Non-myopic Strategies in Prediction Markets

Transcription

1 Non-myopic Strategie in Prediction Market Stanko Dimitrov Department of Indutrial and Operation Engineering Univerity of Michigan, 205 Beal Avenue, Ann Arbor, MI , USA Rahul Sami School of Information Univerity of Michigan, 075 Beal Avenue, Ann Arbor, MI , USA May 8, 2007 Abtract One attractive feature of market coring rule [Hanon 03] i that they are myopically trategyproof: It i optimal for a trader to report her true belief about the likelihood of an event provided that we ignore the impact of her report on the profit he might garner from future trade. Thi doe not rule out the poibility that trader may profit by firt mileading other trader through dihonet trade and then correcting the error made by other trader. Such non-myopic trategie are difficult to analyze, and the empirical reult are inconcluive. In thi paper, we decribe a new approach to analyzing nonmyopic trategie and the exitence of myopic equilibria. We ue a imple model with two partially informed trader in a ingle information market to gain inight into the condition under which different equilibrium behavior emerge. We prove that, under generic condition, the myopically optimal trategy i not a equential equilibrium trategy for the logarithmic market coring rule. We then illutrate how introducing a form of dicounting into the prediction market can peed up information revelation by non-myopic trader. Introduction It ha long been oberved that, becaue market price are influenced by all the trade taking place, they reflect the combined information of all the trader. Prediction market are market deigned and deployed pecifically to aggregate information about future event; in a prediction market, trader trade in ecuritie whoe ultimate value i contingent on the outcome of future event. An informed trader can ue her private information to recognize inaccuracie in the current trading price and execute profitable trade. Thee trade in turn influence the trading price; further, the price provide ignal to other trader about the private

2 information. Other trader learn from thee ignal and adjut their belief about the true value of the ecurity. Ideally, thi will lead to a ituation in which all trader reach a conenu belief that reflect all available information. The ucceful aggregation of information through prediction market thu relie critically on trader adjuting their belief in repone to other trader trade. However, thi reponivene can alo have a drawback in the operation of the market: A trader may attempt to firt milead other trader about the value of the ecurity, and then exploit their inaccurate information in later trade. Awarene of, and reaction to, thi problem can lead trader to be overly cautiou about making inference from market price, thu weakening the aggregative power of the market. A reult, prediction market have alway had to grapple with thi perceived threat of manipulation, even when actual manipulation i abent. It would be very ueful to have a characterization of market ituation in which uch manipulation i poible (or impoible); due to the trategic complexity of traditional double-auction market, uch characterization have been difficult to achieve. With the recent rapid growth of market deigned primarily for information aggregation, reearcher have developed new market deign that are tailored to incentivize informed agent to trade and to reveal their private information in a timely manner. Hanon Market Scoring Rule [Han03] i an innovative tradable ecurity; it i baed on the idea of a proper coring rule [Bri50]. Pennock Dynamic Parimutuel Market [Pen04] i another new market deign that i baed on the traditional parimutuel market form ued in hore racing, but allow for early ale at a dynamically changing price in order to encourage early trade by informed trader. Apart from their other advantage, thee new market form are promiing for another reaon: A one ide of each individual trade i held by an automated market maker with a predetermined (and fairly imple) trategy, thee market form are much more amenable to formal analyi. For the market coring rule, it ha been proven that honet revelation of private information i at leat myopically optimal [Han03]. A imilar (although lightly weaker) characterization of myopically optimal trategie in dynamic parimutuel market i reported in [NS07]. However, much of the concern about manipulation in prediction market i baed on non-myopic trategie; a yet, very little i known theoretically about the exitence and characterization of manipulative non-myopic trategie in thee market. In thi paper, we decribe a new approach we are undertaking to analyzing thee non-myopic trategie. We motivate our idea with numerical exploration of the projection game [NS07], which i a imple analytical model that i trategically equivalent to the market coring rule with a pherical coring rule. Our analytical reult are derived directly uing the logarithmic market coring rule. Related Work There have been everal field and experimental tudie of manipulation in prediction market. Strumpf and Rhode [SR07] conducted experiment on manipulating price in the Iowa Electronic Market. Hanon et al [HOP06] experimentally tudy whether agent with an incentive to manipulate price can influence the trading price of a ecurity. They found that other agent who were aware of potential manipulation adjuted for thi poibility, thu limiting the effect of the manipulation attempt. 2

