The Tsunami

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An Infinite +$EV Opportunity?

Following a family trip to Brunei and Malaysia, I returned to my mother-in-law’s place in Taiwan and found an email from one of my Caltech friends entitled “mildly interesting thought experiment” in my inbox:

How much would you pay to play this game:


Flip a fair coin until you get “tails”.  Let N be the number of flips it
took to get that first “tails”.  I will pay you (2^N)/N dollars.


Of course, if someone wants to make this a real experiment, I’m ready to
accept bids… :)

On the most fundamental level, your goal when making any wager should be to make a profitable one. To deduce the threshold for when this bet is profitable, we need to find the expected value with respect to money ($EV). Finding $EV entails taking an average of payouts across all possible events (weighted by the probability of each event happening). For this particular wager, $EV is the limit as N -> infinity of:

(1/2^1)(2^1/1) + (1/2^2)(2^2/2) + (1/2^3)(2^3/3) +…+(1/2^N)(2^N/N)

This is a divergent series, which means that its value is infinite. I’m atheist, but I’ll admit that I feel blessed to have friends who fill my inbox with infinite +$EV opportunities…or at least theoretically infinite +$EV opportunities. Since my friend doesn’t have an infinite amount of money, there exists some amount of money, beyond which my friend would be forced to default on his bet. As a result, the infinite sum we first considered isn’t really applicable. Instead, we need to cap the payouts at some point to reflect the reality of the situation: we can’t win money that doesn’t exist.

When I talk about poker hands with people, I sometimes refuse to comment on situations that I’d never be in. For example, in no-limit hold’em, I find it awkward to talk about postflop decisions involving hole cards that I think should be folded preflop.

Some people live as though Earth is a limitless bounty of resources, but it’s far from being so. And even if we consider the possibility that our species may one day begin colonizing extraterrestrially, it’s unknown how many resources the universe contains. As a result, considering an infinite +$EV opportunity is also awkward. However, for the sake of a possibly interesting thought experiment, given that you have $D, what percentage would you be willing wager for infinite +$EV?

The percentage of D that you’re willing to wager probably will be a function of D itself. It probably will be a function of your ability to reclaim the amount of your wager via other means if you lose. It probably will be a function of the exact probabilities of various payouts. This is because life isn’t just a series of independent opportunities. Life is a big tree of possible outcomes, and decisions you make place you in completely different parts of the tree. For example, if you won $1,000,000 USD playing poker pre-UIGEA and blew most of it on bling, cocaine, hookers, and degenerate -$EV gambling, the choices currently available to you are probably much different from those available to someone else who won $1,000,000 USD playing poker pre-UIGEA, took good care of himself, and who only spent about $50,000 USD per year.

When considering life as one big tree of possible opportunities and outcomes, the term bankroll management takes on a meaning entirely different from that considered by most poker players. Instead of having a poker bankroll consisting of money only to be used from poker (from which money is occasionally withdrawn for personal use), and instead of compartmentalizing your money in general, think of your money, your assets, and your time, and think about how all of these can be optimally allocated. For example, if you’re a winning poker player who keeps a separate bankroll and withdraws a fixed percentage of net winnings at various times, there’s probably a point at which you’re probably better off taking some of the money from your bankroll and allocating it towards other money-making opportunities (such as starting a business that produces a tangible resource or service).

After all, regardless of what stakes you play, playing poker is always a direct exchange of your time for money. And since poker (as well as investing in things like stocks and currencies) is nothing more than a negative sum game in which money is simply shifted around, it’s always going to be the case that the owners of big businesses that produce tangible goods/services will earn more money per hour worked than successful high stakes poker players.

May Your EV Always Be Positive!

The Tsunami


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2 Responses to “An Infinite +$EV Opportunity?”

  1. Matt says:

    I think I recall reading about a problem similar to the one your friend suggested in a book called “Statistics Hacks,” by Bruce Frey. If it wasn’t that book, it was “The Drunkard’s Walk,” by Leonard Mlodinow. IIRC, you would wager up to an infinite amount of money in that type of situation.

    I think it’s interesting to note that in your last paragraph that while it’s only a money shift, as long as there are individuals willing to make a contribution to that shift that’s not in their favor (i.e., lose money), there will always be money to be earned as long as: A) You know where to look for it, and B) You have the skill to obtain said money. That being said, I agree that businesses that deal in tangible goods and services will always make more money per hour, but there’s something to be said for the amount of work they put in to obtain that higher salary.

  2. This probably only applies to the most profitable businesses, but I feel that the hourly from owning a highly successful business is higher even after accounting for the hard work needed to get such a business off the ground. Of course, this shouldn’t be taken to mean that playing poker is a waste of time…playing poker is fun and not everyone in the world can own a highly profitable business.

    I think the point I really wanted to make in the second half of this post is that there’s life beyond poker. At least some of the really young, successful players probably make the mistake of not respecting their money under the optimistic assumption that easy money will always be available. Additionally, someone with the intelligence required to be a highly successful player is going to be at least somewhat susceptible to boredom from doing the same thing for 30-40 years.

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