An Analysis of Troubled Assets Reverse Auction

Transcription

1 An Analyss of Troubled Assets Reverse Aucton Saeed Alae, Azarakhsh Malekan Unversty of Maryland Abstract. In ths paper we study the Nash-equlbrum and equlbrum bddng strateges of the Pooled Reverse Aucton for troubled assets. The aucton was descrbed n (Ausubel & Cramton 2008[1]). We further extend our analyss to a more general class of games whch we call Summaton Games. We prove the exstence and unqueness of a Nashequlbrum n these games when the utlty functons satsfy a certan condton. We also gve an effcent way to compute the Nash-equlbrum of these games. We show that then Nash-equlbrum of these games can be computed usng an ascendng aucton. The aforementoned reverse aucton can be expressed as a specal nstance of such a game. We also, show that even a more general verson of the well-known olgopoly game of Cournot can be expressed n our model and all of the prevously mentoned results apply to that as well. 1 Introducton In ths paper, we prmarly study the equlbrum strateges of the pooled reverse aucton for troubled assets whch was descrbed n [1]. The US Treasury s purchasng the troubled assets to nfuse lqudty nto the market to recover from the current fnancal crss. Reverse auctons n general have been a powerful tool for njectng lqudty nto the market n places where t wll be most useful. As explaned n [1] a smple and nave approach for the government could be to run a sngle reverse aucton for all the holders of toxc assets as follows. The auctoneer(government) then sets a total budget to be spent. The auctoneer starts at a prce lke 100 on a dollar. All the holders, bd the quantty of ther shares that they are wllng to sell at the current prces. There can be excess supply. The auctoneer then lowers the prce n steps e.g. 95, 90, etc. and bdders ndcate the quanttes that they are wllng to sell at each prce. At some pont (for example at 30 on a dollar) the total supply offered by all the holders for sale equals or falls bellow the specfed budget of the treasury. At that pont the aucton concludes and the auctoneer buys the securtes offered at the clearng prce. As explaned n [1], ths smple approach s flawed as t leads to a severe adverse selecton problem. Note that at the clearng prce the securtes that are offered are only the ones that are actually worth less than 30 on each dollar of face Dept. of Computer Scence, Unversty of Maryland, College Park, MD saeed@cs.umd.edu. Dept. of Computer Scence, Unversty of Maryland, College Park, MD malekan@cs.umd.edu.

2 value. They could as well worth far bellow 30. In other words, the government would pay most of ts budget to buy the worst of the securtes. In [1], the authors propose the followng two type of auctons. A Securty by Securty Reverse Aucton A Pooled Reverse Aucton They are both part of a two phase plan. The frst one can be used to extract prvate nformaton of holders about the true value of the securtes to gve an estmate on how much each securty and smlar securtes are actual worth of. Later, that nformaton can be used to establsh reference prces n the Pooled Reverse Aucton. In ths paper we focus our attenton on the second class of aucton. In a Pooled Reverse Aucton, dfferent securtes are pooled together. The government puts a reference prce on each securty and then runs a reverse aucton on all of them together. We explan ths aucton n more detal n secton 2. In secton 3, we study the Nash-equlbrum and the bddng strateges of the Pooled Reverse Auctons n detal. We then create a more abstract model of t at the end of secton 2. In secton 4 we descrbe a general class of games that can be used to model the Pooled Reverse Aucton as well as other problems. In secton 4, we gve some exctng result on these games. We gve a condton whch s suffcent for the exstence of a Nash-equlbrum. We further explan how the Nash-equlbrum can be computed effcently usng a an ascendng aucton-lke mechansm. Later n secton 5, we show how we can apply our result of secton 4 to Pooled Reverse Auctons. secton 6 explans how a more general verson of the Cournot s olgopoly game can be expressed n our model. 1.1 Related Work We partton the related works to two man groups. The frst group that s closely related to our model, are computng equlbrum n Cournot and publc good provson games. The second one wth smlar model but dfferent objectve are the works related to bandwdth sharng problems and the effcency of computed equlbra. One well known problem that can be consdered as an example of our model s the Cournot s olgopoly game. It can be descrbed as an olgopoly of frms producng a homogeneous good. The strategy of frm s to choose q whch s the quantty t produces. Assumng that the producton cost s c per tem, the utlty of frm s (p(q) c )q for whch Q = q s the total producton and p(q) s the global prce of the good based on the total producton. There s a vast amount of lterature on Cournot games (e.g. [7]). Dfferent aspect of Cournot equlbrum has been studed (For example, n [3] Bergstrom and Varan, studed the effect of taxaton on Cournot equlbrum and also showed some characterstcs of the Cournot equlbrum.) Another set of results, wth smlar model, but wth dfferent crtera are the works related to bandwdth sharng problem. At a hgh level, the problem s to

