Log-time Prediction Markets for Interval Securities
Miroslav Dudík, Xintong Wang, David M. Pennock, David M. Rothschild
LLog-time Prediction Markets for Interval Securities
Miroslav Dudík ∗ Microsoft Research, New York, [email protected]
Xintong Wang ∗ University of Michigan, Ann Arbor, [email protected]
David M. Pennock
Rutgers University, New Brunswick, [email protected]
David M. Rothschild
Microsoft Research, New York, [email protected]
ABSTRACT
We design a prediction market to recover a complete and fully gen-eral probability distribution over a random variable. Traders buyand sell interval securities that pay $1 if the outcome falls into aninterval and $0 otherwise. Our market takes the form of a central automated market maker and allows traders to express intervalendpoints of arbitrary precision. We present two designs in bothof which market operations take time logarithmic in the numberof intervals (that traders distinguish), providing the first computa-tionally efficient market for a continuous variable. Our first designreplicates the popular logarithmic market scoring rule (LMSR), butoperates exponentially faster than a standard LMSR by exploitingits modularity properties to construct a balanced binary tree anddecompose computations along the tree nodes. The second designconsists of two or more parallel LMSR market makers that mediatesubmarkets of increasingly fine-grained outcome partitions. Thisdesign remains computationally efficient for all operations, includ-ing arbitrage removal across submarkets. It adds two additionalbenefits for the market designer: (1) the ability to express utility forinformation at various resolutions by assigning different liquidityvalues, and (2) the ability to guarantee a true constant bounded lossby appropriately decreasing the liquidity in each submarket.
KEYWORDS prediction market; automated market maker; expressive betting
ACM Reference Format:
Miroslav Dudík ∗ , Xintong Wang ∗ , David M. Pennock, and David M. Roth-schild. 2021. Log-time Prediction Markets for Interval Securities. In Proc.of the 20th International Conference on Autonomous Agents and MultiagentSystems (AAMAS 2021), Online, May 3–7, 2021 , 9 pages+5 pages appendix.
Consider a one-dimensional random variable, such as the openingvalue of the S&P 500 index on December 17, 2021. We design amarket for trading interval securities corresponding to predictionsthat the outcome will fall into some specified interval, say between2957.60 and 3804.59, implemented as binary contracts that pay out$1 if the outcome falls in the interval and $0 otherwise. We areinterested in designing automated market makers to facilitate a fully expressive market computationally efficiently. Traders can select ∗ Authors contribute equally.
Proc. of the 20th International Conference on Autonomous Agents and Multiagent Systems(AAMAS 2021), U. Endriss, A. Nowé, F. Dignum, A. Lomuscio (eds.), May 3–7, 2021, Online custom interval endpoints of arbitrary precision corresponding to acontinuous outcome space, whereas the market maker will alwaysoffer to buy or sell any interval security at some price.A form of interval security called the condor spread is commonin financial options markets, with significant volume of trade. Eachcondor spread involves trading four different options, and financialoptions offered by the market may only support a limited subset ofapproximate intervals. As of this writing, S&P 500 options expiringon December 17, 2021, distinguish 56 strike prices, allowing thepurchase of around 1500 distinct intervals of minimum width 25.Moreover, as each strike price trades independently despite thelogical constraints on their relative values, it will require time linearin the number of offered strike prices to remove arbitrage.Outside traditional financial markets, the logarithmic marketscoring rule (LMSR) market maker [15, 16] has been used to elicitinformation through the trade of interval securities. The Gates Hill-man Prediction Market at Carnegie Mellon University operatedLMSR on 365 outcomes, representing 365 days of one year, to fore-cast the opening time of the new computer science building [21].Traders could bet on different intervals by choosing a start and anend date. A similar market was later launched at the University ofTexas at Austin, using a liquidity-sensitive variation of LMSR [20].Moreover, LMSR has been deployed to predict product-sales lev-els [23], instructor ratings [4], and political events [14].LMSR has two limitations that prevent its scaling to marketswith a continuous outcome space. First, LMSR’s worst-case losscan grow unbounded if traders select intervals with prior proba-bility approaching zero [12]. Second, standard implementations ofLMSR operations run in time linear in the number of outcomes ordistinct future values traders define—in our case, arbitrarily many.The constant-log-utility and other barrier-function-based marketmakers [8, 22] achieve constant bounded loss, but still suffer thesecond limitation regarding computational intractability. Thus, pre-vious markets allow only a relatively small set of predeterminedintervals and run in time linear in the number of supported out-comes, limiting the ability to aggregate high-precision trades andelicit the full distribution of a continuous random variable.In this paper, we propose two automated market makers thatperform exponentially faster than the standard LMSR and previousdesigns. Market operations (i.e., price , cost , and buy ) can be exe-cuted in time logarithmic in the number of distinct intervals traded, A call option written on an underlying stock with strike price 𝐾 and expiration date 𝑇 pays max { 𝑆 − 𝐾, } , where 𝑆 is the opening price of the stock on date 𝑇 . For example,25 shares of “$1 iff [2650,2775]” ≈ max { 𝑆 − , } − max { 𝑆 − , } − max { 𝑆 − , } + max { 𝑆 − , } . a r X i v : . [ c s . G T ] F e b r linear in the number of bits describing the outcome space. Ourfirst market maker calculates LMSR exactly, but employs a balancedbinary tree to implement interval queries and trades. We show thatthe normalization constant of LMSR—a key quantity in its price andcost function—can be calculated recursively via local computationson the balanced tree. Our work here contributes to the rich litera-ture that aims to overcome the worst-case constant bounded loss indepen-dent of market precision and prices can be kept coherent efficientlyby removing arbitrages across submarkets. We demonstrate throughagent-based simulation that our second design enjoys more flexibleliquidity choices to facilitate the information-gathering objective:it can get close to the “best of both worlds” displayed by coarse andfine LMSR markets, with prices converging fast at both resolutionsregardless of the traders’ information structure.The two proposed designs, to our knowledge, are the first to si-multaneously achieve expressiveness and computational efficiency.As both market makers facilitate trading intervals at arbitrary pre-cision, they can elicit any probability distribution over a continuousrandom variable that can be practically encoded by a machine. Weuse the S&P 500 index value as a running example, but our frame-work is generic and can handle any one-dimensional continuousvariable, for example, the landfall point of a hurricane along a coast-line or the number of tickets sold in the first week of a movie release. We first review cost-function-based market making [1, 8], and thenintroduce interval markets.
Let Ω denote a finite set of outcomes , corresponding to mutuallyexclusive and exhaustive states of the world. We are interested ineliciting expectations of binary random variables 𝜙 𝑖 : Ω → { , } ,indexed by 𝑖 ∈ I , which model the occurrence of various events,such as “ S&P 500 will open between 2957.60 and 3804.59 on December17, 2021 ”. Each variable 𝜙 𝑖 is associated with a security that pays out 𝜙 𝑖 ( 𝜔 ) when the outcome 𝜔 ∈ Ω occurs, and thus 𝜙 𝑖 is also calledthe payoff function . Binary securities pay out $1 if the specifiedevent occurs and $0 otherwise. The vector ( 𝜙 𝑖 ) 𝑖 ∈I is denoted 𝝓 .Traders trade bundles 𝜹 ∈ R |I | of security with a central marketmaker, where positive entries in 𝜹 correspond to purchases andnegative entries to short sales. A trader holding a bundle 𝜹 receivesa payoff of 𝜹 · 𝝓 ( 𝜔 ) , when 𝜔 occurs.Following [1] and [8], we assume that the market maker deter-mines security prices using a convex and differentiable potentialfunction 𝐶 : R |I | → R , called a cost function . The state of the mar-ket is specified by a vector 𝜽 ∈ R |I | , listing the number of shares ofeach security sold by the market maker so far. A trader who wantsto buy a bundle 𝜹 in the market state 𝜽 must pay 𝐶 ( 𝜽 + 𝜹 ) − 𝐶 ( 𝜽 ) to the market maker, after which the new state becomes 𝜽 + 𝜹 . The vector of instantaneous prices in the corresponding state 𝜽 is 𝒑 ( 𝜽 ) (cid:66) ∇ 𝐶 ( 𝜽 ) . Its entries can be interpreted as the market’scollective estimates of E [ 𝜙 𝑖 ] : a trader can make an expected profitby buying (at least a small amount of) the security 𝑖 if she believesthat E [ 𝜙 𝑖 ] is larger than the instantaneous price 𝑝 𝑖 ( 𝜽 ) = 𝜕𝐶 ( 𝜽 )/ 𝜕𝜃 𝑖 ,and by selling if she believes the opposite. Therefore, risk neutraltraders with sufficient budgets maximize their expected profitsby moving the price vector to match their expectation