If you don't set the upper threshold high enough, sometimes it won't find the root even for positive EV when the EV is barely positive. For your original example, it actually changes from positive to negative when the first prize goes from 60 to 59, not 90 to 89 as I said before. But to not get an error at 60, you need an upper threshold of 0.999999999. That's 9 nines. With only 8 nines, you get an error.

One way to flag the error condition is to set the output of uniroot to a variable, and before calling it, set that variable to some error value like 999. When you get the error, it doesn't change this value, so it will still be there. But when you don't get an error, it returns several values in a structure, so checking it like it is an ordinary variable will generate a warning. It will work, but it's ugly. You could find a more elegant solution, such as checking if the type of the variable changed without generating a warning.

It would be nice if you could just use an upper threshold of 1, and have it give you the lowest root, but it insists on giving you the highest root when the root is 1. In Excel, you give it a starting value lower than the actual root, like 0.5, and then it doesn't have this problem.

Actually you can handle both the case of negative ev and the case of the threshold not being high enough just by making sure that the value of the function at the upper limit is negative, so you wouldn't need the ev.

Hmmm...has got me thinking about rounding issues. I'm not sure if it is my computer or R, but I can only type 7 digits of .9999999. At the 8th digit it treats it as 1.0. Obviously that is somewhat of a problem here.

.999999 # Returns .999999
.9999999 # Returns .9999999
.99999999 # Returns 1
.999999999 # Returns 1

I see that too. I think it has to do with the way the number is displayed by default. You should be able to change that. Internally the number should be more precise than that. There is also double precision. I think the documentation for uniroot shows displaying a certain number of digits.

bump for general utility

This is a little off topic, but the ROR would be less if the payouts were flatter. I took the time make payouts more consistent using a log function to determine 1st place, then another for consisent jumps between payouts. Also, I expanded payouts to 11.1% of field from 10.2%

In Alt B, only 12 spots would pay less and 801 pay more. A much better result for the vast majority of players. I will post some more revealing info regarding ROR more directly when I find my notes.


Place Actual Alt A Alt B
1 $8,944,138 $7,438,180 $6,225,549
2 $5,545,855 $5,490,027 $4,723,021
3 $4,129,979 $4,079,975 $3,603,032
4 $3,092,497 $3,053,067 $2,763,987
5 $2,332,960 $2,300,550 $2,132,243
6 $1,772,939 $1,745,680 $1,654,186
7 $1,356,708 $1,334,003 $1,290,603
8 $1,045,738 $1,026,667 $1,012,687
9 $811,823 $795,802 $799,185
10-12 $635,011 $621,307 $634,340
13-15 $500,165 $488,601 $506,424
16-18 $396,967 $387,056 $406,667
19-27 $317,161 $308,878 $328,482
28-36 $255,242 $248,323 $266,898
37-45 $206,395 $201,136 $218,151
46-54 $168,556 $164,146 $179,375
55-63 $138,285 $134,977 $148,380
64-72 $114,205 $111,842 $123,484
73-81 $94,942 $93,388 $103,393
82-90 $79,806 $78,586 $87,101
91-99 $67,422 $66,649 $73,829
100-171 $57,102 $56,972 $62,969
172-243 $48,847 $49,088 $54,041
244-315 $41,967 $42,635 $46,671
316-387 $36,463 $37,331 $40,561
388-459 $31,647 $32,953 $35,476
460-531 $27,519 $29,328 $31,227
532-603 $24,079 $26,318 $27,664
604-675 $21,327 $23,815 $24,667
676-747 $19,263 $21,732 $22,139
748-813 $0 $20,000 $20,000
Entries 7319
% Paid 10.21% 11.11% 11.11%
Winner
6% Fee Paid More: 642 801
Paid Fewer: 171 12

I found my data for a reality check for ROR in playing the WSOP main event.
What I did was look at the situation where a player would shove allin preflop every hand. An average player would have a 50% chance to win a coin flop. If however, you got your money in on average as a 52.77% favorite, then your chance of winning the tournament would be twice that of an average player and your ROI would go from -6% for an average player to +41%. To have a 4 times better than average chance to win, you would have to shove with a 55.7% chance to win each time resulting in a 117% ROI.

I then took a look at how each type of skilled player's ROI would look if he failed to ever finish at selected high levels. For example, if your skill level was such that you would win the event 3 times more than an average player, your overall ROI would be 80%, but if you never finished in the top 18 spots, your ROI would fall to -15%!

Even a 4 times skilled player with an ROI of 117% would only have an ROI of 12% if he never made the final table! And he would only expect to make the final table once every 286 years!

The final table shows how often in years one can expect to make selected finishing spots based on skill level.

For example, an average player can expect to make top 18 once every 407 years (based on 7319 entries), but a player with 4 times the average skill level would make it once every 159 years.

Even with a skill level 4 times that of an average player (most likely a level only achieved by world class pros), such a player could only expect to make the top 36 once every 89 years on average! In otherwords, even a world class pro is a lifetime underdog to ever make the final 4 tables of the WSOP main event with 7319 players. And thus his ROI would be -16%. ( or -$160 per entry)

I suspect these numbers will be a surprise to many, but tournament poker has extremely high variance resulting in most players going bust, even very good players. An improvement would be to lower the variance by making payouts flatter and paying more places.