3 allocate a fxed amount of an nfntely dvsble good among ratonal competng users. [8] studes ths problem from prcng perspectve. Kelly [6], consdered a generalzed varant of ths problem n the context of routng and chargng (However the equlbrum pont of hs mechansm was not fully effcent) Hs model, for a sngle resource wth fxed supply, s to gve each person proportonal to hs bd from the resource and charge hm hs bd. Later, Johar et al n [4], showed that Kelly s mechansm s at least 75% effcent at the equlbrum pont. In another work, Johar et al show that, Kelly s model mnmzes effcency loss (at the equlbrum pont) when prce dscrmnaton s not allowed and then they present a class of mechansms that has an effcent outcome at the equlbrum pont assumng that prce dscrmnaton s allowed ([5]). 2 Model for Pooled Reverse Aucton In ths secton, we explan the basc model for the reverse aucton of pooled securtes. We wll use ths model throughout the rest of ths paper. We start by explanng our notatons: There are n bdders N = {1,, n}, and m securtes. Government has evaluated a reference prce of r j for each securty j. Also let r = (r 1,, r m ) denote the vector of reference prces for all the m securtes. The reference prces are publc nformaton. These prces are n the form of the rato of the evaluated prce to the face prce and are expressed n cents per dollar. For example r j = 0.75 means every dollar of the face value of the securty s actually worth 75. Each bdder holds q,j shares of securty j. Also let q = ( q,1,, q,m ) denote the vector of the quanttes of shares that bdder holds from each securty. The shares are expressed n quantty of the face value. Each bdder has a prvate valuaton functon v (l) for recevng a lqudty amount of l. In a quas-lnear settng, we would assume that v (l) = l. In our model, we assume the v could be an arbtrary functon. v can capture the bonus for acqurng a needed amount of lqudty or can be negatve to account for the cost ncurred by the shortage thereof. For example consder the followng: { l + (l L ) l L v (l) = (2.1) l l L We could nterpret the above v as the followng. Bdder has a lqudty need of L dollars. She ncurs a cost of L l dollars f she rases only l dollars where l < L. Her value for any lqudty that she receves beyond L s just the same as the amount that she receves. The expermental study of reverse aucton for troubled assets n [2] consders two cases for v. In the frst case, each bdder has a lqudty need L and v (l) = 2l for l L and v (l) = l+l for l > L. In the second case, bdders don t have lqudty needs, so v (l) = l. In ths paper, we consder arbtrary v under some constrants as we wll see later.

4 Each bdder has a prvate value of w,j for each dollar of securty j. Also let w = (w,1,, w,m ) denote the vector of the valuatons of bdder for dfferent securtes. In realty, we should have assumed a sngle common value for each securty whch s unknown and can only be computed by aggregatng all the prvate nformaton of all bdders. However, that model s prohbtvely hard to analyze n case of non trval valuaton functons for lqudty (.e. when v (l) s not the dentty functon). Therefore, we assume that w s the prvate values of bdder for the securtes. Next, we brefly explan the reverse aucton mechansm for pooled securtes as descrbed n [1]. Aucton 1 (Pooled Reverse Aucton) Intally, the auctoneer (government) establshes the reference prces for all the securtes. These reference prces are supposed to be the best estmate of the government about the true value of the securtes. The reference prces are announced publcly. The aucton uses a sngle descendng clock α whch specfes the current prces as a percentage of the reference prces. For example, α = 110% means the current prce of each securty s 110% of ts reference prce. As the clock goes down, partcpants upte ther bds. Bdder submts a bd b = (b,1,, b,m ), where b,j s the quantty of shares from securty j that bdder would lke to sell at the current prces. These quanttes are specfed n terms of dollars of face value. The auctoneer collects all the bds and computes the actvty ponts for each bdder as a = r b (remember r s the vector of reference prces). In other words, the actvty ponts of each bdder s her bd quantty for each securty tmes the reference prce of that securty summed over all the securtes. The auctoneer also computes the total actvty pont A = a. Assumng that M s the total budget of the government, the clock keeps gong down for as long as Aα > M. In practce, the clock goes down n dscrete steps. At each step the auctoneer collects all the bds and computes the aggregate actvty pont. At the frst step that Aα becomes less than or equal to M, the clock stops and the aucton concludes. The auctoneer then buys from each bdder the quantty of shares specfed n her bd. Bdders are pad at the current prces (.e. the reference prce scaled by the current value of the clock). Assumng that α was the fnal value of the clock and for each bdder, b was the fnal bd of bdder, the amount of lqudty that bdder receves s α r b. In the next secton we study the equlbrum of the above aucton. 3 The Equlbrum of Pooled Reverse Aucton In ths secton, we study the Nash-equlbrum of Aucton. 1 and propose a method that can be used to effcently compute that. We also develop a bddng strategy that leads to the Nash-equlbrum. Frst, we show how to compute the utlty of each bdder. Assume that b s the bd of bdder and α s the current value of the clock. Also, as we defned n