of 𝝓 . Anyexpected payoff must lie in the convex hull of the set { 𝝓 ( 𝜔 )} 𝜔 ∈ Ω ,which we denote M and call a coherent price space with its elementsreferred to as coherent price vectors .We assume that the cost function satisfies two standard prop-erties: no arbitrage and bounded loss . The no-arbitrage propertyrequires that as long as all outcomes 𝜔 are possible, there be nomarket transaction with a guaranteed profit for a trader. In thispaper, we use the fact that 𝐶 is arbitrage-free if and only if it yieldsprice vectors 𝒑 ( 𝜽 ) that are always coherent [1]. The bounded-loss property is defined in terms of the worst-case loss of a marketmaker, sup 𝜽 ∈ R |I| sup 𝜔 ∈ Ω (cid:2) 𝜽 · 𝝓 ( 𝜔 ) − 𝐶 ( 𝜽 ) + 𝐶 ( ) (cid:3) , meaning thelargest difference, across all possible trading sequences and out-comes, between the amount that the market maker has to pay thetraders (once the outcome is realized) and the amount that the mar-ket maker has collected (when securities were traded). The propertyrequires that this worst-case loss be a priori bounded by a constant. In a complete market, we have I = Ω . Securities are indicators ofindividual outcomes, 𝜙 𝑖 ( 𝜔 ) = { 𝜔 = 𝑖 } , where 1 {·} denotes thebinary indicator. We denote each market security as 𝜙 𝜔 . A risk-neutral trader is incentivized to move the price of each security 𝜙 𝜔 to her estimate of E [ 𝜙 𝜔 ] = P [ 𝜔 ] , which is her subjective probabilityof 𝜔 occurring. Thus, traders can express arbitrary probability dis-tributions over Ω . We consider variants of LMSR market maker [15]for a complete market, described by cost function and prices 𝐶 ( 𝜽 ) = 𝑏 log (cid:32) ∑︁ 𝜔 ∈ Ω 𝑒 𝜃 𝜔 / 𝑏 (cid:33) , 𝑝 𝜔 ( 𝜽 ) = 𝜕𝐶 ( 𝜽 ) 𝜕𝜃 𝜔 = 𝑒 𝜃 𝜔 / 𝑏 (cid:205) 𝜈 ∈ Ω 𝑒 𝜃 𝜈 / 𝑏 , (1)where 𝑏 is the liquidity parameter, controlling how fast the pricemoves in response to trading and limiting the worst-case loss ofthe market maker to 𝑏 log | Ω | [15].The securities in a complete market can be used to express betson any event 𝐸 . Specifically, one share of a security for the event 𝐸 can be represented by the indicator bundle 𝐸 ∈ R Ω with entries1 𝐸,𝜔 = { 𝜔 ∈ 𝐸 } . We refer to this bundle as the bundle security forevent 𝐸 . The immediate price of the bundle 𝐸 in the state 𝜽 is 𝑝 𝐸 ( 𝜽 ) (cid:66) 𝐸 · 𝒑 ( 𝜽 ) = ∑︁ 𝜔 ∈ 𝐸 𝑝 𝜔 ( 𝜽 ) = (cid:205) 𝜔 ∈ 𝐸 𝑒 𝜃 𝜔 / 𝑏 (cid:205) 𝜈 ∈ Ω 𝑒 𝜃 𝜈 / 𝑏 . (2)The cost of buying the bundle 𝑠 𝐸 , or sometimes referred to as “thecost of 𝑠 shares of 𝐸 ”, can be written as a function of 𝑝 𝐸 ( 𝜽 ) and 𝑠 : 𝐶 ( 𝜽 + 𝑠 𝐸 ) − 𝐶 ( 𝜽 ) (3) = 𝑏 log (cid:32) ∑︁ 𝜔 ∉ 𝐸 𝑒 𝜃 𝜔 / 𝑏 + ∑︁ 𝜔 ∈ 𝐸 𝑒 ( 𝜃 𝜔 + 𝑠 )/ 𝑏 (cid:33) − 𝑏 log (cid:32) ∑︁ 𝜔 ∈ Ω 𝑒 𝜃 𝜔 / 𝑏 (cid:33) = 𝑏 log (cid:16) 𝑝 𝐸 𝑐 ( 𝜽 ) + 𝑒 𝑠 / 𝑏 𝑝 𝐸 ( 𝜽 ) (cid:17) = 𝑏 log (cid:16) − 𝑝 𝐸 ( 𝜽 ) + 𝑒 𝑠 / 𝑏 𝑝 𝐸 ( 𝜽 ) (cid:17) . bove, we write 𝐸 𝑐 for the complementary event 𝐸 𝑐 = Ω \ 𝐸 , anduse the fact 𝑝 𝐸 ( 𝜽 ) + 𝑝 𝐸 𝑐 ( 𝜽 ) =
1, which follows from Eq. (2). [ , ) We consider betting on outcomes within an interval [ , ) . Ourapproach generalizes to outcomes that are in any [ 𝛼, 𝛽 ) ⊆ [−∞ , ∞) by applying any increasing transformation 𝐹 : [ 𝛼, 𝛽 ) → [ , ) . Weassume that the outcome 𝜔 is specified with 𝐾 bits, meaning thatthere are 𝑁 = 𝐾 outcomes with Ω = { 𝑗 / 𝑁 : 𝑗 ∈ { , , . . . , 𝑁 − }} .At the end of Sections 3 and 4, we discuss how the assumption ofpre-specified bit precision can be removed.Example 1 (Complete market for S&P 500). We construct acomplete market for the S&P 500 opening price on December 17, 2021,by setting 𝑁 = = I = { , . , . . . , . , . } , where we cap prices at $5242.87(i.e., larger prices are treated as $5242.87). The transformed outcomeis then 𝜔 = 𝜔 ′ / 𝑁 , where 𝜔 ′ is the 𝑆 & 𝑃 price in cents. In the outcome space Ω , we would like to enable price and costqueries as well as buying and selling of bundle securities for theinterval events 𝐼 = [ 𝛼, 𝛽 ) for any 𝛼, 𝛽 ∈ Ω ∪{ } . For cost-based mar-kets, sell transactions are equivalent to buying a negative amountof shares, so we design algorithms for three operations: price ( 𝐼 ) , cost ( 𝐼, 𝑠 ) , and buy ( 𝐼, 𝑠 ) , where 𝐼 is the interval event and 𝑠 the num-ber of shares. A naive implementation of price and cost followingEqs. (2) and (3) would be linear in 𝑁 . In this paper, we propose toimplement these operations in time that is logarithmic in 𝑁 . We design a data structure, referred to as an
LMSR tree , which re-sembles an interval tree [9, Section 15.3], but includes additionalannotations to support LMSR calculations. We first define the LMSRtree, and show that it can facilitate market operations in time loga-rithmic in the number of distinct intervals that traders define. [ , ) We represent an LMSR tree 𝑇 with a full binary tree , where eachnode 𝑧 has either no children (when 𝑧 is a leaf) or exactly twochildren, denoted left ( 𝑧 ) and right ( 𝑧 ) (when 𝑧 is an inner node). Theroot is denoted root and the parent of any non-root node par ( 𝑧 ) .Definition 1 (LMSR Tree). An LMSR tree is a full binary tree,where each node 𝑧 is annotated with an interval 𝐼 𝑧 = [ 𝛼 𝑧 , 𝛽 𝑧 ) with 𝛼 𝑧 , 𝛽 𝑧 ∈ Ω ∪ { } , a height ℎ 𝑧 ≥ , a quantity 𝑠 𝑧 ∈ R that recordsthe number of sold bundle securities associated with 𝐼 𝑧 , and a partialnormalization constant 𝑆 𝑧 ≥ (defined below in Eq. 6).An LMSR tree is required to satisfy: • Binary-search property : 𝐼 root = [ , ) , and for inner node 𝑧 , 𝛼 𝑧 = 𝛼 left ( 𝑧 ) < 𝛽 left ( 𝑧 ) = 𝛼 right ( 𝑧 ) < 𝛽 right ( 𝑧 ) = 𝛽 𝑧 . • Height balance : ℎ 𝑧 = for leaves, and for inner node 𝑧 , ℎ 𝑧 = + max { ℎ left ( 𝑧 ) , ℎ right ( 𝑧 ) } , | ℎ left ( 𝑧 ) − ℎ right ( 𝑧 ) | ≤ . • Partial-normalization correctness : 𝑆 𝑧 = 𝑒 𝑠 𝑧 / 𝑏 · ( 𝛽 𝑧 − 𝛼 𝑧 ) for leaves,and for inner node 𝑧 , 𝑆 𝑧 = 𝑒 𝑠 𝑧 / 𝑏 · (cid:16) 𝑆 left ( 𝑧 ) + 𝑆 right ( 𝑧 ) (cid:17) . The binary-search property helps to find the unique leaf thatcontains any 𝜔 ∈ Ω by descending from root and choosing left orright in each node based on whether 𝜔 < 𝛽 left ( 𝑧 ) or 𝜔 ≥ 𝛽 left ( 𝑧 ) .The height-balance property ensures that the path length from rootto any leaf is at most O( log 𝑛 ) , where 𝑛 is the number of leaves ofthe tree [17]. We adopt an AVL tree [3] at the basis of our LMSRtree, but other balanced binary-search trees (e.g., red-black trees orsplay trees) could also be used.To facilitate LMSR computations, we maintain a scalar quantity 𝑠 𝑧 ∈ R for each node 𝑧 , which records the number of bundle secu-rities associated with 𝐼 𝑧 sold by the market maker. Therefore, themarket state and its components for each individual outcome 𝜔 represented by the LMSR tree 𝑇 are 𝜽 ( 𝑇 ) = ∑︁ 𝑧 ∈ 𝑇 𝑠 𝑧 𝐼 𝑧 ; 𝜃 𝜔 ( 𝑇 ) = ∑︁ 𝑧 ∈ 𝑇 𝑠 𝑧 𝐼 𝑧 ,𝜔 = ∑︁ 𝑧 ∋ 𝜔 𝑠 𝑧 . (4)The normalization constant in the LMSR price (Eq. 2) is then ∑︁ 𝜔 ∈ Ω 𝑒 𝜃 𝜔 / 𝑏 = ∑︁ 𝜔 ∈ Ω 𝑒 (cid:205) 𝑧 ∋ 𝜔 𝑠 𝑧 / 𝑏 = ∑︁ 𝜔 ∈ Ω (cid:214) 𝑧 ∋ 𝜔 𝑒 𝑠 𝑧 / 𝑏 . (5)We decompose the computation of the above normalization con-stant along the nodes of an LMSR tree, by defining a partial nor-malization constant 𝑆 𝑧 in each node: 𝑆 𝑧 (cid:66) 𝑁 ∑︁ 𝜔 ∈ 𝑧 (cid:214) 𝑧 ′ : 𝑧 ⊇ 𝑧 ′ ∋ 𝜔 𝑒 𝑠 𝑧 ′ / 𝑏 . (6)Thus, we have (cid:205) 𝜔 ∈ Ω 𝑒 𝜃 𝜔 / 𝑏 = 𝑁𝑆 root and obtain the following re-cursive relationship, which we refer to as partial-normalizationcorrectness and is at the core of implementing price and buy : 𝑆 𝑧 = (cid:40) 𝑒 𝑠 𝑧 / 𝑏 · ( 𝛽 𝑧 − 𝛼 𝑧 ) if 𝑧 is a leaf, 𝑒 𝑠 𝑧 / 𝑏 · (cid:16) 𝑆 left ( 𝑧 ) + 𝑆 right ( 𝑧 ) (cid:17) otherwise. (7)Based on the LMSR tree construction, we implement the follow-ing operations for any interval 𝐼 = [ 𝛼, 𝛽 ) : • price ( 𝐼,𝑇 ) returns the price of bundle security for 𝐼 ; • cost ( 𝐼, 𝑠,𝑇 ) returns the cost of 𝑠 shares of bundle security for 𝐼 ; • buy ( 𝐼, 𝑠,𝑇 ) updates 𝑇 to reflect the purchase of 𝑠 shares of bundlesecurity for 𝐼 .For cost , it suffices to implement price and use Eq. (3). Since theprice of [ 𝛼, 𝛽 ) satisfies 𝑝 [ 𝛼,𝛽 ) ( 𝜽 ) = 𝑝 [ 𝛼, ) ( 𝜽 ) − 𝑝 [ 𝛽, ) ( 𝜽 ) , it sufficesto implement price for intervals of the form [ 𝛼, ) . Similarly, buying 𝑠 shares of [ 𝛼, 𝛽 ) is equivalent to first buying 𝑠 shares of [ 𝛼, ) andthen buying (− 𝑠 ) shares of [ 𝛽, ) , as the market ends up in thesame state 𝜽 + 𝑠 [ 𝛼,𝛽 ) . We implement price and buy for one-sidedintervals 𝐼 = [ 𝛼, ) , and the remaining operations will follow. We consider price queries for 𝐼 = [ 𝛼, ) . Let vals ( 𝑇 ) = { 𝛼 𝑧 : 𝑧 ∈ 𝑇 } denote the set of distinct left endpoints in the tree nodes. We startby assuming that 𝛼 ∈ vals ( 𝑇 ) , and later relax this assumption. Weproceed to calculate 𝑝 𝐼 ( 𝜽 ) in two steps. First , we construct a set ofnodes Z whose associated intervals 𝐼 𝑧 are disjoint and cover 𝐼 . Toachieve this, we conduct a binary search for 𝛼 , putting in Z all ofthe right children of the visited nodes that have 𝛼 𝑧 > 𝛼 , as well We write 𝜔 ∈ 𝑧 to mean 𝜔 ∈ 𝐼 𝑧 and 𝑧 ′ ⊆ 𝑧 to mean 𝐼 𝑧 ′ ⊆ 𝐼 𝑧 . Thus, 𝑧 ′ ⊆ 𝑧 meansthat 𝑧 ′ is a descendant of 𝑧 in 𝑇 , and 𝑧 ′ ⊂ 𝑧 means that 𝑧 ′ is a strict descendant of 𝑧 . lgorithm 1 Query price of bundle security for an interval 𝐼 = [ 𝛼, ) . Input:
Interval 𝐼 = [ 𝛼, ) , 𝛼 ∈ Ω , LMSR tree 𝑇 . Output:
Price of bundle security for 𝐼 . Initialize 𝑧 ← root , 𝑃 ← price ← while 𝛼 𝑧 ≠ 𝛼 and 𝑧 is not a leaf do 𝑃 ← 𝑃𝑒 𝑠 𝑧 / 𝑏 if 𝛼 < 𝛼 right ( 𝑧 ) then price ← price + 𝑃𝑆 right ( 𝑧 ) / 𝑆 root 𝑧 ← left ( 𝑧 ) else 𝑧 ← right ( 𝑧 ) return price + 𝛽 𝑧 − 𝛼𝛽 𝑧 − 𝛼 𝑧 · 𝑃𝑆 𝑧 / 𝑆 root as the final node with 𝛼 𝑧 = 𝛼 . Thanks to the height balance, thecardinality of Z is O( log 𝑛 ) , where 𝑛 is the number of leaves of 𝑇 .The resulting set Z satisfies 𝑝 𝐼 ( 𝜽 ) = (cid:205) 𝑧 ∈Z 𝑝 𝐼 𝑧 ( 𝜽 ) . Second , we determine 𝑝 𝐼 𝑧 ( 𝜽 ) for each node 𝑧 ∈ Z . Starting fromthe LMSR price in Eq. (2), we take advantage of the defined partialnormalization constants 𝑆 𝑧 to calculate 𝑝 𝐼 𝑧 ( 𝜽 ) : 𝑝 𝐼 𝑧 ( 𝜽 ) = 𝑁𝑆 root ∑︁ 𝜔 ∈ 𝑧 𝑒 𝜃 𝜔 / 𝑏 = 𝑆 root · 𝑁 ∑︁ 𝜔 ∈ 𝑧 (cid:214) 𝑧 ′ ∋ 𝜔 𝑒 𝑠 𝑧 ′ / 𝑏 (8) = 𝑆 root · 𝑁 ∑︁ 𝜔 ∈ 𝑧 (cid:34)(cid:32) (cid:214) 𝑧 ′ : 𝑧 ⊇ 𝑧 ′ ∋ 𝜔 𝑒 𝑠 𝑧 ′ / 𝑏 (cid:33) (cid:32) (cid:214) 𝑧 ′ ⊃ 𝑧 𝑒 𝑠 𝑧 ′ / 𝑏 (cid:33)(cid:35) (9) = 𝑆 𝑧 𝑆 root (cid:32) (cid:214) 𝑧 ′ ⊃ 𝑧 𝑒 𝑠 𝑧 ′ / 𝑏 (cid:33)(cid:124) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:123)(cid:122) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:125) 𝑃 𝑧 . (10)In Eq. (8), we use that 𝑁𝑆 root = (cid:205) 𝜔 ∈ Ω 𝑒 𝜃 𝜔 / 𝑏 and then expand 𝜃 𝜔 using Eq. (4). In Eq. (9), we use the fact that any node 𝑧 ′ with a non-empty intersection with 𝑧 (i.e., 𝐼 𝑧 ∩ 𝐼 𝑧 ′ ≠ ∅ ) must be either a descen-dant or an ancestor of 𝑧 as a direct consequence of the binary-searchproperty. The product 𝑃 𝑧 in Eq. (10) iterates over 𝑧 ′ on the path fromroot to 𝑧 , and thus can be calculated along the binary-search path.We now handle the case when 𝛼 ∉ vals ( 𝑇 ) . After the leaf 𝑧 on thesearch path is reached, we have 𝛼 𝑧 < 𝛼 < 𝛽 𝑧 . Instead of expandingthe tree, we conceptually create two children of 𝑧 : 𝑧 ′ and 𝑧 ′′ with 𝐼 𝑧 ′ = [ 𝛼 𝑧 , 𝛼 ) and 𝐼 𝑧 ′′ = [ 𝛼, 𝛽 𝑧 ) , and add 𝑧 ′′ in Z . Since 𝜃 𝜔 is constantacross 𝜔 ∈ 𝐼 𝑧 , we obtain 𝑝 𝐼 𝑧 ′′ ( 𝜽 ) = 𝛽 𝑧 − 𝛼𝛽 𝑧 − 𝛼 𝑧 · 𝑝 𝐼 𝑧 ( 𝜽 ) by Eq. (2).Summarizing the foregoing procedures yields Algorithm 1, whichsimultaneously constructs the set Z and calculates the prices 𝑝 𝐼 𝑧 ( 𝜽 ) .Since it suffices to go down a single path and only perform constant-time computation in each node, the resulting algorithm runs in time O( log 𝑛 vals ) , where 𝑛 vals denotes the number of distinct values ap-peared as endpoints of intervals in all the executed transactions.We defer complete proofs from this paper to the appendix, whichis available in the full version of this paper on arXiv.Theorem 1. Algorithm 1 implements price in time O( log 𝑛 vals ) . We next implement buy ([ 𝛼, ) , 𝑠,𝑇 ) while maintaining the LMSRtree properties. The main challenge here is to simultaneously main-tain partial-normalization correctness and height balance . We addressthis by adapting AVL-tree rebalancing. Algorithm 2
Buy 𝑠 shares of bundle security for an interval 𝐼 = [ 𝛼, ) . Input:
Quantity 𝑠 ∈ R , interval 𝐼 = [ 𝛼, ) , 𝛼 ∈ Ω , LMSR tree 𝑇 . Output:
Tree 𝑇 updated to reflect the purchase of 𝑠 shares of 𝐼 . Define subroutines:NewLeaf( 𝛼 , 𝛽 ): return a new leaf node 𝑧 with 𝐼 𝑧 = [ 𝛼 , 𝛽 ) , ℎ 𝑧 = 𝑠 𝑧 = 𝑆 𝑧 = ( 𝛽 − 𝛼 ) ResetInnerNode( 𝑧 ): reset ℎ 𝑧 and 𝑆 𝑧 based on the children of 𝑧ℎ 𝑧 ← + max { ℎ left ( 𝑧 ) , ℎ right ( 𝑧 ) } , 𝑆 𝑧 ← 𝑒 𝑠 𝑧 / 𝑏 ( 𝑆 left ( 𝑧 ) + 𝑆 right ( 𝑧 ) ) AddShares( 𝑧, 𝑠 ): increase the number of shares held in 𝑧 by 𝑠𝑠 𝑧 ← 𝑠 𝑧 + 𝑠 , 𝑆 𝑧 ← 𝑒 𝑠 / 𝑏 𝑆 𝑧 Initialize 𝑧 ← root while 𝛼 𝑧 ≠ 𝛼 and 𝑧 is not a leaf do ⊲ add 𝑠 shares to 𝑧 ∈ Z if 𝛼 < 𝛼 right ( 𝑧 ) then AddShares( right ( 𝑧 ) , 𝑠 ) 𝑧 ← left ( 𝑧 ) else 𝑧 ← right ( 𝑧 ) if 𝛼 𝑧 < 𝛼 then ⊲ split the leaf 𝑧 left ( 𝑧 ) ← NewLeaf( 𝛼 𝑧 , 𝛼 ), right ( 𝑧 ) ← NewLeaf( 𝛼, 𝛽 𝑧 ) 𝑧 ← right ( 𝑧 ) AddShares( 𝑧, 𝑠 ) while 𝑧 is not a root do ⊲ trace the binary-search path back 𝑧 ← par ( 𝑧 ) if | ℎ left ( 𝑧 ) − ℎ right ( 𝑧 ) | ≥ then ⊲ restore height balance Rotate 𝑧 and possibly one of its children(details in Appendix A.2, Algorithm 5) ResetInnerNode( 𝑧 ) ⊲ update ℎ 𝑧 and 𝑆 𝑧 We begin by considering the case 𝛼 ∈ vals ( 𝑇 ) . Similar to pricequeries, we conduct binary search for 𝛼 to obtain the set of nodes Z that covers 𝐼 = [ 𝛼, ) . We update the values of 𝑠 𝑧 across 𝑧 ∈ Z by adding 𝑠 , and obtain 𝑇 ′ that has the same structure as 𝑇 withthe updated share quantities 𝑠 ′ 𝑧 = (cid:40) 𝑠 𝑧 + 𝑠 if 𝑧 ∈ Z 𝑠 𝑧 otherwise.Thus, the resulting market state is 𝜽 ( 𝑇 ′ ) = ∑︁ 𝑧 ∈ 𝑇 ′ 𝑠 ′ 𝑧 𝐼 𝑧 = ∑︁ 𝑧 ∈ 𝑇 𝑠 𝑧 𝐼 𝑧 + ∑︁ 𝑧 ∈Z 𝑠 𝐼 𝑧 = 𝜽 ( 𝑇 ) + 𝑠 𝐼 . We then rely on the recursive relationship defined in Eq. (7) toupdate the partial normalization constants 𝑆 𝑧 . It suffices to updatethe ancestors of the nodes 𝑧 ∈ Z , all of which lie along the searchpath to 𝛼 , and each update requires constant time.When 𝛼 ∉ vals ( 𝑇 ) , we split the leaf 𝑧 that contains 𝛼 ∈ [ 𝛼 𝑧 , 𝛽 𝑧 ) before adding shares to right ( 𝑧 ) . This may violate the height-balanceproperty . Similar to the AVL insertion algorithm [17, Section 6.2.3],we fix any imbalance by means of rotations , as we go back along thesearch path. Rotations are operations that modify small portionsof the tree, and at most two rotations are needed to rebalance thetree [3]. We show in Appendix A.2, Lemma 1, that in each rotation,only a constant number of nodes needs to be updated to preservethe partial-normalization correctness . Thus, the overall running timeof the buy operation, presented in Algorithm 2, is O( log 𝑛 vals ) .Theorem 2. Algorithm 2 implements buy in time O( log 𝑛 vals ) .emarks. We show that price , cost and buy can be implementedin time O( log 𝑛 vals ) , which is bounded above by the log of thenumber of buy transactions O( log 𝑛 buy ) and the bit precision of theoutcome O( log 𝑁 ) = O( 𝐾 ) . We note that none of the operationsrequire the knowledge of 𝐾 , so the market in fact supports querieswith arbitrary precision. However, the market precision affects theworst-case loss bound for the market maker, which is O( log 𝑁 ) = O( 𝐾 ) . Next section presents a different construction that achievesa constant worst-case loss independent of the market precision. We introduce our second design, referred to as the multi-resolutionlinearly constrained market maker (multi-resolution LCMM). Thedesign is based on the LMSR, but it enables more flexibility by assign-ing two or more parallel LMSRs with different liquidity parametersto orchestrate submarkets that offer interval securities at differentresolutions. However, running submarkets independently can cre-ate arbitrage opportunities, as any interval expressible in a coarsermarket can also be expressed in a finer one. To maintain coherentprices, we design a matrix that imposes linear constraints to tie mar-ket prices among different submarkets to support the efficient re-moval of any arbitrage opportunity, following Dudík et al. [10]. Wefirst define the multi-resolution LCMM and its properties, and showthat price , cost and buy can be implemented in time O( log 𝑁 ) . [ , ) A binary search tree remains atthe core of our multi-resolution market construction. Unlike a log-time LMSR that uses a self-balancing tree, it builds upon a static one,where each level of the tree represents a submarket of intervals,forming a finer and finer partition of [ , ) . We start with an exampleof a market that offers interval securities at two resolutions.Example 2 (Two-level market for [ , ) ). We consider a mar-ket composed of two submarkets, indexed by I = { , } and I = { , , , } , which partition [ , ) into interval events attwo levels of coarseness: 𝐼 = (cid:2) , (cid:1) , 𝐼 = (cid:2) , (cid:1) ; 𝐼 = (cid:2) , (cid:1) , 𝐼 = (cid:2) , (cid:1) , 𝐼 = (cid:2) , (cid:1) , 𝐼 = (cid:2) , (cid:1) . The market provides six interval securities 𝜙 , . . . , 𝜙 associatedwith the corresponding interval events, i.e., I = I (cid:210) I and |I| = . We extend Example 2 to multiple resolutions. We represent theinitial independent submarkets with a complete binary tree 𝑇 ∗ ofdepth 𝐾 , which corresponds to the bit precision of the outcome 𝜔 .Let Z ∗ denote the set of nodes of 𝑇 ∗ and Z 𝑘 for 𝑘 ∈ { , , . . . , 𝐾 } the set of nodes at each level. Z contains the root associated with 𝐼 root = [ , ) , and each consecutive level contains the children ofnodes from the previous level, which split their corresponding par-ent intervals in half. Thus, level 𝑘 partitions [ , ) into 2 𝑘 intervalsof size 2 − 𝑘 and the final level Z 𝐾 contains 𝑁 = 𝐾 leaves.We index interval securities by nodes, with their payoffs definedby 𝜙 𝑧 ( 𝜔 ) = { 𝜔 ∈ 𝐼 𝑧 } . We partition securities into submarkets Clearly, 𝑛 vals ≤ 𝑛 buy with each buy transaction introducing at most two newendpoint values. The value of 𝑛 vals is also bounded above by 𝑁 + since the intervalendpoints are always in Ω ∪ { } . corresponding to levels, i.e., I 𝑘 = Z 𝑘 for 𝑘 ≤ 𝐾 , where |I 𝑘 | = 𝑘 and I = (cid:210) 𝑘 ≤ 𝐾 I 𝑘 . For each submarket, we define the LMSR costfunction 𝐶 𝑘 with a separate liquidity parameter 𝑏 𝑘 > 𝐶 𝑘 ( 𝜽 𝑘 ) = 𝑏 𝑘 log (cid:169)(cid:173)(cid:171) ∑︁ 𝑧 ∈Z 𝑘 𝑒 𝜃 𝑧 / 𝑏 𝑘 (cid:170)(cid:174)(cid:172) . (11) Following the abovemulti-resolution construction, the overall market has a direct-sumcost ˜ 𝐶 ( 𝜽 ) = (cid:205) 𝑘 ≤ 𝐾 𝐶 𝑘 ( 𝜽 𝑘 ) , which corresponds to pricing securitiesin each block I 𝑘 independently using 𝐶 𝑘 . However, as there arelogical dependencies between securities in different levels, indepen-dent pricing may lead to incoherent prices among submarkets andcreate arbitrage opportunities.Example 3 (Arbitrage in a two-level market). ContinuingExample 2, we define separate LMSR costs, where 𝑏 = and 𝑏 = : 𝐶 ( 𝜽 ) = log (cid:16) 𝑒 𝜃 + 𝑒 𝜃 (cid:17) ; 𝐶 ( 𝜽 ) = log (cid:16) 𝑒 𝜃 + 𝑒 𝜃 + 𝑒 𝜃 + 𝑒 𝜃 (cid:17) . The direct-sum market ˜ 𝐶 ( 𝜽 ) = 𝐶 ( 𝜽 ) + 𝐶 ( 𝜽 ) allows incoherentprices. For example, after buying some shares of security 𝜙 associatedwith 𝐼 = (cid:2) , (cid:1) in submarket I , the market can have ˜ 𝑝 ( 𝜽 ) = .
