On Manipulation in Prediction Markets When Participants Influence Outcomes Directly

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1 On Manipulation in Prediction Markets When Participants Influence Outcomes Directly Paper XXX ABSTRACT Prediction markets are popular mechanisms for aggregating information about a future event such as the outcome of an election. In situations where market participants may significantly influence the outcome (as in an election with a small number of voters running the prediction market could change the incentives of participants in the process that creates the outcome (e.g. agents may want to change their vote in the election. We propose a new game-theoretic model that captures two aspects of real-world prediction markets: ( agents directly affect the outcome the market is predicting ( some outcome-deciders may not participate in the market. We show that this game has two different types of equilibria; when some outcome-deciders are unlikely to participate in the market equilibrium prices reveal expected market outcomes conditional on market participants private information whereas when all outcomedeciders are likely to participate equilibria are collusive participants effectively coordinate in an uninformative and untruthful way. Finally we suggest an approach towards incentivizing truthfulness by subsidizing agents appropriately using ideas from peer prediction. Categories and Subject Descriptors J. [Social and Behavioral Sciences]: Economics; K.. [Electronic Commerce]: Payment Schemes General Terms Economics Keywords Information Elicitation Prediction Markets Market Scoring Rule Manipulation Peer Prediction. INTRODUCTION Prediction markets are platforms designed to aggregate and disseminate information bearing on some future event dispersed among a potentially diverse crowd. It is generally assumed that market participants may have superior information about the relevant event but have no direct control Appears in: Proceedings of the th International Conference on Autonomous Agents and Multiagent Systems (AA- MAS 05 Bordini Elkind Weiss Yolum (eds. May 8 05 Istanbul Turkey. Copyright c 05 International Foundation for Autonomous Agents and Multiagent Systems ( All rights reserved. over the outcome. However prediction markets are often used in situations where this assumption is violated to a greater or lesser degree. In fact sometimes it is this very potential for violation that gives agents the informational edge that such markets get their value from. If market participants can influence event outcomes it is natural to ask to two questions: ( Are the actions of the outcome-deciders still truthful i.e. do they take the same actions that they would in absence of the prediction market? ( Are market prices still informative or in other words how much do they still tell us about the realized outcome? Consider three canonical real-world examples where prediction markets (or betting markets have demonstrated their forecasting ability to great effect: politics [3] sporting events [39] and software product releases []. In each of these cases it is easy to see how the presence of a prediction market on the event may make agents affecting the outcome act differently than they otherwise would. A congressional staffer or member of congress may know more about the probable outcome of a key vote than the general public but she is also in a position to influence said outcome. A referee or player has substantial ability to influence the outcome of a sporting event. A software engineer has the potential to delay (or speed up the release of a product. While it is acknowledged that prediction markets have value as forecasting tools that may help in making business and policy decisions they have gone through cycles of hype and bust for reasons that include regulatory concerns about manipulation. The emblematic anecdote about this problem is the failure of DARPA s proposed Policy Analysis Markets which were caricatured in the media as terrorism futures [0 3]. There are obviously markets that will not work but stock and futures markets have been used for a long time as predictive tools and prediction markets are no different in essence. The key is to understand when markets may be prone to manipulation and how much to trust them. We are not concerned with markets where an individual has a very small effect on the outcome (like large elections. So for the first part of the paper where we theoretically investigate potential incentives for manipulation and consequences thereof we restrict ourselves to a twoplayer model which helps us understand worst case manipulative behavior. Our model captures the two aspects of real-world prediction markets that we are interested in: ( agents directly affect the outcome that the market is set up to forecast ( some of the agents who have influence over the outcome may not participate in the prediction market (e.g. people who have an impact on the outcome of a basket-

2 ball game such as the players the coaches and the referees typically do not all take part in a betting market for that game. Among other things we study what effect if any the uncertainty around the participation of some outcomedeciders has on the actions of a strategic agent when there is incentive for manipulation.. Contributions The model of prediction markets we formulate in this paper is a two-stage game involving two players or agents whom we call Alice and Bob. Before the game commences each agent receives some true private signal about the underlying entity (e.g. for the vote-share prediction market of a two-candidate election the entity would be one of the two candidates. In the first stage of the game both players have the opportunity to participate (sequentially in a prediction market where trades are mediated by a market scoring rule or MSR [9] market-maker. Alice moves first; Bob may or may not participate