3 We are not aware of prior theoretical analyi of manipulation in prediction market pecifically. However, there i a rich literature on manipulation in financial market, which are cloely related. Thi literature ha tudied manipulation baed on releaing fale information (perhap through trade in other market), a well a manipulation that only require trategic manipulation in a ingle market; the latter form of manipulation i cloely related to our tudy here. Allen and Gale [AG92] decribe a model in which a manipulative trader can make a deceptive trade in an early trading round, and then profit in later round, even though the other trader are aware of the poibility of deception and act rationally. They ue a tylized model of a multi-period market; in contrat, we eek to exactly model a market coring rule model. Apart from other advantage of detailed modeling, thi allow u to contruct impler example of manipulative cenario: The model in [AG92] need to aume trader with different rik attitude to get around no-trade reult, which i rendered unneceary by the inherent ubidy in the market coring rule. Our model require only rik-neutral trader in a prediction-market context. We refer reader to the paper by Chakraborty and Yilmaz [CY04] for reference to other reearch on manipulation in financial market. Feigenbaum et al [FFPS05] alo tudied prediction market in which the information aggregation i ometime low, and ometime fail altogether. In their etting, the aggregation problem arie from a completely different ource: The trader are nontrategic, but extracting individual trader information from the market price i difficult. Here, we tudy cenario in which extracting information from price would be eay if trader were not trategic; the complexity arie olely from the ue of non-myopic trategie. Baed on the intuition provided by the projection game, Nikolova and Sami [NS07] preent an intance in which myopic trategie are not optimal for both player, and ugget (but do not analyze) uing a form of dicounting to reduce manipulative poibilitie in a prediction market. We draw on a generalization of thi intance a the tarting point of our analyi. Structure of the paper The ret of thi paper i tructured a follow: In ection 2 we decribe the 2-player model we ue to highlight deception threat. In ection 2., we analyze thi model to derive condition under which the myopically optimal truthful trategy i optimal even in a non-myopic ene. We explain how thi analyi can be recurively extended to characterize finite equence of move that are in equilibrium and reult in ultimate revelation of true information. In ection 3, we preent preliminary numerical reult that ugget that three cae can arie in a market with dicounting: () The myopic trategie form an equilibrium and information i rapidly incorporated into the market; (2) The myopic trategie do not form an equilibrium, but there i an equilibrium with bluffing in initial round followed by myopic behavior; and, (3) There i no equilibrium profile in which information i incorporated in a finite number of round. Motivated by the numerical reult, we preent our main reult in ection 4: We formally prove that, for an undicounted log-coring rule market with two player in a Bayeian etting, there i generically no myopic equilibrium; in fact, there i generically no equilibrium in which all information i incorporated in a finite amount of time. Our technique, building on reult from claic information theory, could be very ueful for other apect of information market analyi. In ection 5 we dicu how our reult may be generalized and ued to gain inight about more complex 3

4 market. We draw parallel with claical bargaining theory, and decribe ome future tep. 2 A Simple Model In thi ection, we decribe a model of an extremely imple prediction market etting. The etting i a follow: A prediction market i deigned to predict a future event E, by trading in a ecurity F baed on E. Two player, P and P2, are each endowed with ome private information about E. We aume the implet poible cae, in which P and P2 each have a ingle bit of information (x,x 2 repectively) relevant to E. (Equivalently, they each receive a ignal that can take on two poible value: 0 or ). Further, we aume that the trader are rik-neutral, and hare a common (and accurate) prior probability ditribution over P, P2, and E. The prior probability ditribution can then be completely pecified by pecifying the prior probabilitie of the ignal and the conditional probability of E given each combination of ignal. We aume that the two ignal are independent. Further, we aume for implicity that x i 0 or with equal probability. The probability that x 2 i i given by a parameter 0 < q <. Thu, the model can be fully pecified by pecifying four probabilitie p A = p(e/x = 0,x 2 = 0) (or p(e/00) for hort), p B = p(e/), p C = p(e/0) and p D = p(e/0). We tudy the behavior of the market for different value of the parameter p A, p B, p C, p D and q. We aume that the trade in ecurity F i conducted uing a market coring rule [Han03]. Player make a equence of market move; in each move, the player report a probability p i. At the end, when the event E i revealed, the move earn a player a net core (E, p i ) (E, p i ), where i ome proper coring rule. The market maker eed the market with a value p 0 that i irrelevant for our analyi. Here, we primarily conider the pherical coring rule function for. We conider a equence of alternating move in which P move firt, P2 move next, P potentially move again, and o on. Following [NS07], trade in thi market (with a pherical coring rule) can be viualized a play in a retricted projection game, decribed a follow: The game i played on a 2-dimenional board, with two axe correponding to the two poible outcome E and E. The game poition at any point of time i a point on the poitive quadrant of the unit circle. Player take turn in moving from the current poition to another point on the circle. A poition (co θ, in θ) correpond to a probability aement p uch that p p = tanθ. The profit of each move (coθ,inθ ) (coθ 2,inθ 2 ), conditional on a true probability p, i determined a follow: It i the projection of the directed egment from the tarting to the final point onto the line through the origin and the point correponding to the true probability p. Thu, move toward the true probability have a poitive profit, and move away from the true probability have a negative profit. We tre that we are uing the projection game merely a a way to viualize the expected profit of different move in a market coring rule, in order to compare different trategie. The projection game cannot be directly implemented (becaue the true probabilitie are unknown), but the reult of [NS07] how that trategie in thi game are equivalent to trategie in a market with the pherical coring rule, and perhap a lightly different prior ditribution. Thu, although we decribe myopic and bluffing trategie in 4

