Topics in Game Theory - Prediction Markets

Transcription

1 Topics in Game Theory - Prediction Markets A Presentation PhD Student: Rohith D Vallam Faculty Advisor: Prof Y. Narahari Department of Computer Science & Automation Indian Institute of Science, Bangalore Bengaluru, India. Dept of CSA (IISc) Prediction Markets October 14, / 26

2 Outline of the Presentation Organization of the Presentation Prediction Markets: Introduction Research Directions in Prediction Markets Important Prediction Market Mechanisms Automated Market Makers Dynamic Parimutuel Markets Market Scoring Rule Wagering mechanisms Collective revelation Scope for future work Dept of CSA (IISc) Prediction Markets October 14, / 26

3 Prediction Markets: Motivation How do people predict? Popular Prediction Techniques Automated market makers / Prediction Markets - aggregate information from agents with diverse information Wagering mechanisms - players wager and are rewarded based on prediction accuracy Collective revelation - infer private information of all agents Proper scoring rules - single agent, single event scenario Peer-prediction methods - predict about what others are predicting Simple polls - simple tool for aggregating informations Delphi methods - structured discussion through many rounds Dept of CSA (IISc) Prediction Markets October 14, / 26

4 Prediction Markets: Motivation Searching the roots: Origin of Prediction Markets Hayek Hypothesis(1940) Price system in a competitive market is a very efficient mechanism to aggregate dispersed information among market participants. In an efficient market, price of a security almost instantly incorporates all available information. Market price summarizes all relevant information across traders. Market price is market participants consensus expectation about the future value of the security. The Wisdom of Crowds: James Surowiecki (2004) Why the Many Are Smarter Than the Few and How Collective Wisdom Shapes Business, Economies, Societies and Nations Dept of CSA (IISc) Prediction Markets October 14, / 26

5 Prediction Markets: Motivation What is a Prediction Market? Intuition through an example Let us say we want to predict the outcome about who the next Prime Minister of India will be. Specifically, we want to know if Dr. Manmohan Singh will continue to be the next Prime Minister! A prediction market can trade a Dr. Manmohan Singh Security, a share of which pays the following after the election: { Re.1 if Dr. Manmohan Singh becomes the PM Value of 1 share = 0 otherwise Let W represent the event that Dr. Manmohan Singh wins the election. Now, the expected payoff of a share of Manmohan Singh security is p = Pr(W ) 1 + [1 Pr(W )] 0 (1) where p is the price of Manmohan Singh security. For instance, if current price of the security is Re. 0.6, it means that market traders believe that, with probability 0.6, Dr. Manmohan Singh will beat, say, Mr. Narendra Modi. This probability will become consensus among the market participants. If some market traders possess crucial information that leads them to believe that Dr. Manmohan Singh only has half chances to win, they will sell their security holdings at the current price, which in turn drives down the price. Dept of CSA (IISc) Prediction Markets October 14, / 26

6 Prediction Markets: Motivation What is a Prediction Market? Knowledge Integration and Price discovery Eliciting and aggregation of information from diverse and frequently self-interested sources. A market designed primarily for price discovery. For example, the market operator may be happy to pay for the information it seeks, instead of enforcing neutral or positive revenue. Dept of CSA (IISc) Prediction Markets October 14, / 26

7 Prediction Markets: Motivation Advantages of Prediction Markets over other approaches of information aggregation Compared with statistical forecasting methods Can incorporate real-time information, which was not contained in historical data. Compared with eliciting expert opinions Less constrained by space and time. Eliminate the effort of identifying experts and soliciting their participation. Less expensive in practice. They do not need to deal with conflicting opinions. Dept of CSA (IISc) Prediction Markets October 14, / 26

