All--

In PrimordialAA's WSOP 10K HU thread, many claims were made about what winrate one requires in order to have a certain ROI in the tournament. I thought this question was interesting, so I built a tournament simulator.

You can check out that thread for those arguments and for the initial discussion of how to characterize the skill distribution in that field.

I've modified my simulator to run tournaments with 152 entrants and last year's WSOP payout structure. I think it's reasonable to take people's ROI claims in this thread as applying to tournaments with this field and payout formula. Here's a first claim we can test:

I created a fake-Primordial and had him play one million matches against opponents drawn from a distribution JDalla and others recommended.

Primordial won 53.5891% of these matches, so I was a bit over-generous to him.

Then I had him play 250000 152-man tournaments where the field was drawn from opponents with the same skill distribution that fake-Primordial achieved a 53.5891% winrate against. The results:

I've done fairly extensive testing of my simulator. I've had it print out round-by-round results of a tournament, telling me the winrates of the winners and losers. I've run many tests guaranteeing that the prize pool is correct and is being distributed correctly. I've run many tests making sure that fake-Primordial's winrates in the play-in round and rounds 1-7 are plausible, and that the mechanics of the play-in round are correct. One can't ever really guarantee that a program has no bugs, but I've worked to eliminate potential problems.

Perhaps Primordial meant something by "53.5% average winrate vs. the field" other than: he would win 53.5% of the time over a large sample of matches from opponents picked at random from the list of entrants. But I sure can't think of a better interpretation than that.

I agree with Primordial that we should sit down and do the work to figure out the relationship between winrate-vs.-field and ROI as precisely as possible. So that's what I've tried to do.

I can tweak the simulation in any number of ways. I'm very happy to do this in response to people's suggestions here (e.g.: "Hey, Nate, there should be more bad players, but they shouldn't be quite as bad!"). Although I've built a test suite and I've run it after making any significant change to the program, I'd also be very happy to implement any further tests that anyone thinks would allow us to be even more confident that the program is running accurately. I wouldn't be posting this unless I were already pretty sure that things were functioning well, but I'm always happy to make a test suite more reliable.

All my best,

--Nate

Good stuff this is interesting! What about 250000 512 runner fields? And 256?

Good idea. Care to suggest a pay table?

All my best,

--Nate

http://pokerdb.thehendonmob.com/event.php?a=r&n=40122
http://pokerdb.thehendonmob.com/event.php?a=r&n=35080

Something like this I guess for 256

Could you post standard devs for all these as well?

I remember these results look reasonably similar to the HU ones i did when I built a lot of models to compute ROIs and sds for various sizes/edges/structures for different types of mtts a couple years ago to understand this stuff a little better, but I lost that laptop and have been too lazy to rebuild everything

Galen--

Yes, I'd be happy to. On my list after I work on Rupert's question. Likely I'll get to it tonight.

All my best,

--Nate

EDIT: For the love of all that is holy, use GitHub.

Rupert--

I pulled up the page before you added that second link. The first one is a bit funny--if you add up the payouts, it looks like they only took 5.175% out.

Anyway, I threw it in, ran a couple tests, and then did the simulation. The ROI came out as 24.3197432%.

Notes:

(1) I didn't have as much time to test, so it's slightly more likely that there is an error somewhere.
(2) ***The rake is only 5.175% in this sim!*** Important.
(3) It's 256, not 512 as you asked. I only had time to throw these numbers in.

All my best,

--Nate

Good stuff. How was Primordial ranked in the field?

edit: nm found it

Any guesses as to what a 53% winrate equates to in terms of BB/100 winrate?

Ben--

Sorry--I don't. Someone with a good HUSNG database could probably give a good answer (though you doubtless knew that already).

All my best,

--Nate

[see below; sorry.]

All--

Primordial also posted this:

I ran two more 250,000-tournament simulations (with the 2012 field size and WSOP payouts). A guy with a ~55.07% winrate had a 35.08% ROI. A guy with a ~54.03% winrate had a 25.2% ROI.

