Prediction Markets

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

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

Motivation • We’d like to predict an event of interest • Ideally, we’d like tomake agents say the truth, the whole truth and nothing but the truth – and do it NOW • We’re willing to pay for it… • Market Requirements: • Liquidity • Bounded loss • Discourage manipulation • Extract predictions easily • How can we create such a market??

Liquidity: • Liquidity isthe 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. • We’ve already seen last time that by subsidizing the market we increase liquidity. • we’d like to bound that subsidy. we’ll talk about it later…

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 : That’s a scoring rule for p!!

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 market’s 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

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: ,

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

Cost function Market from MSR (Chen, Vaughan’10) • There’s 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, there’s a mapping from CF to SR, not presented here.

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 avoid0 division in price function. • is the market’s prediction

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: • MM’s expected utility: remains constant (Chen, Pennock ‘07)

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 market’s prediction

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

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

Incentive Compatibility – terms • BNE – Bayesian Nash Equilibrium.we’ll 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 we’ll 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 – Astrategy is dominantif, 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.

Incentive Compatibility • How do we encourage agents to say the truth now and nothing but the truth • We’d like agents to reveal their information truthfullyand immediately. Push the market to a truthful equilibrium as fast as possible. • Rewarding truth-tellers is first step: agents don’t waste time calculating strategies before placing their bids. • Picking the right type of market is another step. • Problems: • No-trade theorem(‘82): “Rational traders won’t 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.

Incentive Compatibility – agents interaction • Signaling through trades may lead agents to lie (“bluffing”) to profit by correcting their bluffs later. • In reality, it’s hard to avoid agents interactions… Limiting agents to participate only once may partially helpbut 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.

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|w’s are independent (3) or si’s are independent unconditionally (4). • Analysis shows: • Information is better aggregated when players play sequentially. • If si|w’s are independent, truth telling is the onlyPBE, Agents tell the truth as soon as possible. • If si’s are independent unconditionally, the BNE is unknown. Truth-telling is not even a good strategy, and a BNE might not exist.

Manipulation • An agent can manipulate the market in several ways: • Take action to change event’s outcome. • Send misleading signals inside the market. • Send signals from outside the market.

Manipulation - Changing event’s 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.

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. • Let’s see an example…

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

Expressiveness • How do we encourage them to say the truth (now), the whole truth and nothing but the truth … • Motivation: We’d 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 won’t win and horse B won’t win. • The entire permutation of horses.

Expressiveness (Cont.) • We’ll examine the market’s 2 computational challenges: • Pricing: setting the price of a share such that it’s coherent with events probabilities. • The Auctioneer Problem: Given a set of bids in a combinatorial auction, allocate items to bidders—including the possibility that the auctioneer retains some items—such that the auctioneer’s revenue is maximized.

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.

Expressiveness (cont.) • Problems: • Events are obviously correlated, but it’s 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, it’s not commonly used.

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 Agent’s reports against the reports of its peers.

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, it’s 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).

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 Agent’s true belief regarding the poll’s outcome (S) is reduced, allowing to extract true prediction from the polls outcome.

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.

Questions

Prediction Markets