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

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

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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??

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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…

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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!!

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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

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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:,

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

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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.

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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

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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)

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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

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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

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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…

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Manipulation – correlated markets - example Market A: LMSR, b = 0.1Market B: LMSR b=1 MM seeds both markets with initial prob 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

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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.

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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.

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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.

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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.

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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.

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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).

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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.

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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.

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Questions

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