in trading and if he does participate he goes second. In the second stage the two players simultaneously (and independently take public actions which we term votes for convenience although in general they model each participant s role in determining the outcome. In our model voting is identical to announcing a signal (not necessarily one s true signal. The payoffs from the stage-one prediction market are determined by a simple function specifically the arithmetic mean of the stage two votes. If Bob has not traded his vote is truthful (i.e. consistent with his private signal otherwise he is strategic; Alice is always strategic. The analysis of this simple model yields many interesting insights. Our main result on this model is that the equilibria of the game can be cleanly categorized into two types depending on Bob s probability of participation in the trading stage. Below a threshold on Bob s said participation probability say p we call the equilibrium a low participation probability (LPP equilibrium and above p we call it a high participation probability ( equilibrium. The threshold p itself depends on the MSR used and Alice s belief about Bob s signal. In a LPP equilibrium Alice essentially predicts Bob s vote and then bases her trading on the optimal combination of her own vote and Bob s vote and the prediction market price is reflective of the expected outcome. In a equilibrium on the contrary Alice effectively expects Bob to enter and collude with her and she chooses a prediction market price that allows Bob and her to split the profit (not necessarily evenly. We can examine the characteristics of these equilibria from two perspectives the information conveyed by market prices about the outcome of the vote (keeping in mind that voting may be strategic and whether or not Alice and Bob reveal their true signals in the voting stage. The results are summarized below. Informativeness of prices. If Bob does trade the price he updates the market to will be equal to the average vote and hence an accurate forecast. The price after Alice trades in a LPP equilibrium will be equal to Alice s posterior expectation of the average vote given her signal and her trading action and therefore an effective disseminator of information. In equilibria the price after Alice trades is insufficient for revealing her signal but is still indicative of her own vote and also determines Bob s vote if he participates in the market hence contains partial information about the outcome. Truthfulness of outcomes. The trading actions of either agent do not necessarily reveal their private signals directly: Regardless of whether the equilibrium is LPP or if Bob does end up participating in the trading stage his vote is fully determined already by Alice s trading choice independent of his signal. Alice s trading decision on the other hand is based on her belief about Bob s signal which in our Bayesian setting has an indirect dependence on her own information. In fact under the mild assumption of stochastic relevance [6] of Alice s signal for that of Bob we can recover Alice s true signal from her price-update in a LPP equilibrium. Alice s equilibrium price-update is still uninformative about her private signal a limiting case being the situation when Bob s participation is certain. Value of prediction markets when participants influence outcomes substantially. In situations where the second phase ( voting is already assumed to be truthful a market designer might wish to ensure that she did not make it untruthful just because she added a prediction market. One major implication of our model is that adding the prediction market is less likely to produce damaging incentives when some fraction of the outcome-deciders refrain from market participation and are truthful in their outcome-affecting actions. This is suggested by the previous paragraph: for a sufficiently high probability that the successor (Bob will not trade and will vote truthfully the predecessor though strategic is forced to act in a way that divulges her private information indirectly. Choice of market scoring rule. We present results for three different MSRs from the literature: logarithmic (LMSR quadratic and spherical. The merits of LMSR for prediction market design are well documented. We identify an additional property that is pertinent to our outcome manipulation model: when Alice is highly uncertain about Bob s private signal the threshold p on Bob s participation probability to be crossed for the uninformative collusive equilibrium domain to set in is significantly higher than those for the other two MSRs. Relatedly Alice and Bob split the profits more evenly with LMSR than with the other two MSRs in a equilibrium. Incentivizing truthful trading and voting. In Section 5 we return to the market design question and extend our model to n participants. We propose a remedy for manipulation when the principal can directly incentivize participants by introducing a payment scheme based on the Peer Prediction Method [6] for the voting mechanism. We also analyze the level of subsidy needed in the combined prediction/voting mechanism.. Related Work This paper relates to three major strands of literature: ( Incentives and manipulation in prediction markets ( Insider trading in financial markets (3 Information elicitation when the ground truth is not revealed [35]. The market microstructure we use in our model is a market scoring rule (MSR introduced by Hanson [8] and ex-