5; ˜ 𝑝 ( 𝜽 ) + ˜ 𝑝 ( 𝜽 ) = . . These prices are incoherent, i.e., do not correspond to probabilities of 𝐼 , 𝐼 , 𝐼 , because under any probability distribution over Ω , wemust have P [ 𝐼 ] = P [ 𝐼 ] + P [ 𝐼 ] and P [ 𝐼 ] = P [ 𝐼 ] + P [ 𝐼 ] .Thus, a coherent price vector 𝝁 ∈ R |I | must satisfy linear constraints 𝜇 − 𝜇 − 𝜇 = and 𝜇 − 𝜇 − 𝜇 = , which can be also writtenas a ⊤ 𝝁 = and a ⊤ 𝝁 = where a = ( , , − , − , , ) ⊤ and a = ( , , , , − , − ) ⊤ . We refer to A = ( a , a ) ∈ R |I |× as the constraint matrix. We extend Example 3 to specify price constraints in a multi-resolution market. Later we will show how the constraint matrix canbe used to remove arbitrage arising from the constraint violations.Recall that M denotes a coherent price space , where any ex-pected payoff lies in the convex hull of { 𝝓 ( 𝜔 )} 𝜔 ∈ Ω . For the multi-resolution market, we specify a set of homogeneous linear equalities describing a superset of M . M ⊆ { 𝝁 ∈ R |I | : A ⊤ 𝝁 = } . (12)We design the constraint matrix A to ensure that any pair ofsubmarkets is price coherent, meaning that any interval event 𝐼 ⊆ Ω gets the same price on all levels that can express it. Therefore, foreach inner node 𝑦 ∈ Z 𝑙 where 𝑙 < 𝐾 , we have 𝜇 𝑦 = ∑︁ 𝑧 ∈Z 𝑘 : 𝑧 ⊂ 𝑦 𝜇 𝑧 for any 𝑙 < 𝑘 ≤ 𝐾 .For algorithmic reasons (as we will see in Section 4.3), we furthertie the price of 𝑦 to the prices of all of 𝑦 ’s descendants and weighteach level by its liquidity parameter 𝑏 𝑘 : (cid:16)∑︁ 𝑘 > ℓ 𝑏 𝑘 (cid:17)(cid:124) (cid:32) (cid:123)(cid:122) (cid:32) (cid:125) 𝐵 ℓ 𝜇 𝑦 = ∑︁ 𝑘 > ℓ (cid:16) 𝑏 𝑘 ∑︁ 𝑧 ∈Z 𝑘 : 𝑧 ⊂ 𝑦 𝜇 𝑧 (cid:17) . (13)Now we can formally define the constraint matrix A . Let Y ∗ = Z ∗ \Z 𝐾 be the set of inner nodes of 𝑇 ∗ and let level ( 𝑧 ) denotehe level of a node 𝑧 . The matrix A ∈ R |Z ∗ |×|Y ∗ | contains theconstraints from Eq. (13) across all 𝑦 ∈ Y ∗ : 𝐴 𝑧𝑦 = 𝐵 level ( 𝑧 ) if 𝑧 = 𝑦 , − 𝑏 level ( 𝑧 ) if 𝑧 ⊂ 𝑦 ,0 otherwise. (14)Arbitrage opportunities arise if the price of bundle a 𝑗 differsfrom zero, where a 𝑗 denotes the 𝑗 th column of A . Traders profit bybuying a positive quantity of a 𝑗 if its price is negative, and sellingotherwise. Thus, the constraint matrix A gives a recipe for arbitrageremoval. We provide the intuition for this in the two-level market,and then give the definition of the multi-resolution LCMM.Example 4 (Arbitrage removal in a two-level market). Con-tinuing Example 3, the prices ˜ 𝒑 ( 𝜽 ) violate the constraint A ⊤ 𝝁 = ,because a ⊤ ˜ 𝒑 ( 𝜽 ) = ˜ 𝑝 ( 𝜽 ) − ˜ 𝑝 ( 𝜽 ) − ˜ 𝑝 ( 𝜽 ) = . − . ≠ . Thevector a reveals an arbitrage opportunity: buy the security 𝜙 (atthe initial price . ) and simultaneously sell securities 𝜙 and 𝜙 (atthe initial price . ), i.e., buy bundle a . Since under any outcome 𝜔 ,the payout for the bundle a is , this is initially profitable. However,buying a will increase the price of 𝜙 and decrease the prices of 𝜙 and 𝜙 . Once a sufficiently large quantity 𝑠 of shares of a is bought,this form of arbitrage is removed and we have a ⊤ ˜ 𝒑 ( ˜ 𝜽 ) = in a newstate ˜ 𝜽 = 𝜽 + 𝑠 a = 𝜽 + A 𝜼 , where 𝜼 : = ( 𝑠, ) ⊤ . A linearly constrained market maker (LCMM) [10] leveragesviolated constraints similarly as in Example 4 to remove arbitrage,and then returns the arbitrage proceeds to the trader. Formally, anLCMM is described by the cost function 𝐶 ( 𝜽 ) = inf 𝜼 ∈ R |Y∗| ˜ 𝐶 ( 𝜽 + A 𝜼 ) . (15)It relies on the direct-sum cost ˜ 𝐶 , but with each trader purchase 𝜹 that causes incoherent prices, an LCMM automatically seeks themost advantageous cost for the trader by buying bundles A 𝜹 arb on the trader’s behalf to remove arbitrage. Trader purchases areaccumulated as the state 𝜽 , and automatic purchases made by theLCMM are accumulated as A 𝜼 . We note that the purchase of bundle A 𝜹 arb has no effect onthe trader’s payoff, since ( A 𝜹 arb ) ⊤ 𝝓 ( 𝜔 ) = 𝜔 ∈ Ω thanksto Eq. (12) and the fact that 𝝓 ( 𝜔 ) ∈ M . However, the purchaseof A 𝜹 arb can lower the cost, so optimizing over 𝜹 arb benefits thetraders, while maintaining the same worst-case loss guarantee forthe market maker as ˜ 𝐶 [10]. Consider a fixed 𝜽 and the correspond-ing 𝜼 ★ minimizing Eq. (15). We calculate prices as 𝒑 ( 𝜽 ) = ∇ 𝐶 ( 𝜽 ) = ∇ ˜ 𝐶 ( 𝜽 + A 𝜼 ★ ) . By the first order optimality, 𝜼 ★ minimizes Eq. (15)if and only if A ⊤ (cid:0) ∇ ˜ 𝐶 ( 𝜽 + A 𝜼 ★ ) (cid:1) = . This means that A ⊤ 𝒑 ( 𝜽 ) = ,and thus arbitrage opportunities expressed by A are completelyremoved by the LCMM cost function 𝐶 .To implement an LCMM, we maintain the state ˜ 𝜽 = 𝜽 + A 𝜼 in thedirect-sum market ˜ 𝐶 . After updating 𝜽 to a new value 𝜽 ′ = 𝜽 + 𝜹 , weseek to find 𝜼 ′ = 𝜼 + 𝜹 arb that removes all the arbitrage opportunitiesexpressed by A . The resulting cost for the trader is˜ 𝐶 ( 𝜽 ′ + A 𝜼 ′ ) − ˜ 𝐶 ( 𝜽 + A 𝜼 ) = ˜ 𝐶 ( ˜ 𝜽 + 𝜹 + A 𝜹 arb ) − ˜ 𝐶 ( ˜ 𝜽 ) . We finish this section by pointing out two favorable propertiesof the multi-resolution LCMM. Above, we have established thatLCMM removes all arbitrage opportunities expressed by A . The next theorem shows that this actually removes all arbitrage. The proofshows that consecutive levels are coherent, which by transitivityimplies that the overall price vector is coherent (see Appendix A.3).Theorem 3. A multi-resolution LCMM is arbitrage-free.
