1. Presented By: Ofir Chen Based on: Designing Markets for Prediction by Yilling Chen and David M. Pennock 2010

2. Outline : -Motivation -Market Makers -Reminder+ (SR, CF), DPM, Utility function, SCPM -Incentive compatibility -Agents interaction -Manipulation -Expressiveness -What is Truth -Peer prediction and BTS

3. Motivation Wed like to predict an event of interest Ideally, wed like to make agents say the truth, the whole truth and nothing but the truth – and do it NOW Were willing to pay for it… Market Requirements: -Liquidity -Bounded loss -Discourage manipulation -Extract predictions easily How can we create such a market??

4. Liquidity: Liquidity is the ability to trade instantly with no significant movement in the price How do we encourage agents to talk... -Simple: the Market Maker (MM) pays them. -Weve already seen last time that by subsidizing the market we increase liquidity. -wed like to bound that subsidy. well talk about it later…

5. Bergman Divergence (BD): How do we make them say the truth… Given a convex function y=f(x) the the BD is: Nonlinear, non-negative function. The expected value over, given and : Thats a scoring rule for p!!

6. Scoring Rule (SR) With this we can create our first market – Market Scoring Rule (MSR): -Sequential trading. -updating r to r, requires paying the previous agent -Therefore payoff is -The final r is the markets prediction. -Disadvantages: -Not natural, no real contracts are traded. -Participating only once -These limitations may make the market less appealing to potential agents. - Solution: Cost Functions

7. Cost function (CF) Idea: Trade Arrow-Debreu (AD) contracts (instead of probabilities). AD contract pays $1 if the event happens, and $0 otherwise Notations and Market definition: - is a vector indicating the total number of shares of each type ever sold. - is the amount of shares of type i. -When changing (by buying/selling): Pay -Price of share i:,

8. Cost function (cont.) Desired properties of a CF: Differentiability (to calculate prices) Monotonically increasing in Positive translation invariant

9. Cost function Market from MSR (Chen, Vaughan10) Theres a one to one mapping between CFM and MSR: Such that and, Agent who change p to p in an MSR receives same payoff as changing q to q in a CFM. Agents will profit the same changing q in an Cost Function based Market (CFM) had they changed p in an MSR iff the following holds: Corollary, theres a mapping from CF to SR, not presented here.

10. DPM – Dynamic Parimutuel Market -Parimutuel: Winning agents split the total pool of money at the end. -Dynamic: Prices vary before outcome is determined (same as CFM) -Main difference: contracts are not Arrow-Debreu. Each contract i pays off: The more winners the smaller the profit. Is the final q. -MM has to initially buy contracts to avoid 0 division in price function. - is the markets prediction

11. Utility function Markets -Utility: utility of an outcome is the total satisfaction received by it. -Dynamic, AD contracts, probability price market, like CFM. -MM sets a subjective probability for all events -MM has a money value vector upon possible outcomes -MM has a utility function u(m) -The instantaneous price is defined as the infinitesimal change in the MM utility: -MMs expected utility: remains constant (Chen, Pennock 07)

12. SCPM: Sequential Convex Parimutuel Mechanism (Not detailed) -Agents state their wanted state vector, quantity, and max-price -the MM decides how many AD contracts to sell to maximize its profit by solving a convex optimization problem. -Prices are determined using VCG mechanism. -Prices reflect the markets prediction

13. Bounded loss: Subsidies are limited – MM would like to bound its losses. -MSR: -CFM: -DPM: initial market subsidy -Utility Market: bounded if m is bounded (from below) or u(m) is bounded (from above) -SCPM: bounded

14. So far… In all the markets weve seen, telling the truth should potentially maximize traders profit. But what if… -Agents can talk to/signal each other? -Agents manipulate the market? Wed like to refine our models to incorporate those real-life scenarios.

15. Incentive Compatibility – terms -BNE – Bayesian Nash Equilibrium. well say that a market is in BNE when all agents already maximized their profits, and any further action from any agent will damage his profit. Most importantly: In a BNE rational traders stop trading. -PBE – Perfect Bayesian Equilibrium well say that a market is in PBE if through every step, all agents acted to maximize their (expected) utility, and eventually reached an equilibrium. -Dominant strategy – A strategy is dominant if, regardless of what any other players do, the strategy earns a player a larger payoff than any other. Hence, a strategy is dominant if it is always better than any other strategy. -Equilibrium Strategy – a strategy that leads to an equilibrium.

16. Incentive Compatibility How do we encourage agents to say the truth now and nothing but the truth -Wed like agents to reveal their information truthfully and immediately. Push the market to a truthful equilibrium as fast as possible. -Rewarding truth-tellers is first step: agents dont waste time calculating strategies before placing their bids. -Picking the right type of market is another step. -Problems: -No-trade theorem(82): Rational traders wont trade in an all- rational Continuous-Double-Auction (CDA) market. -Gradual information leakage may be more beneficial when traders can participate more than once (Chakraborty and Yilmaz 04) -Agents may benefit from manipulations/interactions in the market.

17. Incentive Compatibility – agents interaction -Signaling through trades may lead agents to lie (bluffing) to profit by correcting their bluffs later. -In reality, its hard to avoid agents interactions… Limiting agents to participate only once may partially help but keep in mind the problems in the sequential model (MSR). -In markets that allow any interaction between agents, truth telling is not an equilibrium strategy (Chen 09) -Today, researches focus on extracting predictions from a BNEs, even if they are not the truth telling BNE.