5 secton 2, v (l) s the valuaton of bdder for recevng amount l of lqudty and w = (w,1,, w ) s the vector of her valuatons for dfferent securtes. We denote by u, the tentatve utlty of bdder whch s her utlty f the aucton stops at the current value of the clock. u can be computed as the followng: u = v (αr b ) w b (3.1) Before we start wth the bddng strateges, we restate some of the defntons from Aucton. 1. For a bdder wth current bd b, we use a to defne her actvty pont whch s defned as: The total actvty pont of all bdders s defned as: a = r b (3.2) A = n a (3.3) The aucton clock, α, keeps gong down for as long as αa > M where M s the total budget of the auctoneer. If we denote the value of the clock when the aucton stops by α, then α A M. Note that, to smplfy the analyss, we assume quanttes do not need to be ntegers. We also assume that the clock changes contnuously and bdders upte ther bds contnuously as well. Respectvely, we may assume that when the aucton concludes, the auctoneers budget constrant s met wth equalty so: =1 α A = M (3.4) Next, we show that the best strategy for each player can be descrbed by just specfyng the actvty ponts that she needs to generate. In other words, the only thng that bdder has to decde s how much actvty pont to generate and her best bd vector can be specfed as a functon of that. Lemma 1. In order to play her best strategy, bdder only needs to choose her actvty ponts a and then among all the bd vectors b [0, q ] 1 such that r b = a her best strategy s to submt a bd b that mnmzes w b. We wll refer to one such bd vector as b (a ). Formally: b (a) = argmn b w b : b [0, q ] r b = a (3.5) 1 We use the notaton [a, b] to denote all the vectors that are componentwse greater than or equal to a and less than or equal to b

6 Proof. See the full verson. Based on Lemma 1 to descrbe a best strategy for a bdder we only need to specfy the actvty ponts a that she should bd and then Lemma 1 tells us what condton the correspondng bd vector should satsfy. The next lemma descrbes how we can effcently compute b (a ) for any gven a. Lemma 2. For any gven a [0, r q ] we can compute b (a ) by usng the followng procedure. Wthout loss of generalty, assume securtes are sorted n decreasng order r j w,j r j w,j of so that rj+1 w,j+1. To fnd the bd vector, we start from an ntal zero bd vector and ncrease each q,j up to q,j startng at j = 1 untl the generated actvty pont reaches a. The followng s a more formal defnton of b (a): such that: Proof. See the full verson. b (a) = ( q,1,, q,y 1, b,y, 0,, 0) (3.6) y 1 r y b,y + r j q,j = a (3.7) j=1 Intutvely, Lemma 2 s sayng that a strategc bdder should never sell any shares of a securty j unless for any other securty k for whch r k w,k > rj w,j she has already sold all of her shares of securty k. Defnton 1. We can defne a cost functon c (a) : [0, w q ] R for each bdder whch only depends on her actvty ponts: c (a) = w b (a) 0 a r q (3.8) Intutvely, for bdder, c (a) s the mnmum cost of generatng a actvty ponts. At ths pont, we can defne the bd vectors and all the equatons only n terms of a. Bdders only need to specfy ther actvty pont a. We denote the fnal actvty ponts of bdder when the aucton concluded by a and the fnal total actvty pont by A. The utlty of each bdder can now be wrtten as the followng: u = v (α a ) c (a ) (3.9) Also, the aucton concludes at the hghest clock α such that:

7 n α A = M (3.10) =1 Next, we defne the Nash-equlbrum. Before that, notce we can wrte the utlty of each bdder as u (a, A) whch s a functon of her own bd and the total aggregate bd. Formally: u (a, A) = v ( a A M) c (a) (3.11) Now we are ready to descrbe the Nash-equlbrum. Suppose a 1,, a n are the actvty ponts at whch the aucton has concluded. We say the outcome of the aucton s stable or s a Nash-equlbrum f for every bdder, a s a best response to a. For a Nash equlbrum, the frst order and bounry condtons are suffcent. Assume that ā denotes the maxmum possble actvty ponts that bdder can generate (.e., ā = r q ). The frst order and bounry condtons of the Nash-equlbrum are the followng: d u (a, A ) = 0 and 0 < a < ā or d N : u (a, A ) 0 and a = 0 (3.12) or d u (a, A ) 0 and a = ā n A = a (3.13) =1 Note that, to use the frst order condtons, we need u (a, A) to be a contnuous and dfferentable functon n ts doman. We can however relax the dfferentablty requrement and allow u (a, A) to have dfferent left and rght dervatves at a fnte number of ponts. In that case, f assume that ρ s the d left dervatve of u (a, A ) and ρ + s ts rght dervatve, then n the frst d condton, we can replace u (a, A ) = 0 wth ρ 0 ρ +. To keep the proofs smple, we do not use ths general form but we wll refer to t later when we explan how to compute the equlbrum. We further expand the frst order and bounry condtons. Notce that d u (a, A ) = a u (a, A ) + A u (a, A ) d A. Because we always have d A d = 1, we can wrte u (a, A ) = a u (a, A ) + A u (a, A ). So the frst order and bounry condtons can be rephrased as:

8 a u (a, A ) + A u (a, A ) = 0 and 0 a ā or N : a u (a, A ) + A u (a, A ) 0 and a = 0 (3.14) or a u (a, A ) + A u (a, A ) 0 and a = ā n A = a (3.15) =1 Next, we state the man theorem of ths secton whch gves a suffcent condton for the exstence of a Nash-equlbrum and provdes a method for computng t as well as a bddng strategy. Theorem 1. Consder the Aucton. 1, n whch as explaned before, each bdder s utlty s gven by u = v (αr b ) c b f the aucton stops at the current clock α. Assumng that the valuaton functons v are contnuous, dfferentable 2 and concave, there exsts a unque Nash-equlbrum that satsfes the frst order and bounry condtons of (3.14). Furthermore, there are bd functons g (α), such that for every f bdder bds b (a ) where a = g (α), then the outcome of the aucton concdes wth the unque Nash-equlbrum. g (α) s gven bellow (v and c are the dervatves of v and c ): g (α) = argmn a [0,ā] v (αa) M αa M α c (a) (3.16) Furthermore, the g (α) can be computed effcently usng bnary search on a (the parameter of the argmn) because the expresson nsde the absolute value s a decreasng functon of a. Note that the requrement of v functons beng concave s qute natural. It smply means that the dervatve of v should be decreasng whch can be nterpreted as the margnal value of the frst dollar receved beng more than the margnal value of the last dollar. It s worth mentonng that the bd functon g (α), as descrbed n (3.16) s not necessarly an ncreasng functon of α. In other words, as the clock goes down, g (α) may ncrease at some ponts whch means bdder s actually offerng more for sale although the prces are gong down. Ths phenomenon s n fact qute common when bdder has lqudty needs as we wll explan n secton 5. We defer the proof of Theorem 1 to secton 5. Instead of provng Theorem 1 drectly, we prove a more general theorem n the next secton. Later, n secton 5, we show that Theorem 1 s a specal case of that. 2 we may relax ths to allow v to have dfferent left and rght dervatves at a fnte number of ponts