3 panded upon by many researchers ([ 8] inter alia. Incentives for manipulation in prediction markets may arise in a number of ways. There is a plethora of literature on price manipulation tampering with the market price owing to some extra-market incentives ( [ ] e.g. a politically motivated manipulator might make a large investment in an election prediction market to make one of the candidates appear stronger [3]; a related body of work pertains to decision markets a collection of contingent markets set up to predict the outcomes of different decisions such that only markets contingent on decisions that are taken pay off the rest being voided [7 8 0]. Another type of manipulation that has received some attention is outcome manipulation where an agent can take an action that partially influences the outcome to be predicted (e.g. [ 33]. Here we concern ourselves with the second type of manipulation and the model we use is different from those considered in the above publications. Our model directly captures the prediction market experiments of Chakraborty et al. [7] where students in a class gave their instructor an up/down rating at the end of every two-week period during which a prediction market where those students as well as some outsiders could participate tried to forecast the fraction of up ratings received by the instructor. The second major body of literature comes from theoretical finance and market microstructure. Kyle [5] followed up by [3 5] etc. studied the effect of informed insider(s on market price; Glosten and Milgrom [6] presented another view of how asymmetric information affects price formation and their model has been adapted for market making in prediction markets [3 6]. Ostrovsky [7] recently examined information aggregation with differentially informed traders under both Kyle s pricing model and market scoring rules. Again in all of these models the liquidation value is assumed to be exogenously determined unlike in ours. In the last part of this paper we draw from the literature on truth-telling incentives in traditional means of information gathering such as surveys polls etc. Two important contributions in this vein are the Bayesian Truth Serum (BTS [ ] and the Peer Prediction (PP method ([6 38] and references therein. The situations of interest in this paper are not the ones for which BTS or PP were designed; these are intended to solve the problem of getting people to vote or give their opinion. Our focus is on settings where people would already vote / give their opinion / do their work honestly but the introduction of a prediction market may affect their incentive to do so. Therefore while ideas from this literature will prove useful we cannot simply apply BTS or PP in the first place and ignore the existence of the prediction market.. TWO-PLAYER MODEL Let τ T denote the unobservable type of the entity on which the voting system and its associated prediction market are predicated. At t = 0 the two agents Alice and Bob (A and B in subscripts receive private signals s A s B Ω = {0 } respectively. The belief structure comprising the prior distribution Pr(τ on the type and the conditional joint distribution Pr(s A s B τ of the private signals given the type are common knowledge. Let ( denote Alice s posterior probability that Bob received the signal s B = 0 given her own signal and common knowledge i.e. (s Pr (s B = 0 s A = s s {0 }. We need no further assumptions on the belief structure for our main result (Theorem since it depends only on the magnitude of regardless of how it is evaluated. We shall later discuss a specific belief structure. However it is worthwhile to define here the property of stochastic relevance [6] which has no bearing on our main result but is a necessary assumption for one of our important corollaries. Definition. For binary random variables s i s j {0 } s j is said to be stochastically relevant for s i if and only if the posterior distribution of s i given s j is different for different realizations of s j i.e. if and only if Pr(s i = 0 s j = 0 Pr(s i = 0 s j =. We now describe the market and voting mechanisms (common knowledge. Stage (market stage: The market price at any timestep t is public the starting price at t = 0 being p 0 which is the baseline estimate of the liquidation value (the final gross payoff per unit of the prediction market security. We set p 0 = which is the standard initialization for prediction markets thus assuming that the market maker starts off with no strong beliefs one way or the other. The market is implemented using a Market Scoring Rule (MSR [9]. Let s(r ω denote the underlying strictly proper scoring rule where ω is the true outcome or state of the world and r is a forecast/report on it (not the associated probability see []. In this paper we consider three representative market scoring rules namely logarithmic quadratic and spherical defined as follows : LMSR: s(r ω = ω ln r + ( ω ln( r QMSR: s(r ω = ω (ω r SMSR: s(r ω = (rω + ( r( ω / r + ( r. At t = Alice interacts with the market maker and changes the price to p A. At t = Bob has an opportunity to trade but may not show up with a probability π [0 ] which is called Bob s non-participation probability and is also common knowledge; if he does trade he changes the price to p B. Regardless of whether Bob trades the market terminates after t =. Stage (voting stage: Alice and Bob simultaneously declare their votes v A v B Ω = {0 } respectively. We define truthful voting as declaring one s private signal i.e. v k = s k k {A B}. If Bob did not trade in the first stage we assume that he votes truthfully; any agent participating in the prediction market is Bayesian strategic and riskneutral. The liquidation value of the security is given by v = (v A + v B / {0 }. The net payoffs of Alice and Bob are given by r i = s ( p i v A+v B ( s pj v A+v B ( where i {A B} j = 0 for i = A j = A for i = B. These formulations apply to prediction markets where the reports or forecasts are not probabilities but values of continuous random variables in [0 ]. Similar formulations were used by [7] for theoretical analyses and in [7] for experiments. In general we can have v = αv A + ( αv B α (0 where α models Alice s degree of control over the final outcome. In this paper we focus on the special case α = as a starting point where both agents are equally powerful.

