A Strategic Model for Information Markets

Transcription

1 A Strategic Model for Information Markets Evdokia Nikolova MIT CSAIL Cambridge, MA Rahul Sami University of Michigan School of Information ABSTRACT Information markets, which are designed secifically to aggregate traders information, are becoming increasingly oular as a means for redicting future events. Recent research in information markets has resulted in two new designs, market scoring rules and dynamic arimutuel markets. We develo an analytic method to guide the design and strategic analysis of information markets. Our central contribution is a new abstract betting game, the rojection game, that serves as a useful model for information markets. We demonstrate that this game can serve as a strategic model of dynamic arimutuel markets, and also catures the essence of the strategies in market scoring rules. The rojection game is tractable to analyze, and has an attractive geometric visualization that makes the strategic moves and interactions more transarent. We use it to rove several strategic roerties about the dynamic arimutuel market. We also rove that a secial form of the rojection game is strategically euivalent to the sherical scoring rule, and it is strategically similar to other scoring rules. Finally, we illustrate two alications of the model to analysis of comle strategic scenarios: we analyze the recision of a market in which traders have inertia, and a market in which a trader can rofit by maniulating another trader s beliefs. Categories and Subject Descritors J.4 [Comuter Alications]: Social and Behavioral Sciences Economics General Terms Economics, Theory Keywords Prediction markets, Information markets, Strategic analysis, DPM, MSR, Projection game Suorted in art by American Foundation for Bulgaria Fellowshi and NSF grants ANI , ITR Permission to make digital or hard coies of all or art of this work for ersonal or classroom use is granted without fee rovided that coies are not made or distributed for rofit or commercial advantage and that coies bear this notice and the full citation on the first age. To coy otherwise, to reublish, to ost on servers or to redistribute to lists, reuires rior secific ermission and/or a fee. EC 07, June 11 15, 2007, San Diego, CA. Coyright 2007 ACM...$ INTRODUCTION Markets have long been used as a medium for trade. As a side effect of trade, the articiants in a market reveal something about their references and beliefs. For eamle, in a financial market, agents would buy shares which they think are undervalued, and sell shares which they think are overvalued. It has long been observed that, because the market rice is influenced by all the trades taking lace, it aggregates the rivate information of all the traders. Thus, in a situation in which future events are uncertain, and each trader might have a little information, the aggregated information contained in the market rices can be used to redict future events. This has motivated the creation of information markets, which are mechanisms for aggregating the traders information about an uncertain event. Information markets can be modeled as a game in which the articiants bet on a number of ossible outcomes, such as the results of a residential election, by buying shares of the outcomes and receiving ayoffs when the outcome is realized. As in financial markets, the articiants aim to maimize their rofit by buying low and selling high. In this way, the layers behavior transmits their ersonal information and beliefs about the ossible outcomes, and can be used to redict the event more accurately. The benefit of well-designed information markets goes beyond information aggregation; they can also be used as a hedging instrument, to allow traders to insure against risk. Recently, researchers have turned to the roblem of designing market structures secifically to achieve better information aggregation roerties than traditional markets. Two designs for information markets have been roosed: the Dynamic Parimutuel Market (DPM) by Pennock [10] and the Market Scoring Rules (MSR) by Hanson [6]. Both the DPM and the MSR were designed with the goal of giving informed traders an incentive to trade, and to reveal their information as soon as ossible, while also controlling the subsidy that the market designer needs to um into the market. The DPM was created as a combination of a ari-mutuel market (which is commonly used for betting on horses) and a continuous double auction, in order to simultaneously obtain the first one s infinite buy-in liuidity and the latter s ability to react continuously to new information. One version of the DPM was imlemented in the Yahoo! Buzz market [8] to eerimentally test the market s rediction roerties. The foundations of the MSR lie in the idea of a roer scoring rule, which is a techniue to reward forecasters in a way that encourages them to give their best rediction.