5 the context of the projection game, there i a correpondence with trategie in an MSR. In the projection game repreentation of the prediction market, the poible ignal realization lead to the following point on the circle: A B Point Signal 00 A 0 C 0 D B Event happen C D Event doe not happen Figure : Point realization, a generalization of [NS07] One can alo think of each of thee point a a correponding angle the line through the origin and the point make with the x-axi. For example point C correpond with angle θ c, point A correpond to angle θ a, etc. Ideally, we would like the prediction market to converge to the point correponding to the true ignal; thi would give the bet poible etimate of the probability of E. 2. Myopic behavior We now analyze the price dynamic if each trader followed her myopically optimal trategy. There are two additional point X and Y that arie in the analyi of the myopic behavior becaue of P uncertainty of P2 ignal. Suppoe P aw x =. She would then condition her prior on thi information, reulting in a poterior in which he acribe probability q to the poibility that the optimal point i B, and probability q to the poibility that the optimal point i D. In the balance, her belief about the likelihood of E would be in between that implied by B and that implied by D. It i eay to how that (in the projection game) her optimal myopic trategy i to move to a point X defined by θ x = arctan(qinθ d + ( q)inθ b )/(qcoθ d + ( q)coθ b ). Likewie, if P aw x = 0, he would move to a point Y defined in term of θ a and θ c. Now, P2 cannot directly ee x, but he can infer what P myopic action would have been in each cae. We aume that we are in the non-degenerate cae in which X Y ; thi allow u to focu on trategic threat intead of difficultie in extracting ignal from the price. Then, P2 can infer the value of x ; combining thi with the value of x 2 that P2 oberved, he can calculate the bet poible etimate of the conditional probability of E. Due to the myopic trategyproofne of the projection game (and market coring rule), A a anity check, we alo calculated the numerical reult in ection 3 in an MSR with a logarithmic coring rule. The reult are qualitatively imilar; if anything, they how a tronger tendency to manipulation. 5

6 P2 would move to A,B,C,orD repectively. Subequently, neither player would have an incentive to move. Thu, if player followed their myopic trategie, the market would perform remarkably well: All information would be aggregated optimally in jut two trade. Further, in general, both player would make a profit in expectation in thi market. 2.2 Non-myopic behavior and bluffing Now, uppoe that the player were not retricted to myopic behavior. Specifically, a player may deviate from the myopic trategy to exploit the other player reaction, and make a greater total profit through ubequent move. Conider the way in which P can deviate from her original myopic trategy. We retrict our attention to trategie in which P move to either X or Y in the firt round. Thee are the two poition that P2 i expecting to ee the market in, and thu we can reaon about the reaction that P2 would make to the move; thi would be difficult if the move wa to a different point. Thu, we are intereted in the following kind of bluffing trategie for P: Suppoe P ee a. She could move to Y in the firt round, intead of her myopically optimal trategy of moving to X. Now, if P2 i expecting myopic behavior, he would incorrectly infer that x = 0, and correpondingly move to the wrong point: A intead of D, or C intead of B. P can ee where P2 moved to, and make a ubequent correcting move: A D or C B repectively. P incentive to bluff i thu determined by the profitability of thi bluffing trategy relative to the honet (myopic) trategy. Note that if the myopic trategy i uperior to bluffing, for both value of x, P would follow thi trategy. Then, P2 would have no reaon to bluff (becaue P would not move again). Thu, checking if the myopic trategy i an equilibrium i equivalent to checking if P expected profit from bluffing i le than her expected profit from the myopic trategy, auming that P2 will be myopic. Suppoe that the bluffing trategy ha a trictly higher profit than the myopic trategy for player P. It follow that P will bluff with ome probability r. Note that P2 can analyze P profit in different cenario, and thu, can infer that P would not necearily be truthful. Now, we can try to equilibria in which the bluffing probability r i known to P2, who take it into account and react accordingly. It mut be that 0 < r <, becaue otherwie P2 would know x with certainty. Now, from P2 point of view, the market look very imilar to the market we jut analyzed for P: He ee x 2, and aign ome probability r to x =. The myopic optimal repone for P2 taking into account the probability that P i bluffing can be determined: it i a function of r, x 2 and the poition (X or Y ) that P left the market at. Next, we can repeat the analyi from P2 point of view, and determine if the myopic repone i optimal for P2, or if he too would rather bluff with ome probability. The analyi follow exactly a done for P, except that the role of x and x 2 are interchanged, or equivalently, the label C and D are interchanged. With many intance 2 of point location A,B,C, and D, we can how that for all q uch that 0 < q <, the myopic trategy i not optimal; further, thi hold even when C and D are interchanged. The reaoning above 2 In fact, we conjecture that thi i true for all non-degenerate location of the point A,B, C, D; however, we have not been able to prove thi yet. 6

