A Unified Framework for Dynamic Pari-Mutuel Information Market Design
Shipra Agrawal, Erick Delage, Mark Peters, Zizhuo Wang, Yinyu Ye
aa r X i v : . [ q -f i n . T R ] F e b A Unified Framework for Dynamic Pari-Mutuel InformationMarket Design [Extended Abstract]
Shipra [email protected] Erick [email protected] Mark [email protected] [email protected] Yinyu [email protected]
ABSTRACT
Recently, several new pari-mutuel mechanisms have been in-troduced to organize markets for contingent claims. Hanson[7] introduced a market maker derived from the logarith-mic scoring rule, and later Chen & Pennock [5] developeda cost function formulation for the market maker. On theother hand, the SCPM model of Peters et al. [9] is based onideas from a call auction setting using a convex optimiza-tion model. In this work, we develop a unified frameworkthat bridges these seemingly unrelated models for centrallyorganizing contingent claim markets. The framework, de-veloped as a generalization of the SCPM, will support manydesirable properties such as proper scoring, truthful bidding(in a myopic sense), efficient computation, and guaranteeson worst case loss. In fact, our unified framework will al-low us to express various proper scoring rules, existing ornew, from classical utility functions in a convex optimizationproblem representing the market organizer. Additionally, weutilize concepts from duality to show that the market modelis equivalent to a risk minimization problem where a convexrisk measure is employed. This will allow us to more clearlyunderstand the differences in the risk attitudes adopted byvarious mechanisms, and particularly deepen our intuitionabout popular mechanisms like Hanson’s market-maker. Inaggregate, we believe this work advances our understandingof the objectives that the market organizer is optimizing inpopular pari-mutuel mechanisms by recasting them into oneunified framework.
1. INTRODUCTION
Contingent claim markets are organized for a variety ofpurposes. Prediction markets are created to aggregate in-formation about a particular event. Financial markets in-volving contingent claims allow traders to hedge their ex-posure to certain event outcomes. Betting markets are de-signed for entertainment purposes. The participants in these
Permission to make digital or hard copies of all or part of this work forpersonal or classroom use is granted without fee provided that copies arenot made or distributed for profit or commercial advantage and that copiesbear this notice and the full citation on the first page. To copy otherwise, torepublish, to post on servers or to redistribute to lists, requires prior specificpermission and/or a fee.Copyright 200X ACM X-XXXXX-XX-X/XX/XX ...$5.00. markets trade securities which will pay a fixed amount if acertain event occurs. Some examples of these events wouldbe the winner of the World Series, the value of the latestconsumer price index or the release date of Windows Vista.Prediction markets have grown in popularity as research intothe accuracy of their predictions has shown that they effec-tively aggregate information from the trading population.One of the longest-running prediction markets is the IowaElectronic Market which allows real money betting on var-ious elections. Studies by Berg and her coauthors (see [2],[3] and [4]) have shown that the information generated bythese markets often serves as a better prediction of actualoutcomes than polling data. Google has run internal pre-diction markets over a variety of events and Cowgill et al.[6] have shown that their predictions also perform quite well.Despite the potential value created by these markets, therecan be some difficulties with their introduction and devel-opment. First, many nascent markets suffer from liquidityproblems. Occasionally these problems stem from the choiceof mechanism used to operate the market. Organizing mar-kets as a continuous double auction (like the NASDAQ stockmarket) is a popular option and usually performs well. How-ever, in thin markets, Bossaerts et al. [1] have demonstratedthat some problems surface which inhibit the growth of liq-uidity. To overcome this situation, the market organizercould introduce an automated market-maker which will cen-trally interact with the traders. This mechanism will havesome rules for pricing shares. The market organizer mustdetermine these rules with one key question being his owntolerance for risk. Recently, there has been a surge in re-search of these automated market-makers.New market-making mechanisms based on pari-mutuel prin-ciples have recently been developed by Hanson [7], Pennocket al. [8] and Peters et al. [9]. These market-makers al-low contingent claims in nascent market to be immediatelypriced according to rules of the mechanism. The mechanismsare pari-mutuel in the sense that the winners are generallypaid out by the stakes of the losers. The claims being tradedare commitments to pay out a fixed amount if a particularevent occurs in the future. The mechanism developed byHanson has been shown to perform well in simulated mar-kets [10] and has been adopted by many online predictionmarkets.However, the origins of these new mechanisms differ. Peterst al. [9] developed their mechanism by creating a sequentialversion of a call auction problem which is solved by convexoptimization. Their Sequential Convex Pari-mutuel Mech-anism (SCPM) uses an optimization problem to determinewhen to accept orders and how to price accepted orders. Onthe other hand, Hanson’s mechanism is derived from scor-ing rules. Scoring rules are functions often used to comparedistributions. In particular, Hanson uses the logarithmicscoring rule to determine how much to charge a trader for anew order. His mechanism is called the Logarithmic MarketScoring Rule (LMSR). Using a similar approach as Han-son, it is possible to create market-makers for other scoringrules. In contrast to the SCPM, the MSR model doesn’tdirectly provide an optimization problem from the marketorganizer’s standpoint.Recently, there has been some interest in comparing andunifying these mechanisms for prediction markets. Chenand Pennock [5] give an equivalent cost function based for-mulation for the MSR market makers, and relate them toutility-based market makers. They show that a certain class(hyperbolic absolute risk aversion) of utility-based marketis equivalent to a market scoring rule market maker. Peterset al. [10] empirically compare the performances of variousmarket mechanisms. In this work, we provide strong theo-retical foundation for unifying existing market makers likethe SCPM, the MSR and cost function based markets un-der a single convex optimization framework. Our model notonly aids in comparing various mechanisms, but also pro-vides intuitive understanding of the behavior of the marketorganizer in these seemingly different mechanisms. Specifi-cally, our main contributions are as follows: • The unifying framework : we propose a generalized ver-sion of the SCPM as a unified convex optimizationframework for market makers. The new model en-joys many desirable properties like a truthful pricingscheme, efficient cost function formulation, scoring proper-ness, and guarantees on worst case loss bounds. • Equivalence to the MSR market makers: we establishour claim that the new SCPM framework unifies ex-isting market makers by showing that a) any marketmaker based on a proper scoring rule (MSR) can beformulated as a special case of this SCPM model; andconversely, b) an objective function in our frameworkimplicitly corresponds to a (strictly) proper scoringrule under certain easily verifiable conditions. • Intuitive interpretation of the market mechanisms:
Weshow that the market maker’s model in the new frame-work is equivalent to convex risk minimization for themarket maker. Thus, when using this model to acceptincoming orders the market maker is actually takingrational decisions with respect to a risk attitude as de-fined by the choice of utilities of the framework. Fur-ther, we show that for some popular mechanisms likethe LMSR, the implicit risk function turns out to char-acterize precisely how much the market maker is pre-pared to invest in order to learn a distribution ~p whichis very different from his prior belief. This explains thedesign choices of these mechanisms. • New mechanisms:
We take a step forward and demon-strate with examples that various insights provided by the unified framework can be used to guide the de-sign of new mechanisms that have desirable propertieswith respect to various criteria discussed: truthfulness,worst case loss, computational efficiency, risk attitude,properness of the corresponding scoring rule, etc.The rest of the paper is organized as follows. To begin,Section 2 will provide some background on the mechanismswhich we will be studying. In Section 3, we propose ournew framework based on a generalization of the SCPM, anddemonstrate its properties of truthful pricing scheme, costfunction based formulation, and guarantees on worst caseloss bounds. In Section 4, we show that the SCPM frame-work is equivalent to the MSR market makers. Section 5 willfurther explain how the market organizer’s decision problemin our unified framework is actually equivalent to a convexrisk minimization problem. We conclude the paper with adiscussion in Section 6 where we compare the existing mech-anisms, and provide examples illustrating simple guidelinesfor designing new ones.
