A prediction marketA random variable, e.g.

Turned into a financial instrument payoff = realized value of variable$1 if$0 ifI am entitled to:Will US go into recession in 2012? (Y/N)Recession in 2012Recession in 2012

2012 September 30 8:02 p.m. ET

2012 September 30 8:02 p.m. ETBetween 6.0% and 8.0% chance

Kelly bettingYou have $1000Market price mp is 0.05You think probability p is 0.10Q: How much should you bet?

Kelly bettingYou have $1000Market price mp is 0.05You think probability p is 0.10Q: How much should you bet? Wrong A: $1000 P(bankruptcy)1

Kelly bettingYou have $1000Market price mp is 0.05You think probability p is 0.10Q: How much should you bet? Wrong A: fixed $1 No compounding magic

Kelly bettingYou have $1000Market price mp is 0.05You think probability p is 0.10Q: How much should you bet? Optimal A: $52.63 or f* fraction of your wealth where f* = (p-mp)/(1-mp) edge/odds

Why?Kelly betting maximizing log utilityMaximizes compounding growth rate Maximizes geometric mean of wealthMinimizes expected doubling time Minimizes exp time to reach, say, $1MDoes not maximize expected wealth (All in does, but ensures bankruptcy)

CompoundingYou put $10,000 into the bank at 5% rateCompounded annually: 10,000*1.05 = 10,500Monthly: 10,000*(1+.05/12)12 = $10,511.62Secondly: 10,000*(1+.05/31.5M)31.5M =$10,512.71Compounded continuously: 10,000*e.05 =$10,512.71Good way to compare rates:Wt = W0 ertr = 1/t*ln(Wt/W0)

Choosing a bankr =r =r =r =

Choosing an investmentS&P 500, 1988-2011 r=18.45%

Choosing an investmentS&P 500, 1988-2011

YearAnnualReturn198816.61%198931.69%19903.10%199130.47%19927.62%199310.08%19941.32%199537.58%199622.96%199733.36%199828.58%199921.04%20009.10%200111.89%200222.10%200328.69%200410.88%20054.91%200615.79%20075.49%200837.00%200926.46%201015.06%20112.05%Average18.45%

Kelly = maximize E[r]Initial wealth W0 Final wealth W1 W1 = W0 er

Kelly: Dont maximize E[W1] Instead, maximize E[r] = E[ln(W1/W0)] = E[ln WF] - ln W0

The Long RunIgnores variance of r!But in the long (enough) run, variance doesnt matterWT = W0er1er2er3... lnWT = lnW0+r1+r2+r3+... lnWT lnW0+t*E[r] (law of large #s) WT W0et*E[r]

YearAnnualReturn198816.61%198931.69%19903.10%199130.47%19927.62%199310.08%19941.32%199537.58%199622.96%199733.36%199828.58%199921.04%20009.10%200111.89%200222.10%200328.69%200410.88%20054.91%200615.79%20075.49%200837.00%200926.46%201015.06%20112.05%Average18.45%

The Long RunSuppose you could invest for 1000s or millions of years if need beOr imagine blackjack new indep bet every few secondsStrategy with higher E[r] will win out with high probability

YearAnnualReturn198816.61%198931.69%19903.10%199130.47%19927.62%199310.08%19941.32%199537.58%199622.96%199733.36%199828.58%199921.04%20009.10%200111.89%200222.10%200328.69%200410.88%20054.91%200615.79%20075.49%200837.00%200926.46%201015.06%20112.05%Average18.45%

Decision making under uncertainty, an exampleABC TVs Who Wants to be a Millionaire?

Decision making under uncertainty, an example

v15 = $1,000,000if correct $32,000 if incorrect $500,000 if walk away

Decision making under uncertainty, an example

if you answer:E[v15] = $1,000,000 *Pr(correct) +$32,000 *Pr(incorrect)

if you walk away:$500,000

Decision making under uncertainty, an exampleif you answer:E[v15] = $1,000,000 *0.5 $32,000 *0.5= $516,000

if you walk away:$500,000

you should answer, right?