The multi-resolution LCMM also enjoys the bounded-loss prop-erty. For a suitable choice of liquidities, such as 𝑏 𝑘 = O( / 𝑘 . ) , itcan achieve a constant worst-case loss bound. The proof uses thefact that the overall loss is bounded by the sum of losses of levelmarkets, which are at most 𝑏 𝑘 log |Z 𝑘 | = 𝑘𝑏 𝑘 log 2.Theorem 4. Let { 𝑏 𝑘 } ∞ 𝑘 = be a sequence of positive numbers suchthat (cid:205) ∞ 𝑘 = 𝑘𝑏 𝑘 = 𝐵 ∗ for some finite 𝐵 ∗ . Then the multi-resolutionLCMM with liquidity parameters 𝑏 𝑘 for 𝑘 ≤ 𝐾 guarantees the worst-case loss of the market maker of at most 𝐵 ∗ log 2 , regardless of theoutcome precision 𝐾 . We can now formally de-fine the multi-resolution LCMM tree. The market state of a multi-resolution LCMM is represented by vectors 𝜽 ∈ R |Z ∗ | and 𝜼 ∈ R |Y ∗ | , whose dimensions can be intractably large (e.g., on the orderof 2 𝐾 = 𝑁 ). However, since each LCMM operation involves onlya small set of coordinates of 𝜽 and 𝜼 , we only keep track of thecoordinates accessed so far and represent them as an annotatedsubtree 𝑇 of 𝑇 ∗ , referred to as an LCMM tree .Definition 2 (LCMM Tree). An LCMM tree 𝑇 is a full binarytree, where each node 𝑧 is annotated with 𝐼 𝑧 = [ 𝛼 𝑧 , 𝛽 𝑧 ) , 𝜃 𝑧 ∈ R , 𝜂 𝑧 ∈ R , such that 𝐼 root = [ , ) , and for every inner node 𝑧 : 𝛼 𝑧 = 𝛼 left ( 𝑧 ) , 𝛽 left ( 𝑧 ) = 𝛼 right ( 𝑧 ) = 𝛼 𝑧 + 𝛽 𝑧 , 𝛽 right ( 𝑧 ) = 𝛽 𝑧 . The tree 𝑇 contains the coordinates of 𝜽 and 𝜼 accessed so far.Since 𝜽 and 𝜼 are initialized to zero, their remaining entries arezero. We write 𝜽 ( 𝑇 ) ∈ R |Z ∗ | and 𝜼 ( 𝑇 ) ∈ R |Y ∗ | for the vectors rep-resented by 𝑇 . To calculate prices, we maintain 𝜼 ( 𝑇 ) that minimizesEq. (15), or equivalently 𝜼 ( 𝑇 ) that satisfies A ⊤ ˜ 𝒑 (cid:0) 𝜽 ( 𝑇 )+ A 𝜼 ( 𝑇 ) (cid:1) = . If this property holds, we say that an LCMM tree 𝑇 is coherent . There are many ways to decompose an interval 𝐼 in a multi-resolutionmarket, but they all yield the same price thanks to coherence. Theno-arbitrage property also guarantees that the price of [ 𝛼, 𝛽 ) canbe obtained by subtracting the price of [ 𝛽, ) from [ 𝛼, ) . Therefore,we focus on pricing one-sided intervals of the form 𝐼 = [ 𝛼, ) .Let 𝑇 be a coherent LCMM tree and 𝜽 (cid:66) 𝜽 ( 𝑇 ) and 𝜼 (cid:66) 𝜼 ( 𝑇 ) bethe vectors represented by 𝑇 . Let ˜ 𝜽 = 𝜽 + A 𝜼 be the correspondingstate in ˜ 𝐶 , so the current security prices are 𝝁 : = ˜ 𝒑 ( ˜ 𝜽 ) . As before,we identify a set of nodes Z that covers 𝐼 , and then rely on pricecoherence to calculate each 𝜇 𝑧 along the search path.Assume that 𝑧 is not a root node and we know the price of itsparent. Let sib ( 𝑧 ) denote the sibling of 𝑧 and 𝑘 = level ( 𝑧 ) . We canthen relate the price of 𝑧 to the price of par ( 𝑧 ) : 𝜇 𝑧 = 𝜇 𝑧 𝜇 par ( 𝑧 ) · 𝜇 par ( 𝑧 ) = 𝜇 𝑧 𝜇 𝑧 + 𝜇 sib ( 𝑧 ) · 𝜇 par ( 𝑧 ) (16) = 𝑒 ˜ 𝜃 𝑧 / 𝑏 𝑘 𝑒 ˜ 𝜃 𝑧 / 𝑏 𝑘 + 𝑒 ˜ 𝜃 sib ( 𝑧 ) / 𝑏 𝑘 · 𝜇 par ( 𝑧 ) . (17) lgorithm 3 Query price of bundle security for an interval 𝐼 = [ 𝛼, ) . Input:
Interval 𝐼 = [ 𝛼, ) , 𝛼 ∈ Ω , coherent LCMM tree 𝑇 . Output:
Price of bundle security for 𝐼 . Initialize 𝑧 ← root , 𝜇 𝑧 ← price ← while 𝛼 𝑧 ≠ 𝛼 and 𝑧 is not a leaf do 𝑧 𝑙 ← left ( 𝑧 ) , 𝑧 𝑟 ← right ( 𝑧 ) , 𝑘 ← level ( 𝑧 𝑙 ) 𝑒 𝑙 ← exp {( 𝜃 𝑧 𝑙 + 𝐵 𝑘 𝜂 𝑧 𝑙 )/ 𝑏 𝑘 } , 𝑒 𝑟 ← exp {( 𝜃 𝑧 𝑟 + 𝐵 𝑘 𝜂 𝑧 𝑟 )/ 𝑏 𝑘 } , 𝜇 𝑧 𝑙 ← 𝑒 𝑙 𝑒 𝑙 + 𝑒 𝑟 𝜇 𝑧 , 𝜇 𝑧 𝑟 ← 𝑒 𝑟 𝑒 𝑙 + 𝑒 𝑟 𝜇 𝑧 ⊲ calculate prices by Eq. (19) if 𝛼 < 𝛼 right ( 𝑧 ) then 𝑧 ← 𝑧 𝑙 , price ← price + 𝜇 𝑧 𝑟 else 𝑧 ← 𝑧 𝑟 return price + 𝛽 𝑧 − 𝛼𝛽 𝑧 − 𝛼 𝑧 · 𝜇 𝑧 Eq. (16) follows by price coherence and Eq. (17) follows by the pricecalculation in Eq. (1). Thus, we descend the search path to calculateeach price 𝜇 𝑧 , beginning with 𝜇 root =
1. It remains to obtain ˜ 𝜃 𝑧 , forwhich we follow the construction of A in Eq. (14):˜ 𝜃 𝑧 = 𝜃 𝑧 + ∑︁ 𝑦 ∈Y ∗ 𝐴 𝑧𝑦 𝜂 𝑦 = 𝜃 𝑧 + 𝐵 𝑘 𝜂 𝑧 − 𝑏 𝑘 ∑︁ 𝑦 ⊃ 𝑧 𝜂 𝑦 . (18)Plugging the above equation back in Eq. (17), we obtain 𝜇 𝑧 = exp (cid:16) 𝜃 𝑧 + 𝐵 𝑘 𝜂 𝑧 𝑏 𝑘 (cid:17) exp (cid:16) 𝜃 𝑧 + 𝐵 𝑘 𝜂 𝑧 𝑏 𝑘 (cid:17) + exp (cid:16) 𝜃 sib ( 𝑧 ) + 𝐵 𝑘 𝜂 sib ( 𝑧 ) 𝑏 𝑘 (cid:17) · 𝜇 par ( 𝑧 ) . (19)These steps yield Algorithm 3. The final line of the algorithmaddresses the case when the search ends in the leaf 𝑧 with 𝛼 𝑧 < 𝛼 < 𝛽 𝑧 . Rather than expanding the tree to its lowest level 𝐾 , weuse price coherence again: since any strict descendant 𝑧 ′ ⊂ 𝑧 onthe path from 𝑧 to a leaf node 𝑢 ∈ Z 𝐾 has 𝜃 𝑧 ′ = 𝜂 𝑧 ′ = [ 𝛼, 𝛽 𝑧 ) equals 𝛽 𝑧 − 𝛼𝛽 𝑧 − 𝛼 𝑧 · 𝜇 𝑧 .The length of search path for 𝛼 is prec ( 𝛼 ) , which denotes thebit precision of 𝛼 , defined as the smallest integer 𝑘 such that 𝛼 isan integer multiple of 2 − 𝑘 . As the computation at each node onlyrequires constant time, the time to price 𝐼 = [ 𝛼, ) is O( prec ( 𝛼 )) ,which is bounded above by O( 𝐾 ) .Theorem 5. Let 𝐼 = [ 𝛼, ) , 𝛼 ∈ Ω . Algorithm 3 implements price ( 𝐼,𝑇 ) in time O( prec ( 𝛼 )) . Different from LMSR, the cost query for a multi-resolution LCMMcannot be directly derived from prices. We instead augment buy to implement cost by executing buy and then reverting all thechanges. We focus on buy ( 𝐼, 𝑠,𝑇 ) for 𝐼 = [ 𝛼, ) . By buying 𝑠 sharesof [ 𝛼, ) and then (− 𝑠 ) shares of [ 𝛽, ) , we obtain buying [ 𝛼, 𝛽 ) .We summarize the procedure in Algorithm 4, which performs buy ( 𝐼, 𝑠,𝑇 ) and keeps track of cost ( 𝐼, 𝑠,𝑇 ) . Similar to price queries,we start with a set of nodes Z that partition 𝐼 , by searching for 𝛼 and simultaneously calculating prices 𝜇 𝑧 along the way (lines 3–6). The factor exp {− (cid:205) 𝑦 ⊃ 𝑧 𝜂 𝑦 } = exp {− (cid:205) 𝑦 ⊃ sib ( 𝑧 ) 𝜂 𝑦 } appears in both the numeratorand the denominator after plugging Eq. (18) to Eq. (17), so it cancels out. Algorithm 4
Buy 𝑠 shares of bundle security for an interval 𝐼 = [ 𝛼, ) . Input:
Quantity 𝑠 ∈ R , interval 𝐼 = [ 𝛼, ) , 𝛼 ∈ Ω , coherent LCMM tree 𝑇 . Output:
Cost of 𝑠 shares of bundle security for 𝐼 , the updated tree 𝑇 . Define subroutines:NewLeaf( 𝛼 , 𝛽 ): return a new leaf node 𝑧 with 𝐼 𝑧 = [ 𝛼 , 𝛽 ) , 𝜃 𝑧 = 𝜂 𝑧 = 𝑦, 𝜇 other ): restore price coherence amongsubmarkets 𝑘 ≥ level ( 𝑦 ) following Eq. (20) and update costLet ℓ = level ( 𝑦 ) , 𝑦 ′ = sib ( 𝑦 ) , 𝑡 = 𝑏 ℓ 𝐵 ℓ − log (cid:16) − 𝜇 𝑦 𝜇 𝑦 · 𝜇 other − 𝜇 other (cid:17) 𝑆 = 𝜇 𝑦 𝑒 𝑡𝐵 ℓ / 𝑏 ℓ + − 𝜇 𝑦 , 𝑆 other = 𝜇 other 𝑒 − 𝑡 + − 𝜇 other 𝜂 𝑦 ← 𝜂 𝑦 + 𝑡 , 𝜇 𝑦 ← 𝜇 𝑦 𝑒 𝑡𝐵 ℓ / 𝑏 ℓ / 𝑆 , 𝜇 𝑦 ′ ← 𝜇 𝑦 ′ / 𝑆 cost ← cost + ( 𝑏 ℓ log 𝑆 ) + ( 𝐵 ℓ log 𝑆 other ) AddShares( 𝑧, 𝑠 ): increase shares held in 𝑧 by 𝑠 , update cost, andrestore price coherence among submarkets 𝑘 ≥ level ( 𝑧 ) Let ℓ = level ( 𝑧 ) , 𝑧 ′ = sib ( 𝑧 ) , 𝜇 other = 𝜇 𝑧 , 𝑆 = 𝜇 𝑧 𝑒 𝑠 / 𝑏 ℓ + − 𝜇 𝑧 𝜃 𝑧 ← 𝜃 𝑧 + 𝑠 cost ← cost + ( 𝑏 ℓ log 𝑆 ) 𝜇 𝑧 ← 𝜇 𝑧 𝑒 𝑠 / 𝑏 ℓ / 𝑆 , 𝜇 𝑧 ′ ← 𝜇 𝑧 ′ / 𝑆 RemoveArbitrage( 𝑧, 𝜇 other ) Initialize 𝑧 ← root , 𝜇 𝑧 ←
1, a global variable cost ← while 𝛼 𝑧 ≠ 𝛼 do if 𝑧 is a leaf then left ( 𝑧 ) ← NewLeaf( 𝛼 𝑧 , ( 𝛼 𝑧 + 𝛽 𝑧 ) ), right ( 𝑧 ) ← NewLeaf( ( 𝛼 𝑧 + 𝛽 𝑧 ) , 𝛽 𝑧 ) Calculate 𝜇 left ( 𝑧 ) , 𝜇 right ( 𝑧 ) , and update 𝑧 according to 𝛼 (same as Algorithm 3 lines 3-8) AddShares( 𝑧, 𝑠 ) while 𝑧 is not a root do ⊲ remove arbitrage up the search path 𝑧 ′ ← sib ( 𝑧 ) , 𝑦 ← par ( 𝑧 ) if 𝑧 ′ = right ( 𝑦 ) then AddShares( 𝑧 ′ , 𝑠 ) ⊲ add shares to 𝑧 ∈ Z RemoveArbitrage( 𝑦 , 𝜇 𝑧 + 𝜇 𝑧 ′ ) 𝑧 ← 𝑦 return cost We then proceed back up the search path, adding 𝑠 shares tonodes within the cover Z (lines 7–13). Consider one of such node 𝑦 ∈ Z at level ℓ (cid:66) level ( 𝑦 ) . Increasing 𝜃 𝑦 by 𝑠 creates price inco-herence between the submarket at level ℓ and submarkets at allother levels. We design RemoveArbitrage to remove any arbitrageopportunity between level ℓ and all finer levels with 𝑘 > ℓ . We showin Appendix A.6, Lemma 3, that in order to restore coherence, itsuffices to update 𝜂 𝑦 by a closed-form amount: 𝑡 = 𝑏 ℓ 𝐵 ℓ − log (cid:18) − 𝜇 𝑦 𝜇 𝑦 · 𝜇 other − 𝜇 other (cid:19) , (20)where 𝜇 other = 𝜇 left ( 𝑦 ) + 𝜇 right ( 𝑦 ) records the price of 𝑦 in all thefiner levels. This key algorithmic step is enabled by the arbitragebundle a 𝑦 , which corresponds to buying 𝜙 𝑦 on the level ℓ whileselling securities associated with all descendants of 𝑦 , with theirshares appropriately weighted by the respective liquidity values asspecified in the constraint matrix A .The market remains incoherent between ℓ and all coarser levels 𝑘 < ℓ . Since the updates have been localized to the subtree