2 The innovation in the MSR is to use these scoring rules as instruments that can be traded, thus roviding traders who have new information an incentive to trade. The MSR was to be used in a olicy analysis market in the Middle East [15], which was subseuently withdrawn. Information markets rely on informed traders trading for their own rofit, so it is critical to understand the strategic roerties of these markets. This is not an easy task, because markets are comle, and traders can influence each other s beliefs through their trades, and hence, can otentially achieve long term gains by maniulating the market. For the MSR, it has been shown that, if we eclude the ossibility of achieving gain through misleading other traders, it is otimal for each trader to honestly reflect her rivate belief in her trades. For the DPM, we are not aware of any rior strategic analysis of this nature; in fact, a strategic hole was discovered while testing the DPM in the Yahoo! Buzz market [8]. 1.1 Our Results In this aer, we seek to develo an analytic method to guide the design and strategic analysis of information markets. Our central contribution is a new abstract betting game, the rojection 1 game, that serves as a useful model for information markets. The rojection game is concetually simler than the MSR and DPM, and thus it is easier to analyze. In addition it has an attractive geometric visualization, which makes the strategic moves and interactions more transarent. We resent an analysis of the otimal strategies and rofits in this game. We then undertake an analysis of traders costs and rofits in the dynamic arimutuel market. Remarkably, we find that the cost of a seuence of trades in the DPM is identical to the cost of the corresonding moves in the rojection game. Further, if we assume that the traders beliefs at the end of trading match the true robability of the event being redicted, the traders ayoffs and rofits in the DPM are identical to their ayoffs and rofits in a corresonding rojection game. We use the euivalence between the DPM and the rojection game to rove that the DPM is arbitrage-free, deduce rofitable strategies in the DPM, and demonstrate that constraints on the agents trades are necessary to revent a strategic breakdown. We also rove an euivalence between the rojection game and the MSR: We show that lay in the MSR is strategically euivalent to lay in a restricted rojection game, at least for myoic strategies and small trades. In articular, the rofitability of any move under the sherical scoring rule is eactly roortional to the rofitability of the corresonding move in the rojection game restricted to a circle, with slight distortion of the rior robabilities. This allows us to use the rojection game as a concetual model for market scoring rules. We note that while the MSR with the sherical scoring rule somewhat resembles the rojection game, due to the mathematical similarity of their rofit eressions, the DPM model is markedly different and thus its euivalence to the rojection game is esecially striking. Further, because the restricted rojection game corresonds to a DPM with a natural trading constraint, this sheds light on an intriguing connection between the MSR and the DPM. 1 In an earlier version of this aer, we called this the segment game. Lastly, we illustrate how the rojection game model can be used to analyze the otential for maniulation of information markets for long-term gain. 2 We resent an eamle scenario in which such maniulation can occur, and suggest additional rules that might mitigate the ossibility of maniulation. We also illustrate another alication to analyzing how a market maker can imrove the rediction accuracy of a market in which traders will not trade unless their eected rofit is above a threshold. 1.2 Related Work Numerous studies have demonstrated emirically that market rices are good redictors of future events, and seem to aggregate the collected wisdom of all the traders [2, 3, 12, 1, 5, 16]. This effect has also been demonstrated in laboratory studies [13, 14], and has theoretical suort in the literature of rational eectations [9]. A number of recent studies have addressed the design of the market structure and trading rules for information markets, as well as the incentive to articiate and other strategic issues. The two aers most closely related to our work are the aers by Hanson [6] and Pennock [10]. However, strategic issues in information markets have also been studied by Mangold et al. [8] and by Hanson, Orea and Porter [7]. An ucoming survey aer [11] discusses costfunction formulations of automated market makers. Organization of the aer The rest of this aer is organized as follows: In Section 2, we describe the rojection game, and analyze the layers costs, rofits, and otimal strategies in this game. In Section 3, we study the dynamic arimutuel market, and show that trade in a DPM is euivalent to a rojection game. We establish a connection between the rojection game and the MSR in Section 4. In Section 5, we illustrate how the rojection game can be used to analyze non-myoic, and otentially maniulative, actions. We resent our conclusions, and suggestions for future work, in Section THE PROJECTION GAME In this section, we describe an abstract betting game, the rojection game; in the following sections, we will argue that both the MSR and the DPM are strategically similar to the rojection game. The rojection game is concetually simler than MSR and DPM, and hence should rove easier to analyze. For clarity of eosition, here and in the rest of the aer we assume the sace is two dimensional, i.e., there are only two ossible events. Our results easily generalize to more than two dimensions. We also assume throughout that layers are risk-neutral. Suose there are two mutually eclusive and ehaustive events, A and B. (In other words, B is the same as not A.) There are n agents who may have information about the likelihood of A and B, and we (the designers) would like to aggregate their information. We invite them to lay the game described below: At any oint in the game, there is a current state described by a air of arameters, (,y), which we sometimes write in vector form as. Intuitively, corresonds to the 2 Here, we are referring only to maniulation of the information market for later gain from the market itself; we do not consider the ossibility of traders having vested interests in the underlying events.

