Liquidity-Sensitive Automated Market Makers via Homogeneous Risk Measures

Transcription

1 Lqudty-Senstve Automated Market Makers va Homogeneous Rsk Measures Abraham Othman and Tuomas Sandholm Computer Scence Department, Carnege Mellon Unversty Abstract. Automated market makers are algorthmc agents that provde lqudty n electronc markets. A recent stream of research n automated market makng s the desgn of lqudty-senstve automated market makers, whch are able to adjust ther prce response to the level of actve nterest n the market. In ths paper, we ntroduce homogeneous rsk measures, the general class of lqudty-senstve automated market makers, and show that members of ths class are (necessarly and suffcently) the convex conjugates of compact convex sets n the non-negatve orthant. We dscuss the relaton between features of ths convex conjugate set and features of the correspondng automated market maker n detal, and prove that t s the curvature of the convex conjugate set that s responsble for mplctly regularzng the prce response of the market maker. We use our nsghts nto the dual space to develop a new famly of lqudty-senstve automated market makers wth desrable propertes. 1 Introducton Automated market makers are algorthmc agents that provde lqudty n electronc markets. Markets wth large event spaces or sparse nterest from traders mght fal because buyers and sellers have trouble fndng one another. Automated market makers can prevent ths falure by steppng n and provdng a counterparty for prospectve traders; nstead of makng bets wth each other, traders place bets wth the automated market maker. Automated market makers have been the object of theoretcal study nto market mcrostructure [Ostrovsky, 2009, Othman and Sandholm, 2010b] and successfully mplemented n practce n large electronc markets [Goel et al., 2008, Othman and Sandholm, 2010a]. A broad ntroducton to the mechancs of automated market makng can be found n Pennock and Sam [2007]. Othman et al. [2010] ntroduce a lqudty-senstve automated market maker. Ths market maker s able to adapt ts prce response to ncreasng actvty wthn the market; wth ths market maker bets wll not move prces very much when there s lots of money already wagered wth the market maker. Ths s n contrast to tradtonal market-makng agents that provde dentcal prce responses regardless of whether there are tens of dollars or tens of mllons of dollars wagered wth the market maker. Ths s an extended verson of the WINE 2011 paper that ncludes fgures and proofs.

2 Unfortunately, ths lqudty-senstve market maker does not generalze easly. In Othman et al. [2010] t s referred to as a technque to contnuously channel profts nto lqudty, a vew echoed by Abernethy et al. [2011]. Whle ths vew may be accurate, t s not prescrptve: t offers no nsght about how to create other lqudty-senstve market makers, or of the relaton between lqudty-senstve market makers and the other market makers of the lterature. In ths paper, we solve the puzzle of how lqudty-senstve market makers work, and ther relaton to other market makers from the lterature. We are able to contextualze, generalze, and expand the dea of lqudty-senstve market makers. In order to do ths, we frst stuate lqudty-senstve market makers wthn the same framework as ther lqudty-nsenstve counterparts. Usng a set of desderata taken from the predcton market and fnance lterature we ntroduce a new class of automated market makers, homogeneous rsk measures, whch we argue correctly embody the noton of lqudty senstvty, and we prove that the market maker of Othman et al. [2010] s a member of ths class. Our prncpal result s a necessary and suffcent characterzaton of the complete set of homogeneous rsk measures: they are the support functons of compact convex sets n the non-negatve orthant. Most ntrgungly, ths dual vew allows us to acheve a synthess between homogeneous rsk measures and the experts algorthm perspectve of Chen and Vaughan [2010], another recent vew of automated market makng. In ths perspectve, homgeneous rsk measures are unregularzed follow-the-leader algorthms that (generally) put non-unt total weght on the set of experts. We show t s the shape of the convex conjugate set (partcularly, that set s curvature) that mplctly acts as a regularzer for the homogeneous rsk measure. Furthermore, the bulge of the convex set away from the probablty smplex defnes notons lke the maxmum sum of prces. We use these nsghts to create a new famly of lqudty-senstve automated market makers, the unt ball market makers, that have desrable propertes: defned costs for any possble bet, defned bounds on sums of prces, and tghtly bounded loss. 2 Background In ths secton we provde a bref ntroducton to automated market makng, wth emphass on the recent results that gude the remander of the work. 2.1 Cost functons and rsk measures We consder a general settng n whch the future state of the world s exhaustvely parttoned nto n events, {ω 1,..., ω n }, so that exactly one of the ω wll occur. Ths model apples to a wde varety of settngs, ncludng fnancal models on stock prces and nterest rates, sports bettng, and tradtonal predcton markets. In our notaton, x s a vector and x s a scalar, 1 s the n-dmensonal vector of all ones, and f represents the -th element of the gradent of a functon f. The non-negatve orthant s gven by R n + {x mn x 0}.

