## Wednesday, June 13, 2012

### Selling a Spot at the Big One Poker Tournament

Some of the spots at a \$1 million buy-in poker tournament called the Big One were won by players playing and winning satellite events. These players are then faced with a difficult choice: play the Big One for a shot at many millions, or sell the spot for as much as they can get. The dollar amounts are definitely life-changing, and it can be difficult to make a rational choice. How much is a spot in this tournament worth?

If there are 45 players, the payouts for this tournament go to the top 9 players:

\$17,200,002.15
\$9,480,001.19
\$4,080,000.51
\$2,480,000.31
\$1,720,000.22
\$1,520,000.19
\$1,320,000.17
\$1,160,000.15
\$1,040,000.13

The bottom 36 players get nothing which makes the average payout \$888,889 (the rest of the buy-in, \$111,111, goes to the charity One Drop). So, one answer to the question is that a spot in the tournament is worth \$888,889. But this ignores two important factors:

1. Not all players have equal skill.
2. Looking at the arithmetic average payout ignores risk.

Let’s begin with the simplifying assumption that the player considering selling his or her spot is equally likely to end up in any position from first to 45th. This takes skill out of the equation.

To account for risk, let’s use a method that is generally considered to be the least risk-averse that anyone should be: maximize the compound return. (For the mathematically-inclined, this amounts to maximizing the expected value of the logarithm of your net worth.)

It turns out that a person’s net worth is an important factor in valuing risk. It makes sense that a billionaire would be less concerned about losing a million dollars than an average person would.

Using these assumptions, a person with a net worth of \$100,000 would value a spot in this poker tournament at only \$94,799. This is a substantial discount to the average payout, but it reflects the fact that taking this amount almost doubles net worth.

If we add human capital (future earnings) to net worth, then most of us would assume a net worth of at least \$1 million. But at this net worth, the spot in the tournament is still only worth \$316,465. At a net worth of \$10 million, the spot’s value goes up to \$657,143, and at \$100 million, the value goes up to \$850,044.

All this shows that poker tournaments have high risk because so much of the prize money is loaded into the top few spots. If they are being rational, most people should be willing to part with their spot in this tournament for less than \$400,000.