2. BACKGROUND
In this section, we will provide some background on thekey mechanisms for prediction markets discussed in thiswork – the market scoring rule mechanisms (MSR), cost-function formulation of markets, and the Sequential ConvexPari-mutuel mechanism (SCPM).Let ω represent a discrete or discretized random eventto be predicted, with N mutually exclusive and exhaustiveoutcomes. We consider a contingent claims market whereclaims are of the form “Pays $1 if the outcome state is i ”.A new trader arrives and submits an order which essentiallyspecifies the claims over each outcome state that the traderdesires to buy. The market maker then decides what priceto charge for the new order. Various mechanisms treat anew order in the following seemingly different manners: Let ~r = ( r , r , ..., r N ) represents a probability estimatefor the random event ω . A scoring rule is a sequence of scor-ing functions, S = S ( ~r ) , S ( ~r ) , ..., S N ( ~r ), such that a score S i ( ~r ) is assigned to ~r if outcome i of the random variable ω is realized. A proper scoring rule [16] is a scoring rule thatmotivates truthful reporting. Based on proper scoring rules,Hanson[7] developed a Market Scoring Rule(MSR) mecha-nism. In the MSR market, the market maker with a properscoring rule S begins by setting an initial probability esti-mate, ~r . Every trader can change the current probabilityestimate to a new estimate of his choice as long as he agreesto pay the market maker the scoring rule payment associ-ated with the current probability estimate and receive thescoring rule payment associated with the new estimate.Some examples of market scoring rules are the logarithmicmarket scoring rule (LMSR)[7]: S i ( ~r ) = b log( r i ) ( b > S i ( ~r ) = 2 br i − b X j r j ( b > truthful bids from the market traders. Recently, Chen & Pennock [5] proposed a cost functionbased implementation of market makers. Let the vector ~q ∈ R N represents the number of claims on each state cur-rently held by the traders. In the cost function formulation,the total cost of all the orders ~q is calculated via some costfunction C ( ~q ). A trader submits an order characterized bythe vector ~a ∈ R N where a i reflects the number of claimsover state i that the trader desires. The market organizerwill charge the new trader C ( ~q + ~a ) − C ( ~q ) for his order. Atany time in the market, the going price of a claim for state i , p i ( ~q ), equals ∂C/∂q i . The price is the cost per share forpurchasing an infinitesimal quantity of security i .Chen & Pennock [5] show that any scoring rule has an equiv-alent cost function formulation. For example, below are thespecific cost and pricing functions for LMSR: C ( ~q ) = b log “P j e q j /b ” and p i ( ~q ) = e qi/b P j e qj/b For general market scoring rules, they proposed three equa-tions that the cost function C should satisfy so that the costfunction based market maker is equivalent to the marketbased on a given scoring rule S : S i ( ~p ) = q i − C ( ~q ) + K ∀ i P i p i = 1 p i = ∂C∂q i ∀ i (1)We will use this formulation later to prove equivalence ofMSR and SCPM mechanisms. The SCPM was designed to require traders to submit or-ders which include three elements: a limit price ( π ), a limitquantity ( l ) and a vector ( ~a ) that represents which statesthe order should contain. The components of the vector ~a will contain either a 1 (if a claim over the specified state isdesired) or a 0 (if it is not desired). The limit price refers tothe maximum amount that the trader wishes to pay for oneshare. The limit quantity represents the maximum numberof shares that the trader is willing to buy. The market makerdecides the actual number of shares x to be granted to a neworder, and the price to be charged for the order. The marketmaker solves the following optimization problem for makingthis decision:maximize x,z,~s πx − z + P i θ i log( s i )subject to ~ax + ~s + ~q = z~e ≤ x ≤ l, (2)where parameters ~q stands for the numbers of shares cur-rently held by the traders prior to the new order ( π, l,~a )arrives, and ~e represents the vector of all 1s. Each time anew order arrives, the optimization problem (2) is solvedand the state prices are defined to be the optimal dual vari-ables corresponding to the first set of constraints denoted as ~p . The trader is then charged according to the inner productof the final price and the order filled ( ~p T ~a ). This optimization problem, without the utility function P i θ i log s i , has the following interpretation for the marketmaker: besides x , decision variable z represents the maxi-mum number of accumulated shares, including ~s represent-ing the contingent numbers of surplus shares would be keptby the market maker, over all states; and ( πx − z ) in theobjective represents the profit that would be made from thenew order. Thus, market organizer aims at maximizing hisworst case profit. As we establish later in this paper, addingthe utility function P i θ i log( s i ) enhances the risk takingability of the market maker.