3 Let U be a convex subset of R n. Our work concerns functons C : U R whch map vector payouts over the events to scalar values. A state refers to a vector of payouts. Traders make bets wth the market maker by changng the market maker s state. To move the market maker from state x to state x, traders pay C(x ) C(x). For nstance, f the state s x 1 = 5 and x 2 = 3, then the market maker needs to pay out fve dollars f ω 1 s realzed and pay out 3 dollars f ω 2 s realzed. If a new trader wants a bet that pays out one dollar f event ω 1 occurs, then they change the market maker s state to be {6, 3}, and pay C({6, 3}) C({5, 3}). There are two broad research streams that explore these functons. The predcton market lterature, where they are called cost functons, and the fnance lterature, where they are called rsk measures. We use the terms cost functon and rsk measure nterchangeably. The most popular cost functon used n Internet predcton markets s Hanson s logarthmc market scorng rule (LMSR), an automated market maker wth partcularly desrable propertes, ncludng bounded loss and a smple analytcal form [Hanson, 2003, 2007]. The LMSR s defned as ( ) C(x) = b log exp(x /b) for fxed b > 0. b s called the lqudty parameter, because t controls the magntude of the prce response of the market maker to bets. 1 For nstance, f the LMSR s used wth b = 10 n our example above, C({6, 3}) C({5, 3}).56, and so the market maker would quote a prce of 56 cents to the agent for ther bet. If b = 1, the same bet would cost 92 cents. The prces p of a dfferentable rsk measure are gven by the gradent of the cost functon the margnal cost on each event: p = exp(x /b) j exp(x j/b) Observe that the prces n the LMSR sum to one. The noton of sum of prces s crucal to our work. The market maker s proft cut (or vgorsh n gamblng contexts) can be thought of as the dfference between the sum of prces and unty [Othman et al., 2010]. Ths proft cut serves to compensate the market maker for takng bets wth traders, and typcal values for the vgorsh n real applcatons are small, rangng from one percent to 20 percent. Snce the LMSR and many other cost functons of the lterature [Chen and Pennock, 2007, Peters et al., 2007, Agrawal et al., 2009, Abernethy et al., 2011] do not have a proft cut, they can be expected to run at a loss n practce [Pennock and Sam, 2007]. 2.2 Lnk to onlne learnng One of the most ntrgung recent developments n automated market makng s the lnk between cost functons and onlne learnng algorthms, partcularly 1 Wth b = 1, the LMSR s equvalent to the entropc rsk measure of the fnance lterature [Föllmer and Sched, 2002].

4 between cost functons and onlne follow-the-regularzed-leader algorthms. Ths lnk frst appeared n a supportng role n Chen et al. [2008], and was sgnfcantly expanded n later work by those authors [Chen and Vaughan, 2010, Abernethy et al., 2011]. Any loss-bounded convex rsk measure (Secton 3 wll make ths precse) s equvalent to a no-regret follow-the-regularzed-leader onlne learnng algorthm. These onlne learnng algorthms are conventonally expressed not as cost functons (or, n the machne learnng lterature, potental functons), but rather n dual space [Shalev-Shwartz and Snger, 2007]. The dual-space formulaton s a powerful way of nterpretng and constructng automated market makers. Let Π be the probablty smplex. Chen and Vaughan [2010] show that we can wrte any convex rsk measure n terms of a convex optmzaton over a follow-the-leader term and a convex regularzer term. Ths optmzaton s n fact a conjugacy operaton restrcted to the probablty smplex: C(x) = max y Π x y f(y) Here, x y s the follow-the-leader term, and f s a regularzer. 2.3 The OPRS cost functon The Othman-Pennock-Reeves-Sandholm cost functon (OPRS) was orgnally ntroduced n Othman et al. [2010] as a lqudty-senstve extenson of the LMSR. The OPRS s defned as ( ) C(x) = b(x) log exp(x /b(x)) where b(x) = α x for α > 0. The OPRS can be contrasted wth the LMSR, for whch b(x) b. Unlke the LMSR, the OPRS s only defned over the non-negatve orthant (for contnuty we can set C(0) = 0). Also unlke the LMSR, the sum of prces n the OPRS s always greater than 1. The OPRS has several desrable propertes. These nclude a concse analytcal closed form and outcome-ndependent proft, the ablty to (for certan fnal quantty vectors) book a proft regardless of the realzed outcome. Perhaps the most practcal property of the OPRS s ts scale-nvarant lqudty senstvty: ts consstent prce reacton over dfferent scales of market actvty. (Ths scale-nvarance s a consequence of the OPRS cost functon beng postve homogeneous.) For large lqud markets, say wth mllons of dollars, a one-dollar bet wll have a much smaller mpact on prces than n a less-lqud market. Ths s not the case for the LMSR, where a one dollar bet moves prces the same amount n both heavly- and lghtly-traded markets.

