# Challenge: Derive the Kelly criteria for play money

The Kelly criteria is a money management strategy for gamblers and investors. The strategy says that, when faced with a positive-expectation bet, you should invest a fraction of your budget that is proportional to your expected profit. The more your expect to gain, the more you should risk, but you never risk your entire budget.

The Kelly strategy is optimal in several senses: (1) it minimizes your “doubling time”, or the time it takes to go from having X dollars to having 2X dollars; (2) it minimizes the time it takes to achieve any given level of wealth; (3) it maximizes your long-run wealth.

(It turns out that the Kelly strategy is equivalent to maximizing a logarithmic utility function.)

A key reason the Kelly strategy is optimal is that it is very careful to never take you completely bankrupt: you spend only a fraction of your money, always reserving a bit for tomorrow, however small. This is sound advice when dealing with real money. (Aside: this all assumes you have a strict budget cap, which is not entirely realistic: you can almost always borrow at least some amount, even in today’s economy.)

But what about maximizing your virtual “wealth” inside a play-money game like NewsFutures, InklingMarkets, HubDub, or MediaPredict? The problem is not quite the same, precisely because you cannot really go bankrupt. Almost every game offers an option to “recharge” your account if you go bust. Even if the option is not explicit, you can always just abandon your account and start a new one with a fresh initial bankroll they typically give to new players.

So what is the Kelly criteria for play money? What is the optimal strategy that minimizes your doubling time when you’re always allowed to recharge back to a fixed starting value any time you go bankrupt? The answer is not obvious to me, so I’m crowdsourcing the problem: can readers derive the right rule?

My only conjecture is that it might become optimal to go “all in” on every single bet. But I’m not sure. [Update: I’ve convinced myself this is not optimal. Imagine two sequential bets, the first with minuscule expected profit and the second with huge expected profit: surely you should not go “all in” on the first.]

Note that finding the optimal solution may not just help you win more bragging rights in online games. There is a fascinating sports betting site called CentSports that gives everyone ten real cents to start with. If you can turn that ten cents into twenty dollars, they’ll cut you a check. Moreover, if you ever go to zero, they’ll restore you right back to ten cents. In other words, the system works just like play-money games except the potential for profit is real. So another way to phrase the challenge question is: what strategy in CentSports minimizes the time it takes you to go from ten cents to twenty dollars?

# Yoopick: A sports prediction contest on Facebook with a research twist

I’m happy to announce the public beta launch of Yoopick, a sports prediction contest with a twist.

You pick any range you think the score difference or point spread of the game will fall into, for example you might pick Pittsburgh wins by between 2 and 11 points.

The more your prediction is viewed as unlikely by others, and the more you’re willing to stake on your prediction, the more you stand to gain. Of course it’s all for fun: you win and lose bragging rights only.

You can settle a pick even before the game is over, much like selling a stock in the stock market. Depending on what other players have done in the interim, you may be left with a gain or loss. You gain if you were one of the first to pick a popular outcome.

If you run out of credit, you can “work off your debt” by helping to digitize old books via the recaptcha project.

Those are the highlights if you want to go play the game. If you’re interested in more details, read on…

### Motivation, Design, and Research Goals

There are a great many sources of sports predictions, including expert communities, statistical number crunchers, bookmakers, and betting exchanges. Many of these sources are highly accurate, however they typically focus on predicting the outright or spread-adjusted winner of the game. Our goal is to obtain more information about the final score, including the relative likelihood of each point spread. For example, if our system is working, on average there should be more weight put on point spreads of 3 and 7 in NFL games than on 2,4,6, or 8.

We chose sports as a test domain to tap into the avid fan base and the armies of arm chair (and Aeron chair) prognosticators out there. However, the same approach should translate well to any situation where you’d like to predict a number, for example, the vote share of a politician or the volume of sales of your company’s widget. In addition to giving you the expected value of the number, our approach gives you the confidence or variance of the prediction — in fact, it gives you the entire probability distribution, or the likelihood of every possible value of the number.

Underneath the hood, Yoopick is a type of combinatorial prediction market where the possible outcomes are the values of the point spread, and each pick is a purchase of a bundle of outcomes in a given interval. We use Hanson’s logarithmic market scoring rules market maker to price the picks — that is, to set the risk/reward ratio. This pricing mechanism also determines the gain or loss when picks are settled early.

Wins and losses on Yoopick are measured in milliyootles, a social currency useful for expressing thanks.

Our market maker can — and we expect will — lose yootles on average. Stated another way, we expect players as a whole to gain on average. At the same time, we actively work to improve our market maker to limit its losses to control inflation in the game.

Because the outcomes of a game are tied together in a unified market, picks in one region automatically affect the price of picks in other regions in a logically consistent way. Players have considerable flexibility in how and what information they can inject into the market. In particular, players can replicate the standard picks like outright winner and spread-adjusted winner if they want, or they can go beyond to pick any interval of the point spread. No matter the form of the pick, all the information flows into a single market that aggregates everything in a unified prediction. In contrast, at venues from Wall Street to Churchill Downs to High Street to Las Vegas Boulevard, markets with many outcomes are usually split into independent one-dimensional markets.

Our goal is to test whether our market design is indeed able to elicit more information than traditional methods. We hope you have fun playing in our Petri dish.