4 Note that unlike in a traditional prediction market the underlying type τ is never revealed. Nevertheless if every agent were to vote truthfully (and this were common knowledge then it is easy to see from the properties of market scoring rules that an agent s expected net payoff would be maximized by moving the market price to their posterior expectation of the liquidation value v. With this in mind p A and p B will be sometimes referred to as the reports (on the market outcome of Alice and Bob respectively. 3. EQUILIBRIUM ANALYSIS OF THE TWO- PLAYER GAME We now present our main result for the case p 0 = ; they generalize easily (with different numerical values to cases where p 0. Theorem. For any π [0 ] and (0 \{ } the game described above has a unique perfect Bayesian equilibrium. For every there exists a fixed value of Bob s non-participation probability say π c( which we call the crossover probability (dependent on the MSR on either side of which the equilibria are qualitatively different. We call the sub-interval π < π c the high participation probability ( equilibrium domain and the sub-interval π > π c the low participation probability (LPP equilibrium domain. In a equilibrium: In Stage Alice moves the market price to p A = p L if > and to pa = ph if < ; pl ( 0 p H ( are functions of the MSR only 3 and independent of π. If Bob trades his price-update is p B = 0 if p A = p L and p B = if p A = p H. In Stage Alice votes v A = 0 if she set p A = p L v A = if p A = p H. If Bob participated in Stage he votes v B = 0 if he set p B = 0 and v B = if he set p B =. In a LPP equilibrium: In Stage Alice s price-report p LP A P is equal to her posterior expection of the market liquidation value (average vote given the parameters π and her report p LP A P i.e. p LP A P = E [ ] v π p A = p LP A P. Moreover p LP P A < if q0 > p LP A P > if q0 <. If Bob trades his price-update is p B = 0 if p L p A pb = if < pa ph and p B = otherwise. In Stage Alice votes v A = 0 if p A > va = if pa <. If Bob traded he votes v B = 0 if p A [ ] ( p L p H ] v B = otherwise. More specifically p LP A P is one of µ 00 = π( µ 0 = π µ 0 = +π( µ = π where µ 00 < µ 0 < < µ0 < µ π q0. Tables and 3 detail the dependence of Alice s equilibrium trading action p P A BE on π for LMSR QMSR and SMSR respectively. Figure depicts π c as a function of for each of the three MSRs. Proof. The following lemmas spell out the effect of Alice s price-update on her vote and Bob s best response to the trading choice made by Alice. The proofs involve simple algebra and hence are omitted. Lemma. If p A > p 0 = (resp. pa < then Alice s rational vote is v A = (resp. v A = 0 regardless of p B v B. 3 ( p L p H are ( 5 5 ( 3 ( for LMSR QMSR SMSR respectively. π p P A BE (LMSR Domain 0 < < 0 π < x L ( 5 5 x L ( < π < 5 µ 0 LPP 5 < π µ LPP = 0 π q0 < 0 π < < π µ LPP = 0 π < < π µ00 or µ LPP 5 q0 < 3 0 π < 5 5( p L = 5 < π µ00 LPP 5( = π 3 5 q0 < 0 π < y L ( 5 5 yl ( < π µ 0 LPP x ( is the unique solution to the fixed point equation x = ln 6 ln(5( x x (+x +x ( q0 ln ( q0 ln in ( 3 5 q0 for q0 < 5 ; yl ( is the unique solution to the fixed point equation y = ln 6 ln(5( y y (+y +y in ( 3 5 q0 for q0 > 3. 5 Table : Alice s PBE price-report for LMSR market p 0 =. π p P A BE (QMSR Domain 0 < < 0 π < x Q ( 3 x Q ( < π µ 0 LPP = 0 π or 3 < q0 < 0 π < y Q ( y Q ( < π µ 0 LPP x Q ( = y Q ( = 3+v v v = 3+u u u =. ; Table : Alice s PBE price-report for QMSR market p 0 =. Lemma. If p A > (resp. pa < then Bob s rational vote is v B = if p A p H also (resp. if p A < p L also and is v B = 0 otherwise regardless of s B. Since Bob knows v A from p A and also his best v B he can set p B = (v A + v B/. Now that we have figured out Alice s vote as well as Bob s report-vote pair given Alice s trading choice it remains to show that Table gives Alice s expected payoff-maximizing actions for different parameter values. We sketch the proof for LMSR those for the other two being analogous. A little thought reveals that our maximization problem requires the analysis of a piecewise smooth function described by f uv(p = ln ( p µuv ( p µuv where (u v = (0 (0 0 ( ( 0 over the intervals [ 0 5 [ 5 ] ( 5 ] ( 5 ] respectively. These (u v pairs are the rational votes of Alice and Bob in the respective regions and the µ uv s are as defined above. It is easy to show that these µ uv s are the global maximizers of the corresponding f uv s over