Decision making under uncertainty, an exampleMost people wont answer: risk averseU($x) = log($x)if you answer:E[u15] = log($1,000,000) *0.5 +log($32,000) *0.5= 6/2+4.5/2 = 5.25if you walk away:log($500,000) = 5.7

Decision making under uncertainty, an exampleMaximizing E[ui] for i

Decision making under uncertainty, in general =set of all possible future states of the world

Decision making under uncertainty, in general are disjoint exhaustive states of the world

i: rain tomorrow & Bush elected & Y! stock up & car not stolen & ...

j: rain tomorrow & Bush elected & Y! stock up & car stolen & ...

123i||

Decision making under uncertainty, in generalEquivalent, more natural:

Ei: rain tomorrow Ej: Bush electedEk: Y! stock upEl: car stolen

||=2nE1E2EiEjEn

Decision making under uncertainty, in generalPreferences, utilityi>j u(i) > u(j)Expected utilityE[u] = Pr()u()Decisions (actions) can affect Pr()What you should do: choose actions to maximize expected utilityWhy?: To avoid being a money pump [de Finetti74], among other reasons...

Preference under uncertaintyDefine a prospect, = [p, 1; 2]Given the following axioms of :orderability:(1 2) (1 2) (1 ~ 2)transitivity:(1 2) (2 3) (1 3)continuity:1 2 3 p. 2 ~ [p, 1; 3] substitution:1 ~ 2 [p, 1; 3] ~ [p, 2; 3]monotonicity:1 2 p>q [p, 1; 2] [q, ; 2]decomposability:[p, 1; [q, 2; 3]] ~ [q, [p, 1; 2]; [p, 1; 3]]Preference can be represented by a real-valued expected utility function such that:u([p, 1; 2]) = p u(1) + (1p)u(2)

Utility functions( a probability distribution over )E[u]: represents preferences,E[u]() E[u]() iff Let () = au() + b, a>0.Then E[]() = E[au+b]() = a E[u]() + b.Since they represent the same preferences, and u are strategically equivalent ( ~ u).

Utility of moneyOutcomes are dollarsRisk attitude:risk neutral: u(x) ~ xrisk averse (typical): u concave (u(x) < 0 for all x)risk prone: u convexRisk aversion function:r(x) = u(x) / u(x)

Risk aversion & hedging

E[u]=.01 (4)+.99 (4.3) = 4.2980Action: buy $10,000 of insurance for $125

E[u]=4.2983Even better, buy $5974.68 of insurance for $74.68

E[u] = 4.2984 Optimal1: car stolen u(1) = log(10,000)2: car not stolen u(2) = log(20,000)u(1) = log(19,875)u(2) = log(19,875)u(1) = log(15,900)u(2) = log(19,925)

What utility function?Economist: Only you can chooseEngineer: Lets find out

Performance of Prediction Markets with Kelly BettorsDavid Pennock, Microsoft*John Langford, Microsoft*Alina Beygelzimer (IBM)

*work conducted at Yahoo!

Wisdom of crowdsMore: http://blog.oddhead.com/2007/01/04/the-wisdom-of-the-probabilitysports-crowd/ http://www.overcomingbias.com/2007/02/how_and_when_to.htmlWhen to guess: if youre in the 99.7th percentile

Can a prediction market learn?TypicalMkt >> avg expertStaticThis paperMkt >? best expertDynamic/Learning (experts algos)Assumption: Agents optimize growth rate (Kelly)

$1 if E1$1 if E2$1 if ESMarket ModelCompetitiveequilibrium pricesmp, mp ,qi(mp)=0

$1 if E1$1 if E2$1 if ESMarket ModelCompetitiveequilibrium pricesmp, mp ,qi(mp)=0qi(mp)=wi/mp*(pi-mp)/(1-mp)

Wealth-weighted averageAssume all agents optimize via Kelly and are price takersThen mp = wi pi

Fractional Kelly bettingBet f* fraction of your wealth, [0,1]Why? Ad hoc reasonsFull Kelly is too risky (finite horizon)Im not confident of p (2nd-order belief)Our (new) reason fraction Kelly behaviorally equiv to full Kelly with revised belief p+(1-)mpHas Bayesian justification

Wealth-weighted averageAssume all agents optimize via Kelly and are price takersThen mp = wi pi Fractional Kelly: mp = i wi pi

Wealth dynamicsAgents trade in a binary pred market Event outcome is revealed Wealth is redistrib. (losers pay winners) RepeatAfter each roundIf event happens: wi wi pi/pmIf event doesnt: wi wi(1-pi)/(1-pm)Wealth is redistributed like Bayes rule!