5 3 Desderata, dual spaces, and an mpossblty result Ths secton expands upon the dual-space approach to automated market makng [Agrawal et al., 2009, Chen and Vaughan, 2010, Abernethy et al., 2011], partcularly as a vehcle for contextualzng and generalzng the OPRS. 3.1 Desderata and ther combnatons In ths secton we ntroduce fve desderata for cost functons. Each of these propertes has been acknowledged as desrable n the market makng lterature [Agrawal et al., 2009, Othman et al., 2010, Abernethy et al., 2011]. The market makers from the lterature satsfy varous subsets of these desderata. Desderatum 1 (Monotoncty) For all x and y such that x y, C(x) C(y). Monotoncty prevents smple arbtrages lke a trader buyng a zero-cost contract that never results n losses but sometmes results n gans. Desderatum 2 (Convexty) For all x and y and λ [0, 1] C(λx + (1 λ)y) λc(x) + (1 λ)c(y). Convexty can be thought of as a condton that encourages dversfcaton. The cost of the blend of two payout vectors s not greater than the sum of the cost of each ndvdually. Consequently, the market maker s ncentvzed to dversfy away ts rsk. The acknowledgment of dversfcaton as desrable goes back to the very begnnng of the mathematcal fnance lterature [Markowtz, 1952]. Desderatum 3 (Bounded loss) sup x [max (x ) C(x)] <. A market maker usng a cost functon wth bounded loss can only lose a fnte amount to nteractng traders, regardless of the traders actons and the realzed outcome. Desderatum 4 (Translaton nvarance) For all x and scalar α, C(x + α1) = C(x) + α. Translaton nvarance ensures that addng a dollar to the payout of every state of the world wll cost a dollar. Desderatum 5 (Postve homogenety) For all x and scalar γ > 0, C(γx) = γc(x). Postve homogenety ensures a scale-nvarant, currency-ndependent prce response, as n the OPRS. From a rsk measurement perspectve, postve homogenety ensures that doublng a rsk doubles ts cost. A cost functon that satsfes all of these desderata except bounded loss s called a coherent rsk measure. Coherent rsk measures were frst ntroduced n Artzner et al. [1999].

6 Defnton 1 A coherent rsk measure s a cost functon that satsfes monotoncty, convexty, translaton nvarance, and postve homogenety. When we relax postve homogenety from a coherent rsk measure, we get a convex rsk measure. Convex rsk measures were frst ntroduced n Carr et al. [2001] and feature promnently n the predcton market lterature [Hanson, 2003, Ben-Tal and Teboulle, 2007, Hanson, 2007, Chen and Pennock, 2007, Peters et al., 2007, Agrawal et al., 2009, Abernethy et al., 2011]. When we nstead relax translaton nvarance from a coherent rsk measure, we get what we dub a homogeneous rsk measure. Defnton 2 A homogeneous rsk measure s a cost functon satsfyng monotoncty, convexty, and postve homogenety. To our knowledge, the only homogeneous rsk measure of the lterature that s not also a coherent rsk measure s the OPRS. Proposton 1 The OPRS s a homogeneous rsk measure (for vectors n the non-negatve orthant). The desderata are global propertes that need to hold over the entre space the cost functon s defned over. It s often dffcult to verfy that a gven cost functon satsfes these desderata drectly, and nversely, t s dffcult to construct new cost functons that satsfy specfc desderata. Remarkably, each of these desderata have smple representatons n Legendre-Fenchel dual space. We proceed to descrbe these equvalences n the next secton, whch gves us a way to descrbe the entre space of homogeneous rsk measures. 3.2 Dual space equvalences The rest of the paper reles on the well-developed theory of convex conjugacy. Defnton 3 The Legendre-Fenchel dual (aka convex conjugate) of a convex cost functon C s a convex functon f : Y R over a convex set Y R n such that C(x) = max [x y f(y)] y Y We say that the cost functon s conjugate to the par Y and f. Convex conjugates exst unquely for convex cost functons defned over R n [Rockafellar, 1970, Boyd and Vandenberghe, 2004]. We wll refer to the convex optmzaton n dual space as the optmzaton or optmzaton problem, and the maxmzng y as the maxmzng argument. One way of nterpretng the dual s that t represents the prce space of the market maker, as opposed to a cost functon whch s defned over a quantty space [Abernethy et al., 2011]. The only prces a market maker can assume