7 lead to a tartling conequence that no finite equence of move by the two player leading to the optimal value (where the ith move involve bluffing with ome probability r i,0 r i ) can be an equilibrium! Thi follow by a backward induction argument: If there i an equilibrium trategy that lead to the exact optimal value, the lat move in that trategy hould reflect both x and x 2. Suppoe, without lo of generality, that the lat (nth) move wa by P; it follow that P mut know x 2 with certainty at the time of her lat move, and thu, the econd-lat move mut reveal x 2 with certainty. By aumption, if P2 had any uncertainty about x when he made thi move, it would be profitable for him to bluff with ome probability 0 < r n < and correct the market after P made the nth move. It follow that, for thi trategy to be an equilibrium, P2 mut know x with certainty before the (n )th move. Repeating thi argument, we reach a contradiction, and therefore, no uch equilibrium trategy exit. 3 Numerical Exploration In thi ection we decribe our approach to numerically analyzing the profitability of addreing non-myopic trategie. The reult in thi ection motivate our theoretical reult in ection 4, and give inight into the required dicounting factor. We analyze the incentive to bluff in two way: () The relative profit from bluffing i influenced by the relative poition of the point A,B,C,D. We conduct a numerical analyi of the profit for pecific poition of four point, and all permutation of the four label A,B,C,D over thee poition, with different value of q. We ue thi to qualitatively infer how the propenity to bluff change in different ituation. Our reult how many ituation in which the myopic trategy, or any finite trategy profile terminating in the optimal poition, cannot be in equilibrium. In fact, they ugget that thi i alway true. We then conider a very natural olution to thi problem, which i to introduce an explicit form of time-dicounting into the market rule. The numerical reult how that even a mall dicount factor could tabilize the myopic equilibrium. 3 (2) We can analytically tudy the problem for a pecial cae of Boolean function: The cae in which p A, p B, p C, p D are each either 0 or, o that the event under conideration i a determinitic function of the bit x,x 2. Such function were tudied by Feigenbaum et al [FFPS05]; in their model, which did not include the poibility of manipulation, they found that the market would converge rapidly for certain Boolean function, uch a the AND function. Here, we how that bluffing i profitable even in thi imple cae of the AND function, and hence, the market convergence to the true price i not guaranteed to be rapid if agent are trategic in a non-myopic way. 3 It could be argued that there i alway an implicit dicount factor influencing trader, due to impatience or uncertainty over whether the game will be terminated abruptly, a i tandard in bargaining model. However, thi i le natural in information market etting: player do not obtain their profit until the market cahe out, o there i little jutification for impatience, and automation of trading might lead to long equence of preciely timed move in a hort amount of time. For thi reaon, we prefer to introduce an explicit dicount factor. Controlling the magnitude of the dicounting could alo control the peed of convergence of the market. 7

8 3. Expreing bluffing profit A noted earlier it may not be alway advantageou for the market player to act myopically. In particular if the expected core or profit from bluffing in the market i greater than the expected core of acting myopically then a rational player will bluff. Equivalently if the difference in bluffing and acting myopically i poitive then a rational player will bluff. Therefore, during the firt move in the market the firt player ha to decide if he will bluff or will not bluff. Without lo of generality, we aume that player P ee x = ; thu, the myopically optimal poition to move to i X, wherea a move to Y correpond to a bluffing move. P ha to decide whether to move to X or Y in the firt round. If P move to Y in the firt round, thu bluffing, he will play a corrective move in the third round after P2 ha moved. The corrective move be either from A to D (if P2 revealed that x 2 = 0) or from C to B (if P2 revealed that x 2 = ). We want to analyze the difference between the profit P earn from the truthful and bluffing trategie. We analyze the firt-round and third-round profit difference eparately, and then combine them. Due to the path-independence property of market coring rule, the expected profit difference between the truthful and bluffing firt-round move i equal to the expected profit of a move from X to Y. (Thi will be negative, a X i the optimal point given P belief in the firt round). In the bluffing trategy, P make an additional corrective move, wherea in the myopic trategy he doe not. The additional corrective move alway yield a poitive profit to P, a it end at the optimal point given the true information. Thu, we can compare the relative profit of myopic and bluffing trategie by tudying when the expected combined profit of the move from X to Y and the corrective move i poitive. 3.2 Numerical calculation of profitability of bluffing Here we dicu ome of the numerical analyi applied to non-myopic trategie in information market. Following our dicuion that myopic trategie do not alway lead to a market equilibrium, we firt preent one olution, namely a dicount factor, to thi iue we then preent a numerical analyi for a et of game. Firt we try to characterize under what condition player P will bluff. She will bluff if he loe le from the initial mileading tep than he gain, in expectation, from the final corrective tep. We can look at the negative of the ratio, m, of the expected cot of mileading to the expected payoff of the corrective tep to ee if player P ha an incentive to bluff during her firt move. Given the definition of m, if m < player P ha at incentive to bluff during her firt move, if m = player P i indifferent to bluffing, and if m > player P ha no incentive to bluff initially. Given the poition of the point A,B,C,D, and the probability q that x 2 = 0, we can calculate the ratio m Dicounted Market and Dicount Factor The ratio m ha another, inightful interpretation. We conider modifying the rule of trading in the market to introduce dicounting over time, a follow: Define a factor δ,0 δ. When cahing out the market, the profit or lo made by player P in her ith move i caled by δ i. In each individual move, a player 8