9 4 Summaton Games In ths secton, we descrbe a general class of games whch we wll refer to as summaton games. Later, we show that the reverse aucton explaned n the prevous secton and some well known problems lke the Courant-Nash equlbrum of an olgopoly game [7] can be expressed n ths model. Next, we defne a Summaton Game: Defnton 2 (Summaton Game). There are n players N = {1,, n}. Each player can choose a number a from the nterval [0, ā ] where ā s a constant. The utlty of each bdder depends only on her own number as well as the sum of all the numbers. In other words, assumng that A = n =1 a, the utlty of each bdder s gven by u (a, A). We next show that f the utlty functons u (a, A) meet a certan requrement, the summaton game has a unque Nash-equlbrum that can be computed effcently. Before that, we defne the followng notaton Defnton 3. For each player, assumng that u (a, A) s her utlty functon, defne her characterstc functon h (x, T ) as the followng: h (x, T ) = a u (xt, T ) + A u (xt, T ) (4.1) Theorem 2. If all the characterstc functons h (x, T ) 3 are strctly decreasng functons n both x and T, then the game has a unque Nash-equlbrum 4 and n that equlbrum, the bd of each player s a = x (A)A where x s defned as the followng: x (T ) = argmn x [0,mn(1, ā T )] h (x, T ) (4.2) Furthermore, because h (x, T ) s decreasng n both x and T, x (T ) s also decreasng n T and the equlbrum can be computed effcently usng two nested bnary searches or usng an aucton-lke mechansm wth an ascendng clock T n whch the each bdder submts a = x (T )T and the clock T keeps gong up as long as a > T. Proof. See the full verson. To fnd the values of a s at the Nash-equlbrum we can use the followng algorthm: 3 Note that we allow h (x, T ) to be dscontnuous at a fnte number of ponts (e.g. a step functon). 4 f we relax the requrement of h s beng strctly decreasng to just beng nonncreasng then there s a contnuum of Nash-equlbra n whch there s one Nashequlbrum that s strctly preferred by some players and s just as good as other Nash equlbra for other players.

10 Algorthm 1 Start wth T = 0 (or a suffcently small postve T ). Keep ncreasng T for as long as n =1 x (T ) > 1. Stop as soon as n =1 x (T ) 1 and then set each bd a = x (T )T 5 Proof. See the full verson. Note that Alg. 1 can be mplemented ether usng bnary search on T or as an ascendng aucton-lke mechansm n whch each player submts the bd a = x (T )T where T s the ascendng clock and n whch the clock stops once a T. In the next secton we fnsh our analyss of the pooled reverse aucton of 1. Later, n secton 6, we gve example of a well-known problem that can be expressed n our model and ts Nash-equlbrum can be computed usng Alg Back to Pooled Reverse Aucton In the prevous secton we descrbed a more general class of games and n Theorem 2 we gave suffcent condtons for the exstence of a Nash-equlbrum. We explaned when t s unque and how to compute t. In ths secton we contnue our analyss of Aucton. 1. We frst gve a proof for Theorem 1 be reducng t to a specal case of Theorem 2. Next, we gve a proof for Theorem 1 whch s based on a reducton to Theorem 2. Proof (Proof of. Theorem 1). To be able to apply Theorem 2, we frst need to show that the utlty functon of each bdder n Aucton. 1 meets the requrement of Theorem 2. More specfcally, we have to show that h (x, T ) = a u (xt, T ) + A u (xt, T ) s a decreasng functon n both x and T. Remember that n our model for Aucton. 1, we can wrte the utlty of bdder as u (a, A) = v ( a A M) c (a). Frst, we show that c (a) s a convex functon. Lemma 1. The cost functon c (x) as defned n (3.8) s always a convex functon and has an non-decreasng frst order dervatve n [0, r. q] although ts dervatve mght be dscontnuous n at most m ponts. Proof. See the full verson. Lemma 2. The h (x, T ) functons for bdders n Aucton. 1 are decreasng n both x and T : h (x, T ) = a u (xt, T ) + A u (xt, T ) (5.1) = v (xm) 1 x + c T (T x) (5.2) 5 If n =1 x(t ) < 1 then arbtrarly choose each a from the nterval [lm ɛ 0+ x (T ɛ)t, x (T )T ] such that n =1 a = T (It s easy to show that each player s ndfferent to all a [lm ɛ 0+ x (T ɛ)t, x (T )T ]).