8 Prediction Markets: Motivation Prediction Markets At Work Online Prediction Market Examples The Iowa Electronic Markets (IEM) are real-money futures markets to predict economic and political events such as presidential elections Hollywood Stock Exchange (HSX) trades securities to forecast future box office proceeds of new movies MIT s Innovation Futures predict important business and technology trends Tech Buzz Game aims at both forecasting high-tech trends and testing a new market mechanism Case study: Iowa Electronic Markets Dept of CSA (IISc) Prediction Markets October 14, / 26

9 Prediction Markets: Motivation Broad Areas of Research in Prediction Markets Accuracy, Effectiveness and Scope of Prediction Markets Compare with Expert Aggregation (Experiments with 2003 US National Football League games) Computational Aspects of Prediction Markets Equilibrium price of a financial security reflects all of the information regarding the security s value Design of Prediction market mechanisms Mechanisms are usuallly pari-mutuel in the sense that the winners are generally paid out by the stakes of the losers. Truthful prediction market mechanisms Dept of CSA (IISc) Prediction Markets October 14, / 26

10 Automated Market Makers Through the looking glass: Thin Markets-An issue in Prediction markets Thin Markets - A chicken-or-egg problem Continuous Double Auction: A double auction is a process of buying and selling goods when potential buyers submit their bids and potential sellers simultaneously submit their ask prices to an auctioneer, and then an auctioneer chooses some price p that clears the market: all the sellers who asked less than p sell and all buyers who bid more than p buy at this price p. A CDA only matches willing traders, and so poses no risk whatsoever for the market institution. But a CDA can suffer from illiquidity if trading is light and thus markets are thin. Thin markets lead to a chicken and egg problem where few traders care to participate because other traders are scarce, potentially spiraling the market into failure. Dept of CSA (IISc) Prediction Markets October 14, / 26

11 Automated Market Makers Through the looking glass: Thin Markets-An issue in Prediction markets Automated Market Makers An automated market maker can improve the liquidity of a prediction market. The market maker continually announces prices offering both to buy and to sell some quantity of the security, adjusting his prices in programmatic response to trader demand. Should run into predictable /bounded loss Informed traders should have incentive to trade whenever their information would change the price After any trade, computing new prices should be tractable A prediction market maker may subsidize a market maker that expects to lose some money, in return for improving trader incentives, liquidity, and price discovery. Approaches to forming Market makers Continuous Double Auction with Market Maker (CDAwMM): have built-in liquidity, but the market maker is exposed to significant risk of large monetary losses. Hanson Scoring Rule (HSR) Dynamic Pari-mutuel markets(dpm) Dept of CSA (IISc) Prediction Markets October 14, / 26

12 Dynamic Pari-mutuel markets(dpm) Automated Market Makers: Dynamic Pari-mutuel markets(dpm) An artist s impression Pari-mutuel markets Pari-mutuel markets effectively have infinite liquidity: Pari-mutuel markets also involve no risk for the market institution. Since there is a strong disincentive for placing bets until either (1) all information is revealed, or (2) the market is about to close. Pari-mutuel market participants cannot buy low and sell high Dynamic Pari-mutuel markets(dpm) Hybrid between mutuel market and CDA. Pari-mutuel by nature - acts to redistribute money from some traders to another. A subsidy needed to start the market Has infinite liquidity: The essential characteristic of a liquid market is that there are ready and willing buyers and sellers at all times. Dept of CSA (IISc) Prediction Markets October 14, / 26

13 Dynamic Pari-mutuel markets(dpm) DPM - Key points Mechanism for wagering on a future uncertain event Satisfies 3 important properties Guaranteed No Risk Continuous incorporation Liquidity for the market maker information CDA No Yes Yes CDAwMM Yes No Yes MSR Yes No Yes DPM Yes Yes Yes DPM acts as an automated market maker willing to accept infinite buying orders at some price Price functions derived which encode how prices change continuosly as shares are purchased Possibility of an aftermarket wherein traders can cash out of the market early to lock in their gains or limit their losses. Dept of CSA (IISc) Prediction Markets October 14, / 26