9 profit from a move i proportional to her profit from the ame move in an uncaled market, and therefore, the myopic propertie of the prediction market are not altered. However, the bluffing trategie decribed above are altered, becaue the profit from the corrective move i caled by δ, while the bluff move i not caled. It i eay to oberve that in any ituation with ratio m <, m i exactly the maximum value of δ for which the myopic trategy i at leat a good a the bluffing trategy for P, auming that P2 will play myopically. Thu, we plot the value of m for different ituation, a q i varied. To make thi notion more concrete, aume that player oberve zero a her ignal, we are auming that player i equally likely to oberve either ignal. For the notation below θ i, i = {a,b,c,d} correpond to the point aociated with the ignal in Figure. Therefore in expectation her net profit for the bluffing move i: E{NP[bluff move]} = [q((coθ x coθ y )coθ c + (inθ x inθ y )inθ c )+ ( q)((coθ x coθ y )coθ a + (inθ x inθ y )inθ a )] Similarly player ha an expected net profit from the corrective move of: E{NP[corrective move]} = δ[q( co(θ b θ c ))+ Let θ x,θ y be the myopically optimal poition. A noted above player will behave myopically if: Let := be the expected ( lo due to bluffing. Define the ratio m = ( q)( co(θ d θ a ))] E{NP[corrective move]} + E{NP[bluff move]} 0 qco(θ b θ c ) ( q)co(θ d θ a ) ). If m i non-negative for all realization of θ a,θ b,θ c,θ d, and q then there i alway a δ uch that player will behave myopically in the firt move. Claim: m 0 q,θ i. Proof:. qco(θ b θ c ) ( q)co(θ d θ a ) 0 Note that co(θ). A q [0,] it mean that qco(θ b θ c ) ( q)co(θ d θ a ) giving the concluion. 2. < 0 thi follow from the fact that both of thee move are deviation away from the true p-line. Therefore the dot product mut be both negative. The above imply that m 0. In the cae that qco(θ b θ c ) ( q)co(θ d θ a ) = 0 we have a cae of an unbounded m auming that we are not the degenerate cae of the firt player move not revealing any information, i.e. θ x = θ y. A we have jut how that m 0 for all θ a,θ b,θ c,θ d, and q we know that for any realization of the problem we can find a dicount factor uch that the expected net profit of bluffing i le than the expected net profit of acting myopically. 9

10 3.2.2 How much to Dicount Below we dicu a three cenario, a nominal projection game cenario and the AND game. The dicuion will be motivated through an numerical plot of the different cenario. Each of the plot below will have a title with a permutation of the letter ABCD. Aume that a > b > c > d, the the title correpond to the aignment of the angle θ a through θ d to the value a through d. For example the title of ABCD implie that θ a = a,θ b = b,θ c = c,θ d = d and ACDB implie that θ a = a,θ b = d,θ c = b,θ d = c. Without lo of generality, we can aume that A i the point with maximum θ a ; if it in t, thi can alway be engineered by relabeling the ignal value for x and/or x 2. Thu, we look at configuration with A fixed. We now conider a nominal intance of the projection game. In particular let a = 75,b = 60,c = 30, and d = 5. Looking at the graph in Figure 2 we note that m < for all value of q 0 or. What i more urpriing i that in ome cenario, in particular ACDB, with a dicount factor of approximately 0.99 player will behave myopically regardle of what probability player 2 oberve her ignal. Moreover in ome of the other cenario, ABDC for example, a dicount factor of make player behave myopically for a large range of q value, [0,0.5] and [5,]. ABCD ABDC ACDB q q q ADCB ACBD ADBC q q q Figure 2: m v q for the Nominal projection game with a = 75,b = 60,c = 30, and d = 5 Immediately, we can ee that there are cae in which, in the abence of dicounting, there i no finite equilibrium trategy in which the market converge to the correct poition. For example, conider the market defined by configuration ABCD, with player playing firt. We can ee that m < whenever q i between 0 and, o player would bluff with ome probability 0 < r <. Next, from player 2 point of view, the market would be in configuration ABDC. Again, we ee that m < whenever P2 ha ome upicion that P bluffed in the previou round. Thu, P2 would alo bluff with ome probability. Following the reaoning in ection 2, no finite revealing equilibrium trategy profile i poible. Thi reult appear to hold generically, for all poition of the four point and all value of q between 0 and. 0