18. Incentive Compatibility example model (Chen 2009) -Market: LMSR (Logarithmic MSR) -Event w with 2 outcomes -n players, each gets si correlated to the event w -Distribution of si and w is common knowledge -Players play sequentially (1) or when they decide (2). -si|ws are independent (3) or sis are independent unconditionally (4). Analysis shows: -Information is better aggregated when players play sequentially. -If si|ws are independent, truth telling is the only PBE, Agents tell the truth as soon as possible. -If sis are independent unconditionally, the BNE is unknown. Truth-telling is not even a good strategy, and a BNE might not exist.

19. Manipulation -An agent can manipulate the market in several ways: -Take action to change events outcome. -Send misleading signals inside the market. -Send signals from outside the market.

20. Manipulation - Changing events outcome -Consider a company with n employees that uses a PM to predict its product delivery date. -An employee can affect the outcome by acting from within the company. -Note that the company has a desired outcome. -Shi, Conitzer and Guo (09) showed the following: -Allowing one time participation in an MSR market will encourages the agents to play truthfully, and prevent sending misleading signals between agents. -The MM can incentivize the agents to not manipulate the outcome by paying times more than in a normal MSR.

21. Manipulation – correlated markets Consider 2 correlated markets: -Alice trades in Market A -Bob makes his trading decision in Market B -Alice can now trade in market B and potentially benefit from her decision in market A, even if the latter was not truthful. -Lets see an example…

22. Manipulation – correlated markets - example Market A: LMSR, b = 0.1Market B: LMSR b=1 MM seeds both markets with initial prob. 0.5 0.5 Alice changes prob A to 0.4 -Alice believes event w happens with probability 0.9 -Bob is not sure… hes looking for easy profit (like most of us). 0.4 Bob follows her and changes prob B to 0.4 Alice changes prob B to 0.9 0.9

23. Expressiveness How do we encourage them to say the truth (now), the whole truth and nothing but the truth … Motivation: Wed like agents to put as much data as possible in the market. But How? Combinatorial bids – bids on more than one outcome. Improves expressiveness! -Example – horse race: -Horse A will finish before horse B. -Horse A wont win and horse B wont win. -The entire permutation of horses.

24. Expressiveness (Cont.) Well examine the markets 2 computational challenges: -Pricing: setting the price of a share such that its coherent with events probabilities. -The Auctioneer Problem: Given a set of bids in a combinatorial auction, allocate items to biddersincluding the possibility that the auctioneer retains some itemssuch that the auctioneers revenue is maximized.

25. Expressiveness – known results -Permutation betting: horse racing both auctioneer problem and pricing are hard. auctioneer problem under specific settings can be possible. -Boolean betting: vector of {0,1}s both auctioneer problem and pricing are hard. -Tournament betting: sport teams in a playoff tree, leaves are teams Pricing team A advances to round k is possible. the auctioneer problem is still hard -Taxonomy betting: summing tree, leaves are base elements LMSR pricing is possible auctioneer problem and general pricing are hard.

26. Expressiveness (cont.) -Problems: -Events are obviously correlated, but its hard to price them as such. -Even if we could price events properly, analyzing the results is hard -Recall that polynomial in the number of outcomes is actually exponential number of base events. -In real life: -Under some settings and when number of all possible outcomes is bounded and low, it is feasible to allow combinatorial bids. -In practice, its not commonly used.

27. But what is truth?? Problem: -Truth may be subjective or non-verifiable: -Rating the quality of a movie -Determine extinction year of the human race. Solution: -Peer prediction: determine a relative truth. -Idea (Miller, Resnick, Zeckhauser 05) : evaluate Agents reports against the reports of its peers.

28. Peer prediction - (Miller, Resnick, Zeckhauser 05) Consider the following setting: -Each agent gets a signal si on event w. distributions of w and si|w are common knowledge, but w is not verifiable. -Agent i reports si. -MM randomly picks a reference agent j and calculates -Agent i will be rewarded according to. -At the case mentioned, truth telling will lead to a BNE. -Unfortunately, its not the only BNE… -Requires a mass of truth-tellers -Further research shows that there are ways to make truth telling a unique equilibrium under this setting (Jurca and Faltings 07).

29. BTS: Bayesian Truth Serum (Prelec 04) Consider the following setting: -A simple poll – each agent states her opinion -In addition – each agent is asked to estimate the final distribution over possible answers denoted by S. -Agents score: -Opinion score: the more common it is the higher the score is. -Poll estimation score: the denominator is the statistical distance between S and P. -Truthful reporting is a BNE with these settings! -When allowing to reveal partial poll results, this is not the only BNE…. -But even then, the gap between the updated poll (affected by ) and the Agents true belief regarding the polls outcome (S) is reduced, allowing to extract true prediction from the polls outcome.

30. Summary -We saw prediction markets of different kinds -We understood some of the setbacks when those markets are used in reality, including some interesting ideas on how to overcome those -You might have noticed most of the quotation brought here are from last decade, many new results, fast development. -In reality some those markets can outperform regular polls and surveys.

31. Questions