Let's Make a (Fair) Deal

The final table of the tournament is fourhanded. The top four places will pay $80,000, $40,000, $20,000, and $16,000. You're the short stack with about $100,000 in tournament chips. Your opponents have $400,000, $300,000, and $250,000, and the guy with $300,000 asks, "You guys wanna deal?" He then proposes a particular split, and you want to know if it's fair.



This question comes up all the time in tournaments. A discussion of how to determine a fair deal was a recent topic on the BARGE mailing list. That's the Big August Rec.Gambling Excursion, an annual convention of poker enthusiasts who communicate throughout the year by e-mail postings, discussing, among other things, strategy and how their members are doing in various tournaments. (This year, BARGE members did particularly well at the World Series of Poker; notable among them was math Ph.D. Bill Chen.)



Steve "Scoop" Evans posted about the seven-card stud eight-or-better tournament he was in at The Orleans, when it was fourhanded at the final table. The antes, bring-ins, and betting levels were getting high, at $400, $600, and $2,000-$4,000, respectively. Steve was down to about 10 big bets, and … "We went on break, and when I arrived back at the table, I was asked by the two chip leaders if I wanted to make a deal. Here is what happened, and I ask the group to scrutinize this and inform me if I did the right thing. I think I made a stellar deal, but we all know that math is hard! [That's a BARGE catchphrase.] The payouts were slated to be $12,500, $8,500, $5,500, and $3,500, making a prize pool of $30,000.



"The chip counts, with $234,000 in play, were approximately $90,000 for the leader, $80,000 for the next stack, $42,000 for me, and the short stack had $22,000. The chip leader wanted to avoid a crapshoot and was willing to give up a large chunk of first.



The second-place guy really wanted the trophy. Here is the deal I took.



"The chip leader got $9,000. Second got $8,500 and the trophy. I got $7,000, and we gave the short stack third-place money, $5,500.



"I was very proud to make the final table two years in a row, and while I did not win this year, I had a much tougher time with the competition, so I was happy with the outcome. But did I do the right thing?"



Stephen Landrum responded, "The Burns-Landrum estimate is that your fair share of the tournament prize pool is $7,029.91. This estimator is known to bias slightly against large stacks and slightly favor small stacks. I think you made an OK deal, but you had another stellar performance at the Orleans Open. Congratulations! A full rundown of the chip positions using a Burns-Landrum estimate is $8,397.44, $8,112.54, $7,029.91, and $6,460.11. The chip leaders got good deals, and the short stack got shafted." (Burns-Landrum is named for East Coaster Jazbo Burns and West Coaster Stephen Landrum. Both started posting erudite mathematical poker analyses on RGP starting in the mid-'90s.)



Another BARGE member suggested using a straight "percentage of the remainder" calculation to come up with $9,653, $8,970, $6,371, and $5,006.



Landrum responded, "Unless you are a tall stack, you should never let someone talk you into a 'percentage of the remainder' settlement. It's grossly unfair to short stacks, and can even end up in situations with some players getting more than first-place money. On the other hand, if you want to rip off your opponents, it sounds 'fair,' so you can offer it up if you do happen to be one of the tall stacks.



"The Burns-Landrum estimator that I referenced in my earlier reply, while slightly biased for the stack extremes, is far more accurate an estimator than the 'percentage of the remainder' method in all circumstances. … The formulation for the Burns-Landrum tournament equity estimator is as follows:



"Assume that your stack size relative to the total chips in play represents your chance of taking first. If you don't take first, assign equal probabilities to finishing in all other places. The second part is a simplification that's known to be inaccurate, but since most tournament payout structures are very top-heavy, it's not a horrible approximation."



Here's the math:



Let n be the number of players remaining (including you).



Let C be the total chips in play.



Let S be your stack size.



Let P1 be the payout for first place, P2 the payout for second, and so on.



Let Pr be the sum of P2, P3, … Pn (in other words, the remaining prize pool excluding first place).



Then, your estimated equity in the tournament is:



((S/C) x P1) + (((C-S)/C) x (Pr/(n-1)))



"This assumes equal skill among players. [This] assumption [is] obviously false in real tournament situations, but the estimation of your equity is useful to know for starting negotiation."



Other estimators exist, but this is a very good one. I'll look into others another time.



Michael Wiesenberg's The Ultimate Casino Guide, published by Sourcebooks, is available at fine bookstores and at Amazon.com and other online book purveyors. Send praise, protests, and problems to [email protected].