3. THE UNIFYING FRAMEWORK
In this paper, we illustrate that a generalized formula-tion of SCPM provides a unifying framework for pari-mutuelmarket mechanisms. We propose the following convex op-timization model with a concave continuous utility function u ( ~s ) : maximize x,z,~s πx − z + u ( ~s )subject to ~ax + ~s + ~q = z~e ≤ x ≤ l (3)Note that utility function u ( ~s ) = P i θ i log s i used in theoriginal SCPM model of Peters et al. [10] is a special caseof (3). From here on, “SCPM” refers to the above gener-alized SCPM model. When required, we disambiguate byreferring to the original model with u ( ~s ) = P i θ i log s i as“Log-SCPM”. The optimization model (3) has exactly thesame meaning for the market maker as the Log-SCPM, andinherits many desirable properties like intuitive interpreta-tion, convex formulation, global optimality, Lagrange du-ality, polynomial computational complexity, etc., as in theoriginal Log-SCPM model. Next, we demonstrate some newdesirable properties of the new SCPM framework includingtruthfulness of the pricing scheme, efficient cost functionbased scoring rules, and easily computable guarantees onworst case loss. The original SCPM model does not provide incentivesfor the traders to bid truthfully [10]. On the other hand,market scoring rules such as LMSR ensure truthful bidding.We show that this difference in incentives is attributed to adifference between the implementation of SCPM and MSRpricing scheme. In the SCPM model, the market organizerwill typically charge the trader for an accepted number ofshares based on the final price calculated by the mechanism.However, in the market scoring rules such as LMSR, thetrader is actually charged by a cost function which is equiv-alent to the integral of the pricing function over the numberof shares accepted. Thus, as the price increases while the or-der is filled, the trader is charged the instantaneous price foreach infinitesimally small portion of his order that is filled.We show that our general SCPM framework equally ad-mits truthfulness if the market maker charge traders accord-ing to the integral of the price function over infinitesimallysmall accepted shares. More specifically, the new chargingscheme solves the optimization problem for infinitesimally Indeed, there is another technical condition on u ( · ) thatis required for this model to be feasible and bounded, thatis, ∀ ~q ≥ ∃ t , ∇ u ( t~e − ~q ) T ~e = 1, where ∇ u ( · ) denotes the(sub-)gradient function.mall orders, thus computing incremental prices. Define ~p ( ǫ ) to be the dual prices corresponding to the followingoptimization problem (with l = ǫ ):maximize x,z,~s πx − z + u ( ~s )subject to ~ax + ~s + ~q = z~e ≤ x ≤ ǫ (4)The SCPM model (3) can be equivalently viewed as gradu-ally increasing ǫ until the price become more than the bidoffered, or the trader reaches his limit l . Let ¯ x be the optimalsolution to (3), i.e. the largest ǫ such that the last constraintin (4) is tight at optimality. Now, instead of charging thetrader the final price ~p (¯ x ) T ~a as in the conventional SCPM,we charge an accepted order by the following formula:( R ¯ x ~p ( ǫ ) dǫ ) T ~a (5)Below, we establish the integrability of the pricing func-tion and some important properties of the new pricing mech-anism: Lemma
1. The price vector sums up to 1 for every ǫ ;2. The price is non-negative if the utility function u ( · ) isnon-decreasing;3. The price is consistent, i.e., the market organizer willaccept another infinitesimal order if π is greater thanthe instantaneous price of the order and vice versa;4. The instantaneous price ~p ( ǫ ) T ~a is non-decreasing in ǫ ;5. The price function is integrable. Proof.
The proof of this theorem is provided in Ap-pendix A.1.The above properties of the pricing scheme lead to the fol-lowing strong result about the truthfulness of this scheme:
Theorem
Irrespective of the choice of utility func-tion u ( · ) , the optimal bidding strategy in the SCPM is my-opically truthful when the traders are charged according topricing scheme (5). Proof.
Assume that the trader has a valuation of γ forhis desired order ~a with a quantity limit of l . The trader’sprofit can be expressed as: R ( x ) = γx − ( R x ~p ( ǫ ) dǫ ) T ~a . Thetrader seeks to maximize R by choosing a proper π . Fromthe previous lemma, we know that ~p ( ǫ ) T ~a is non-decreasingin ǫ . Thus, it is easy to see that an optimal strategy when ~p (0) T ~a ≥ γ is to not place an order. Thus, we will assumethat ~p (0) T ~a < γ .Also, since p ( x ) is non-decreasing in x , R ( x ) is a concavefunction in x . Therefore, the optimal x ≤ l (that maximizestrader’s profit R ( x )) is given by the following optimalityconditions ( x − l ) ` γ − ~p ( x ) T ~a ´ = 0 γ ≥ ~p ( x ) T ~a On the other hand, the market organizer will accept a bidif and only if ~p ( ǫ ) T ~a is less than π . Hence, setting π = γ gives the optimal x , and thus optimal profit for the trader.This is equivalent to bidding truthfully since the trader willessentially bid such that the instantaneous price is driveneither to their valuation or as close to their valuation aspossible before hitting the limit quantity constraint. Therefore, we have shown that the truthfulness of thismechanism does not depend on the particular choice of util-ity function but rather on the implementation of the charg-ing method. By revising the charging method to an ‘incre-mental’ one, we can create a truthful implementation for thegeneral SCPM. As we show in the next section, the integralin the expression (5) does not need to be explicitly calculatedin order to compute it; given ¯ x , we can compute the totalcharge efficiently using a convex cost function formulation. Section 2.2 discussed a cost function based implementa-tion for market makers, introduced by Chen and Pennock[5]. In this section, we derive a convex cost function forthe SCPM, which will reduce the problem of computing theintegral (5) to a simple convex optimization problem.
Theorem
Let ~q be the number of shares on eachstate held by the traders in the SCPM market, and a neworder ( π, l,~a ) is accepted up to level ¯ x and is charged ac-cording to pricing scheme (5). Then, the charge is „Z ¯ x ~p ( ǫ ) dǫ « T ~a = C ( ~q + ~a ¯ x ) − C ( ~q ) , where C ( ~q ) is a convex cost function defined by C ( ~q ) = min t t − u ( te − ~q ) , (6) and has the property ~e T ∇ C ( ~q ) = 1 , ∀ ~q ≥ . Proof.