11 With even light dicounting, we can identify ituation in which the myopic equilibrium i poible: for example, the ACDB configuration with a dicount factor of Further, with ome dicount factor δ, we can contruct example in which there i an equilibrium in which P bluff with ome probability r, but P2 play myopically even knowing thi, and thu the market converge in 3 round (after P corrective move). For example, conider the market in configuration ADBC, with q = and δ =. Inpection of the graph how that P till benefit from ome bluffing in the firt round. However, from P2 point of view, the market configuration will be modeled by the ACBD configuration, and thu, bluffing will not be profitable with δ =. AND function : Next, we conider a pecial cae, the AND game, defined a follow: The ecurity payoff i the logical AND of the two player ignal. We how that a dicount factor of one i adequate only when q i either zero or one. Below in Figure 3 we ee the plot of m v. q for the AND game. To model the AND game we et a = d = c = 0 and b = 90 where the role of a,b,c,d are decribed above. A the plot in Figure 3 decribe, a player in the AND game will alway bluff a long a q 0 or. ABCD ABDC ACDB q q q ADCB ACBD ADBC q q q Figure 3: m v q for the AND game 4 Analyi of the logarithmic MSR Building on the intuition developed in ection 3, we now conider an analytical proof to how that bluffing i alway profitable. It turn out to be eaiet to analyze the logarithmic market coring rule in thi cae, becaue we can reduce it to a tandard reult on information-theoretic (Shannon) entropy. At the end of the ection, we dicu why we expect thi reult to hold for other coring rule (including the pherical coring rule, and thu, the egment game.)

12 4. Generic Bluffing Conider an order of play with P playing before P2. A player i playing firt we define the following: p x = qp b + ( q)p d P expectation if he ee x = p y = qp c + ( q)p a P expectation if he ee x = 0 The probabilitie p x and p y determine the optimal myopic move for player. Lemma. Conider any equilibrium trategy profile. If player ha a determinitic trategy of playing p u when he ee x = and p v p u when he ee x = 0, then p u = p x and p v = p y. Proof. Aume that p u p x. Whenever player play p u player 2 will deduce that player ha oberved a ; then, player 2 will capture all the remaining urplu. Player thu get at mot the profit he derive from her firt move. Therefore, by definition of p x, player ha a poitive deviation, in expected payoff, from p u to p x. A imilar argument hold for p v and p y. Aume that the market tart at an arbitrary point p a player i moving firt, when he move firt we can think of her alway moving to p x and if he decided to bluff he will then move to p y. Therefore when comparing the two trategie the initial move p to p y cancel out therefore we can aume that the market tart at p x and therefore the core of the initial move i zero. We now expre the expected profit in term of information-theoretic entropy. While the connection between proper coring rule and entropy i well known, it ha, to the bet of our knowledge, not been ued in the prediction market literature. To thi end, we introduce the following notation: S(p i, p j ) The expected log core from moving from poition p i to p j, with p j being the true belief: S(p i, p j ) = p j log p j p i + ( p j )log p j p i D(p i p j ) The relative entropy of two probability ma function p(x) and q(x) i defined in [CT9] a: D(p q) = p(x)log p(x) q(x) Lemma 2. Aume that Player 2 expect player to play honetly, and react accordingly. If Player oberve x = and bluff, her expected increae in core (over following the myopic trategy) i qd(p b p c )+ ( q)d(p d p a ) D(p x p y ). Proof. We analyze the change in Player core due to her two move (deviation and ubequent correction) eparately. Given P information (x = ), the expected probability of the event happening i p x. Thu, the expected deviation move core for player i: S(p x, p y ) = p x log p y p x + ( p x )log p y p x = D(p x p y ) 2