5 Cross over probability π c LMSR QMSR LPP 0.8 LPP 0.8 LPP Alice s posterior probability of Bob s signal being 0 ( SMSR Figure : Dependence of crossover probability on Alice s posterior belief about Bob for the three MSRs; e.g. for QMSR if = 0.5. then π c so we have a LPP equilibrium with p A = ( + π( / for π > and a equilibrium with p A = p H = 0.75 for π < Note that for LMSR when 0. < < 0.6 the crossover probability actually decreases with Alice s increasing uncertainty unlike for the other two! This can be attributed to a difference in the shapes of the MSRs. π p P A BE (SMSR Domain 0 < < 0 π < x S ( x S ( < π µ 0 LPP = 0 π < π S 0.75 or πs < π µ 0 or µ 0 LPP q0 < 0 π < y S ( 0.75 ys ( < π µ 0 LPP K +K v K 3 v x S ( = K v = v ; ys K ( = +K u K 3 u K u = u ; K = ( ; K = (3 0.76; K 3 = 3 + ( ; πs = ( K + K K 3 /( K Table 3: Alice s PBE price-report for SMSR market p 0 =. 0 p. However the overall function is ill-behaved with jump discontinuities at and since f00(0. > f0(0. and 5 5 f (0.8 > f 0(0.8 in general; moreover it changes its form drastically depending on the relative magnitudes of π and. It is thus impossible to analyze the function without considering the plethora of cases enumerated in Table. This makes the complete proof lengthy though straightforward and so we relegate it to a full version of the paper owing to paucity of space. Here we present part of the proof for the representative case < q0 < 3 π > 5 5( which adequately demonstrates the methodology we employ for prov- ing the relevant results for all such cases. Notice that the constraint < 3 ensures < 5 5( so that the condition π > is achievable. Now π > 5( implies 5( µ00 = π( > so that 5 µ0 > µ00 > 5 too. So f 00 is maximized at p = µ 00 ( 5 whereas f 0 is monotonically increasing in [ 0 5 and on this interval f 0(p < f 0(0. < f 00(0. < f 00(µ 00. Thus the local maximum of Alice s payoff for p < is attained at p = µ 00 and is given by ˆf 00 = h(µ 00 where h(x = b ln ( µ µ ( µ00 µ 00 x (0. Since > we have > 5( 5. So π > implies π > 5( 5. Hence by arguments similar to the above we obtain µ 0 < µ < which in turn leads to the conclusion that the local maximum of Alice s payoff for p > is ˆf = h(µ. But since 5 the function h(x is symmetric about its unique minimizer x = q0 > µ00 > µ h(µ00 > h(µ. ( Thus the best trading action for Alice is p A = µ 00 5 followed by the vote va = 0. This also implies from Lemma that v B = 0 if Bob trades. Hence Bob s expected vote from Alice s perspective is ˆv B = π( 0 + ( + ( π 0 = π( making the expected average vote from Alice s perspective ˆv = (0 + ˆv B/ = µ 00 = p A. Proceeding along similar lines for other possible parameter values (π we can complete the proof. 3. Implications Corollary. If Alice s signal s A is stochastically relevant for Bob s signal s B then the value of s A can be recovered from Alice s price-report in a LPP equlibrium p LP A P = µ uv (π u v {0 } regardless of whether v A = s A. Proof. By observing which sub-interval ( 0 p L [ p L ( ph] or ( p H contains Alice s LPP price-report we can infer v A and v B (Lemmas and and hence which µ uv u v {0 } equals p A. Since π is common knowledge we can solve for from the expression for µ uv. Under the assumption of stochastic relevance (s A is one-to-one so we can deduce s A uniquely from its value. Private signal revelation. Unfortunately if Bob trades his report-vote pair is fully determined by Alice s report and does not depend on s B. There is no guarantee that Alice s vote will be truthful either even in a LPP equilibrium (in general she is likely to guess which way Bob will vote and vote the same way.