Note that ~p ( ǫ ) is the optimal dual solution asso-ciated with the optimization problem:maximize z,~s πǫ − z + u ( ~s )s . t . ~aǫ + ~s + ~q = z~e. Let V ( ~q, ǫ ) denote the optimal objective value of the aboveproblem. Then, V ( ~q, ǫ ) = max z,~s πǫ − z + u ( ~s )s . t . ~aǫ + ~q + ~s = z~e = πǫ − min z n z − u ( z~e − ~q − ~aǫ ) o = πǫ − C ( ~q + ~aǫ )Next, using local sensitivity analysis results (e.g., discussedin [15]), the optimal dual variables ~p ( ǫ ) p ( ǫ ) i = − ∂V ( ~q, ǫ ) ∂q i = ∂C ( ~q + ~aǫ ) ∂q i Thus, we can conclude the statement presented in our the-orem by performing the integration Z ¯ x ~p ( ǫ ) T ~adǫ = Z ¯ x ∇ q C ( ~q + ~aǫ ) T ~adǫ = Z ¯ x dC ( ~q + ~aǫ ) dǫ dǫ = C ( ~q + ~a ¯ x ) − C ( ~q )Finally, we can verify that C ( ~q ) is a convex function of ~q since it is the minimum over t of a function that is jointlyconvex in both t and ~q .Note that not only is the cost function convex, but it canalso be simply computed for any given ~q by solving a singleariable convex optimization problem. This is in contrastto the market scoring rules, where computing the cost func-tion is non-trivial and requires solving a set of differentialequations (refer equation (1) in section 2.2). An interesting consequence of the cost function represent-ing the SCPM developed in the previous section, is that theworst case loss can be formulated as a convex optimizationproblem:
Theorem
Assuming the market starts with sharesinitially, then the worst case loss for the market maker usingthe SCPM mechanism is given by B + C (0) where B = max i { max ~s u ( ~s ) − s i } and C ( · ) is the cost function defined by (6). Proof.
Let the number of shares held by the tradersat time t is ( ~q ) t . By Theorem 3.3, assuming we startedwith 0 shares initially, the total money collected at time t is C (( ~q ) t ) − C (0). On the other hand, if state scenario i occurs,the market maker needs to pay amount ( q i ) t , Thus, we canfind for each state scenario i , the worst case loss by solvingthe optimization problem :max ~q ≥ q i − ( C ( ~q ) − C (0)) = max t,~q ≥ { ( q i − t ) + u ( t~e − ~q ) } + C (0)= max ~s { u ( ~s ) − s i } + C (0) . Then, we take the maximum among all scenarios to concludethe proof.As a corollary of the above theorem, we have that:
Corollary
Computing the worst case loss boundfor the SCPM is a convex optimization problem. Further-more, a necessary and sufficient condition on utility func-tion u ( · ) in order to guarantee a bounded loss is that thedifference u ( ~s ) − s i is bounded from above. Below, we illustrate the application of the above theoremthrough some examples. Detailed proofs for these examplesare available in Appendix D.
Example
For u ( ~s ) = − b log `P i exp( − s i /b ) ´ , wecan compute B = 0 , C (0) = b log N , giving worst case lossas b log N . Example
For u ( ~s ) = ~e T ~sN − b ~s T ( I − ~e~e T N ) ~s , we canderive B = b (1 − N ) , C (0) = 0 giving bound ( N − bN . Later in Section 4, we demonstrate that the above two choicesof utility functions are equivalent to the LMSR and theQuadratic market scoring rules respectively. Observe thatthe derived bounds match those known in the literature forthese scoring rules [5].
Example
For the Log-SCPM, u ( ~s ) = P i θ i log( s i ) ,we can show that B is unbounded by using s = 1 , s i = α ,and letting α → ∞ . B ≥ lim α →∞ θ log 1 + X i =1 θ i log( α ) − ∞ and C (0) = P i θ i − P i θ i log P i θ i . Thus, the worst caseloss is unbounded in this case. Example
For u ( ~s ) = min i s i , the worst case lossis ; since for all values of ( ~s, t ) : u ( ~s ) ≤ s i and t = u ( t~e ) .Observe that here the utility of the surplus profit ~s is equal tothe minimum or “worst case” profit. This represents extremerisk averseness of the market organizer. The last example provided a glimpse of how the utility func-tion relates to the risk averseness of the market maker. InSection 5, we will further build this intuition for the behav-ior of the organizer by recasting our optimization frameworkas a risk minimization problem.
4. RELATIONSHIP OF THE SCPM AND THEMSR
The Market scoring rules (MSR) form a large class of pop-ular pari-mutuel mechanisms. In this section, we demon-strate a strong equivalence between the SCPM and the MSRmarkets. Particularly, we show that:
Theorem
Any proper market scoring rule with costfunction C ( · ) of (1) can be formulated as an SCPM model(3) with the ‘concave’ utility function u ( ~s ) = − C ( − ~s ) , andthe two models are equivalent in terms of the orders acceptedand the price charged for a submitted order. Theorem
The SCPM with any utility function u ( · ) gives an implicit proper scoring rule, as long as the utilityfunction has the property that its derivative spans the sim-plex { ~r : ~e T ~r = 1 , ~r ≥ } , that is, for all vectors ~r in thesimplex: ∇ u ( ~s ) = ~r, ∃ ~s. (7)Thus, the SCPM framework subsumes the class of properscoring rule mechanisms. Moreover, a proper scoring rulebased market can be created by simply choosing a utilityfunction u ( · ) that satisfies condition (7). As we shall demon-strate later in this section, this condition is not difficult tosatisfy or validate, thus providing a useful tool to designmarket mechanisms that correspond to a proper scoring rule.We first prove the above theorems for general MSR basedmarket, and then illustrate with specific examples of theLMSR and the Quadratic market scoring rules. To establish Theorem 4.1, we use the cost function for-mulation of market scoring rules discussed in [5], and brieflyexplained in Section 2.2. We first establish the following im-portant properties of the cost function for any proper scoringrule:
Lemma
The cost function C ( · ) for any proper scor-ing rule has following properties:1. C ( ~q ) is a convex function of ~q
2. For any vector ~q and scalar d , it holds that: C ( ~q + d~e ) = d + C ( ~q ) Proof.
The proof is referred to Appendix B. Note thatin [5], the second property above was treated as an assump-tion based on the principle of no arbitrage. Here, we showthat it can actually be derived from the properties of costfunction formulation itself.ow, we are ready to prove Theorem 4.1:
Proof.