Wealth dynamicsSingle securityMultiperiod marketAgents with log utilFixed beliefsbelieftimesuccesses:0 trials:0successes:0 trials:1successes:1 trials:2successes:2 trials:3successes:2 trials:4successes:3 trials:5successes:5 trials:10successes:10 trials:20successes:17 trials:30successes:24 trials:40successes:30 trials:50wealthBeta(1,2)wealthBeta(2,2)wealthBeta(3,2)wealthBeta(3,3)wealthBeta(4,3)wealthBeta(6,6)wealthBeta(11,11)wealthBeta(18,14)wealthBeta(25,17)wealthBeta(31,21)

Wealth dynamics

Wealth dynamics: RegretTheorem:Mkt log loss < mini agent i log loss - ln wi Applies for all agents, all outcomes, even adversarialSame bound as experts algorithms No price of anarchy

Learning the Kelly fractionCompetitive equilibrium: = 1Rational expectations equil: = 0Practical (wisdom of crowds): = A proposal: Fractional Kelly as an experts algorithm btw yourself and the marketStart with = 0.5If right, increase ; If wrong, decreaseWont do much worse than market (0) Wont do much worse than original prior p

Summary and QuestionsSelf-interested Kelly bettors implement Bayes rule, minimize markets log lossIf mkt & agents care about log loss: win-winQuestionsCan non-Kelly bettors be induced to minimize log loss?Can Kelly bettors be induced to minimize squared loss, 0/1 loss, etc.?Can/should we encourage Kelly betting?

Recommended ReadGreat science writing (Pulitzer prize winner)Fun readAccessible yet accurateWell researchedMob meets politics meets Time Warner meets Wall Street meets Vegas meets Bell Labs scientists

****Standard utility axioms. The key one is substitution.Justifies numeric representation of utility and taking expectations (necessary to avoid always looking at entire probability distributions). In principle, we could do everything in terms of preference order.*Strategic equivalence: represents same preference order, entails same choices.In fact, these are necessary as well as sufficient conditions for strategic equivalence.Certain outcomes:u: represents preferences,u(w) u(w) iff w wFor increasing, (u) is strategically equivalent (u~(u)).

**The basic setup for my market model is straightforward. It consists of individual agents, each with with subjective belief and risk-averse utility for money. S securities are available, each paying off contingent on some event. Each agent chooses how much of each security to buy or sell by maximizing its own expected utility. Equilibrium is reached when all agents are optimizing, and supply and demand are equalized at current prices. These equilibrium prices are interpreted as the agents consensus probabilities.*The basic setup for my market model is straightforward. It consists of individual agents, each with with subjective belief and risk-averse utility for money. S securities are available, each paying off contingent on some event. Each agent chooses how much of each security to buy or sell by maximizing its own expected utility. Equilibrium is reached when all agents are optimizing, and supply and demand are equalized at current prices. These equilibrium prices are interpreted as the agents consensus probabilities.*I have also investigated a multiperiod market through both simulation and analysis. In this example, a single security is traded repeatedly over several time periods. All agents have GLU and begin with one dollar. Their beliefs are uniformly distributed between 0 and 1 and remain fixed throughout. This is a graph of wealth versus belief in the initial period. In this period, agents trade in the security until equilibrium; then outcome of the event is revealed, and payoffs are distributed accordingly. In this example, the event does not occur and agents that sold the security make money while agents that bought lose money. Then the agents trade until equilibrium again, the event this time does occur, and payoffs are collected. Agents trade again, the event occurs again, and wealth is redistributed accordingly. This solid line is a Beta distribution, scaled only by a constant, that reflects the number of successes and trials observed so far. This Beta distribution is the correct posterior probability of success when the prior is uniform between 0 and 1. The curve does not depend on the temporal order of successes or failures. I have proven that the fit is exact as the number of agents goes to infinity. After 50 periods, the event has occurred 60% of the time and the agents with belief around 0.6 have fared best. The market reward structure in this case naturally supports adaptation; accurate agents gain relative wealth and influence on prices over time, at the expense of the inaccurate agents. The bottom graph shows the price at each period along with the observed frequency, or the number of successes divided by the number of periods. Since price is a wealth-weighted average, the price equals the expectation of the Beta distribution, which is approximately equal to the observed frequency. Even though each individual agents belief remains fixed, the market as a whole appears to calculate rational Bayesian updates over time.