6 What is still somewhat interesting is that the very possibility of Bob not trading but voting truthfully engenders a situation (LPP domain in which Alice s trading action indirectly reveals her private information (Corollary! This stands in contrast to the situation where Bob s participation is certain the extreme case of the domain. In a equilibrium we can only tell whether > (if pa = pl or < (if pa = ph which is insufficient for recovering s A without further assumptions about the signal structure. profit sharing. The equilibria are a world where collusion appears with Alice as the leader picking the vote that both will coordinate on and pushing the price to just the level where it makes sense for Bob if he transacts with the market maker to push the price all the way to 0 or and vote the same way as Alice. In this way they extract the maximum profit from the market maker and split it between the two of them in a ratio that is MSR-dependent: Profit if Bob trades LMSR QMSR SMSR Alice s share in total profit 67.8% 75% 78.3% Bob s share in total profit 3.9% 5%.68% Thus for all three MSRs considered Alice makes more profit than Bob in a collusive equilibrium with the discrepancy being the least for LMSR. Informativeness of market prices about final outcome. Finally we can put all our results together into the following table (note that if Bob does not trade the the final price p B = p A and ih does then p B = v where Bayes. Est. (Bayesian estimate is Alice s expectation of the average vote before Bob trades; Pre. (Predetermined signifies p A {p L p H }; Correct denotes the actual outcome. Bob trades Bob does not trade LPP LPP p A Bayes. Est. Pre. Bayes. Est. Pre. p B Correct Correct Bayes. Est. Pre. 3. A specific belief structure Thus far we have been non-specific about the signal structure proving general results; we now consider a concrete example scenario to illustrate our findings: The type space of the entity is identical to the signal space i.e. T = Ω = {0 } the prior probability of type 0 being ρ 0 (0. Given the type τ the agents signals are independently and identically distributed: for any true type of the entity each participant gets the correct signal (identical to said true type with probability ( ρ e otherwise gets the wrong signal; the error probability ρ e (0 \{ }. Then (0 = ( ρ e ρ 0 +ρ e ( ρ 0 ( ρ ( ρ eρ 0 +ρ e( ρ 0 and q 0( = eρ e ρ eρ 0 +( ρ e( ρ 0. This belief structure has multiple interesting information-revealing characteristics: First for any admissible ρ 0 and ρ e we have (0 ( i.e. Alice s signal is stochastically relevant for that of Bob. Hence Corollary applies. Second it is easy to show that if ρ 0 = (a uniform common prior then Alice s vote is always truthful since for any ρ e (0 s A = 0 > va = 0. Note that iff ρ e = q0(0 = q0( = regardless of ρ 0 hence signals are not informative [8] i.e. the prior and posterior probabilities are equal. Figure shows Alice s equilibrium report in a LMSR market and her expected liquidation value vs. π for fixed ρ 0 ρ e (hence a fixed. The and LPP regions are clearly visible to the left and right of the cross-over probability. The corresponding plots for the other two MSRs are qualitatively similar hence omitted. Alice s best response and expected liquidation value LMSR; Alice s signal s =0; error probability ρ =0.; prior ρ = A e Alice s price update p A Alice s posterior expected liquidation value Probability of Bob not participating Figure : Crossover from to LPP equilibria regions for LMSR over 0 < π < for fixed prior and likelihood structure ( = 0.5. For any π < π c = 0./( Alice updates price to p HP A P = 0.; for π > she moves the price p LP A P her posterior expectation of average vote (given Bob votes truthfully if he does not trade.. DISCUSSION AND EXTENSIONS We have presented a stylized model above that captures the essence of several outcome manipulation scenarios in prediction markets. Here we briefly sketch how it can apply to some more realistic scenarios and discuss the insights our results offer for these scenarios. First Alice and Bob s effect on the outcome could be stochastic rather than deterministic the outcome could be drawn from a probability distribution whose mean is their average vote. For example whether or not a product gets finished on time could be a stochastic function of the effort put in by the key players. It is easy to see that our analysis still holds as the expressions for expected payoffs remain unchanged. A second generalization is a scenario where there are traders in the prediction market other than Alice and Bob but these traders have no control over the relevant outcome. To test how the equilibrium strategies implied by Theorem fare in such scenarios we ran some simple simulations. Again Alice moves first (the market starting at p 0 = 0.5; preceding traders only matter in how they would move the previous price but she is followed by a sequence of 0 zerointelligence budget-limited traders; Bob then arrives at the market with probability π and trades if he does arrive. As before the outcome is Alice and Bob s average vote (note that successors of Bob if any do not matter in a MSR-based market since they have no effect on the payoffs of Alice or Bob. Each intervening trader obtains her private signal from the same source as the outcome-deciders and then draws her belief from a beta distribution with mean equal to her own posterior expectation of the outcome which is