Using the result in part 1 of Lemma 4.3, clearlythe proposed utility function u ( ~s ) = − C ( − ~s ) is concave.In light of part 2 of Lemma 4.3, we have C ( − ~s + z~e ) = C ( − ~s ) + z . Therefore, the proposed utility function u ( ~s ) = − C ( − ~s ) = z − C ( z~e − ~s ). Incorporating u ( ~s ) = z − C ( z~e − ~s )in the SCPM model (3), we get the following optimizationproblem: maximize x,z,~s πx − C ( z~e − ~s )subject to z~e − ~ax − ~s = ~q ≤ x ≤ l Since, C ( · ) was proven to be convex in part 1 of Lemma 4.3,this is a convex optimization problem with KKT conditions: ~p T ~a + y ≥ π, x · ( ~p T ~a + y − π ) = 0 ,p i = ∂C ( ~q + ~ax ) ∂q i , y · ( l − x ) = 0 , y ≥ , ≤ x ≤ l. Thus, x is increased until x = l , or the price ~p T ~a = ∇ C ( ~q + ~ax ) T ~a becomes greater than the bid price π . In particu-lar, under continuous charging scheme, where a series ofoptimization problems are solved with small l = ǫ , thiscorresponds to charging the increasing instantaneous price ~p ( ǫ ) T ~a = ∇ C ( ~q + ~aǫ ) T ~a until it is greater than the price of-fered by the bidder. For infinitesimally small ǫ , lim ǫ → p ( ǫ ) i = ∂C∂q i . Thus, the orders accepted and price charged becomeequivalent to the market scoring rule with cost function C ( ~q ). In this section, we prove that the SCPM mechanism isproper, that is, it implicitly corresponds to scoring the re-ported beliefs with a proper scoring rule, as long as the util-ity function satisfies the spanning condition (7).
Proof.
By definition, a scoring rule S ( · ) is proper if andonly if given any outcome distribution ~r , an optimal strategyof a selfish trader is to report belief ~r , that is: ~p ∗ ∈ arg max ~p P i r i S i ( ~p ) = ~r In cost function based markets like the SCPM, the tradersdo not directly report a belief ~p . Instead, they buy shares ~q paying a price that equals to the difference of the cost func-tion, and thus indirectly reporting the belief as the resultingprice vector ~p . For these markets, an implicit scoring rule isdefined in the following manner [5]: S i ( ~p ) = q i − C ( ~q ) + K ∀ i where p i = ∂C∂q i ∀ i (also refer equation (1) in Section 2.2). Therefore, the proper-ness condition in terms of ~q is represented as: ~p ∗ := ∇ C ( ~q ∗ ) = ~r (8)where ~q ∗ ∈ arg max ~q ≥ P i r i ( q i − C ( ~q )) (9)for all distributions ~r .Intuitively, since the traders receive $1 for each share onthe actual outcome, the profit of traders for outcome state i is q i − C ( ~q ). Thus, the “properness” condition ensures thatan optimal strategy for selfish traders is to buy orders ~q sothat the resulting price vector is equal to their actual belief.Now, the optimality conditions for (9) are: r i − ∂C ( ~q ∗ ) ∂q ∗ i + η ∗ i = 0 , η ∗ i ≥ , q ∗ i ≥ , η ∗ i q ∗ i = 0 , ∀ i Thus, condition (8) is satisfied if there exists a positive op-timal solution to (9). As derived in Theorem 3.3, the costfunction of the SCPM mechanism is given by (6). Therefore,the optimization problem (9) is equivalent tomax ~q ≥ P i r i ( q i − min t { t − u ( t~e − ~q ) } ) ⇔ max ~q ≥ ,t P i r i ( q i − t + u ( t~e − ~q )) ⇔ max ~q ≥ ,t ~r T ( ~q − t~e ) + u ( t~e − ~q ) ⇔ max ~s : ~s = t~e − ~q,~q ≥ u ( ~s ) − ~r T ~s ⇔ max ~s u ( ~s ) − ~r T ~s As long as there exists an optimal solution ~s ∗ to the aboveproblem, we can set t ∗ as a large positive value and set ~q ∗ = t ∗ ~e − ~s ∗ >
0. Thus, the condition (7), that is, ∇ u ( ~s )spans the simplex, ensures the properness. It is easy to seethat this is also a necessary condition. This proves Theorem4.2.A concern however is that the price vector ~p ∗ that max-imizes trader’s expected profit may not be unique. Thiscould be either because there are multiple sub-gradients ofthe cost function C ( ~q ) at optimal ~q ∗ resulting in multipleprice vectors ~p ∗ , or because there are multiple optimal ~q ∗ and they all result in different corresponding price vectors ∇ C ( ~q ∗ ). This is typically undesirable since in this case, ei-ther buying the orders ~q ∗ associated with the true belief ~r isnot the only optimal strategy for the traders, or even in thecase that the traders acquire ~q ∗ , the market maker is stillunable to recover the true belief. This situation is avoidedby the concept of strictly proper scoring rules. A scoringrule is called “strictly proper” if the only optimal strategyfor traders is to honestly report the belief [16]. In termsof our market mechanism, it means that the optimal pricevector ~p ∗ that satisfies condition (8) − (9) must be unique.Since u ( · ) is concave, it is easy to see that a sufficient con-dition to ensure strict properness in the SCPM is that u ( · )is a smooth function, that is, ∇ u ( · ) is continuous (over thesimplex).Next, we illustrate the equivalence between the SCPMand proper scoring rules using some popular mechanisms asexamples. Example
The LMSR market maker is equivalent tothe SCPM framework with utility function u ( ~s ) = − C ( − ~s ) = − b log ( P i e − s i /b ) . This scoring rule is known to be strictlyproper [7]. Note that our condition for properness is satisfiedas well since u ( · ) is smooth and ∇ u ( ~s ) = » e − s i /b P i e − s i /b – which clearly spans the simplex. Example
A market maker using Quadratic-Scoring-Rule is equivalent to the SCPM framework with utility func-tion u ( ~s ) = − C ( − ~s ) = ~e T ~sN − b ~s T ( I − ~e~e T N ) ~s . This scoringrule is known to be strictly proper [5]. Our condition forproperness is satisfied since u ( · ) is smooth and ∇ u ( ~s ) = » N + ¯ s − s i b – where ¯ s = ~e T ~s/N . Thus, for any ~r in simplex, we can set s i = − br i to get ∇ u ( ~s ) = ~r xample For the Log-SCPM [10], u ( ~s ) = P i θ i log s i , ∇ u ( ~s ) i = θ i /s i , which clearly spans the interior of the sim-plex for any positive ~θ . Also, u ( · ) is smooth, thus this mech-anism is strictly proper. Example
For u ( ~s ) = min i s i , the set of sub-gradientsat ~s = ~e is the convex hull of orthogonal vectors { ~e i } i =1 ,...,n where ~e i denotes a vector with at position i and else-where. This convex hull is exactly the simplex. Thus, thescoring rule in the SCPM with this utility is proper but notstrictly proper. Example
Consider a linear utility function u ( ~s ) = ~c T ~s . This mechanism is not proper, since the derivative ofthe function is a constant vector, and does not span the sim-plex. So far, we established the SCPM as a general market mecha-nism that includes all scoring rules which are proper and pos-sess common desired properties. A natural question wouldbe: what is the difference among the different utility objec-tive functions adopted in the SCPM? Next, we will showthat they represent different risk measures for the marketmaker. Thus, when using this model to accept incoming or-ders the market maker is actually taking rational decisionswith respect to a specific risk attitude defined in terms of u ( · ).