3 total holding of shares in A, and y corresonds to the holding of shares in B. In each move of the game, one layer (say i) lays an arrow (or segment) from (, y) to (, y ). We use the notation [(, y) (, y )] or [, ] to denote this move. The game starts at (0,0), but the market maker makes the first move; without loss of generality, we can assume the move is to (1, 1). All subseuent moves are made by layers, in an arbitrary (and otentially reeating) seuence. Each move has a cost associated with it, given by C[, ] =, where denotes the Euclidean norm, = 2 + y 2. Note that none of the variables are constrained to be nonnegative, and hence, the cost of a move can be negative. The cost can be eressed in an alternative form, that is also useful. Suose layer i moves from (, y) to (, y ). We can write (, y ) as ( + le, y + le y), such that l 0 and e 2 + e y 2 = 1. We call l the volume of the move, and (e, e y) the direction of the move. At any oint (ˆ, ŷ), there is an instantaneous rice charged, defined as follows: c((ˆ, ŷ),(e, e y)) = ˆe + ŷey (ˆ, ŷ) = ˆ e ˆ. Note that the rice deends only on the angle between the line joining the vector (ˆ, ŷ) and the segment [(, y),(, y )], and not the lengths. The total cost of the move is the rice integrated over the segment [(, y) (, y )], i.e., C[(, y) (, y )] = Z l w=0 c((+we, y+we y),(e, e y))dw We assume that the game terminates after a finite number of moves. At the end of the game, the true robability of event A is determined, and the agents receive ayoffs for the moves they made. Let = (, y) = (,1 ). The ayoff (,1 ) to agent i for a segment [(, y) (, y )] is given by: P([(, y) (, y )]) = ( )+ y(y y) =.( ) We call the line through the origin with sloe (1 )/ = y/ the -line. Note that the ayoff, too, may be negative. One drawback of the definition of a rojection game is that imlementing the ayoffs reuires us to know the actual robability. This is feasible if the robability can eventually be determined statistically, such as when redicting the relative freuency of different recurring events, or vote shares. It is also feasible for one-off events in which there is reason to believe that the true robability is either 0 or 1. For other one-off events, it cannot be imlemented directly (unlike scoring rules, which can be imlemented in eectation). However, we believe that even in these cases, the rojection game can be useful as a concetual and analytical tool. The moves, costs and ayoffs have a natural geometric reresentation, which is shown in Figure 1 for three layers with one move each. The layers aend directed line segments in turn, and the ayoff layer i finally receives for a move is the rojection of her segment onto the line with sloe (1 )/. Her cost is the difference of distances of the endoints of her move to the origin. 2.1 Strategic roerties of the rojection game We begin our strategic analysis of the rojection game by observing the following simle ath-indeendence roerty. y MM move 1 s move 1 s ayoff 2 s move 2 s ayoff 3 s move 3 s ayoff Figure 1: A rojection game with three layers Lemma 1. [Path-Indeendence] Suose there is a seuence of moves leading from (,y) to (, y ). Then, the total cost of all the moves is eual to the cost of the single move [(, y) (, y )], and the total ayoff of all the moves is eual to the ayoff of the single move [(, y) (, y )]. Proof. The roof follows trivially from the definition of the costs and ayoffs: If we consider a ath from oint to oint, both the net change in the vector lengths and the net rojection onto the -line are comletely determined by and. Although simle, ath indeendence of rofits is vitally imortant, because it imlies (and is imlied by) the absence of arbitrage in the market. In other words, there is no seuence of moves that start and end at the same oint, but result in a ositive rofit. On the other hand, if there were two aths from (, y) to (, y ) with different rofits, there would be a cyclic ath with ositive rofit. For ease of reference, we summarize some more useful roerties of the cost and ayoff functions in the rojection game. Lemma The instantaneous rice for moving along a line through the origin is 1 or 1, when the move is away or toward the origin resectively. The instantaneous rice along a circle centered at the origin is When moves along a circle centered at the origin to oint on the ositive -line, the corresonding ayoff is P(, ) =, and the cost is C[, ] = The two cost function formulations are euivalent: C[, ] = Z l w=0 1 cos( + we,e)dw =,, where e is the unit vector giving the direction of move. In addition, when moves along the ositive -line, the ayoff is eual to the cost, P(, ) =. Proof. 1. The instantaneous rice is c(,e) = e/ = cos(,e), where e is the direction of movement, and the result follows. 2. Since is on the ositive -line, = =, hence P(, ) = ( ) = ; the cost is 0 from the definition.

4 3. From Part 1, the cost of moving from to the origin is C[,0] = Z l w=0 cos( + we,e)dw = Z l w=0 ( 1)dw =, where l =, e = /. By the ath-indeendence roerty, C[, ] = C[,0] + C[0, ] =. Finally, a oint on the ositive -line gets rojected to itself, namely = so when the movement is along the ositive -line, P(, ) = ( ) = = C[, ]. y rofit =. z rofit =. rofit = 0 z 1 We now consider the uestion of which moves are rofitable in this game. The eventual rofit of a move [, ], where = + l.