11 Proof. See the full verson. Snce n Lemma 2 we proved that h (x, T ) s decreasng n both x and T, we can now apply Theorem 2 and all of the clams of Theorem 1 follow from Theorem 2. Also note that g (α) whch was defned n (3.16) s actually the same as x (T )T, where T = M α It s nterestng to notce that the aucton-lke mechansm of Theorem 2 and Aucton. 1 are actually equvalent. In fact, the x (T ) where T = M/α, has a very natural nterpretaton n Aucton. 1. It specfes the fracton of the budget of the auctoneer that the bdder s demandng at the clock α. In fact we may modfy the aucton to ask the bdders to submt the amount of lqudty that they are demandng drectly at each step of the clock and then the aucton stops when the demand becomes less than or equal to the budget of the auctoneer. Then, each bdder wll be requred to sell enough quantty of her shares at the current prces to pay for the lqudty that she had demanded. It s easy to see that the lqudty that each bdder demands may only decrease as the α ncreases. However, the value of the bd, x (T )T, may actually ncrease because bdder I may want to mantan her demand for the lqudty. 6 Applcaton to Cournot s Olgopoly In ths secton, we show how the well-known problem of Cournot s Olgopoly can be expressed n our model of a summaton game and all the results of Theorem 2 can therefore be appled: Defnton 4 (Cournot s Olgopoly). There are n frms. The frms are olgopolst supplers of a homogenous good. At each perod, each frm chooses a quantty q to supply. The total supply Q on the market s the sum of all frms supples: Q = q (6.1) All frms receve the same prce p per unt of the good. The prce p on the market depends on the total supply Q as: p(q) = p 0 (Q max Q) (6.2) Each frm ncurs a cost c per unt of good. These costs can be dfferent for dfferent frms and are prvate nformaton Each frm s proft s gven by: u (q, Q) = (p(q) c )q (6.3) After each market perod, frms are nformed of the total quantty Q and the market prce p(q) of the prevous perod.

12 If we wrte down the h (x, T ) for each frm we get: h (x, T ) = a u (xt, T ) + A u (xt, T ) (6.4) = p(t ) c + p (T )T x (6.5) = p 0 (Q max T ) c p 0 T x (6.6) Notce that clearly the above h (x, T ) s a decreasng functon of both x and T and therefore all of the nce results of Theorem 2 can be appled. Notce n fact that as long as p(q) s concave and a decreasng functon of Q, h (x, T ) s stll a decreasng functon of both x and T and all of the results of Theorem 2 stll holds. 7 Concluson In ths paper we studed the Nash-equlbrum and equlbrum bddng strateges of the troubled assets reverse aucton. We further generalzed our analyss to a more general class of games wth non quas-lnear utltes. We proved the exstence and unqueness of a Nash-equlbrum n those games and we also gave an effcent way to compute the equlbrum of those games. We also showed that fndng the Nash equlbrum can be mplemented usng an ascendng mechansm so that the partcpants don t need to reveal ther utlty functons. We also, showed that even a more general verson of the well-known problem of Cournot s Olgopoly can be expressed n our model and all of the prevously mentoned results apply to that as well. References 1. L. Ausubel and P. Cramton. A troubled asset reverse aucton. In Workng paper, L. M. Ausubel, P. Cramton, E. Flz-Ozbay, N. Hggns, E. Ozbay, and A. Stockng. Common-value auctons wth lqudty needs:an expermental test of a troubled assets reverse aucton. Unversty of maryland, economcs workng paper seres, Department of Economcs, Unversty of Maryland, T. Bergstrom and H. Varan. Two remarks on cournot equlbra. Unversty of calforna at santa barbara, economcs workng paper seres, Department of Economcs, UC Santa Barbara, Feb R. Johar, S. Mannor, and J. N. Tstskls. Effcency loss n a network resource allocaton game. Mathematcs of Operatons Research, 29: , R. Johar and J. Tstskls. Effcency of scalar-parameterzed mechansms. Operaton Research, F. Kelly. Chargng and rate control for elastc traffc. European Transactons on Telecommuncatons, 8:33 37, A. Mas-colell, M. D. Whnston, and G. J. R. 8. S. Shenker, D. Clark, D. Estrn, and S. Herzog. Prcng n computer networks: Reshapng the research agen. ACM Computer Communcaton Revew, 26: , 1996.