14 Proper Scoring rules Scoring rules Ω = {1, 2,..., m} - Outcome space P = {p R m : 0 < p i < 1, m i=1 p i = 1} Reported Probability distribution : ˆp ; True Probability distribution : p Definition A scoring rule is a function s : P Ω R. For each report ˆp P and each outcome i Ω, it specifies a payment s(ˆp, i). The expected payment s under the scoring rule is given by Definition s(ˆp, p) = m s(ˆp, i) p i A scoring rule s : P Ω R is (weakly) proper if p, ˆp P, i=1 s(p, p) s(ˆp, p). Dept of CSA (IISc) Prediction Markets October 14, / 26

15 Scoring rules Proper Scoring rules - Single Event Scenario Suppose we want to incentivize a single agent to truthfully report its subjective probability p E that event E will take place. We need to pay the agent some amount of money that depends on the reported probability ˆp E and on whether the event actually happens. Proper Scoring Rule : s(ˆp E, x E ) where x E = 1 if the event occurs and x E = 0 otherwise p = arg maxˆp (p s(ˆp, 1) + (1 p) s(ˆp, 0)) where p is the true estimate of the agent s probability. Quadratic Scoring Rule : s(ˆp E, x E ) = 1 (x E ˆp E ) 2 Logarithmic Scoring Rule : s(ˆp E, x E ) = (x E logˆp E ) + (1 x E ) log(1 ˆp E ) Dept of CSA (IISc) Prediction Markets October 14, / 26

16 Market Scoring rules Scoring rules Basically using proper scoring rule in the setting of multiple agents. Current Estimate of the probability - ˆp E At any point of time, agent can change it to ˆp E After event is realized, agent will be paid: s(ˆp E, x E) s(ˆp E, x E ) (may be negative) In some sense, this gives right incentive to agent as it cannot affect s(ˆp E, x E ) Nice property: Net payment made by the rule : s(ˆp E f, x E) s(ˆp E 0, x E) where ˆp E f is the final probability and ˆp E 0 is the initial probability. Dept of CSA (IISc) Prediction Markets October 14, / 26

17 Truthful Mechanisms Truthful Mechanisms for Prediction Markets Wagering Mechanisms - Setting Operate in 2 steps Each player announces a report chosen from a certain set of possible reports R and wagers any positive amount of money After realization of experiment, the common pot is divided among the players based on their performance Defnition A Wagering mechanism is a tuple (R, Ω, Π) where R is set of allowed reports, Ω is outcome space, and Π = (Π i (r, m, ω)) i N is the vector of payout functions Π i : R N [0, + ) N Ω [0, + ) with Π i (r, m, ω) = 0 if m i = 0 Dept of CSA (IISc) Prediction Markets October 14, / 26

18 Truthful Mechanisms Truthful Mechanisms for Prediction Markets Examples - Wagering Mechanisms Pari-mutuel betting markets Wager on mutually exclusive and exhaustive events i.e., {E 1,..., E m } Players lose their wagers when the true outcome is not what they bet. Winning players share money in proportion to their own wager. Such a market is a wagering mechanism with R = {E 1,..., E m } and payout Π i = 1 ω ri m i j m j1 ω rj j m j Dept of CSA (IISc) Prediction Markets October 14, / 26

19 Truthful Mechanisms Truthful Mechanisms for Prediction Markets Distribution properties A distribution property Γ(P) is a function that assigns a real value to any probability distribution P in a given convex domain. Examples Probability of an event the expectation, the variance medians/quantiles, moments, skewness, kurtosis, etc. Strictly proper scoring rules - Recap A score function for a vector of distribution properties Γ = (Γ 1, Γ 2,..., Γ k ) is a real-valued function s(r, ω) with r = (r 1, r 2,..., r k ) and r i the report for property Γ i. Strictly proper when E P [s(r, ω)] < E P [s(γ(p), ω)] Dept of CSA (IISc) Prediction Markets October 14, / 26