5. RISKS FOR THE MARKET MAKER
Each time he or she is offered an order, the market makermust consider the risks involved in accepting it. This isdue to the fact that the monetary return generated fromthe market depends on the actual state outcome. In theearlier pari-mutuel market introduced in [12], this risk waseffectively handled in terms of maximizing the worst casereturn generated by the market relative to the set of out-comes (i.e., u ( ~s ) = min i s i , refer Example 3.9). Unfortu-nately, this risk attitude is somewhat limiting as it will cre-ate a market which is likely to accept very few orders andextract little information. In what follows, we consider thereturn generated by the market to be a random variable Z and demonstrate that, when accepting orders based on theSCPM with a non-decreasing utility function, the marketmaker effectively takes rational decisions with respect to arisk attitude. We use duality theory to gain new insightsabout how this attitude relates to the concept of prior beliefabout the true probability of outcomes. In a finitely discrete probability space (Ω , F ), the set ofrandom variables Z can be described as the set of functions Z : Ω → ℜ . A convex risk measure on the set Z is definedas follows: Definition
When the random outcome Z representsa return, a risk measure is a function ρ : Z → ℜ that de-scribes one’s attitude towards risk as : random return Z ispreferred to Z ′ if ρ ( Z ) ≤ ρ ( Z ′ ) . Furthermore, a risk mea-sure is called convex if it satisfies the following: • Convexity : ρ ( λZ + (1 − λ ) Z ′ ) ≤ λρ ( Z ) + (1 − λ ) ρ ( Z ′ ) , ∀ Z, Z ′ ∈ Z , and ∀ λ ∈ [0 , • Monotonicity : If
Z, Z ′ ∈ Z and Z ≥ Z ′ then ρ ( Z ) ≤ ρ ( Z ′ ) • Translation Equivariance : If α ∈ ℜ and Z ∈ Z , then ρ ( Z + α ) = ρ ( Z ) − α Convex risk measures are intuitively appealing. First,even in a context where the decision maker does not knowthe probability of occurrence for the different outcomes, itis still possible to describe a risk function ρ ( Z ). The threeproperties of convex risk measures are also natural ones toexpect from such a function. Convexity states that diversi-fying the returns leads to lower risks. Monotonicity statesthat if the returns are reduced for all outcomes then the riskis higher. And finally, translation invariance states that ifa fixed income is added to random returns then it is irrele-vant wether this fixed income is received before or after therandom return is realized. We refer the reader to [11] for adeeper study of convex risk measures.Next, we formulate the SCPM model for prediction mar-kets as a convex risk minimization problem. In context ofprediction markets, the random return Z will represent therevenue for the market organizer, which depends on the ac-tual outcome of the random event in question. Let ~q repre-sent the total orders held by the traders, and c represent thetotal money collected so far from the traders in the market.Since the market organizer has to pay $1 for each acceptedorder that matches the outcome, his revenue for outcomestate i is c − q i . When a new trader enters with a bid of π ,based on the number of accepted orders x , the total revenuefor state i is given by ( c − q i + πx − a i x ). The risk minimiza-tion model seeks to choose the number of accepted orders x to minimize the risk on total revenue. Below, we formallyshow that the SCPM model is equivalent to a convex riskminimization model. Theorem
Let
Ω = { ω , ω , ..., ω m } , ~Z ∈ ℜ m be thevector representation of Z such that ~Z i = Z ( ω i ) , and Z x ( ω i ) = c − q i + πx − a i x . Then, given that u ( · ) is non-decreasing,the SCPM optimization model (3) is equivalent in terms ofset of optimal solutions for x to the risk minimization model minimize x ρ ( Z x )s . t . ≤ x ≤ l with convex risk measure ρ ( Z ) = min t { t − u ( ~Z + t~e ) } . Proof.
The equivalence can be obtained by first elim-inating ~s in (3), and then performing a simple change ofvariable t = z − πx − c :max z πx − z + u ( z~e − ~ax − ~q )= max t − t + u ( t~e + ( πx + c ) ~e − ~ax − ~q ) − c = − min t { t − u “ ~Z x + t~e ” } − c = − ρ ( Z x ) − c . Since maximizing − ρ ( Z x ) − c over x is equivalent to minimiz-ing ρ ( Z x ) in terms of optimal solution set, the equivalencefollows directly.It remains to show that, when u ( · ) is concave and non-decreasing, the proposed measure satisfies the three prop-erties (convexity, monotonicity, and translational equivari-ance) of a convex risk measure. The convexity and themonotonicity follow directly from concavity and monotonic-ity of u ( · ). We refer the reader to Appendix C for moredetails on this part of the proof. emark More importantly, Theorem 5.2 essentiallyshows that any convex risk measure ρ ( Z ) can potentially beused to create a version of the SCPM market which acceptsorders according the risk attitude described by ρ ( Z ) . Thisis achieved by simply choosing the utility function u ( ~s ) = − ρ ( Y ~s ) where Y ~s : Ω → ℜ is a random variable defined as Y ~s ( ω i ) = s i . Such a constructed u ( · ) is necessarily concaveand increasing. We just showed that the SCPM actually represents a riskminimization problem for the market maker when u ( · ) isnon-decreasing. In fact, we can get more insights about thespecific risk attitude by studying the dual representation ofrisk measure ρ ( Z ): ρ ( Z ) = min ~p ∈{ ~p | ~p ≥ , P i p i =1 } E ~p [ Z ] + L ( ~p ) , (10)where L ( ~p ) = max ~s u ( ~s ) − ~p T ~s , and E ~p [ Z ] = P i p i ~Z i . Werefer the reader to [11] for more details on the equivalence ofthis representation. Note that ρ ( Z ) is evaluated by consid-ering the worst distribution ~p in terms of trading off betweenreducing expected return and reducing the penalty function L ( ~p ).In terms of the SCPM, this representation equivalenceleads to the conclusion that orders are accepted accordingto: max ≤ x ≤ l min ~p ∈{ ~p | ~p ≥ , P i p i =1 } X i p i ~Z xi + L ( ~p ) ! . (11)In this form, it becomes clearer how L ( ~p ) encodes the intentsof the market maker and relates it to his belief about thetrue distribution of outcomes. For instance, we know thatthe first order is accepted only if ∀ ~p ∈ { ~p | ~p ≥ , X i p i = 1 } , X i p i ~Z xi ≥ − ( L ( ~p ) − L (ˆ p )) , where ˆ p = argmin ~p ∈{ ~p | ~p ≥ , P i p i =1 } L ( ~p ). For any given ~p ,the penalty L ( ~p ) −L (ˆ p ) therefore reflects how much the mar-ket maker is willing to lose in terms of expected returns inthe case that the true distribution of outcomes ends up be-ing ~p . It is also the case that after accepting ~q orders, thedistribution described by ~p ∗ = argmin ~p ~p T ( c~e − ~q ) + L ( ~p ) isactually the vector of dual prices computed in the SCPM. Inother words, the price vector in the SCPM market reflectsthe distribution that is being considered as the outcome dis-tribution by the market organizer in order to determine hisexpected return. This confirms the interpretation of pricesas a belief consensus on outcome distribution generated fromthe market.As we will see next, the function L ( ~p ) will typically bechosen so that L ( ~p ) − L (ˆ p ) is large if ~p is far from ˆ p , andˆ p will reflect a prior belief of the market organizer. Thatis, the market organizer is willing to lose on the expectedreturn in order to learn a distribution ~p that is very differentfrom his prior belief. This is in accordance with the factthat the market is being organized as a prediction marketrather than a pure financial market, and one of the goals ofmarket organizer is to learn beliefs even if at some risk tothe generated returns.We make the above interpretations clearer through thefollowing examples. Example
The utility function u ( ~s ) = min i { s i } cor-responds to cost L ( ~p ) = min i { s i } − ~p T ~s = 0 for all ~p . Thatis, the market organizer is purely maximizing his worst casereturn. Example
For the LMSR, u ( ~s ) = − b log P i exp ( − s i /b ) ,which is equivalent to using L ( ~p ) as the Kullback-Leibler di-vergence of ~p from uniform distribution L KL ( ~p ; U ) . Thisis minimized at ˆ p = U reflecting a uniform prior. The cor-responding risk measure is also known as the Entropic RiskMeasure and its level of tolerance is measured by b . Example
The Log-SCPM uses u ( ~s ) = P i θ i log( s i ) ,which is equivalent to choosing the penalty function to bethe negative log-likelihood of ~p being the true distributiongiven a set of observations described by the vector ~θ . Morespecifically, L LL ( ~p ; ~θ ) = − log “Q i p θ i i ” + K , which is mini-mized at ˆ p i = θ i P i θ i and tolerance to risk is measured through P i θ i . These examples illustrate how the risk minimization rep-resentation provides insights on how to choose u ( · ). In thecase of the Min-SCPM (Example 5.4), the associated penaltyfunction leads to a market where trades that might generatea loss for the market maker are necessarily rejected. Hence,the traders have no incentive for sharing their belief. On theother hand, both the LMSR and the Log-SCPM are mech-anism that will accept orders leading to negative expectedreturns under a distribution ~p , as long as this distribution is“far enough” from ˆ p . Effectively, a trader with a belief thatdiffers from ˆ p will have his order accepted given that hesubmits it early enough. In practice, choosing between theLMSR and the Log-SCPM involves determining whether theKullback − Leibler divergence or a likelihood measure bettercharacterizes the market maker’s commitment to learningthe true distribution.
6. DISCUSSION
In this work, we introduced a unified convex optimiza-tion framework for constructing prediction market mecha-nisms. We first showed that in this new framework, thepricing mechanism always allows truthful bidding (in a my-opic sense) to be an attitude that is optimal for the traders.Also, the pricing mechanism always leads to prices that canbe computed efficiently using a convenient convex cost func-tion formulation. We showed how the original SCPM mech-anism and mechanisms that are based on scoring rules couldbe cast in this unifying framework. These mechanisms actu-ally differ only in terms of the choice of utility function. Thisfact led to the analysis of properties of this utility functionwhich are of particular interest when designing a predictionmarket. We first proposed a way to compute the potentialworst case loss obtained from the market in this framework.We also discussed conditions on the utility function to en-sure ‘properness’ of the mechanism inline with the definitionof a proper scoring rule. Later, we showed that when select-ing a non-decreasing utility function, the market maker isimplicitly defining his risk attitude with respect to the po-tential revenues obtained from the market. able 1: A summary of properties of various market mechanisms u ( ~s ) Truthful WorstCost ConvexRisk Measure L ( ~p ) PropernessLMSR − b log( P i exp( − s i b )) Yes b log N Yes b L KL( ~p || U ) Strictly ProperLog-SCPM b P i log( s i ) Yes ∞ Yes b L LL( ~p || U ) Strictly ProperMin-SCPM min i s i Yes 0 Yes 0
ProperQuad. Scoring Rule N ~e T ~s − b ~s T ( I − N ~e~e T ) ~s Yes b N − N No -
Strictly ProperExponetial-SCPM b (1 − N P i exp( − s i b )) Yes b log N Yes b L KL( ~p || U ) Strictly ProperQuad-SCPM max ~v ≤ ~s N ~e T ~v − b ~v T ~v Yes b N − N Yes b k ~p − U k Strictly Proper
