|
|
2006-12-18
Calculating Your ‘Risk of Ruin’ in Big-Money Tournaments
When I played the first open event at the World Series of Poker at the Rio in 2005, I was amazed. Maybe it was the three-hour wait to get on the list for the event or the jitters from deciding only the day before to fly to Vegas and play the entire WSOP.
Looking back, I think it was the huge room with thousands of players all playing no-limit. Of course, the Main Event this year had even more players, and this year I had the added bonus of being sponsored by PokerStars.com.
One thing I was thinking about was how big my bankroll had to be to really effectively play these big events. Of course, the Main Event of the WSOP is a unique event; I mean, it is hard to figure what the return on investment is in a Main Event with 9,000 entrants.
Let’s discuss how much money you need to play on the major tournament circuit. Let’s say you play a bunch of $10,000 buy-in events, like the main event of the Borgata Poker Open that I just played. There were 540 players. The prize structure was such that about 10 percent of the field was paid. Here are the payoffs:
1 $1,519,020
2 $ 802,985
3 $ 419,040
4 $ 366,660
5 $ 314,280
6 $ 261,901
7 $ 209,521
8 $ 157,141
9 $ 104,761
10 to 12 $ 68,094
13 to 15 $ 52,380
16 to 18 $ 33,000
19 to 27 $ 26,190
28 to 36 $ 15,714
37 to 45 $ 14,142
46 to 54 $ 13,095
The main way we look at risk in my upcoming book, The Mathematics of Poker with Jerrod Ankenman, is the “Risk of Ruin” function. This function takes a bankroll (B) and returns the probability of going broke if we keep playing the same event repeatedly. Let us call this function f(B). There are a couple of interesting properties of this function. The first property we see is that f(0)=1, that is, if your bankroll is down to zero. (The numeral “1” meaning that you have a 100 percent chance of going broke.) We also have a multiplicative rule, that is:
f(A+B) = f(A)f(b)
That is, if you have two bankroll amounts A and B, then the Risk of Ruin for the sum of the two bankrolls is the products of the Risks of Ruin. That makes intuitive sense since we must lose both bankrolls sequentially to lose the sum of the two bankrolls and the probabilities are multiplicative.
In fact, the two properties plus some assumptions about f being continuous are all that is needed to derive the fact that f follows an exponential rule, that is:
f(B)=exp(-aB)
The ‘Half-Bankroll’
Another related idea that we have is the half-bankroll. That is the value of B such that f(B)=1/2, or the point at which the probability of going broke is exactly a half (50 percent). This is an important reference point because if we double the bankroll, the probability of going broke is ¼, if we triple the bankroll, the probability goes to 1/8 and so forth, ten times the half-bankroll and the risk of ruin falls to 1/1024.
Using the prize structure above and the methods described in our book, I calculated the half-bankroll with the assumption that the player in question is a fairly good player, that he has double the chance of finishing in each of the cash places. That is, there is a 2/540 chance of finishing in each of places 1-54 and only an 80 percent chance of finishing out of the money. (Since the top 10 percent are paid, an average player would finish in the money just 10 percent of the time, and out of the money 90 percent.)
With these favorable assumptions the half-bankroll is around $185,000. The reader may be surprised that even for such a relatively strong player, he has a 50 percent chance of going broke. Now, most people cannot stomach a 50 percent rate of failure, so a more reasonable bankroll may be around half a million, where the risk of ruin is 15 percent.
However, if we reduce the hypothetical player’s win rate from twice the average to only 1.5 times the average, the half-bankroll becomes $451,000.
(Note: The probability of going broke refers to the probability with repeated play, indefinitely. Of course, in the long run we have a winning player in this
scenario, so going broke usually occurs relatively soon if it happens. For example, if you were to win one of the events and $1.5 million, you would add several half-bankrolls, so your chance of going broke would be small.)
One variable you really need to do these computations is the player’s win rate. In fact, you need the distribution of wins of the player. To collect data to any statistical significance, you need thousands of samples. So you may have to spend a lot of money playing $10,000 tournaments before you can gauge how much money you really need. Not to mention that it will take a lifetime of playing to collect the data.