rootedat 𝑦 , we use Lemma 3 again to update 𝜂 par ( 𝑦 ) and restore coherencemong all levels 𝑘 ≥ ℓ − buy transaction byevaluating Eq. (3) in the component submarkets. Note that costs inall submarkets with 𝑘 > ℓ can be evaluated simultaneously thanksto the restored coherence. Since the computations in each accessednode are constant time, Algorithm 4 runs in time O( prec ( 𝛼 )) .Theorem 6. Let 𝐼 = [ 𝛼, ) , 𝛼 ∈ Ω . Algorithm 4 implements asimultaneous buy ( 𝐼, 𝑠,𝑇 ) and cost ( 𝐼, 𝑠,𝑇 ) in time O( prec ( 𝛼 )) .Remarks. In Algorithms 3 and 4, we assume that each node 𝑧 can store a scalar 𝜇 𝑧 , which can be modified during the run tosupport price calculations but is disposed afterwards. The onlypart of our algorithms that depends on 𝐾 are the cumulative liq-uidities 𝐵 ℓ = (cid:205) 𝐾𝑘 = ℓ + 𝑏 𝑘 . To remove such dependence, we can use 𝐵 ′ ℓ = (cid:205) ∞ 𝑘 = ℓ + 𝑏 𝑘 = 𝐵 ∗ − (cid:205) ℓ𝑘 = 𝑏 𝑘 , where 𝐵 ∗ = (cid:205) ∞ 𝑘 = 𝑏 𝑘 . This has noimpact on the correctness of our algorithms: if at a given time thelargest level in the tree 𝑇 is 𝐿 , we can simply view 𝑇 as a multi-resolution LCMM with 𝐾 = 𝐿 + 𝑏 , 𝑏 , . . . , 𝑏 𝐿 , 𝐵 ′ 𝐿 .The last level 𝐾 = 𝐿 + { 𝐶 𝑘 } ∞ 𝑘 = 𝐿 + . Thus, a multi-resolution LCMMcan achieve a constant loss bound regardless of 𝐾 and supportmarket operations for 𝐼 = [ 𝛼, 𝛽 ) in time O( prec ( 𝛼 ) + prec ( 𝛽 )) . We have proposed two cost-function-based market makers that sup-port trading interval securities of arbitrary precision and executemarket operations exponentially faster than previous designs. Inwhat situations is one preferable over the other?The log-time LMSR enjoys better storage and runtime efficiency,because search paths in LMSR tree are shorter thanks to its height-balance property. The log-time LMSR would therefore be com-putationally preferable, for example, when the designer expectsbetting interest to be concentrated on a smaller set of intervals.However, the log-time LMSR implements a standard LMSR, whichfaces well-known design challenges, such as the requirement toset a suitable liquidity value and the precision of bets in advance.Correctly setting these parameters often requires a good estimateof trader interest even before trading in the market starts.On the other hand, the multi-resolution LCMM does not requirea hard specification of the betting precision. Flexible pricing al-lows the designer to attenuate liquidity across different precisionsin a way that best reflects the designer’s information-gatheringpriorities. For example, an LMSR that operates at precision 𝑘 = 𝑏 can be represented by an LCMM with the levelliquidity values b = ( , , , 𝑏, , , . . . ) . Moreover, if the marketdesigner expects most of the information at precision 4 but alsowants to support bets up to precision 8, they could run an LCMMwith the liquidity placed at two levels as b = ( , , , 𝑏 , , , , 𝑏 ) .By choosing different values 𝑏 and 𝑏 , the market designer canexpress utility for information at different precision levels.We empirically highlight such flexibility by showing how LCMMcan interpolate between LMSRs at different resolutions, allowingthe market to match the coarseness of traders’ information. We P r i c e c o n v e r g e n c e e rr o r LMSR k =4 LMSR k =8 LCMM (a) 𝑘 = . P r i c e c o n v e r g e n c e e rr o r LMSR k =4 LMSR k =8 LCMM (b) 𝑘 = . Figure 1: The price convergence error as a function of thenumber of trades, measured at two resolution levels. conduct agent-based simulation using the trader model with expo-nential utility and exponential-family beliefs [2, 11]. We defer thedetailed trader model to Appendix B.1. Agents trade with either anLMSR or a multi-resolution LCMM, and we are interested in evaluat-ing market makers’ performance in terms of price convergence error ,calculated as the relative entropy between the market-clearing price (that is the price reached when agents only trade among themselves)and the price maintained by the market maker.We operate in a market over [ , ) and the outcome is specifiedwith 𝐾 =
10 bits. We consider budget-limited market makers, whoseworst-case loss may not exceed a budget constraint 𝐵 . For LMSRat precision 𝑘 , this means setting the liquidity parameter to 𝑏 = 𝐵 / log ( 𝑘 ) . Following our motivating example, we compare twoLMSR markets at precision levels 4 and 8, denoted as LMSR 𝑘 = and LMSR 𝑘 = , to an LCMM that evenly splits budget to precision levels4 and 8, denoted as LCMM / . Fig. 1 shows the price convergence as a function of the numberof trades. As one may expect,
LMSR 𝑘 = achieves a faster price con-vergence at the coarser precision level 𝑘 = LMSR 𝑘 = (Fig. 1a), but fails to elicit information at any finer granularity bydesign. The proposed
LCMM / , by equally splitting the budgetbetween 𝑘 = 𝑘 =
8, is able to interpolate between the per-formance of
LMSR 𝑘 = and LMSR 𝑘 = and achieves the “best of bothworlds”: it can elicit forecasts at the finer level 𝑘 = LMSR 𝑘 = , but also obtain a fast convergence at the coarser level 𝑘 =
4, almost matching the convergence speed of
LMSR 𝑘 = .Two immediate questions arise from our work. First, do ourconstructions generalize to two- or higher-dimensional outcomes?One promising avenue is to combine the ideas from our log-timeLMSR market maker with multi-dimensional segment trees [19]to obtain an efficient multi-dimensional LMSR based on a statictree. However, it is not clear how to generalize our balanced LMSRtree construction or the multi-resolution LCMM. Second, does ourapproach extend to non-interval securities, such as call options?We leave these questions open for future research. We note that while our market makers support agents with any beliefs and utilities,the exponential trader model is convenient, as it allows a closed-form derivationof market-clearing price [2, 11], which can be viewed as a “ground truth” for theinformation elicitation. The LCMM has an infinite number of choices for its liquidity at each level. We choose
LCMM / as an instance here to showcase its interpolation ability. In Fig. 1b, to facilitate comparisons, we assume that
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DEFERRED PROOFSA.1 Proof of Theorem 1
The binary-search property implies that the nodes 𝑧 included in the price calculation (lines 5 and 9) form the cover of 𝐼 , so the algorithmcorrectly returns the price of 𝐼 . The running time follows thanks to height balance, which implies the depth of the tree is O( log 𝑛 vals ) . A.2 Proof of Theorem 2
We start by showing that a rotation at node 𝑧 preserves its partial normalization correctness . There are two kinds of rotations, depicted inFig. 2. The left rotation takes as input a node 𝑧 , with children denoted 𝑧 and 𝑧 , and children of 𝑧 denoted 𝑧 and 𝑧 , and rearranges theserelationships by removing the node 𝑧 and creating a node 𝑧 , such that 𝑧 now has children 𝑧 and 𝑧 , and 𝑧 has children 𝑧 and 𝑧 . The right rotation is the symmetric operation. Figure 2: Left and right rotations with node 𝑧 as an input. Depicted update corresponds to the left rotation. The full procedure of
RotateLeft is described in Algorithm 5. When performing rotations, we need to ensure that the node removal (i.e.,removal of 𝑧 in left rotation and of 𝑧 in right rotation) does not impact the market state. We achieve this by moving the shares from theremoved node into its children, so at the time of removal it holds zero shares (see the right-hand side of Fig. 2, and line 4 of Algorithm 5). Algorithm 5
Left rotation at node 𝑧 (right rotation is symmetric). Define subroutines:ResetInnerNode( 𝑧 ): reset ℎ 𝑧 and 𝑆 𝑧 based on the children of 𝑧 and the value 𝑠 𝑧 : ℎ 𝑧 ← + max { ℎ left ( 𝑧 ) , ℎ right ( 𝑧 ) } , 𝑆 𝑧 ← 𝑒 𝑠 𝑧 / 𝑏 ( 𝑆 left ( 𝑧 ) + 𝑆 right ( 𝑧 ) ) AddShares( 𝑧, 𝑠 ): increase the number of shares held in 𝑧 by 𝑠 : 𝑠 𝑧 ← 𝑠 𝑧 + 𝑠 , 𝑆 𝑧 ← 𝑒 𝑠 / 𝑏 𝑆 𝑧 procedure RotateLeft( 𝑧 ): Let 𝑧 = left ( 𝑧 ) , 𝑧 = right ( 𝑧 ) , 𝑧 = left ( 𝑧 ) , 𝑧 = right ( 𝑧 ) AddShares( 𝑧 , 𝑠 𝑧 ), AddShares( 𝑧 , 𝑠 𝑧 ), delete node 𝑧 Let 𝑧 be a new node with: left ( 𝑧 ) = 𝑧 , right ( 𝑧 ) = 𝑧 , 𝐼 𝑧 = 𝐼 𝑧 ∪ 𝐼 𝑧 , 𝑠 𝑧 = ResetInnerNode( 𝑧 ) Update node 𝑧 : left ( 𝑧 ) ← 𝑧 , right ( 𝑧 ) ← 𝑧 , ResetInnerNode( 𝑧 ) Lemma 1.