We believe these properties to be very valuable for selectingthe most effective market mechanism in a given application.Table 1 summarizes the conclusions we derived for a set ofpopular mechanisms: the LMSR, the original SCPM (Log-SCPM), the riskless Market (Min-SCPM), and the marketbased on the quadratic scoring rule. The table relates truth-fulness and properness of the mechanisms to the worst caseloss and the relationship to market maker’s risk attitude.We believe that these results can provide guidance in de-signing cost effective prediction markets. The table alsointroduces new versions of the SCPM model described inExample 6.1 and 6.2. These models are valuable since theyextend very naturally the landscape of structures for char-acterizing the market maker’s risk attitude through the def-inition of penalty functions L ( ~p ). Example
Consider the “Exponential-SCPM” obtainedfrom using the following utility: u ( ~s ) = b (1 − N X i e − s i /b ) This utility function is concave, non-decreasing and separa-ble. It has a bounded worst case loss equal to b log N . And,the function L ( ~p ) is the Kullback-Leibler divergence of ~p fromuniform distribution L KL ( ~p ; U ) . Using the pricing schemedescribed in Section 3.1, it leads to a prediction market thatis myopically truthful. Also, the corresponding scoring ruleis strictly proper. And, the orders can be priced using convexcost function C ( ~q ) = log P i e qi/b N . The above example shows that we can achieve propertiessame as the LMSR using an alternative separable utilityfunction in the SCPM. Our unified analysis of the SCPMmodel also allows us to suggest a modification to the pre-diction market that uses a quadratic scoring rule. Althoughthis rule is known to be myopically truthful, in practice amarket maker that uses this rule needs to explicitly restrictprices to be between [0,1] at all times. A solution to thisproblem is to use an SCPM market with the following semi-quadratic non-decreasing utility function:
Example
Consider the “Quad-SCPM” obtained fromusing the following utility: u ( ~s ) = max ~v ≤ ~s N ~θ T ~v − b~v T ~v for some ~θ such that P i θ i = 1 . This utility function is non-decreasing and concave which ensures that resulting pricesare non-negative and sum to . It has bounded worst caselost given by b ( k ~θ k + 1 − i θ i ). And, the distance fromthe prior ˆ p = ~θ is measured by L ( ~p ) = b k ~p − ~θ k , that is the 2-norm distance. The resulting prediction market is myopi-cally truthful and leads to orders that can be priced using thecost function: C ( ~q ) = min t,~v ≤ t~e − ~q t − ~θ T ~v + 14 b~v T ~v . which requires solving a quadratic program. Also, the corre-sponding scoring rule is strictly proper. The above market is closely related to the market associatedwith the quadratic scoring rule, since C ( ~q ) reduces to thepopular quadratic cost function when the constraint ~v ≤ t~e − ~q is replaced with ~v = t~e − ~q . However, the slightmodification ensures that prices are non-negative, and hasan intuitive interpretation in terms of distance to the priorbelief.Detailed proofs for the properties summarized in Table 1are available in Appendix D.
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Journal of the AmericanStatistical Association . APPENDIXA. PROPERTIES OF PRICING SCHEMEA.1 Proof of Lemma 3.1
Consider the KKT condition for (4). ~e T ~p = 1 p i = ∇ u ( ~s ) i ~p T ~a + y ≥ πx · ( ~p T ~a + y − π ) = 0 y ≥ y · ( ǫ − x ) = 0 ~ax + ~q + ~s = z~e (12) The first and second assertions follow from thetop two equations in (12), and the assumption that u ( · ) isnon-decreasing. The consistency of price follows from constraints 3-6 in(12). Let ( x , z , ~s ) , ( x , z , ~s ) be the optimal solution of (4)for the case of ǫ = ǫ and ǫ respectively, where ¯ x > ǫ > ǫ .Observe that from the problem formulation, x > x . Since u is concave, we know that u ( z ~e − ~q − x ~a ) − u ( z ~e − ~q − x ~a ) ≤ ∇ u ( z ~e − ~q − x ~a ) T (( z − z ) e − ( x − x ) ~a ) u ( z ~e − ~q − x ~a ) − u ( z ~e − ~q − x ~a ) ≤ ∇ u ( z ~e − ~q − x ~a ) T (( z − z ) e − ( x − x ) ~a )Therefore by adding the above inequalities, we get( ∇ u ( z ~e − ~q − x ~a ) −∇ u ( z ~e − ~q − x ~a )) T (( z − z ) e − ( x − x ) ~a ) ≥ ∇ u ( z ~e − ~q − x ~a ) T ~e = 1 , ∇ u ( z ~e − ~q − x ~a ) T ~e = 1Also, x > x . Therefore, we have( ∇ u ( z ~e − ~q − x ~a ) − ∇ u ( z ~e − ~q − x ~a )) T ~a ≥ ~p T ( ǫ ) ~a ≥ ~p T ( ǫ ) ~a . This proves the claim. To prove this claim, we use the following result from realanalysis: a bounded increasing or decreasing function on afinite interval is integrable . Using the observation made inpart 4, the proof follows.
Remark
A.1.
For ease of presentation, in above proofwe assumed u to be differentiable. Otherwise, the proof stillholds with the gradients replaced by sub-gradients. B. EQUIVALENCE OF THE MSR AND THESCPMB.1 Proof of Lemma 4.3 C ( ~q ) is a convex function of ~q Proof.
To prove C ( ~q ) is convex, it suffices to showthat for any ~q and ~q , C ( ~q + λ~q ) is convex in λ . Wehave the following: dCdλ = ~q · ∇ C | ( ~q + λ~q ) = ~q · p ( ~q + λ~q )Therefore, it suffices to show that ~q · ~p ( ~q + λ~q ) is in-creasing in λ .We will denote ~p ( ~q + λ~q ) by ~p ( λ ) and C ( ~q + λ~q )by C ( λ ). By the first condition of (1) and the proper-ness of S , we have for any λ > λ P i p i ( λ )( ~q + λ ~q ) i − C ( λ )= P i p i ( λ ) S i ( p ( λ )) − K ≥ P i p i ( λ ) S i ( p ( λ )) − K ≥ P i p i ( λ )( ~q + λ ~q ) i − C ( λ )So, we are left with the following relation: P i p i ( λ )( ~q + λ ~q ) i ≥ P i p i ( λ )( ~q + λ ~q ) i (13)Similarly, P i p i ( λ )( ~q + λ ~q ) i ≥ P i p i ( λ )( ~q + λ ~q ) i (14)Then (13)-(14) yields P i ( λ − λ )( p i ( λ ) − p i ( λ )) q i ≥ C ( ~q ) is convex in ~q for every cost function corre-sponding to a proper scoring rule.2. For any vector ~q and scalar d : C ( ~q + d~e ) = d + C ( ~q ) ∀ d Proof.
We prove by contradiction. If there exists ~q and d such that C ( ~q + d~e ) > d + C ( ~q ). Then we set ¯ q = ~q + d~e . Then S i ( ~p ) −S i (¯ p ) = ( q i − ¯ q i ) − ( C ( ~q ) − C (¯ q )) > i , which is impossible since there must be atleast one p ′ i > p i and S i is increasing function of p i .Similarly, we can prove a contradiction for C ( ~q + d~e )