13 A player 2 ha a probability of q of eeing a one, player will have to have a corrective tep from p c to p b with probability q. Similarly with probability q player will make her corrective tep from p a to p d. Therefore the expected core from the corrective tep i qs(p c, p b ) + ( q)s(p a, p d ). S(p c, p b ) = p b log p b p c + ( p b )log p b p c = D(p b p c ) S(p a, p d ) = p d log p d p a + ( p d )log p d p a = D(p d p a ) Theorem 3. Suppoe two player are trading in a market with alternate move; without lo of generality, uppoe P make the firt move. Suppoe that the following condition hold at the tart of trading: C: Player acribe ome belief q 0, to player 2 having een a C2: Player 2 acribe ome belief q 0, to player having een a 0 C3: The optimal probabilitie p a, p b, p c, p d are in general poition, i.e., no two are equal. Further, retrict attention to equilibria in which P firt round move i not determinitic trategy that doe not depend on x. (In other word, we ignore equilibria in which P alway move to ome value of p, regardle of her information. Such trategie merely pa the buck to P2.) Then, there i no finite equential equilibrium trategy profile in which P alway move to ome p u in the firt round when he ee x =, and P alway move to p v p u in the firt round when he ee x = 0. Proof. Let (σ,σ 2 ) be a equential equilibrium trategy. For contradiction, uppoe that σ require P to follow the myopic trategy in the firt round. By lemma, P mut move to p x when x = and p y when x = 0. Now, in equilibrium, P2 will take thi into account, and will therefore know both bit after the firt round. She can capture all the remaining urplu by moving to p a, p b, p c, p d depending on x 2 and the inferred value of x. Thu, in any equilibrium, he will eventually move to the optimal point. Now, conider a deviation from thi trategy in which P bluff in the firt round and correct P2 final move at the end. When x =, Lemma 2 how that the expected additional core increae if P bluffed i given by: qd(p b p c ) + ( q)d(p d p a ) D(p x p y ) Now, from a well-known convexity property of the relative entropy [CT9, pp.32], we have: qd(p b p c ) + ( q)d(p d p a ) D(qp b + ( q)p d qp c + ( q)p a ) () qd(p b p c ) + ( q)d(p d p a ) D(p x p y ) (2) qd(p b p c ) + ( q)d(p d p a ) D(p x p y ) 0 (3) 3

14 Inequality 2 follow from the definition of p x and p y. Thu, inequality 3 implie that bluffing i alway at leat a profitable a behaving myopically by lemma 2. Moreover, inequality i trict when q 0, and p d p a, p b. Thu, bluffing will be a trictly profitable deviation under the condition of the theorem, and hence the myopic trategy for P cannot be part of an equilibrium profile. We oberve that the general poition condition we aumed i ufficient but not neceary. All that appear to be neceary i that the optimal value i not independent of either player ignal. Thi explain why there are no myopic equilibria even in the cae of the AND function, for which the point are not in general poition. 4.2 Nonexitence of finite equilibrium Theorem and Lemma how that there i no equilibrium in which player follow a determinitic trategy that i dependent on her ignal. If there wa uch an equilibrium, then, in equilibrium, player 2 would infer player bit and move to the optimal point. Now, it follow that there i no equential equilibrium trategy profile for the extended trading game, under the ame aumption, that atifie the condition that the market i in the optimal tate with certainty after ome finite number n of round. For thi reult, we make the implifying aumption that all mixed trategie are dicrete, i.e., a player can only mix trategie over finitely many point in any one move. Thi i a very mild aumption; it hould be poible to relax it with careful analyi. Theorem 4. Under condition [C]-[C3] of theorem 3, there i no equential equilibrium trategy profile in which the market i certain to be in the optimal tate after n round, for any finite n. Proof. Suppoe the condition [C]-[C3] of theorem 3 are met. Let (σ,σ 2 ) be any equential equilibrium. We now argue that, with nonzero probability over P mixed trategy, the condition [C]-[C3] of the theorem will be met after P firt move, with the role of P and P2 interchanged. Firt, oberve that the value of the optimal probabilitie p a, p b, p c, p d never change during the coure of play; witching role from P firt to P2 firt correpond to imply wapping the label on thee point. Thu, they remain in general poition throughout the execution of the trategy. (The point p x and p y may change with the player belief, but thi doe not matter.) It i therefore ufficient to how that the other two propertie are preerved (with nonzero probability). From theorem 3, there are two cae that could arie in equilibrium. Cae (i): P play ome trategy p u with certainty regardle of the value of x. In thi cae, P2 ha learned nothing about P bit, and thu, the condition of the theorem alway hold after round. Cae (ii): P play a mixed trategy for at leat one value of x. Now, we claim that there i a poition p u uch that P moved to p u with nonzero probability t when x = 0 and with nonzero probability t when x =. Suppoe there wa no uch p u. Then, the upport of P firt-round trategy for different value of x would be completely dijoint, and thu, P2 could infer P bit exactly. Thu, P would effectively have 4

