we set x i = z i (T) then x i and T satisfy the equilibrium condition for bidder i We show that z i (T) is a decreasing function Because z i (T) is decreasing, we can find T* such that  z i (T*) = 1 Final solution a i = z i (T*) T*, A = T*

38. 38 Step 3 :Computing Nash equilibrium for Summation Game How?  Assume z i (T) is decreasing.   z i (T) is also decreasing in T  We want to compute T such that   z i (T) = 1 Start from an arbitrary value T  If  z i (T) >1 increase T  till  z i (T) = 1 We can prove that zi(T) is also decreasing. T 1 z(T)

39. 39 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 x i for each player and T for each player such that the equilibrium conditions are satisfied We show that there is a function z i (T) such that if we set x i = z i (T) then x i and T satisfy the equilibrium condition for bidder i We show that z i (T) is a decreasing function Because z i (T) is decreasing, we can find T* such that  z i (T*) = 1 Final solution a i = z i (T*) T*, A = T*

40. 40 h i (x,T 1 )h i (x,T 2 ) Step 3 :Computing Nash equilibrium for Summation Game Finally: We show that z i (T) is decreasing Consider T 2 >T 1  x 1 = z(T 1 ), x 2 = z(T 2 )  We show that x 2 < x 1 For a moment ignore the boundary cases and assume that h(x 1,T 1 ) =0  h(x 1, T 2 ) < 0  To push it back to 0  We need to decrease x  X 2 < x 1 Since h(x,T) is decreasing in both x and T Increasing one results in decreasing the other x1x1 T1T1 T2T2 x2x2 x1x1 x2x2