7 are those y Y, whle the functon f serves as a measure of market senstvty and a way to lmt how quckly prces are adjusted n response to bets. As we have dscussed, n the predcton market lterature prces denote the partal dervatves of the cost functon [Pennock and Sam, 2007, Othman et al., 2010]. When t s unque, the maxmzng argument of the convex conjugate s the gradent of the cost functon, and when t s not unque, then the maxmzng arguments represent the subgradents of the cost functon. A fuller dscusson of the relaton between convex conjugates and dervatves s avalable n convex analyss texts [Rockafellar, 1970, Boyd and Vandenberghe, 2004]. Another nterpretaton of the dual space s from onlne learnng, specfcally onlne regularzed follow-the-leader algorthms [Chen and Vaughan, 2010]. We dscussed the lterature relatng to ths lnk n Secton 2.2. Here, the set Y represents the allowable weghts we can assgn to experts, and the functon f s a regularzer that determnes how quckly we adjust the weght between experts n response to returns whch are the same as payouts n ths nterpretaton. Generally speakng when the set Y exceeds the probablty smplex Π, then the weghts placed on the experts wll not be guaranteed to sum to unty. Wth these nterpretatons n mnd, we proceed to show the power of the dual space: we can represent homogeneous rsk measures wth a compact convex set n the non-negatve orthant. The relatons between convex and monotonc cost functons, convex and postve homogeneous cost functons, and ther respectve duals are a consequence of well-known results n the convex analyss lterature [Rockafellar, 1966, 1970]. Proposton 2 A rsk measure s convex and monotonc f and only f the set Y s exclusvely wthn the non-negatve orthant. Proposton 3 A rsk measure s convex and postve homogeneous f and only f ts convex conjugate has compact Y and has f(y) = 0 for every y Y. In the lterature ths latter result relates ndcator sets (here, the set Y) to support functons (here, the cost functon). Snce f(y) = 0 for all y Y, the cost functon conjugacy s defned only by the set Y. Consequently, we wll abuse termnology slghtly and refer to the cost functons as conjugate to the convex compact set alone. A necessary and suffcent condton on the set of homogeneous rsk measures follows. Corollary 1 A cost functon s a homogeneous rsk measure f and only f t s conjugate to a compact convex set n the non-negatve orthant. The followng results can be derved from convex analyss and the work of Abernethy et al. [2011]. Proposton 4 A rsk measure s convex, monotonc, and translaton nvarant f and only f the set Y les exclusvely on the probablty smplex. Proposton 5 A rsk measure s convex and has bounded loss f and only f the set Y ncludes the probablty smplex.

8 The only market maker that satsfes all fve of our desderata s max. Proposton 6 The only coherent rsk measure wth bounded loss s C(x) = max x. The max market maker corresponds to an order-matchng, rsk-averse cost functon that ether charges agents nothng for ther transactons, or exactly as much as they could be expected to gan n the best case. For nstance, a trader wshng to move the max market maker from state {5, 3} to state {7, 3} would be charged 2 dollars, exactly as much as they would wn f the frst event happened whch means takng the bet s a domnated acton. On the other hand, a trader wshng to move the market maker from state {5, 3} to state {5, 5} pays nothng! These two small examples suggest that max s a poor rsk measure n practce, and therefore Proposton 6 should be vewed as an mpossblty result. Combnng all of our dual-space equvalences, we have that the conjugate of max s defned exclusvely on the whole probablty smplex, where t s dentcally 0. 2 There are two ways to smooth out the prce response of max: One way s to use a regularzer, so that prce estmates do not mmedately jump to the axes (.e., zero or one). Ths corresponds to a regularzed onlne follow-the-leader algorthm, whch s a convex rsk measure [Chen and Vaughan, 2010]. We ntroduce a dfferent approach, to expand the shape of the vald prce, so that the shape of the space tself serves as an mplct regularzer over the prce estmates. Ths wll generally result n prces that are not probablty dstrbutons, and as we explore n the next secton, ths approach leads to homogeneous rsk measures. 4 Shapng the dual space Recall that only the convex conjugate set Y of a homogenenous rsk measure s responsble for determnng the market maker s behavor, because the conjugate functon f takes value zero everywhere n that set. In ths secton, we explore two features of the conjugate set that produce desrable propertes: ts curvature and ts dvergence from the probablty smplex. (We dscuss these propertes n relaton to the OPRS n Appendx B.) 4.1 Curvature We would lke for our cost functon to always be dfferentable (outsde of 0, where a dervatve of a postve homogeneous functon wll not generally exst). The OPRS s dfferentable n the non-negatve orthant (agan, exceptng 0) whle max s dfferentable only when the maxmum s unque. In ths secton, 2 In dual prce space, the maxmzng argument to the max cost functon can always be represented as one of the axes. In the onlne learnng vew, max represents an unregularzed follow-the-leader algorthm, puttng all of ts probablty weght on the current best expert (.e., the event wth largest current payout).