FWIW, I am starting to feel like this guy, so if I'm clogging this forum I'd be happy to take this to my blog or somewhere else. I'll let the community (and the mods) decide.

All my best,

--Summer Glau

What edge does he need over the general field to justify a 1.25 mu?

I'm a completely neutral observer and I'm certainly finding this all very interesting! Nice work

It'd be better if you signed it River Tam.


Anyway, I remember it was way easier to build a HU model than for MTTs (which was super ****ing hard), but I think it's still educational. The MTT model I tried to build had basically a list of levels with win rates at each level in terms of bb/s100 and sd bbs/100 and then average stack sizes, then the percentage of the field that made it to each level then tried to like figure out expected chip gain rates and extrapolate from there, but it just became a huge mess.

Out of sheer curiosity, rake aside I wonder what win rate you'd need against the starting field to break even in this.

I've long maintained that I think the marketplace undervalues "decent regs" and undervalues elite players in well structured events (like Bay 101, say), because decent regs have a reasonable edge on the starting field, but will have a negative edge a lot of the time at the final 2 tables or final table, shorthanded and HU and actually end up being breakeven or slightly losing overall, while elite players have an edge the whole way through and so it compounds like here.

For example, in a really sick structured HU tournament where win probabilities for the best and worst players would be much higher, I think you'd see a lot of guys who are above average against the field be losers overall.



Last (unrelated) comment:

Can you re-do the calculations to see what happens if this tournament is played best 2/3? Obviously this drives up the ROI of players who have a hypothetical 54 or 55% chance of winning each match, and drives down the ROI of the weaker players. Would be curious to see how much it affects overall ROI.

EPT HUs are 2/3, but WSOP plays with bullets which isn't quite a true 2/3

Galen--

Looks like 51.5% or so, though assumptions about field structure might really punish this guy.

All my best,

--Nate

Turbulence--

Depends what ROI justifies a 25% markup! Opinions differ on what's expected/ethical. Primordial himself suggested that he wouldn't have felt comfortable charging 1.25 if he only had a 26% ROI.

Although my simulation isn't perfect, I think it's a lot better than what almost anyone else is using to estimate these things. I'd be surprised if investing at 1.25 in a 53.5%-against-the-field player was even +EV. If you're expected to assign roughly 1/3 of your expected dollar edge to the backer, it is possible that you would need an edge that many people claim nobody has (>55% against the field).

But again, my simulator is directly working on ROI questions, and only indirectly on ethical ones.

All my best,

--Nate

I'm an idiot so sorry for the stupid question, but does your simulation account for each successive opponent getting tougher to beat?

Champ--

No worries! Sorry if I haven't explained myself clearly enough.

It works the way you might expect it to: it chooses 152 players from the skill distribution JDalla and others advocated. Then it randomly pairs them up and makes them play. It keeps track of who wins, pairs them up, and makes them play, repeating until there is a winner. Then it assigns payouts according to the WSOP pay schedule.

So I didn't encode any explicit formula to represent opponent skill from round to round--what you see is the results of how the tournaments actually play out in the sim.

All my best,

--Nate

Galen--

I just played around with this. A guy who is ~53.5% to win any given match out of 3 was roughly 55.2% to win a 2-of-3 vs. a random opponent (this simulation result agrees with back-of-the-envelope approximation, which FWIW is a simple extra check but lends a little more evidence that I'm not overlooking an obvious bug).

I don't know if you want to call that guy a 53.5ish% player, because that's his chances of winning a match, or a 55.2ish% player, because that's his chances of winning the whole round.

Anyway, that guy's ROI was ~28.3% over 350,000 tournaments. The effect of magnifying the edge over worse players overwhelmed the effect of the best players' edges over him also being magnified.

All my best,

--Nate

Btw, thx Nate! This is great.

OK yeah that's pretty much what I thought. Thanks.

Awesome. Github link?

Galen--

After the first few sims, it looks like you're right. Unless I've done something wrong, the ROI of a guy who's ~51% in any given game (out of 3 games) against a random opponent has something like a -5.25% ROI over the tournament you describe.

All my best,

--Nate

Keep posting