A rotation operation preserves partial-normalization correctness.
Proof. We prove that the original partial normalization value of node 𝑧 , 𝑆 𝑧 , is the same as the updated value, 𝑆 ′ 𝑧 , after a left rotation. Aright rotation follows symetrically. 𝑆 𝑧 = 𝑒 𝑠 𝑧 / 𝑏 · (cid:0) 𝑆 𝑧 + 𝑆 𝑧 (cid:1) = 𝑒 𝑠 𝑧 / 𝑏 · (cid:16) 𝑆 𝑧 + 𝑒 𝑠 𝑧 / 𝑏 · (cid:0) 𝑆 𝑧 + 𝑆 𝑧 (cid:1)(cid:17) = 𝑒 𝑠 𝑧 / 𝑏 · (cid:16) 𝑆 ′ 𝑧 + 𝑆 ′ 𝑧 + 𝑆 ′ 𝑧 (cid:17) = 𝑒 𝑠 𝑧 / 𝑏 · (cid:16) 𝑒 𝑠 ′ 𝑧 / 𝑏 · (cid:16) 𝑆 ′ 𝑧 + 𝑆 ′ 𝑧 (cid:17) + 𝑆 ′ 𝑧 (cid:17) (since 𝑠 ′ 𝑧 = = 𝑒 𝑠 𝑧 / 𝑏 · (cid:16) 𝑆 ′ 𝑧 + 𝑆 ′ 𝑧 (cid:17) = 𝑆 ′ 𝑧 □ Proof of Theorem 2. The correctness of the buy operation follows because the shares are added to the nodes that form the cover of 𝐼 (lines 5 and 12 in Algorithms 2), and the updates up the search path restore the properties of the LMSR tree (lines 13–17 in Algorithms 2).The running time follows from height balance, which implies that the length of the search path is O( log 𝑛 ) = O( log 𝑛 vals ) . □ .3 Proof of Theorem 3 We first show that the constraints A ⊤ 𝝁 = imply that all levels ℓ = , , . . . , 𝐾 in 𝝁 are mutually coherent. To do this, it suffices to show thatall pairs of consecutive levels ℓ and ℓ + 𝜇 𝑦 = 𝜇 𝑦 l + 𝜇 𝑦 r for all 𝑦 ∈ Z ℓ where we let 𝑦 l = left ( 𝑦 ) and 𝑦 r = right ( 𝑦 ) .We proceed by induction, beginning with ℓ = 𝐾 −
1. In this base case, the constraint a ⊤ 𝑦 𝝁 =
0, expressed in Eq. (13), states that 𝑏 𝐾 𝜇 𝑦 = 𝑏 𝐾 𝜇 𝑦 l + 𝑏 𝐾 𝜇 𝑦 r , implying levels 𝐾 − 𝐾 are coherent.Now assume that all the levels 𝑘 > ℓ are mutually coherent. We aim to show that levels ℓ and ℓ + 𝑦 ∈ Z ℓ . Thenthe constraint a ⊤ 𝑦 𝝁 =
0, expressed in Eq. (13), implies that (cid:18) ∑︁ 𝑘 > ℓ 𝑏 𝑘 (cid:19) 𝜇 𝑦 = ∑︁ 𝑘 > ℓ 𝑏 𝑘 ∑︁ 𝑧 ∈Z 𝑘 : 𝑧 ⊂ 𝑦 𝜇 𝑧 = ∑︁ 𝑘 > ℓ 𝑏 𝑘 (cid:18) ∑︁ 𝑧 ∈Z 𝑘 : 𝑧 ⊆ 𝑦 l 𝜇 𝑧 + ∑︁ 𝑧 ∈Z 𝑘 : 𝑧 ⊆ 𝑦 r 𝜇 𝑧 (cid:19) = ∑︁ 𝑘 > ℓ 𝑏 𝑘 (cid:16) 𝜇 𝑦 l + 𝜇 𝑦 r (cid:17) . (21)Eq. (21) follows because 𝑦 l and 𝑦 r are in level ℓ +
1, which is coherent with all levels 𝑘 ≥ ℓ + 𝜇 𝑦 = 𝜇 𝑦 l + 𝜇 𝑦 r for all 𝑦 ∈ Z ℓ , establishing the coherence between levels ℓ and ℓ + 𝐾 are determined by 𝐶 𝐾 , so they describe a probability distribution over Ω .Since A ⊤ 𝒑 ( 𝜽 ) = , all the levels in 𝒑 ( 𝜽 ) are coherent with level 𝐾 , which means that they correspond to the expectation of 𝝓 under theprobability distribution described by the prices at level 𝐾 . Thus, 𝒑 ( 𝜽 ) is a coherent price vector and the multi-resolution LCMM is thereforearbitrage-free. A.4 Proof of Theorem 4
The worst-case loss of an LCMM is bounded by the sum of the worst-case losses of the component markets 𝐶 𝑘 [10]. In our case, these areLMSR submarkets with losses bounded by 𝑏 𝑘 log |Z 𝑘 | , so the worst-case loss of the resulting LCMM is at most 𝐾 ∑︁ 𝑘 = 𝑏 𝑘 log ( 𝑘 ) = 𝐾 ∑︁ 𝑘 = 𝑏 𝑘 ( 𝑘 log 2 ) ≤ 𝐵 ∗ log 2 , proving the theorem. A.5 Proof of Theorem 5
Algorithm 3 returns the correct price of 𝐼 , because prices are coherent among submarkets and the nodes included in price calculations forma cover of 𝐼 . The running time is proportional to the length of the search path, which terminates, at the latest, once the first node 𝑧 with 𝛼 𝑧 = 𝛼 is reached. The level of this node coincides with the precision of 𝛼 . A.6 Proof of Theorem 6 and Additional Deferred Material from Section 4.3
We begin by deriving an identity that will be useful in the following analysis. For this derivation, let 𝐶 be an LMSR with the liquidityparameter 𝑏 , defined over an outcome space Ω . We will derive a relationship between the price vector in a state 𝜽 and the price vector ina new state 𝜽 ′ = 𝜽 + 𝜹 , where 𝜹 is any bundle restricted to securities in 𝐸 , i.e., 𝛿 𝜔 = 𝜔 ∉ 𝐸 . Denoting 𝝁 = 𝒑 ( 𝜽 ) , 𝜇 𝐸 = 𝑝 𝐸 ( 𝜽 ) , and 𝝁 ′ = 𝒑 ( 𝜽 ′ ) , we have 𝜇 ′ 𝜔 = 𝑒 𝜃 𝜔 / 𝑏 𝑒 𝛿 𝜔 / 𝑏 (cid:205) 𝜈 ∉ 𝐸 𝑒 𝜃 𝜈 / 𝑏 + (cid:205) 𝜈 ∈ 𝐸 𝑒 𝜃 𝜈 / 𝑏 𝑒 𝛿 𝜈 / 𝑏 = 𝜇 𝜔 𝑒 𝛿 𝜔 / 𝑏 − 𝜇 𝐸 + (cid:205) 𝜈 ∈ 𝐸 𝜇 𝜈 𝑒 𝛿 𝜈 / 𝑏 , (22)where Eq. (22) follows by dividing the numerator as well as denominator by (cid:205) 𝜈 ∈ Ω 𝑒 𝜃 𝜈 / 𝑏 .We next establish correctness of the arbitrage removal procedure from Algorithm 4. The following lemma provides a critical step:Lemma 2. Fix a level ℓ < 𝐾 . Let ˜ 𝜽 be a market state in ˜ 𝐶 such that the associated prices, 𝝁 = ˜ 𝒑 ( ˜ 𝜽 ) , are coherent among all levels 𝑘 > ℓ . Then,for any 𝑡 ∈ R and any node 𝑦 with level ( 𝑦 ) ≤ ℓ , the prices after buying 𝑡 shares of a 𝑦 , i.e., 𝝁 ′ = ˜ 𝒑 ( ˜ 𝜽 + 𝑡 a 𝑦 ) , remain coherent among all levels 𝑘 > ℓ . To use Lemma 2 for arbitrage removal, we start with a market state ˜ 𝜽 where all levels are coherent. When a trader buys some shares of asecurity 𝜙 𝑦 , the level ℓ = level ( 𝑦 ) loses coherence with other levels. By buying a certain number of shares of a 𝑦 , it is possible to restorecoherence between ℓ and ℓ +
1, and Lemma 2 then implies that coherence with all further levels 𝑘 > ℓ + 𝑦 and the bundle a par ( 𝑦 ) as implemented in Algorithm 4.roof. Consider two arbitrary levels 𝑘 and 𝑚 with ℓ < 𝑘 < 𝑚 . Since prices are coherent between levels 𝑘 and 𝑚 before buying 𝑡 shares of a 𝑦 , we have, for any 𝑧 ∈ Z 𝑘 , 𝜇 𝑧 = ∑︁ 𝑢 ∈Z 𝑚 : 𝑢 ⊂ 𝑧 𝜇 𝑢 . (23)Let 𝜋 𝑦 denote the price of 𝜙 𝑦 according to the securities in Z 𝑘 and Z 𝑚 , that is, 𝜋 𝑦 = (cid:205) 𝑧 ∈Z 𝑘 : 𝑧 ⊂ 𝑦 𝜇 𝑧 = (cid:205) 𝑢 ∈Z 𝑚 : 𝑢 ⊂ 𝑦 𝜇 𝑢 . Note that 𝜋 𝑦 mightdiffer from 𝜇 𝑦 , because level ℓ is not necessarily coherent with levels 𝑘 and 𝑚 . Let ˜ 𝜽 ′ = ˜ 𝜽 + 𝑡 a 𝑦 . From the definition of matrix A , the updated˜ 𝜃 ′ 𝑧 and ˜ 𝜃 ′ 𝑢 for any 𝑧 ∈ Z 𝑘 and 𝑢 ∈ Z 𝑚 are˜ 𝜃 ′ 𝑧 = (cid:40) ˜ 𝜃 𝑧 − 𝑡𝑏 𝑘 if 𝑧 ⊂ 𝑦 ,˜ 𝜃 𝑧 otherwise, ˜ 𝜃 ′ 𝑢 = (cid:40) ˜ 𝜃 𝑢 − 𝑡𝑏 𝑚 if 𝑢 ⊂ 𝑦 ,˜ 𝜃 𝑢 otherwise.We calculate the new price 𝜇 ′ 𝑧 of any node 𝑧 ∈ Z 𝑘 and show it equals to the price derived from its descendants 𝑢 ∈ Z 𝑚 . First, if 𝑧 ⊂ 𝑦 , thenby Eq. (22) and Eq. (23), 𝜇 ′ 𝑧 = 𝜇 𝑧 𝑒 − 𝑡 𝜋 𝑦 𝑒 − 𝑡 + − 𝜋 𝑦 = (cid:205) 𝑢 ∈Z 𝑚 : 𝑢 ⊂ 𝑧 𝜇 𝑢 𝑒 − 𝑡 𝜋 𝑦 𝑒 − 𝑡 + − 𝜋 𝑦 = ∑︁ 𝑢 ∈Z 𝑚 : 𝑢 ⊂ 𝑧 𝜇 ′ 𝑢 . If 𝑧 ⊄ 𝑦 , then we similarly have 𝜇 ′ 𝑧 = 𝜇 𝑧 𝜋 𝑦 𝑒 − 𝑡 + − 𝜋 𝑦 = (cid:205) 𝑢 ∈Z 𝑚 : 𝑢 ⊂ 𝑧 𝜇 𝑢 𝜋 𝑦 𝑒 − 𝑡 + − 𝜋 𝑦 = ∑︁ 𝑢 ∈Z 𝑚 : 𝑢 ⊂ 𝑧 𝜇 ′ 𝑢 . Thus, prices remain coherent among all levels 𝑚 > 𝑘 > ℓ . □ Building upon Lemma 2, the following lemma provides the precise trade required to restore coherence after an update.Lemma 3.
