Presented By: Ofir Chen

Prediction Markets

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 to make 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 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.

- 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 :

( , ) ( ) ( ) ( )( )fD p q f p f q f q p q

~

~

[ ( , )] ( , ) ( )

argmax ( [ ( , )])

argmax ( ( , ) ( ))

argmax ( ( , ))

i q f i f

p i q f i

p f

p f

E D e p D q p g q

E D e p

D q p g q

D q p q

That’s a scoring rule for p!!

ie p ~i q

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

( ) ( , )i f i iS p D e p h

( )iS r( ')iS r

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: ,

( ) ( )new oldC q C q

q

q

iq

( )( )i

i

C qp q

dq

1

( ) 1m

ii

p q

Cost function (cont.)

Desired properties of a CF:

• Differentiability (to calculate prices)

• Monotonically increasing in

• Positive translation invariant

q

( 1) ( )C q k C q k

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.

, ', , 'p p q q ( )C q p ( ') 'C q p

1 1

( ) sup ( )m

m m

i i i ip i i

C q p q p s p

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 market’s prediction

( )( )

ff

i fi

C qo q

q

p

2( )

( )i i

i

ij

q qp q

C q q

fq

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)

1n

( )

( )i i

ij j

u mp

u m

m

( )j jj

u m

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

0( ) ( )i i iS e S r

sup (sup ( ( ) (0)))q i iq C q C

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 – 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.

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 truthfully and 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 only PBE, 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.( )n

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

Market A: LMSR, b = 0.1 Market B: LMSR b=1

0.1 log0.5 0.1 log0.4 0.4 log0.9

0.1 log0.5 0.1 log0.9 0.9 log0.9

i i i i

pays A gets from A pays Bob gets fromB

i i i i

a a a log a

a a a log a

MM seeds both markets with initial prob. 0.5

0.5 0.5

Alice changes prob A to 0.4

- Alice believes event w happens with probability 0.9 - Bob is not sure… he’s looking for easy profit (like most of us).

0.4

0.4• Bob follows her

and changes prob B to 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).

* ( ' | )i i js P s s*is

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.

1

S P

ip

ip

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