20 Truthful Mechanisms Truthful Mechanisms for Prediction Markets Weighted Score mechanisms A weighted-score mechanism is a wagering mechanism (Ω, R, Π) asscociated with a vector of properties Γ = (Γ 1, Γ 2,..., Γ k ). Π is the vector of payout functions with the payout of player i defined as Π i (r, m, ω) = m i (1 + s(r i, ω) j s(r j, ω)m j j m j where s is a strictly proper score function for Γ taking values in [0, 1] ) Reference N. S. Lambert, J. Langford, J. Wortman, Y. Chen, D. Reeves, Y. Shoham, and D. M. Pennock, Self-financed wagering mechanisms for forecasting, in Proceedings of the 9th ACM Conference on Electronic Commerce (EC 08). New York, NY, USA: ACM, 2008, pp Dept of CSA (IISc) Prediction Markets October 14, / 26

21 Truthful Mechanisms Truthful Mechanisms for Prediction Markets Desirable Properties Budget-balance : if market generates neither profit or loss Anonymity : if payouts does not depend on the player Truthfulness : if players maximize their expected payout when reporting true property values. Normality : relative performance of a player should increase if player s absolute performance increases or when absolute performance of another player decreases. Sybilproofness : if immune to multiple identities. Individual rationality : Monotonicity : if increase in wagers lead to increase in expected payoff. Dept of CSA (IISc) Prediction Markets October 14, / 26

22 Truthful Mechanisms Truthful Mechanisms for Prediction Markets Theorem All weighted score mechanisms satisfy these properties. Proof: (Truthfulness) ( E P [Π i (r, m, ω)] = m i (1 + E P [s(r i, ω)] 1 m ) i j i j m E P [s(r j, ω)m j ] ) j j m j Since s is a strictly proper for Γ, E P [s(r i, ω)] is maximized only at r i = Γ(P), so E P [Π i ((r i, r i ), m, ω)] < E P [Π i ((r i, Γ(P)), m, ω)] for all r i Γ(P) Theorem Weighted score mechanisms are unique mechanisms which satisfies these properties. Dept of CSA (IISc) Prediction Markets October 14, / 26

23 Truthful Mechanisms Truthful Mechanisms for Prediction Markets Goel et al. have recently proposed a mechanism which satisfies key properties such as Reference Incentive compatibility - incentivizes the participants to be truthful. Information weighted - incorporates the fact that some agents are better than the other. Self verifying - the payoff of agents are decided before the objective observations. Budget balanced - external subsidy is not required S. Goel, D. M. Reeves, and D. M. Pennock, Collective revelation: a mechanism for self-verified, weighted, and truthful predictions, in Proceedings of the 10th ACM Conference on Electronic Commerce (EC 09). New York, NY, USA: ACM, 2009, pp Dept of CSA (IISc) Prediction Markets October 14, / 26

24 Future Work The Path Ahead Research Directions - A broad perspective Theoretical Examination: Why do information market work? Will an information market converge to a consensus equilibrium? What is the best possible equilibrium? Experimental Evaluation: To what extent, are theoretical models of information markets valid? Empirical Analysis: How well do information market work? How well do information markets perform compared with other forecasting methods, especially opinion pools? Design and Development: How to develop an effective information markets? When to choose information markets over other forecasting methods? Dept of CSA (IISc) Prediction Markets October 14, / 26

25 Future Work The Path Ahead Research Directions - A narrow perspective Developing and understanding multi-round wagering mechanisms by using techniques like delphi method, etc. Study performance of prediction market mechanisms like market scoring rules, etc through experiments using some open source prediction markets like Zocalo. There exists deep mathematical connections between market scoring rules, prediction markets and no-regret learning. There exists potential to define new prediction mechanisms based on learning algorithm. Dept of CSA (IISc) Prediction Markets October 14, / 26

26 Thank You THANK YOU!! Dept of CSA (IISc) Prediction Markets October 14, / 26