(e, e y), is rofit[, ] = P[, ] C[, ] = l.e C[, ] Differentiating with resect to l, we get d(rofit) dl =.e c( + le,e) =.e + le + le.e We observe that this is 0 if (y + le y) = (1 )( + le ), in other words, when the vectors and ( + le) are eactly aligned. Further, we observe that the rice is non-decreasing with increasing l. Thus, along the direction e, the rofit is maimized at the oint of intersection with the -line. By Lemma 2, there is always a ath from to the ositive -line with 0 cost, which is given by an arc of the circle with center at the origin and radius. Also, any movement along the -line has 0 additional rofit. Thus, for any oint, we can define the rofit otential φ(, ) by φ(, ) =. Note, the otential is ositive for off the ositive -line and zero for on the line. Net we show that a move to a lower otential is always rofitable. Lemma 3. The rofit of a move [, ] is eual to the difference in otential φ(,) φ(, ). Proof. Denote z = and z =, i.e., these are the oints of intersection of the ositive -line with the circles centered at the origin with radii and resectively. By the ath-indeendence roerty and Lemma 2, the rofit of move [, ] is rofit(, ) = rofit(,z) + rofit(z,z ) + rofit(z, ) = ( ) ( ) = φ(,) φ(, ). Thus, the rofit of the move is eual to the change in rofit otential between the endoints. This lemma offers another way of seeing that it is otimal to move to the oint of lowest otential, namely to the -line. Figure 2: The rofit of move [, ] is eual to the change in rofit otential from to. 3. DYNAMIC PARIMUTUEL MARKETS The dynamic arimutuel market (DPM) was introduced by Pennock [10] as an information market structure that encourages informed traders to trade early, has guaranteed liuidity, and reuires a bounded subsidy. This market structure was used in the Yahoo! Buzz market [8]. In this section, we show that the dynamic arimutuel market is also remarkably similar to the rojection game. Couled with section 4, this also demonstrates a strong connection between the DPM and MSR. In a two-event DPM, users can lace bets on either event A or B at any time, by buying a share in the aroriate event. The rice of a share is variable, determined by the total amount of money in the market and the number of shares currently outstanding. Further, eisting shares can be sold at the current rice. After it is determined which event really haens, the shares are liuidated for cash. In the total-money-redistributed variant of DPM, which is the variant used in the Yahoo! market, the total money is divided eually among the shares of the winning event; shares of the losing event are worthless. Note that the ayoffs are undefined if the event has zero outstanding shares; the DPM rules should reclude this ossibility. We use the following notation: Let be the number of outstanding shares of A (totalled over all traders), and y be the number of outstanding shares in B. Let M denote the total money currently in the market. Let c A and c B denote the rices of shares in A and B resectively. The rice of a share in the Yahoo! DPM is determined by the share-ratio rincile: c A c B = y The form of the rices can be fully determined by stiulating that, for any given value of M,, and y, there must be some robability A such that, if a trader believes that A is the robability that A will occur and the market will liuidate in the current state, she cannot eect to rofit from either buying or selling either share. This gives us c A c B i h M = A h M i = B y (1)

5 Since A + B = 1, we have: c A + yc B = M (2) Finally, combining Euations 1 and 2, we get c A c B M = 2 + y 2 M = y 2 + y 2 Cost of a trade in the DPM Consider a trader who comes to a DPM in state (M,, y), and buys or sells shares such that the eventual state is (M,, y ). What is the net cost, M M, of her move? Theorem 4. The cost of the move from (,y) to (, y ) is M M = M 0[ 2 + y y 2 ] for some constant M 0. In other words, it is a constant multile of the corresonding cost in the rojection game. Proof. Consider the function G(, y) = M 0[ 2 + y 2 ]. The function G is differentiable for all, y 0, and it s artial derivatives are: G = M0[ G(, y) ] = 2 + y2 2 + y 2 G y = M0[ y G(, y) ] = y 2 + y2 2 + y 2 Now, comare these euations to the rices in the DPM, and observe that, as a trader buys or sells in the DPM, the instantaneous rice is the derivative of the money. It follows that, if at any oint of time the DPM is in a state (M,, y) such that M = G(, y), then, at all subseuent oints of time, the state (M,, y ) of the DPM will satisfy M = G(, y ). Finally, note that we can ick the constant M 0 such that the euation is satisfied for the initial state of the DPM, and hence, it will always be satisfied. One imortant conseuence of Theorem 4 is that the dynamic arimutuel market is arbitrage-free (using Lemma 1). It is interesting to note that the original Yahoo! Buzz market used a different ricing rule, which did ermit arbitrage; the rice rule was changed to the share-ratio rule after traders started eloiting the arbitrage oortunities [8]. Another somewhat surrising conseuence is that the numbers of outstanding shares, y comletely determines the total caitalization M of the DPM. Constraints in the DPM Although it might seem, based on the costs, that any move in the rojection game has an euivalent move in the DPM, the DPM laces some constraints on trades. Firstly, no trader is allowed to have a net negative holding in either share. This is imortant, because it ensures that the total holdings in each share are always ositive. However, this is a boundary constraint, and does not imact the strategic choices for a layer with a sufficiently large ositive holding in each share. Thus, we can ignore this constraint from a first-order strategic analysis of the DPM. Secondly, for ractical reasons a DPM will robably have a minimum unit of trade, but we assume here that arbitrarily small uantities can be traded. Payoffs in the DPM At some oint, trading in the DPM ceases and shares are liuidated. We assume here that the true robability becomes known at liuidation time, and describe the ayoffs in terms of the robability; however, if the robability is not revealed, only the event that actually occurs, these ayoffs can be imlemented in eectation. Suose the DPM terminates in a state (M,, y), and the true robability of event A is. When the dynamic arimutuel market is liuidated, the shares are aid off in the following way: Each owner of a share of A receives M, and each owner of a share of B receives (1 ) M, for each share y owned. The ayoffs in the DPM, although given by a fairly simle form, are concetually comle, because the ayoff of a move deends on the subseuent moves before the market liuidates. Thus, a fully rational choice of move in the DPM for layer i should take into account the actions of subseuent layers, including layer i himself. Here, we restrict the analysis to