9 we show that only curved conjugate sets produce homogeneous rsk measures that are dfferentable. 3 Defnton 4 A closed, convex set Y s strctly convex f ts boundary does not contan a non-degenerate lne segment. Formally, let Y denote the boundary of the set. Let 0 λ 1 and x, x Y. Then λx + (1 λ)x Y holds only for x = x. Snce strctly convex sets are never lnear on ther boundary they can be thought of as sets wth curved boundares. Proposton 7 A homogeneous rsk measure s dfferentable on R n \0 f and only f ts conjugate set s strctly convex. 4.2 Dvergence from probablty smplex The amount of dvergence from the probablty smplex governs the market maker s dvergence from translaton-nvarant prces (.e., prces that sum to unty). Recall that max s the homogeneous rsk measure that s defned only over the probablty smplex. Proposton 8 Let Y be the dual set of a dfferentable homogeneous rsk measure. Then the maxmum sum of prces (the most a trader would ever need to spend for a unt guaranteed payout) s gven by max y Y y, and the mnmum sum of prces (the most the market maker would ever pay for a unt guaranteed payout) s gven by mn y Y y. Gven any (effcently representable) convex set correspondng to a dfferentable homogeneous rsk measure, the extreme prce sums can be solved for n polynomal tme, snce t s a convex optmzaton over a convex set. It was shown n Othman et al. [2010] that the OPRS acheved ts maxmum sum of prces for quantty vectors that are scalar multples of 1. A corollary of the above result s that ths property holds for every homogeneous rsk measure. (Other vectors may also acheve the same sum of prces.) Corollary 2 In a homogeneous rsk measure every vector that s a postve multple of 1 acheves the maxmum sum of prces. 3 It mght be argued that what we are really nterested n, partcularly f we clam that curved sets act as a regularzer n the prce response, s whether or not curved sets also mply contnuous dfferentablty of the cost functon. Contnuous dfferentablty would mean that prces both exst and are contnuous n the quantty vector. These condtons are n fact the same for convex functons defned over an open nterval (such as R n \0), because for such functons dfferentablty mples contnuous dfferentablty [Rockafellar, 1970].

10 In addton to maxmum prces, the shape of the convex set also determnes the worst-case loss of the resultng market maker. The noton of worst-case loss s closely related to our desderatum of bounded loss a market maker wth unbounded worst-case loss does not have bounded loss, and a market maker wth fnte worst-case loss has bounded loss. Defnton 5 The worst-case loss of a market maker s gven by max x C(x) + C(x 0 ) where x 0 R n + s some ntal quantty vector the market maker selects. In homogeneous rsk measures, the amount of lqudty senstvty s proportonal to the market s state. Snce n practce there s some latent level of nterest n tradng on the event before the market s ntaton, t s desrable to seed the market ntally to reflect a certan level of lqudty. It s desrable to have a tght bound on that worst-case loss, reflectng that n practce, market admnstrators are lkely to have bounds on how much the market maker could lose n the worst case. Tght bounds on worst-case loss assure the admnstrator that that bound wll be satsfed wth maxmum lqudty njected at the market s ntaton. Proposton 9 Let Y be a convex set conjugate to a homogeneous rsk measure that ncludes the unt axes but does not exceed the unt hypercube. Then the worst-case loss of the rsk measure s tghtly bounded by the ntal cost of the market s startng pont. By brngng x 0 as close as desred to 0, we have the followng corollary, whch s a generalzaton of a smlar result for the OPRS. Corollary 3 Let Y be a convex set conjugate to a homogeneous rsk measure that ncludes the unt axes. Then the worst-case loss of the rsk measure can be set arbtrarly small. A bound on prces also emerges from ths result. Corollary 4 Let Y be a convex set conjugate to a homogeneous rsk measure that ncludes the unt axes but does not exceed the unt hypercube. Then the maxmum prce on any event s 1. 5 A new famly of lqudty-senstve market makers We proceed to use our theoretcal results constructvely, to create a famly of homogeneous rsk measures wth desrable propertes that the OPRS, the only pror homogeneous rsk measure, lacks. These nclude tght bounds on mnmum sum of prces and worst-case losses, and defnton over all of R n. Our new famly of market makers s parameterzed (n much the same way as the OPRS) by the maxmum sum of prces. The OPRS s not a member of ths new famly.

