Bill Chen on Tournament Structure

Alex Weldon : July 23rd, 2015

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Bill Chen was playing poker with Matt Glantz yesterday and discussing his ideas about tournament payout structure. Glantz took out his smartphone and recorded the conversation to stream it on Periscope for the benefit of the community.

CLICK HERE TO VIEW MATT & CHEN’S VIDEO

Payout structures are a hot topic these days, in the aftermath of the World Series of Poker. A few controversies arose out of structures at the WSOP; early on, people were upset about the relatively low final table payouts for the Colossus, with first place only getting 6% of the prize pool. The $1 million guaranteed final table payouts for the Main Event also caused a fuss, because the inconsistent pay jumps created as a result led to stalling and a disadvantage for the players at the smaller table when only 11 players remained.

Bill Chen is a quantitative analyst with a PhD in mathematics, so he seems like the sort of guy who could fix these sort of problems once and for all. What he proposed has some problems of its own, however.

The Chen Structure

Despite his background in advanced mathematics, Chen’s proposal is extremely simple, and for him, that’s the main selling point. “Everybody knows what they’re getting paid, there are no weird jumps, it’s all very gradual,” he says.

The principle is this: you decide what percentage of the pool is going to the winner. Second place then gets half of that, third place gets a third, fourth place gets a quarter, and so on. For instance, if you were giving 16% to the winner, that would give you 8% to second, 5.33% to third, 4% to fourth, etc.

It’s easy to see the appeal, in that the reward for advancing through the field is consistent and easy to calculate, once you’re in the money: Outlast half of the players remaining, and you’ll be making twice as much. Dead simple, as Chen says.

Scalability issues with min-cashes

Chen goes on to say that if you have a tournament with about 1000 players and want to pay 100 seats, the min-cash is going to be about 1.8x the buy-in (1.93x by my calculation), which sounds about right. First place, meanwhile, is naturally getting 100x that – 19.3% of the prize pool, or 193x the buy-in. That may sound a bit high to people like me who are used to playing online tournaments, but it’s in line with what the WSOP pays in 1000-player fields.

The trouble is that the Chen Structure is not that easily scalable; the number of seats paid depends only on the percentage of the pool which is going to first, and not on the field size at all. If you’re giving 19.3% to 1st, you will always be paying 100 seats, regardless of how many people are playing. With fewer than 500 players, this leads to the min-cashes being less than a buy-in. Conversely, if you have 10,000 players, it means that 99% of the field is busting out, while a min-cash is for nearly 20 buy-ins.

Of course, you can adjust the first-place payout in order to pay the number of seats you want, but this gives you no control over what the minimum cash will be. You can only ever choose one variable at your discretion: top payout, number of seats paid, or the minimum cash. Everything else is determined by the formula.

So, while you can pay out 1.9x to 10% of a 1000-player field, if you move up to a Main Event-sized field, the min-cash ends up being only 1.4x, and that’s assuming you’re committed to paying only 10% of seats; if you try to pay 1000 seats out of 6500 as they did this year, you end up with min-cashes below the buy-in. For a Colossus-sized event, the min-cash for 10% paid would be around 1.2x, while at the other extreme, a 90-man tournament paying only the final table would give you a 3.5x money bubble.

A solid starting point

Aside from the scalability issues, my main issue with the Chen Structure is the 2x jump from 2nd to 1st, which is way more than players are used to. The typical 2nd-to-1st jump in a WSOP event is more like 1.6x. In a way, though, this is convenient, because it gives us a second bird to kill with the same stone.

Although I was initially dubious, in playing around with Chen’s idea, I’ve become convinced that it’s actually a very solid starting point for building a sensible structure. All that’s needed is to shift a little of the money from 1st place to the bottom of the payout structure in order to fix the min-cash issues. This could be done by hand, or according to some more complicated formula, but either way, if you start with a Chen Structure then take away a quarter of the 1st place payout, you create a WSOP-style pay jump for the heads-up finish, while giving yourself plenty of spare prize money to adjust the bottom end of the structure.

In the case of a 90-man tournament, for instance, starting with a Chen Structure with 35.3% for 1st and nine seats paid, you can reduce that to 26.5%, which gives you a more normal 2nd-to-1st jump. That 8.8% difference is about 8 buy-ins, which you can then use to give 2.64x payouts to 10th through 12th.

Conversely, for the larger tournaments, instead of adding extra payouts, you could add the money to the min-cashes. In a 5000-player field, paying 500 seats, then in a straightforward Chen Structure, you’re paying about 14.7% to 1st and the min-cash is 1.47x the buy-in. If you want to round all cashes less than 2x up to 2x, you really only need about 0.7% of the prize pool to do it; shaving that 14.7% down to 14% is no problem. If that still leaves too big a jump between 1st and 2nd for your liking, you can then either add a bunch of additional 1.5x cashes, or round everyone between 2.00x and 2.99x up to 3x, or any number of other options.

Alex Weldon (@benefactumgames) is a freelance writer, game designer and semipro poker player from Montreal, Quebec, Canada.

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