myoic, infinitesimal strategies: Given the market osition is (M,,y), in which direction should a layer make an infinitesimal move in order to maimize her rofit? We show that the infinitesimal ayoffs and rofits of a DPM with true robability corresond strategically to the infinitesimal ayoffs and rofits of a rojection game with odds /(1 ), in the following sense: Lemma 5. Suose layer i is about to make a move in a dynamic arimutuel market in a state (M,, y), and the true robability of event A is. Then, assuming the market is liuidated after i s move, If <, layer i rofits by buying shares in A, y 1 or selling shares in B. If >, layer i rofits by selling shares in A, y 1 or buying shares in B. Proof. Consider the cost and ayoff of buying a small uantity of shares in A. The cost is C[(, y) ( +, y)] = M, and the ayoff is M. Thus, 2 +y 2 buying the shares is rofitable iff M < M 2 + y y 2 < 2 + y 2 2 > ( y )2 > 1 y r 1 > r < y 1 Thus, buying A is rofitable if <, and selling A y 1 is rofitable if >. The analysis for buying or selling y 1 B is similar, with and (1 ) interchanged. It follows from Lemma 5 that it is myoically rofitable for layers to move towards the line with sloe that there is a one-to-one maing between 1 1. Note and 1

6 in their resective ranges, so this line is uniuely defined, and each such line also corresonds to a uniue. However, because the actual ayoff of a move deends on the future moves, layers must base their decisions on some belief about the final state of the market. In the light of Lemma 5, one natural, rational-eectation style assumtion is that the final state (M,, y ) will satisfy =. (In other y 1 words, one might assume that the traders beliefs will ultimately converge to the true robability ; knowing, the traders will drive the market state to satisfy =.) y 1 This is very lausible in markets (such as the Yahoo! Buzz market) in which trading is ermitted right until the market is liuidated, at which oint there is no remaining uncertainty about the relevant freuencies. Under this assumtion, we can rove an even tighter connection between ayoffs in the DPM (where the true robability is ) and ayoffs in the rojection game, with odds : 1 Theorem 6. Suose that the DPM ultimately terminates in a state (M, X, Y ) satisfying X =. Assume without loss of generality that the constant M 0 = 1, so M = Y 1 X2 + Y 2. Then, the final ayoff for any move [ ] made in the course of trading is ( ) (, 1 ), i.e., it is the same as the ayoff in the rojection game with odds. 1 Proof. First, observe that X M = and Y M = 1. The final ayoff is the liuidation value of ( ) shares of A and (y y) shares of B, which is Payoff DPM [ ] = M X ( ) + (1 ) M Y (y y) = 1 ( 1 ) + (1 ) (y y) 1 = ( ) + 1 (y y). Strategic Analysis for the DPM Theorems 4 and 6 give us a very strong euivalence between the rojection game and the dynamic arimutuel market, under the assumtion that the DPM converges to the otimal value for the true robability. A layer laying in a DPM with true odds /(1 ), can imagine himself laying in the rojection game with odds, because both the costs and the 1 ayoffs of any given move are identical. Using this euivalence, we can transfer all the strategic roerties roven for the rojection game directly to the analysis of the dynamic arimutuel market. One articularly interesting conclusion we can draw is as follows: In the absence of any constraint that disallows it, it is always rofitable for an agent to move towards the origin, by selling shares in both A and B while maintaining the ratio /y. In the DPM, this is limited by forbidding short sales, so layers can never have negative holdings in either share. As a result, when their holding in one share (say A) is 0, they can t use the strategy of moving towards the origin. We can conclude that a rational layer should never hold shares of both A and B simultaneously, regardless of her beliefs and the market osition. This discussion leads us to consider a modified DPM, in which this strategic loohole is addressed directly: Instead of disallowing all short sales, we lace a constraint that no agent ever reduce the total market caitalization M (or, alternatively, that any agent s total investment in the market is always non-negative). We call this the nondecreasing market caitalization constraint for the DPM. This corresonds to a restriction that no move in the rojection game reduces the radius. However, we can conclude from the receding discussion that layers have no incentive to ever increase the radius. Thus, the moves of the rojection game would all lie on the uarter circle in the ositive uadrant, with radius determined by the market maker s move. In section 4, we show that the rojection game on this uarter circle is strategically euivalent (at least myoically) to trade in a Market Scoring Rule. Thus, the DPM and MSR aear to be deely connected to each other, like different interfaces to the same underlying game. 4. MARKET SCORING RULES The Market Scoring Rule (MSR) was introduced by Hanson [6]. It is based on the concet of a roer scoring rule, a techniue which rewards forecasters to give their best rediction. Hanson s innovation was to turn the scoring rules into instruments that can be traded, thereby roviding traders who have new information an incentive to trade. One ositive effect of this design is that a single trader would still have incentive to trade, which is euivalent to udating the scoring rule reort to reflect her information, thereby eliminating the roblem of thin markets and illiuidity. In this section, we show that, when the scoring rule used is the sherical scoring rule [4], there is a strong strategic euivalence between the rojection game and the market scoring rule. Proer scoring rules are tools used to reward forecasters who redict the robability distribution of an event. We describe this in the simle setting of two ehaustive, mutually eclusive events A and B. In the simle setting of two ehaustive, mutually eclusive events A and B, roer scoring rules are defined as follows. Suose the forecaster redicts that the robabilities of the events are r = (r A, r B), with r A + r B = 1. The scoring rule is secified by functions s A(r A, r B) and s B(r A, r B), which are alied as follows: If the event A occurs, the forecaster is aid s A(r A, r B), and if the event B occurs, the forecaster is aid s B(r A, r B). The key roerty that a roer scoring rule satisfies is that the eected ayment is maimized when the reort is identical to the true robability distribution. 4.1 Euivalence with Sherical Scoring Rule In this section, we focus on one secific scoring rule: the sherical scoring rule [4]. Definition 1. The sherical scoring rule [4] is defined by s i(r) def = r i/ r. For two events, this can be written as: s A(r A, r B) = r A r 2 A + r 2 B ; s B(r A, r B) = r B r 2 A + r 2 B The sherical scoring rule is known to be a roer scoring rule. The definition generalizes naturally to higher dimensions. We now demonstrate a close connection between the rojection game restricted to a circular arc and a market scoring rule that uses the sherical scoring rule. At this oint, it is