7 Strategic Bob Truthful Bob (if trades (if trades Strategic Alice Truthful Alice Table : Alice and Bob s expected profits for being strategic vs. truthful with 0 intervening traders. The signal struture is similar to that in Section 3. with ρ 0 = 0.5 ρ ɛ = 0. π = 0.. The variance of each intermediate trader s beta distribution is m( m/ where m is her posterior expected outcome. The budget B = 0.5. The results are averaged over 0 5 simulations. The first entry in each cell corresponds to Alice the second to Bob. Note that strategic in the above table denotes the use of strategies implied by Theorem and described in Section the table obviously does not list all strategies available to the players and does not depict the equilibrium; it merely shows that Theorem strategies outperform (for both players any strategy-pair in which at least one is truthful. We observed similar results for the other parameter conbinations we tried. equal to her posterior expectation of another agent s signal assuming that she is oblivious to the existence of manipulators; she then moves the market price as much towards her belief as her budget allows (for further details on the budget constraint see Section 5. This budget constraint captures the intuition that agents who do not influence an outcome and/or have poor knowledge thereof should be conservative in their trading decisions. As in the above sections it is easy to show that Alice s vote is revealed immediately after she trades (in practice insiders are often big players so it is not unrealistic to suppose that they are identified by other traders hence Bob (if he trades can base his actions on Alice s inferred vote and his immediate predecessor s report. The problem is still easy for Bob since he knows the previous report and can infer Alice s vote. How should Alice play? This is a strategically more complex game and finding equilibrium strategies may be difficult. However one possibility is for Alice to simply ignore the existence of the intermediate traders and use her strategy from Theorem. While this is not necessarily an equilibrium (or a priori even a good strategy in our simulations (Table we see that it significantly improves upon another simple alternative the truthful strategy (which means declaring one s true signal as the vote after updating the price to one s posterior expectation of the average signal. Thus the model may well have predictive value even in this more complex setting. Finally our theoretical results apply to situations where Alice the first mover controls exactly half of the outcome but in general we would expect Alice to have less than 50% control over the outcome and for there to be several participants in determining the outcome. Moreover traders could also strategically pick the time-points at which they interact with the market. These are important generalizations and we believe that our model can serve as a foundation for further research on them. 5. INCENTIVIZING TRUTHFUL TRADING AND VOTING We now propose a design for the combined two-stage tradingvoting mechanism ( voting again being a metaphor for an agent s outcome-affecting action for the general case of n agents (n that disincentivizes manipulative behavior. The basic model is a natural extension of that described in Section but this section is independent of the analysis in Section 3. In the first stage of the game the n agents trade in a prediction market in some pre-determined order and then simultaneously choose their actions v i Ω (not necessarily binary in the second stage the market liquidation value v being some function of {v i} n i= (not necessarily the average. Each agent has zero non-participation probability in the market and trades exactly once. Thus although the myopic assumption still holds the scenario is adversarial in the sense that all outcome-deciders participate deterministically in the prediction market. The number of participants n as well as the belief structure on the underlying type τ and the agents private signals {s i} n i= is common knowledge. Let us denote the agents price reports {p i} n i= the starting market price being p 0. Here we demonstrate our approach with a LMSR market with liquidity parameter b the treatment for other scoring rules being similar. Although the worst-case loss in a traditional LMSR market is bounded it is theoretically possible for an individual trader to earn an unbounded profit from it. To circumvent this issue we place a fixed budget B on every agent (i.e. each agent trades in such a way that their loss can never exceed B. 5 From the properties of LMSR it can be easily shown that agent i can move the market price from their observed value p i to a maximum of p max = ( γ p i and to a minimum of p min = γp i where γ = e B/b p i = p i. Thus agent i s net payoff from the prediction market is ri P M (p i p i v = b [ v ln ( /pi /p i ( ] pi ln p i. ( Simple algebra shows that for each i the maximum and minimum possible values of the above under the budget constraint are r P M imax =b ln ( { γpi max γ } pi p i p i r P M imin = B = b ln γ. (3 Let ˆl i(v i denote agent i s posterior expectation of the liquidation value based on her decision to vote v i (declared signal and her inference from the common prior her private signal s i and the prices p 0 p... p i (assuming every other agent is Bayesian and truthful in both voting and trading and ˆr i P M ( p p i v i be her posterior expected net payoff from the market mechanism on updating the price from p i to any reachable p. From the above linear dependence of ri P M on v ˆr i P M is readily seen to be a simple linear function of ˆl i(v i. Our key idea is to introduce a compensation scheme for the voting mechanism such that the combined payoff from the market and the voting system when an agent is truthful 5 With the budget constraint the market may lose some of its expressiveness [] i.e. a trader may not be able to update the market price so as to coincide with their estimate of the liquidation value but it is still directionally expressive in the sense that it is still rational for an agent to shift the price as close to said estimate as possible.