Online Data
Fortunately, Jerrod, Matt Hawrilenko and I have about 5,000 combined tournaments in our database. Online poker presents an opportunity to collect a lot of data quickly, since many tournaments can be played at once, and
tournaments start all the time. We have shown about a 1-buy-in overlay, the same as our theoretical player. Unlike the distribution mentioned above, where the player has simply twice the chance of finishing in each of the paying spots, we seem to have around a 1-in-7 (14 percent) chance of cashing, and our finishes are top-heavy. That is why we have more chance of finishing first than second, and more chance of finishing second than third, etc.
Even so, our 95 percent confidence interval for our win rate is between 0.6 and 1.5, which as you can see above makes a big difference in our risk of ruin. Now, one problem with our data is that we include tournaments of
different sizes and different games. The problem is that if we restrict our attention to say no-limit tournaments of between 300 and 600 players, the sample size would be too small; that is, the confidence interval would be even looser.
We know that in theory, tournaments with more players increase not only the variance, but also the win rate of a good player. But another problem is how well we can extrapolate our win rate to live tournaments or, indeed, to $10,000 events. I do have records dating from eight years ago that show me winning at a slightly higher rate (overlay of 1.6 buy-ins) over 600 tournaments. How this translates to the top events in the circuit is questionable.
We do have records of around 150 WSOP events, but I suspect for me I may have experienced some luck that is a bit higher than the median. Also, WSOP events from even two years ago are different than events from today.
The problem of not enough data in the relevant situation is a general problem. We have spoken with some of the successful players on the circuit, but besides the problems of scant data or second-hand information, there are some measurement biases. First is the problem of self-selection. Not in talking to us, but in being successful. One problem is that success correlates to both skill and luck. In the latter case, this means that when we take the most
successful players, they probably over-performed a bit. Also, success correlates to skill. Do we have any right to expect to match the win rates of the top players? This seems a bit presumptuous. In any case, one buy-in is probably near the top return for these tournaments.
Satellite Implications
What if someone plays satellites or supers for these events—would
hundreds of thousands or millions of dollars be required for a winning player? Well, actually, it helps a little bit. The problem is that it then becomes a parlay bet. A very strong satellite player can win about one out of seven satellites played, but then the bet is parlayed into the bet in the main event.
One way of thinking about it is that instead of paying the $10,000 entry fee for the main event, the player puts up $7,000 on average, instead. This does one of two things: First the return on investment increases and the entry fee required decreases. For example, take our player at a 0.5 win rate, whose half bankroll was $451,000. Now his average entry fee is $7,000 instead of $10,000, producing a win rate of 1.1 instead of 0.5. This reduces the half-bankroll to $114,000. For a skilled satellite player who also has an overlay in the main event, this is a way to significantly reduce the bankroll required.
Another possibility or similar idea is to play a lot of the smaller tournaments in a series. A skilled player who has
a strong win-rate in the smaller events in a sense can supplement the risk of the big events by showing a strong record in the smaller events. The same goes for a strong cash-game player who also enters these tournaments.
That is how I view the WSOP Main Event. The $10,000 every year is simply an entry into a very high expectation and high variance event that will, on average, take centuries or millennia to have any statistical significance, but the smaller WSOP events with 400 players have a much shorter time horizon. I would simply look at the risk of ruin counting the Main Event as a $10,000 loss. If your bankroll can take this hit, then, in a sense, any success in the Main Event is a freeroll.
One final word of caution. If you are a losing player in the big tournaments, then your bankroll requirement is infinite; that is, you will eventually lose whatever you have. Most of the players in these events fall into this category.
Now, I may fancy myself better than the average pro in these events, but I am aware that that is an opinion that the vast majority of pros would have. It may be discomforting to not have a good idea of your win rate in these events, bearing in mind that this rate is also a dynamic variable (always subject to change), but that is part of the excitement of tournament poker.
- Bill Chen
William Chen cashed six times at this year’s WSOP, including three final tables and wins in limit hold’em and short-
handed no-limit hold’em. |
|
发表于 2007-4-8 12:16
发表于 2010-11-5 15:51