7 convenient to use vector notation. Let = (, y) denote a osition in the rojection game. We consider the rojection game restricted to the circle = 1. Restricted rojection game Consider a move in this restricted rojection game from to. Recall that = (, 1 ), where is the true robability 2 +(1 ) 2 2 +(1 ) 2 of the event. Then, the rojection game rofit of a move [, ] is [ ] (noting that = ). We can etend this to an arbitrary collection 3 of (not necessarily contiguous) moves X = {[ 1, 1],[ 2, 2],,[ l, l]}. X SEG-PROFIT (X) = [ ] [, ] X 2 = 4 X [, ] X 3 [ ] 5 Sherical scoring rule rofit We now turn our attention to the MSR with the sherical scoring rule (SSR). Consider a layer who changes the reort from r to r. Then, if the true robability of A is, her eected rofit is SSR-PROFIT([r,r ]) = (s A(r ) s A(r))+(1 )(s B(r ) s B(r)) Now, let us reresent the initial and final osition in terms of circular coordinates. For r = (r A, r B), define the corresonding coordinates = ( A r r, B ). Note that r 2 A +r 2 B r 2 A +r 2 B the coordinates satisfy = 1, and thus corresond to valid coordinates for the restricted rojection game. Now, let denote the vector [,1 ]. Then, eanding the sherical scoring functions s A, s B, the layer s rofit for a move from r to r can be rewritten in terms of the corresonding coordinates, as: SSR-PROFIT([, ]) = ( ) For any collection X of moves, the total ayoff in the SSR market is given by: X SSR-PROFIT (X) = [ ] [, ] X 2 = 4 X [, ] X 3 [ ] 5 Finally, we note that and are related by = µ, where µ = 1/ 2 + (1 ) 2 is a scalar that deends only on. This immediately gives us the following strong strategic euivalence for the restricted rojection game and the SSR market: Theorem 7. Any collection of moves X yields a ositive (negative) ayoff in the restricted rojection game iff X yields a ositive (negative) ayoff in the Sherical Scoring Rule market. Proof. As derived above, SEG-PROFIT (X) = µ SSR-PROFIT (X). For all, 1 µ 2, (or more generally for an n- dimensional robability vector, 1 µ = 1 n, by the arithmetic mean-root mean suare ineuality), and the result follows immediately. 3 We allow the collection to contain reeated moves, i.e., it is a multiset. Although theorem 7 is stated in terms of the sign of the ayoff, it etends to relative ayoffs of two collections of moves: Corollary 8. Consider any two collections of moves X, X. Then, X yields a greater ayoff than X in the rojection game iff X yields a greater ayment than X in the SSR market. Proof. Every move [, ] has a corresonding inverse move [,]. In both the rojection game and the SSR, the inverse move rofit is simly the negative rofit of the move (the moves are reversible). We can define a collection of moves X = X X by adding the inverse of X to X. Note that SEG-PROFIT (X ) = SEG-PROFIT (X) SEG-PROFIT (X ) and SSR-PROFIT (X ) = SSR-PROFIT (X) SSR-PROFIT (X ); alying theorem 7 comletes the roof. It follows that the e ost otimality of a move (or set of moves) is the same in both the rojection game and the SSR market. On its own, this strong e ost euivalence is not comletely satisfying, because in any non-trivial game there is uncertainty about the value of, and the different scaling ratios for different could lead to different e ante otimal behavior. We can etend the corresondence to settings with uncertain, as follows: Theorem 9. Consider the restricted rojection game with some rior robability distribution F over ossible values of. Then, there is a robability distribution G with the same suort as F, and a strictly ositive constant c that deends only on F such that: (i) For any collection X of moves, the eected rofits are related by: E F(SEG-PROFIT(X)) = ce G(SSR-PROFIT(X)) (ii) For any collection X, and any measurable information set I [0, 1], the eected rofits conditioned on knowing that I satisfy E F(SEG-PROFIT(X) I) = ce G(SSR-PROFIT(X) I) The converse also holds: For any robability distribution G, there is a distribution F such that both these statements are true. Proof. For simlicity, assume that F has a density function f. (The result holds even for non-continuous distributions). Then, let c = R 1 µf()d. Define the density 0 function g of distribution G by g() = µf() c Now, for a collection of moves X, E F(SEG-PROFIT(X)) = = = Z Z Z SEG-PROFIT (X)f()d SSR-PROFIT (X)µ f()d SSR-PROFIT (X)cg()d = ce G(SSR-PROFIT(X))

8 y log scoring rule uadratic scoring rule Figure 3: Samle score curves for the log scoring rule s i(r) = a i +blog r i and the uadratic scoring rule s i(r) = a i + b(2r i P k r2 k). To rove art (ii), we simly restrict the integral to values in I. The converse follows similarly by constructing F from G. Analysis of MSR strategies Theorem 9 rovides the foundation for analysis of strategies in scoring rule markets. To the etent that strategies in these markets are indeendent of the secific scoring rule used, we can use the sherical scoring rule as the market instrument. Then, analysis of strategies in the rojection game with a slightly distorted distribution over can be used to understand the strategic roerties of the original market situation. Imlementation in eectation Another imortant conseuence of Theorem 9 is that the restricted rojection game can be imlemented with a small distortion in the robability distribution over values of, by using a Sherical Scoring Rule to imlement the ayoffs. This makes the rojection game valuable as a design tool; for eamle, we can analyze new constraints and rules in the rojection game, and then imlement them via the SSR. Unfortunately, the result does not etend to unrestricted rojection games, because the relative rofit of moving along the circle versus changing radius is not reserved through this transformation. However, it is ossible to etend the transformation to rojection games in which the radius r i after the ith move is a fied function of i (not necessarily constant), so that it is not within the strategic control of the layer making the move; such games can also be strategically imlemented via the sherical scoring rule (with distortion of riors). 