15 a determinitic trategy; a imple extenion of Lemma 2 how that the myopic trategy would be a good a thi. Oberve that p u i played with finite probability. Further, conditioning on P moving to p u in the firt round, P2 would aign ome probability ˆq 0, to x =. P belief about x 2 haven t changed at all after the firt round. Thu, the condition of theorem 3 till hold after the firt round, conditional on p u being played. Repeating thi argument for each of the firt n round, and conditioning on one of the trategie in the upport of each round, how that the condition of the theorem till hold with ome nonzero (albeit mall) probability. Thu, the equilibrium cannot converge with certainty after n round. 4.3 Poible extenion to the baic reult In thi ection, we briefly dicu our approach to generalizing the core reult. Our reult are obtained for the logarithmic market coring rule. We believe that the reult will immediately be valid for many (if not all) proper coring rule. The key element of the proof i the convexity of the relative entropy. Correponding to any proper coring rule, there i a different meaure of relative entropy (ee the article by Grunwald and Dawid [GD04]); if it alo atifie the convexity propertie, our reult would till be valid. We believe that the convexity arie naturally from the propertie of proper coring rule, but we have not yet verified thi. Another natural extenion i to permit each player ignal to have more than two poible value. Again, we conjecture that the core reult will till be valid; a imilar convexity argument eem attainable. However, the general poition condition we rely on become increaingly retrictive a the number of ignal grow, and o it will be ueful to identify the weaket condition that be ued. A generalization to more than two player alo eem feaible. Suppoe a market ha n player playing in turn. Then, from the firt player point of view, all the remaining player can be thought of a a ingle player with 2 n poible ignal. Thu, the incentive to bluff hould till be preent. Again, it will be important to find a ueful genericity condition. 5 Dicuion and Future Work Thi paper report preliminary reult on analyzing non-myopic trategie in a two-player prediction market etting. We find, urpriingly, that the myopic trategie are almot never optimal in a non-myopic ene. Of coure, in a real market, there may be other reaon why player prefer the myopically optimal trategie: In particular, they are much impler to play, and more robut, and the potential gain from bluffing are often very mall. Thu, our reult are not in any way meant to imply that market coring rule are not a ueful microtructure for organizing a market. Intead, we believe that the analyi uggeted here will be ueful in clarifying when market might be epecially uceptible to long-range manipulative trategie. We invetigated a imple modification, which include a form of dicounting, to ameliorate thi potential 5

16 problem. The need for dicounting how a connection to claical bargaining etting, in which player bargain over how to divide a urplu they can jointly create. In a prediction market, informed player can extract a ubidy from the market player; moreover, player can pool their information together to make harper prediction than either could alone, and thu extract an even larger ubidy. They might engage in bluffing trategie to bargain over how thi ubidy i divided. Explicit dicounting can make thi bargaining more efficient. Thi i work in progre, and there are everal direction for future work. Analyzing ituation with more than two player, and more than two ignal, would be very ueful. Finally, it would be intereting to derive a connection between the dicount factor and the peed of convergence of the prediction market price to their true value. We peculate that a reduction to concept from information theory would be ueful for thi reult a well, perhap to bound the rate at which the entropy i reduced over time. Reference [AG92] Franklin Allen and Dougla Gale. Stock-price manipulation. Review of Financial Studie, 5(3): , 992. [Bri50] [CT9] [CY04] G. Brier. Verification of forecat expreed in term of probability. Monthly Weather Review, 78: 3, 950. Thoma M. Cover and Joy A. Thoma. Element of information theory. Wiley-Intercience, New York, NY, USA, 99. Archihman Chakraborty and Bilge Yilmaz. Informed manipulation. Journal of Economic Theory, 4:32 52, [FFPS05] Joan Feigenbaum, Lance Fortnow, David M. Pennock, and Rahul Sami. Computation in a ditributed information market. Theoretical Computer Science, 343:4 32, (A preliminary verion appeared in the 2003 ACM Conference on Electronic Commerce). [GD04] Peter Grunwald and A. Dawid. Game theory, maximum entropy, minimum dicrepancy and robut bayeian deciion theory. Annal of Statitic, 32: , [Han03] Robin Hanon. Combinatorial information market deign. Information Sytem Frontier, 5():07 9, [HOP06] Robin Hanon, Ryan Oprea, and Dave Porter. Information aggregation and manipulation in an experimental market. Journal of Economic Behavior and Organization, 60(4): , [NS07] Evdokia Nikolova and Rahul Sami. A trategic model for information market. In Proceeding of the Seventh Annual ACM Conference on Electronic Commerce (EC 07) (to appear),

17 [Pen04] David Pennock. A dynamic parimutuel market for information aggregation. In Proceeding of the Fourth Annual ACM Conference on Electronic Commerce (EC 04), June [SR07] Koleman Strumpf and P. Rhode. Manipulating political tock market: A field experiment and a century of obervational data, Working paper. Available from cigar/paper/maniphit Jan2007.pdf. 7