Many say payouts have already been flattened. While generally true, in the old days they started at winner-take-all and the flattening process still needs a long way to go. For example, the main event in 2009 paid 8.9 million to the winner, getting that to 6 million or even 5 million would be a much better situation, and expanding payouts from 10.2% of field to as high as 12.5% would help as well.




2010 Main Event $10,000 NL Holdem
Return on Investment (ROI)
All in % WSOP
50.00% Average Player -6%
52.77% 2x Average Player 41%
54.47% 3x Average Player 80%
55.70% 4x Average Player 117%

Average Player ROI If :
WSOP
No Top 3 Finishes -19%
No Top 9 Finishes -38%
No Top 18 Finishes -44%
No Top 27 Finishes -46%
No Top 36 Finishes -49%

2x Average Player ROI If:
WSOP
No Top 3 Finishes 16%
No Top 9 Finishes -17%
No Top 18 Finishes -27%
No Top 27 Finishes -31%
No Top 36 Finishes -34%

3x Average Player ROI If:
WSOP
No Top 3 Finishes 45%
No Top 9 Finishes -1%
No Top 18 Finishes -15%
No Top 27 Finishes -19%
No Top 36 Finishes -24%

4x Average Player ROI If:
WSOP
No Top 3 Finishes 70%
No Top 9 Finishes 12%
No Top 18 Finishes -5%
No Top 27 Finishes -10%
No Top 36 Finishes -16%

How often in years of expected finishing positions:
read as: 7319 Players Top 3 Top 9 Top 18 Top 27 Top 36
once Average Player 2440 813 407 271 203
every 2x Average Player 1329 482 255 175 134
x years 3x Average Player 931 356 194 136 105
4x Average Player 723 286 159 113 89

Nice work, redoak. Wannabe tourney pros really should take a good look at this.

Yes RedOak, this is a very cool and interesting addition.

I didn't even mention the effect of taxes as well. In the 2010 main event, of the 68 million prize pool, (7319 entries), 19 million would go to pay taxes reducing payouts even further to only 49 million. This is an average tax rate of 28%. So you need to subtract this from the ROI as well. Thus we saw that for a 4 times better than average player with a 117% ROI, it now becomes a 89% ROI, and if they do not ever finish in the top 3 their ROI will be negative!

4x Average Player ROI If:
WSOP (assume 0% tax rate)
No Top 3 Finishes 70%
No Top 9 Finishes 12%
No Top 18 Finishes -5%
No Top 27 Finishes -10%
No Top 36 Finishes -16%

4x Average player:
After tax rate of 28% ROI becomes:
No Top 3 Finishes 42%
No Top 9 Finishes -16%
No Top 18 Finishes -33%
No Top 27 Finishes -38%
No Top 36 Finishes -44%

Great thread and good stuff, RedOak, I was just about to mention taxes, too. You might want to consider the case where a WSOP ME entrant has over $10,000 in gambling winnings for the year, and hence that they will get a $10,000 tax deduction when they lose, as is probably the case for most serious players in the event. However, the net effect of taxes will still be to hurt the EV; for any given player who isn't already in the top tax bracket, their final table finishes will be taxed at the maximum tax rate, whereas their deduction when they lose is at a lesser tax rate. I had made up some numbers and found that the average net amount of extra tax that a typical player pays when entering the WSOP ME is about $600, which *includes* all of the cases where the player finishes out of the money and ends up with a negative amount of extra tax (i.e. a tax deduction) from the event.

The vast majory of the 7319 players at the main event DO NOT have gambling winnings of $10,000 or more. The tax law only allows deductions of gambling losses if they offset gambling wins. So most likley those that lose 10k in the main event WILL NOT be able to reduce their taxable income by that amount due to not having 10k of wins to offset it.

The tax code is unfair in this regard and is designed to discourage gambling. Unless of course you gamble in the stock market. That is encouraged by giving you a $3,000 per year loss allowance against ordinary income and a 15% tax rate on winnings if held 1 year or more even if you make 300 million. What a country!

Yeah, it's pretty unfair that poker doesn't provide essential liquidity to the capital markets and provide a means for efficient allocation of capital which is vital to an economy. Stupid laws.

Yeah, I know what you mean. Just today I bet er I mean I invested that the sp500 would close over 700 come Dec 21, 2012 by selling 10 spy put options with a strike of 70. I will win $697 if my bet er I mean investement holds. spy is 135 right now. I am sure my bet er I mean investement is helping to create jobs and efficiently allocating capital to where it needs to go. what a country.

Of course I could have also invested that the price of oil would go up tomorrow, but if I shove all-in with my AA preflop that is gambling. See the difference?

You miss the entire point. An individual's bets, strategy, and motivation is irrelevant to the functioning of the market, it only matters that they participate in it. Comparing poker to financial markets is absurd. I won't distract this thread further with this point.