4.2 Connection to other scoring rules In this section, we show a weaker similarity between the rojection game and the MSR with other scoring rules. We rove an infinitesimal similarity between the restricted rojection game and the MSR with log scoring rule; the result generalizes to all roer scoring rules that have a uniue local and global maimum. A geometric visualization of some common scoring rules in two dimensions is deicted in Figure 3. The score curves in the figure are defined by {(s 1(r), s 2(r)) r = (r,1 r), r [0, 1]}. Similarly to the rojection game, define the rofit otential of a robability r in MSR to be the change in rofit for moving from r to the otimum, φ MSR(s(r),) = rofit MSR[s(r),s()]. We will show that the rofit otentials in the two games have analogous roles for analyzing the otimal strategies, in articular both otential functions have a global minimum 0 at r =. Theorem 10. Consider the rojection game restricted to the non-negative unit circle where strategies have the natural one-to-one corresondence to robability distributions r = (r,1 r) given by = ( r, 1 r ). Trade in a log market r r scoring rule is strategically similar to trade in the rojection game on the uarter-circle, in that d dr φ(s(r),) < 0 for r < d φ(s(r),) > 0 dr for r >, both for the rojection game and MSR otentials φ(.). Proof. (sketch) The derivative of the MSR otential is d dr φ(s(r),) = d dr s(r) = X i is i(r). For the log scoring rule s i(r) = a i + b log r i with b > 0, d b dr φmsr(s(r),) = r, b 1 r = b r 1 = b r 1 r r(1 r). Since r = (r,1 r) is a robability distribution, this eression is ositive for r > and negative for r < as desired. Now, consider the rojection game on the non-negative unit circle. The otential for any = ( r ) is given by φ((r),) = (r),, 1 r r r It is easy to show that d φ((r),) < 0 for r < and the dr derivative is ositive for r >, so the otential function along the circle is decreasing and then increasing with r similarly to an energy function, with a global minimum at r =, as desired. Theorem 10 establishes that the market log-scoring rule is strategically similar to the rojection game layed on a circle, in the sense that the otimal direction of movement at the current state is the same in both games. For eamle, if the current state is r <, it is rofitable to move to r+dr since the effective rofit of that move is rofit(r, r ) = φ(s(r),) φ(s(r + dr),) > 0. Although stated for logscoring rules, the theorem holds for any scoring rules that induce a otential with a uniue local and global minimum at, such as the uadratic scoring rule and others. 5. USING THE PROJECTION-GAME MODEL The chief advantages of the rojection game are that it is analytically tractable, and also easy to visualize. In Section 3, we used the rojection-game model of the DPM to rove the absence of arbitrage, and to infer strategic roerties that might have been difficult to deduce otherwise. In this section, we rovide two eamles that illustrate the ower of rojection-game analysis for gaining insight about more comle strategic settings.

9 5.1 Traders with inertia The standard analysis of the trader behavior in any of the market forms we have studied asserts that traders who disagree with the market robabilities will eect to gain from changing the robability, and thus have a strict incentive to trade in the market. The eected gain may, however, be very small. A lausible model of real trader behavior might include some form of inertia or ǫ-otimality: We assume that traders will trade if their eected rofit is greater than some constant ǫ. We do not attemt to justify this model here; rather, we illustrate how the rojection game may be used to analyze such situations, and shed some light on how to modify the trading rules to alleviate this roblem. Consider the simle rojection game restricted to a circular arc with unit radius; as we have seen, this corresonds closely to the sherical market scoring rule, and to the dynamic arimutuel market under a reasonable constraint. Now, suose the market robability is, and a trader believes the true robability is. Then, his eected gain can be calculated, as follows: Let and be the unit vectors in the directions of and resectively. The eected rofit is given by E = φ(, ) = 1. Thus, the trader will trade only if 1 > ǫ. If we let θ and θ be the angles of the -line and -line resectively (from the -ais), we get E = 1 cos(θ θ ); when θ is close to θ, a Taylor series aroimation gives us that E (θ θ ) 2 /2. Thus, we can derive a bound on the limit of the market accuracy: The market rice will not change as long as (θ θ ) 2 2ǫ. Now, suose a market oerator faced with this situation wanted to sharen the accuracy of the market. One natural aroach is simly to multily all ayoffs by a constant. This corresonds to using a larger circle in the rojection game, and would indeed imrove the accuracy. However, it will also increase the market-maker s eosure to loss: the market-maker would have to um in more money to achieve this. The rojection game model suggests a natural aroach to imroving the accuracy while retaining the same bounds on the market maker s loss. The idea is that, instead of restricting all moves to being on the unit circle, we force each move to have a slightly larger radius than the revious move. Suose we insist that, if the current radius is r, the net trader has to move to r + 1. Then, the trader s eected rofit would be E = r(1 cos(θ θ )). Using the same aroimation as above, the trader would trade as long as (θ θ ) 2 > 2ǫ/r. Now, even if the market maker seeded the market with r = 1, it would increase with each trade, and the incentives to sharen the estimate increase with every trade. 