11 Our scheme s to take as our dual set the ntersecton of two unt balls n dfferent L p norms, one ball at 0 and the other ball at 1. For 1 < p <, the ntersecton of the two balls s a strctly convex set that ncludes the unt axes but does not exceed the unt hypercube. (At p = 1, we get the probablty smplex, whch s not strctly convex. At p = we get the unt hypercube, whch s also not strctly convex.) Let p denote the L p norm. Then we can defne the vectors n the ntersectons of the unt balls, U(p), as U(p) {y y R n, y 1 p 1, y p 1} Fgure 1 provdes graphcal ntuton for the set U(p) n a smple two-event market. Ths set gves us a cost functon C(x) = max y U(p) x y. We dub ths the unt ball market maker. Snce we can easly test whether a vector s wthn both unt balls (.e., wthn U(p)), the optmzaton problem for the cost functon can be solved n polynomal tme Fg. 1. The conjugate set of the unt ball market maker n dual space s gven by the ntersecton of two unt balls (the dark regon) n a certan L p norm. Here, n = 2 and p 1.08, so by the formula below the maxmum sum of prces s Ths famly of market makers s parameterzed by the L p norm that defnes whch vectors n dual space are n the convex set. By choosng the value of p correctly, we can engneer a market maker wth the desred maxmum sum of prces. The outer boundary of the set s defned by the unt ball from 0 n L p space. Its boundary along 1 s gven by the k that solves p nk p = 1. Solvng for k we get k = n 1/p, and so the maxmum sum of prces s nk = n ( n 1/p) = n 1 1/p. For prces that are at most 1 + v, we can set 1 + v = n 1 1/p. Solvng

12 ths equaton for p yelds p = log n log n log(1 + v) Gven any target maxmum level of vgorsh, ths formula provdes the exponent of the unt ball market maker to use. Consderng that only small dvergences away from unty are natural to the settng, the p we select for our L p norm should be qute small. The norm ncreases n the maxmum sum of prces, and for larger n the same norm produces larger sums of prces. One of the advantages of the unt ball market maker s that t s defned over all of R n, as opposed to just the non-negatve orthant. Its behavor n the postve orthant s to charge agents more than a dollar for a dollar guaranteed payout, because the outer boundary s dverges outwards from the probablty smplex. Its behavor n the negatve orthant, where ts ponts on the nner boundary are selected n the maxmzaton, s to pay less than a dollar for a dollar guaranteed payout. Its behavor n all other orthants s equvalent to max, as the unt axes are selected as maxmzng arguments. Fnally, f we restrct the unt ball market maker to only the non-negatve orthant (lke the OPRS), the sum of prces s tghtly bounded between 1 and n 1 1/p. 6 Conclusons and future work Usng fve desderata that have appeared n the fnance and predcton market lterature, we contextualzed a new class of cost functons, whch we dubbed homogeneous rsk measures. We showed that the OPRS [Othman et al., 2010] s a member of ths class, because t s convex, monotonc, and postve homogeneous. We proved only the max cost functon satsfes all fve of our desderata, but t does not have a dfferentable prce response. To produce a dfferentable prce response, one can add a regularzer, leadng to the regularzed onlne learnng algorthms explored by Chen and Vaughan [2010]. Another approach s to curve the conjugate dual space, relaxng t from the probablty smplex. We dscussed how the propertes of the convex set nduce desrable propertes n ts conjugate homogeneous rsk measure. Fnally, usng our nsghts, we developed a new famly of homogeneous rsk measures, the unt ball market makers, wth desrable propertes. Our work centered on cost functons that are postve homogeneous, because these are the only cost functons that dsplay dentcal relatve prce responses at dfferent levels of lqudty. However, another drecton s to explore cost functons that dsplay some characterstcs of lqudty senstvty (more muted prce responses at hgh levels of lqudty) wthout necessarly beng homogeneous. Fnally, we are attracted to the work of Agrawal et al. [2009] because t provdes a framework to smply add functonalty to handle lmt orders (orders of the form I wll pay no more than p for the payout vector x ) nto a cost functon market maker. That framework reles on convex optmzaton and so would also be able to run n polynomal tme, a sgnfcant gan over naïve mplementatons