8 in both stages (and believes that everyone else is going to be similarly truthful exceeds the largest profit she can make by deviating from truth-telling. A promising technique already proposed in the literature for achieving this end is the peer prediction scheme introduced by [6]. The following is a brief description thereof tailored to our setting. We choose a reference participant f(i a priori for agent i such that i s vote is stochastically relevant for that of f(i (the posterior is different for different realizations of the signal. At the end of voting the transfer made to participant i by the center is a function of the posterior on participant f(i s vote v f(i under the common prior likelihood and agent i s vote v i the function being a strictly proper scoring rule. Suppose we use a strictly proper scoring rule R( (not necessarily logarithmic so that agent i s peerprediction score is given by ri P P (v f(i v i = α ( ir g ( v f(i v i where g( is said posterior. Thus agent i s expected peerprediction score is ˆr i P P (s i v i = α iφ(s i v i where φ(s i v i = v f(i Ω R ( g ( ( v f(i v i g vf(i s i and g( is said posterior α i > 0 i = n. Miller et al show that truthful voting is a strict Nash equilibrium for this mechanism. The constants α i have no effect on the truth-telling incentives of the voting stage alone and we show below how to tune these free parameters to ensure honest behavior. Note that while the LMSR prediction market has a single parameter b that determines all its properties the proposed combined mechanism has n additional parameters. Now we can state our result: there is a way to set the parameters b and α i (which are under the designer s control that guarantees that the overall expected earnings ˆr i = P M (ˆr i + ˆr i P P for every i is uniquely maximized under the imposed constraints when their trading and voting are both consistent with their private information (in equilibrium. Let φ m = min ss Ω s s [φ(s s φ(s s ] which is strictly positive by the incentive compatibility of the peer-prediction method and is a function of the signal structure. Theorem. Let there be n participants in the above mechanism each with budget B. If we promise every agent i when she arrives to trade a peer- prediction payment with some α i satisfying α i > b φ m ln ( { } γpi max γ p i γ( p i γ( p i where all symbols have the meanings stated above then there exists an ex interim Bayes-Nash equilibrium where each agent i announces v i = s i in the outcome-deciding stage after having updated the market price as close as possible to her to truthful expected liquidation value i.e. p min if ˆli(s i < p min p i = ˆli(s i if p min ˆl i(s i p max otherwise. p max Proof. Part I: For any voting choice v i = v agent i s expected market liquidation value assuming everyone else to be truthful is ˆl i(v and since ˆr i P P is independent of p i from Equation ( it follows that ˆr i p i = ˆrP M i p i = ˆl i (v p i p i ( p i 0 pi ˆl i(s i. Part II: For proving that when subsidies are set to yield the conditions on α i above it is in an agent s best interest to pick her honest vote it suffices to show that for any possible signal values s s where s s and any feasible prices p p p ˆr i(s i = s p i = p p i = p v i = s > ˆr i(s i = s p i = p p i = p v i = s which reduces to α i[φ(s s φ(s s ] > ˆr i P M ( p p s ˆr i P M ( p p s. The greatest lower bound on the L.H.S. is by definition α i( φ m where φ m is a known constant and always strictly positive. An upper bound on the R.H.S. is obviously the range of all possible payoffs from the prediction market of agent i with budget B whose predecessor s price-report is p i i.e. (r P M imax r P M imin given by Equations (3. Thus setting α i to a value exceeding the (finite positive bound specified in the theorem statement is a sufficient condition for the desired inequality to hold. The two parts together complete the proof. If R( ln( then the raw peer-prediction scores ri P P are always negative so there is no incentive for voluntary participation. This problem can be solved as in [6] simply by subtracting from the raw score of agent i the constant α i min ss Ω(ln(s s which is a function of the prior and likelihood structures and independent of actual trader behavior. This ensures positive peer-prediction payments but also necessitates subsidization of the mechanism. For our budget-constrained LMSR market b ln is a (perhaps loose upper bound on its loss so the market subsidy is linear in b and independent of n. The amount of subsidy for the voting phase is proportional to n ( { i= αi. A reasonable } choice for α i is α i = κb γpi φ m ln max γ p i γ( p i γ( p i where κ is a constant slightly greater than. It is straightforward to show that ( ln ( ( φ m ( i+ αi ln ( γ γ κb γ + i γ assuming that the starting market price is 0.5. Since γ = e B/b it is clear that for fixed b α i is Ω(B and O(iB and for fixed i B it is Θ(. Hence the total peer prediction subsidy is linear in B independent of b and Ω(n and O(n. 6. DISCUSSION We have introduced a new formal model for studying incentive issues in prediction markets whose participants can affect the outcome by taking actions external to the market. An interesting feature of our model is the impact of uncertainty about the participation of some outcome-deciders. We have characterized the equilibria of the induced game and discussed their properties. Finally we have initiated the study of a methodology for incentivizing truthful behavior in the combined prediction-market/voting mechanism using ideas from the literature on peer prediction. This paper is a first step in exploring the crucial incentive issues that have the potential to derail the effectiveness or perceived effectiveness of prediction markets for various forecasting tasks. We believe that the model is widely applicable and one can use both the intuitions derived from this model and the analysis techniques to study specific situations of interest that might go beyond the literal definitions in the model when designing prediction (or other markets where participants directly influence outcomes. Interesting avenues for future work include generalizing our results to other MSRs investigating liquidation values other than the mean vote and broader investigation of the properties of subsidy mechanisms to achieve incentive compatibility in the combined game.

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