Fix a level ℓ < 𝐾 and a node 𝑦 ∈ Z ℓ and let 𝑦 l = left ( 𝑦 ) and 𝑦 r = right ( 𝑦 ) . Let ˜ 𝜽 and ˜ 𝜽 = ˜ 𝜽 + 𝜹 be market states in ˜ 𝐶 , withassociated prices 𝝁 = ˜ 𝒑 ( ˜ 𝜽 ) and 𝝁 = ˜ 𝒑 ( ˜ 𝜽 ) such that: • prices 𝝁 are coherent among all levels 𝑘 ≥ ℓ ; • 𝜹 is a vector that is zero outside descendants of 𝑦 , i.e., 𝛿 𝑧 = whenever 𝑧 ⊈ 𝑦 ; • prices 𝝁 are coherent among all levels 𝑘 > ℓ .Let ˜ 𝜽 ′ = ˜ 𝜽 + 𝑡 a 𝑦 where 𝑡 = 𝑏 ℓ 𝐵 ℓ − log (cid:18) − 𝜇 𝑦 𝜇 𝑦 · 𝜇 𝑦 l + 𝜇 𝑦 r − 𝜇 𝑦 l − 𝜇 𝑦 r (cid:19) . Then the associated prices 𝝁 ′ = ˜ 𝒑 ( ˜ 𝜽 ′ ) are coherent among all levels 𝑘 ≥ ℓ . Proof. By Lemma 2, adding 𝑡 a 𝑦 to ˜ 𝜽 maintains coherence among levels 𝑘 > ℓ , so it suffices to show that levels ℓ and ℓ + 𝝁 ′ . Thus, we have to show that 𝜇 ′ 𝑧 = 𝜇 ′ left ( 𝑧 ) + 𝜇 ′ right ( 𝑧 ) for all 𝑧 ∈ Z ℓ .First note that by the assumption on 𝜹 and the definition of a 𝑦 , we have˜ 𝜃 𝑧 = ˜ 𝜃 𝑧 = ˜ 𝜃 ′ 𝑧 for all 𝑧 ∈ Z ℓ \{ 𝑦 } ˜ 𝜃 𝑢 = ˜ 𝜃 𝑢 = ˜ 𝜃 ′ 𝑢 for all 𝑢 ∈ Z ℓ + \{ 𝑦 l , 𝑦 r } .Therefore, by Eq. (22), we have for all 𝑧 ∈ Z ℓ \{ 𝑦 } 𝜇 ′ 𝑧 − 𝜇 ′ 𝑦 = 𝜇 𝑧 − 𝜇 𝑦 , and 𝜇 ′ left ( 𝑧 ) + 𝜇 ′ right ( 𝑧 ) − 𝜇 ′ 𝑦 l − 𝜇 ′ 𝑦 r = 𝜇 left ( 𝑧 ) + 𝜇 right ( 𝑧 ) − 𝜇 𝑦 l − 𝜇 𝑦 r . (24)Since the vector 𝝁 satisfies 𝜇 𝑧 = 𝜇 left ( 𝑧 ) + 𝜇 right ( 𝑧 ) for all 𝑧 ∈ Z ℓ \{ 𝑦 } , Eq. (24) implies that we also have 𝜇 ′ 𝑧 = 𝜇 ′ left ( 𝑧 ) + 𝜇 ′ right ( 𝑧 ) for all 𝑧 ∈ Z ℓ \{ 𝑦 } as long as 𝜇 ′ 𝑦 = 𝜇 ′ 𝑦 l + 𝜇 ′ 𝑦 r . Thus, in order to show that levels ℓ and ℓ + 𝝁 ′ , it suffices to show that 𝜇 ′ 𝑦 = 𝜇 ′ 𝑦 l + 𝜇 ′ 𝑦 r .We begin by explicitly calculating ˜ 𝜃 ′ 𝑧 and ˜ 𝜃 ′ 𝑢 for any 𝑧 ∈ Z ℓ and any 𝑢 ∈ Z ℓ + :˜ 𝜃 ′ 𝑧 = (cid:40) ˜ 𝜃 𝑧 + 𝑡𝐵 ℓ if 𝑧 = 𝑦 ,˜ 𝜃 𝑧 otherwise, ˜ 𝜃 ′ 𝑢 = (cid:40) ˜ 𝜃 𝑢 − 𝑡𝑏 ℓ + if 𝑢 ∈ { 𝑦 l , 𝑦 r } ,˜ 𝜃 𝑢 otherwise.Therefore, 𝜇 ′ 𝑦 = 𝜇 𝑦 𝑒 𝑡𝐵 ℓ / 𝑏 ℓ 𝜇 𝑦 𝑒 𝑡𝐵 ℓ / 𝑏 ℓ + − 𝜇 𝑦 = + − 𝜇 𝑦 𝜇 𝑦 𝑒 − 𝑡𝐵 ℓ / 𝑏 ℓ nd similarly, 𝜇 ′ 𝑦 l + 𝜇 ′ 𝑦 r = ( 𝜇 𝑦 l + 𝜇 𝑦 r ) 𝑒 − 𝑡 ( 𝜇 𝑦 l + 𝜇 𝑦 r ) 𝑒 − 𝑡 + − 𝜇 𝑦 l − 𝜇 𝑦 r = + − 𝜇 𝑦 l − 𝜇 𝑦 r 𝜇 𝑦 l + 𝜇 𝑦 r 𝑒 𝑡 . Thus, it remains to show that − 𝜇 𝑦 𝜇 𝑦 𝑒 − 𝑡𝐵 ℓ / 𝑏 ℓ = − 𝜇 𝑦 l − 𝜇 𝑦 r 𝜇 𝑦 l + 𝜇 𝑦 r 𝑒 𝑡 , or equivalently: − 𝜇 𝑦 𝜇 𝑦 · 𝜇 𝑦 l + 𝜇 𝑦 r − 𝜇 𝑦 l − 𝜇 𝑦 r = 𝑒 𝑡 ( + 𝐵 ℓ / 𝑏 ℓ ) . But this follows from our choice of 𝑡 and the fact that 𝐵 ℓ − = 𝐵 ℓ + 𝑏 ℓ , completing the proof. □ We finish the section with the proof of Theorem 6.Proof of Theorem 6. Algorithm 4 correctly updates the tree (and returns the cost), because the shares are added to the nodes that forma cover of 𝐼 , and coherence is then restored by applying Lemma 3 up the search path. Running times of both algorithms are proportional tothe length of the search path to the first node 𝑧 with 𝛼 𝑧 = 𝛼 , whose level coincides with the precision of 𝛼 . □ B TRADING DYNAMICS AND ADDITIONAL RESULTSB.1 Trading Dynamics
We simulate a market consisting of ten traders. The outcome space is [ , ) , discretized at the precision 𝐾 =
10. Traders, indexed as 𝑖 ∈ { , . . . , } , have noisy access to the underlying true signal 𝑝 = .
4. Trader 𝑖 ’s belief takes form of a beta distribution Beta ( 𝑎 𝑖 , 𝑏 𝑖 ) with 𝑎 𝑖 ∼ Binomial ( 𝑝, 𝑛 𝑖 ) , 𝑏 𝑖 = 𝑛 𝑖 − 𝑎 𝑖 , and 𝑛 𝑖 = 𝑖 representing the quality of the agent’s observation of the signal 𝑝 . Each trader 𝑖 has an exponentialutility 𝑢 𝑖 ( 𝑊 ) = − 𝑒 − 𝑊 , where 𝑊 is the trader’s wealth. We consider budget-limited cost-based market makers, whose worst-case loss maynot exceed a budget constraint 𝐵 . For LMSR at precision 𝑘 , this means setting the liquidity parameter to 𝑏 = 𝐵 / log ( 𝑘 ) . In our experiments,we consider two LMSR markets at precision levels 4 and 8, denoted as LMSR 𝑘 = and LMSR 𝑘 = . On the other hand, a multi-resolution LCMMhas an infinite number of choices for its liquidity at each precision level. To showcase its interpolation ability, we consider LCMM thatevenly splits its budget to precision levels 4 and 8, and denote it as LCMM / .Each market starts with the uniform prior, i.e., the initial market prices for all outcomes are equal. In each time step, a uniformlyrandom agent is picked to trade. The selected agent considers a set of 50 interval securities, with endpoints randomly sampled accordingto the agent’s belief. The candidate intervals are rounded to the precision of the corresponding market. The agent considers trading theexpected-utility-optimizing number of shares for each interval, and ultimately picks the best interval and executes the trade. The marketmaker updates prices accordingly, until the market equilibrium is reached (no trader in the market has the incentive to trade).Following the described protocol, we run markets mediated by the three respective market makers,
LMSR 𝑘 = , LMSR 𝑘 = , and LCMM / , overa range of budget constraints. To decrease variance, we generate 40 controlled simulation traces (described by a sequence of agent arrivalsand their draws of the candidate intervals) and run the market makers on those same traces. Therefore, any change in agent behavior andprice convergence is caused by the different cost functions that market makers adopt to aggregate trades. B.2 Additional Experiments
In Section 5, we demonstrated that by splitting the budget between submarkets that offer interval securities at different precisions, the multi-resolution LCMM is able to interpolate the performance of LMSR market makers. It can aggregate information at the coarser level efficiently,while also achieving accurate belief elicitation at the finer resolution (after sufficiently many trades). Here we provide numerical results overa wider range of market maker’s budget constraints, validating how the multi-resolution LCMM can balance the price convergence behaviorof LMSR markets.Fig. 3 shows the price convergence error as a function of budget constraint (thus, the liquidity parameter) and the number of trades forthe three respective market makers. Results are averaged over forty random but controlled trading sequences. The solid lines depict the priceconvergence error at precision level 𝑘 =
8, and the dashed ones for precision level 𝑘 =
4. The minimum point on each curve indicates theoptimal budget, or the optimal value of the liquidity parameter to adopt, for the particular cost function and a specific number of trades.Intuitively, when the budget for running a market is sufficient, a market operator can support interval securities at any fine-grainedprecision level, or use only a portion of the budget to achieve optimal performance. However, when the budget for running a market islimited, say B less than 8, the market designer can preferably aggregate information faster at a coarser resolution by limiting the precision ofinterval endpoints (e.g., adopting
LMSR 𝑘 = ). However, by design, it cannot accurately elicit beliefs at finer resolutions, even when the marketis run for a sufficiently long period of time. The LMSR 𝑘 = , on the other hand, benefits from a larger number of trades to aggregate morefine-grained information. Running the two LMSR markets independently may balance this convergence trade-off, but inevitably results As the number of available interval securities grows exponentially as the supported precision increases, we assume agents have a computational limit and can only consider a(sub)set of available securities. B P r i c e c o n v e r g e n c e e rr o r Num of trades (a)
LMSR 𝑘 = . B P r i c e c o n v e r g e n c e e rr o r Num of trades (b)
LMSR 𝑘 = . B P r i c e c o n v e r g e n c e e rr o r Num of trades (c)
LCMM / . Figure 3: The price convergence error as a function of liquidity and the number of trades (indicated by the color of the line)for the three respective market makers. Solid lines record price convergence error at the finer precision level 𝑘 = , and dashedones at the coarser level 𝑘 = . in inconsistent prices between the markets. Given the different convergence properties of separate LMSRs, a multi-resolution LCMM canallocate its budget accordingly to achieve a desired convergence performance, while maintaining coherent prices. For example, a marketdesigner, who considers information at precision levels 𝑘 = 𝑘 ==