5.2 Analyzing long-term strategies U to this oint, our analysis has been restricted to trader strategies that are myoic in the sense that traders do not consider the imact of their trades on other traders beliefs. In ractice, an informed trader can otentially rofit by laying a subotimal strategy to mislead other traders, in a way that allows her to rofit later. In this section, we illustrate how the rojection game can be used to analyze an instance of this henomenon, and to design market rules that mitigate this effect. The scenario we consider is as follows. There are two traders seculating on the robability of an event E, who each get a 1-bit signal. The otimal robability for each 2- bit signal air is as follows. If trader 1 gets the signal 0, and trader 2 gets signal 0, the otimal robability is 0.3. If trader 1 got a 0, but trader 2 got a 1, the otimal robability is 0.9. If trader 1 gets 1, and trader 2 gets signal 0, the otimal robability is 0.7. If trader 1 got a 0, but trader 2 got a 1, the otimal robability is 0.1. (Note that the imact of trader 2 s signal is in a different direction, deending on trader 1 s signal). Suose that the rior distribution of the signals is that trader 1 is eually likely to get a 0 or a 1, but trader 2 gets a 0 with robability 0.55 and a 1 with robability The traders are laying the rojection game restricted to a circular arc. This setu is deicted in Figure 4. Event haens A B X Event does not haen Signals Ot. Pt 00 C 01 A 10 B 11 D Figure 4: Eamle illustrating non-myoic decetion Suose that, for some eogenous reason, trader 1 has the oortunity to trade, followed by trader 2. Then, trader 1 has the otion of lacing a last-minute trade just before the market closes. If traders were laying their myoically otimal strategies, here is how the market should run: If trader 1 sees a 0, he would move to some oint Y that is between A and C, but closer to C. Trader 2 would then infer that trader 1 received a 0 signal and move to A or C if she got 1 or 0 resectively. Trader 1 has no reason to move again. If trader 1 had got a 1, he would move to a different oint X instead, and trader 2 would move to D if she saw 1 and B if she saw 0. Again, trader 1 would not want to move again. Using the rojection game, it is easy to show that, if traders consider non-myoic strategies, this set of strategies is not an euilibrium. The eact osition of the oints does not matter; all we need is the relative osition, and the observation that, because of the erfect symmetry in the setu, segments XY, BC, and AD are all arallel to each other. Now, suose trader 1 got a 0. He could move to X instead of Y, to mislead trader 2 into thinking he got a 1. Then, when trader 2 moved to, say, D, trader 1 could correct the rating to A. To show that this is a rofitable deviation, observe that this strategy is euivalent to laying two additional moves over trader 1 s myoic strategy of moving to Y. The first move, Y X may either move toward or away from the otimal final osition. The second move, DA or BC, is always in the correct direction. Further, because DA and BC are longer than XY, and arallel to XY, their rojection on the final -line will always be greater Y C D

10 in absolute value than the rojection of XY, regardless of what the true -line is! Thus, the decetion would result in a strictly higher eected rofit for trader 1. Note that this roblem is not secific to the rojection game form: Our euivalence results show that it could arise in the MSR or DPM (erhas with a different rior distribution and different numerical values). Observe also that a strategy rofile in which neither trader moved in the first two rounds, and trader 1 moved to either X or Y would be a subgame-erfect euilibrium in this setu. We suggest that one aroach to mitigating this roblem might be by reducing the radius at every move. This essentially rovides a form of discounting that motivates trader 1 to take his rofit early rather than mislead trader 2. Grahically, the right reduction factor would make the segments AD and BC shorter than XY (as they are chords on a smaller circle), thus making the myoic strategy otimal. 6. CONCLUSIONS AND FUTURE WORK We have resented a simle geometric game, the rojection game, that can serve as a model for strategic behavior in information markets, as well as a tool to guide the design of new information markets. We have used this model to analyze the cost, rofit, and strategies of a trader in a dynamic arimutuel market, and shown that both the dynamic arimutuel market and the sherical market scoring rule are strategically euivalent to the restricted rojection game under slight distortion of the rior robabilities. The general analysis was based on the assumtion that traders do not actively try to mislead other traders for future rofit. In section 5, however, we analyze a small eamle market without this assumtion. We demonstrate that the rojection game can be used to analyze traders strategies in this scenario, and otentially to hel design markets with better strategic roerties. Our results raise several very interesting oen uestions. Firstly, the ayoffs of the rojection game cannot be directly imlemented in situations in which the true robability is not ultimately revealed. It would be very useful to have an automatic transformation of a given rojection game into another game in which the ayoffs can be imlemented in eectation without knowing the robability, and reserves the strategic roerties of the rojection game. Second, given the tight connection between the rojection game and the sherical market scoring rule, it is natural to ask if we can find as strong a connection to other scoring rules or if not, to understand what strategic differences are imlied by the form of the scoring rule used in the market. Finally, the eistence of long-range maniulative strategies in information markets is of great interest. The eamle we studied in section 5 merely scratches the surface of this area. 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