13 of lmt orders wthn cost functon market makers. However, that work reled heavly on smplfcatons to the optmzaton that could be made because of translaton nvarance, so t s unclear how to embed a market maker whose convex conjugate s defned over more than the probablty smplex nto a lmt order framework. Acknowledgements We thank Peter Carr, Ylng Chen, Geoff Gordon, Dlp Madan, Dave Pennock, Steve Shreve, and Kevn Waugh for helpful dscussons and suggestons. Ths work was supported by NSF grants CCF , IIS , IIS , and by the Google Fellowshp n Market Algorthms. Bblography J. Abernethy, Y. Chen, and J. W. Vaughan. An optmzaton-based framework for automated market-makng. In ACM Conference on Electronc Commerce (EC), S. Agrawal, E. Delage, M. Peters, Z. Wang, and Y. Ye. A unfed framework for dynamc par-mutuel nformaton market desgn. In ACM Conference on Electronc Commerce (EC), pages , P. Artzner, F. Delbaen, J. Eber, and D. Heath. Coherent measures of rsk. Mathematcal fnance, 9(3): , A. Ben-Tal and M. Teboulle. An old-new concept of convex rsk measures: The optmzed certanty equvalent. Mathematcal Fnance, 17(3): , S. Boyd and L. Vandenberghe. Convex Optmzaton. Cambrdge Unversty Press, P. Carr, H. Geman, and D. Madan. Prcng and hedgng n ncomplete markets. Journal of Fnancal Economcs, 62(1): , Y. Chen and D. M. Pennock. A utlty framework for bounded-loss market makers. In Proceedngs of the 23rd Annual Conference on Uncertanty n Artfcal Intellgence (UAI), pages 49 56, Y. Chen and J. W. Vaughan. A new understandng of predcton markets va no-regret learnng. In ACM Conference on Electronc Commerce (EC), pages , Y. Chen, L. Fortnow, N. Lambert, D. M. Pennock, and J. Wortman. Complexty of combnatoral market makers. In ACM Conference on Electronc Commerce (EC), pages , H. Föllmer and A. Sched. Stochastc Fnance (Studes n Mathematcs 27). De Gruyter, S. Goel, D. Pennock, D. Reeves, and C. Yu. Yoopck: a combnatoral sports predcton market. In Proceedngs of the Natonal Conference on Artfcal Intellgence (AAAI), pages , R. Hanson. Combnatoral nformaton market desgn. Informaton Systems Fronters, 5(1): , R. Hanson. Logarthmc market scorng rules for modular combnatoral nformaton aggregaton. Journal of Predcton Markets, 1(1):1 15, 2007.

14 H. Markowtz. Portfolo Selecton. The Journal of Fnance, 7(1):77 91, M. Ostrovsky. Informaton aggregaton n dynamc markets wth strategc traders. In ACM Conference on Electronc Commerce (EC), pages , A. Othman and T. Sandholm. Automated market-makng n the large: the Gates Hllman predcton market. In ACM Conference on Electronc Commerce (EC), pages , 2010a. A. Othman and T. Sandholm. When Do Markets wth Smple Agents Fal? In Internatonal Conference on Autonomous Agents and Mult-Agent Systems (AAMAS), pages , Toronto, Canada, 2010b. A. Othman, D. M. Pennock, D. M. Reeves, and T. Sandholm. A practcal lqudty-senstve automated market maker. In ACM Conference on Electronc Commerce (EC), pages , D. Pennock and R. Sam. Computatonal Aspects of Predcton Markets. In Algorthmc Game Theory, chapter 26, pages Cambrdge Unversty Press, M. Peters, A. M.-C. So, and Y. Ye. Par-mutuel markets: Mechansms and performance. In Internatonal Workshop On Internet And Network Economcs (WINE), pages 82 95, R. T. Rockafellar. Level sets and contnuty of conjugate convex functons. Transactons of the Amercan Mathematcal Socety, 123(1):46 63, R. T. Rockafellar. Convex Analyss. Prnceton Unversty Press, S. Shalev-Shwartz and Y. Snger. A prmal-dual perspectve of onlne learnng algorthms. Machne Learnng, 69: , December 2007.

15 A Proofs The OPRS s a homogeneous rsk measure (for vectors n the non-negatve orthant). Proof. We must prove that the OPRS satsfes postve homogenety, convexty, and monotoncty. Of these, t has already been establshed the OPRS satsfes postve homogenety n Othman et al. [2010]. Monotoncty also holds, because t s possble to wrte ndvdual prces n the OPRS as the sum of non-negatve components, and a functon wth well-defned postve partal dervatves satsfes monotoncty. Convexty of the OPRS follows from the relaton of the OPRS to the perspectve functon of the convex log-sum-exp functon: log ( exp x ). The perspectve functon g of a convex functon f s defned as g(z, t) tf(z/t) for t > 0. The perspectve functon s convex n both z and t [Boyd and Vandenberghe, 2004]. Now let f be the log-sum-exp functon, whch s convex. Then consder the relaton between g(x, α x ) and g(y, α y ). Snce the perspectve functon s convex n both of ts arguments, we have for all λ [0, 1]: ( ( λg x, α ) x + (1 λ)g y, α ) y ( g λx + (1 λ)y, α ) λx + (1 λ)y But observe that C(z) g ( z, α z ) and so λc(x) + (1 λ)c(y) C(λx + (1 λ)y) whch proves convexty of the OPRS. The only coherent rsk measure wth bounded loss s C(x) = max x. Proof. Suppose there exsts a coherent rsk measure C and a vector x wth max x = x but C(x) x. Snce C s convex, t s contnuous. Therefore C(0) = 0, because the functon s postve homogeneous and for every z lm C(γz) = 0 γ 0 So by translaton nvarance: C( x1) = x, and so by monotoncty, C(x) x. However, f C(x) < x

16 then the loss s unbounded, because So snce the loss must be bounded, whch s a contradcton. lm k x C(kx) = lm k ( x C(x)) =. k k C(x) = x A homogeneous rsk measure s dfferentable on R n \0 f and only f ts conjugate set s strctly convex. Proof. Frst recall that the maxmzng argument of the optmzaton for vector x 0 s gven by an extreme hyperplane normal to x that ntersects wth the set Y. It follows that the maxmzng argument s always on the boundary of Y. For the forward drecton, consder a set that s convex but not strctly convex. Then there exsts a hyperplane H connectng two ponts x and x on the boundary of the set such that λx + (1 λ)x Y Now consder the set of ponts normal to H whch go through 0. Observe that these ponts le n precsely two orthants. The ponts n one of those orthants wll fnd that the set of ponts between x and x are the maxmzng arguments n the optmzaton, whch means the subgradent at those ponts s not untary, and so the cost functon s not dfferentable. Now consder a strctly convex set Y. Snce the optmzaton s convex, the maxmzng arguments must form a convex set. Therefore f more than one vector s n a maxmzng argument, the lne connectng those vectors must be on the boundary of Y. But then Y s not strctly convex, a contradcton. Let Y be the dual set of a dfferentable homogeneous rsk measure. Then the maxmum sum of prces (the most a trader would ever need to spend for a unt guaranteed payout) s gven by max y Y and the mnmum sum of prces (the most the market maker would ever pay for a unt guaranteed payout) s gven by mn y Y Proof. Recall that the maxmzng argument n the maxmzaton yelds the gradent of the cost functon. The pont n Y wth the largest sum of components (and therefore the largest sum of prces) s selected as the maxmzng argument for x = k1, k > 0. The mnmum sum of prces result holds by smlar logc; the pont n Y wth mnmum sum of components s selected for x = k1. y y

17 Let Y be a convex set conjugate to a homogeneous rsk measure that ncludes the unt axes but does not exceed the unt hypercube. Then the worst-case loss of the rsk measure s tghtly bounded by the ntal cost of the market s startng pont. Proof. Frst, note that any convex set that ncludes the unt axes s conjugate to a cost functon that s at least as large as max, because max s conjugate to the mnmal convex set that ncludes the unt axes and ncreasng the sze of the feasble regon never decreases the value of a maxmzaton. Therefore, the worst-case loss s bounded from above by C(x 0 ). To show that ths bound s tght, we need to show that there exsts a termnal state C(x) where max x = C(x) Such a termnal state s gven by the axes.

18 B The OPRS and ts conjugate set In a broad sense, the overall approach of ths paper s to move from convex ndcator sets n the non-negatve orthant to homogeneous rsk measures wth desrable propertes. For the OPRS, however, we already have a homogeneous rsk measure, and n ths secton we wll explore how to produce ts conjugate convex set. Recall that the OPRS s gven by ( ) C(x) = b(x) log exp(x /b(x)) where b(x) = α x for α > 0 and x R n +. Because the OPRS s monotonc and convex, recall from the prevous secton that t must be conjugate to a convex set n the non-negatve orthant, Y, so C(x) = max y Y x y Observe that we cannot solve for ths convex set drectly, because we only know x and C(x). However, for every x, we can fnd a hyperplane on whch at least one pont s the outer boundary of the convex set. We defne h(x) as h(x) { p p R n + and x p C(x) }. For each x, ths partton dvdes the non-negatve orthant nto two sets those ponts that could be part of the convex set (all the ponts n h(x), under the separatng hyperplane), and those ponts that could not be part of the convex set (or else C(x) would be larger). In order to fully recover the convex set Y, we need to take the ntersecton of every h(x): Y = h(x) x R n + We can smplfy ths operaton consderably, however. Snce the OPRS s homogeneous, we need only consder the ntersecton over the x n the probablty smplex. Ths s because the same pont y Y wll solve the maxmzaton problem for all γx, γ > 0. Therefore, t suffces to only consder the ntersecton of a set of ponts X such that for all x R n +, there exsts an x X such that γx = x for some γ > 0. One such set X s the probablty smplex. Usng ths result, we proceed to plot the convex conjugate ndcator set of the OPRS wth α =.05 n two dmensons as Fgure 2. Because the OPRS s only defned n the non-negatve orthant, t s only the outer boundary of the convex set that s relevant to the prce response.

19 Fg. 2. The convex set n dual space supported by the OPRS market maker n a smple two-event market. (Ths s not the case for cost functons defned over all of R n, because the outer boundary s never selected for vectors n the negatve orthant.) In order to show the dvergence of the outer boundary, Fgure 2 dsplays the nner boundary of the conjugate set as the probablty smplex.