A local charity hosts an annual backgammon Round-Robin competition. The field is set up for 100 players maximum, but they usually get less than half of that. However, you can buy in multiple times (if the tourney doesn't fill).

This is a round robin, where everyone plays everyone one time (it doesn't take place in a single day of course). Your win/loss record is computed at the end of 99 games, and prizes are awarded based on some PayTable (like in an MTT).

If you buy in multiple times, in the games you would be playing v. yourself you get a draw. So if I buy in 3 times, I would play 97 other opponents (where I go, say, 60-37, and I would draw v myself 3 times, for a record of 60-37-3.) Win % is the way scores are ranked, draws basically don't count.

Here are the considerations/variables:

Entrants: 100 (including myself)
Buy-In: $120
Prize Pool: $10000 ($2k removed for charity)
Theoretical ROI based on just one entry: 10%

PayTable:
1st: 30%
2nd: 20%
3rd: 12%
4th: 10%
5th: 8%
6th: 6%
7th: 5%
8th: 4%
9th: 3%
10th: 2%

Questions are:

1. Optimal number of times to buy-in, to maximize EV?
   
2. If I have a Bankroll of $10k, what is the optimal number of times to buy-in, assuming we want to use full-Kelly? (I.e, the last buy-in that is +EV is going to add something like $0.01 to our EV, so perhaps we should not actually buy in that last time...)

I'm most interested in how to solve this. I'm not specifically interested in the exact answers. I set up a spreadsheet in Excel to solve it, but didn't like the answer I came up with as it seemed to suggest I should buy in far too many times, so I believe I made an error.

Thanks in advance for any thoughts.

Neat problem. Three quick questions and maybe more later....

(1) Are ties (playing vs. yourself) considered 1/2 win and 1/2 loss in terms of calculating your win pct, or are they omitted from the calculation (as you sort of suggest above)?

That is, is 60-30-10 (W-L-T) a 65% win pct [(60+(0.5*10))/100] or a 66.67% win pct [60/90]?

For deriving the optimal buy-in strategy, this may matter. I realize that in your 100-player tourney you'd only have 99 matches, but I wanted to keep the example mathematically simple.

(2) You say that your one-entry ROI is 10%. Does it matter how this 10% is achieved? I would assume that your optimal buy-in strategy may depend upon how your expected per opponent win pct is distributed. When you did your analysis, what distribution did you use (if you did it that way)?

(3) Did you use a deterministic or probabilistic model to determine expected per opponent win pcts (and/or match outcomes)? Again, how this is modeled may well affect the optimal buy-in strategy.

If I were doing this, I would assume a "talent" distribution (including where my talent lies), a probabilistic match outcome model based upon the talents of the two opponents (excepting that me vs myself is always a draw) such as the log-5 model, and then simulate the crap out of different number of buy-ins.

Of course, the underlying assumptions would need to be calibrated to achieve an expected 10% ROI on a single entry.

Anyway, this is neat question.

Wait a minute. I just re-read OP. Maybe I am missing something here.

Are you saying that you can have two entries but only play as if you are one player (playing everybody else just once)? With just one chance at a prize.

Or are you essentially two different players, playing everyone else twice? With the chance of getting two different payouts from the prize pool.

Under the first scenario, I don't think it will be profitable to ever enter more than once. You are lowering your score (or, at least, not raising it) but paying more (a higher pct of the prize pool).

I think I have seriously confused myself.

Thanks for the replies.

1) Ties are basically nothing. So 60-30-10 = 60-30 = ~66.67%
2 and 3) Let's assume these are unknown. All we know is the ROI is 10%, and we don't know anything about our opponents, only that we are 10% vs the entire field. In my work before posting this, I assumed the other players were all equal in ability. I understand this is not the case, but it greatly simplifies the problem. (Perhaps later we could add in some distribution of opponents' abilities (in terms of ROIs); and we could add our specific probabilities of finishing at various places.)**

Regarding the 2nd post:
Each entry is an individual entry and you play everyone one time PER entry. So if you enter once, you have 99 matches. If you enter 4 times, you have 396 matches (including the draws against yourself). In theory, if you enter 4 times, you could take home 1st, 2nd, 3rd, and 4th place prizes. It's as if me and my three identical brothers (we are quadruplets in this hypothetical) all entered (and tied ourselves somehow).

I think you understood it correctly initially -- apologies if my OP wasn't clear on this.

Again, thank you for the thoughts so far.


** -- These are all unknowns of course so really I should be giving a range of possible ROIs for myself, and a range for each other person. But, the answer to this was more interesting to me as a theoretical than to actually use the solution to a specific problem, if that makes sense.

I coded up a simulation that I described above. For this initial attempt, I assumed 50 opponents in the tournament and a very wide, symmetrical, and fairly bunched distribution of opponents' talents. On a 0-1 scale, opponents' average talent is 0.5, lowest is 0.1, highest is 0.9, and five opponents have 0.5. Clearly this is purely ad hoc and can be tweaked or significantly modified in subsequent simulations.

I used the log5 method to probabilistically determine who wins each match (except me vs. myself). I calibrated the initial assumptions so that Hero has roughly a 10% ROI on a single entry. This occurs at around a Hero talent level of 0.71.

I find that Hero's optimum number of entries is 2. His ROI with two entries is around 2% and his ROI with three entries is around -2%.

I will continue to run these simulations with more trials to getter greater confidence in these results. I will also tweak the opponent talent distribution since I have a feeling that may influence the result.

Do these results make sense??

P.S. I used draw scores as a half-win and a half-loss. Seeing your post just above, I will need to tweak the simulation to ignore draws, but I am not sure how much if any effect that will have on the optimal buyin strategy.

Firstly, thank you so much for taking the time to do that. Are you using R?

You say that Hero's optimum is 2 entries, which is 2%. But 1 entry is 10%?

Aside from that, 2 entries seems much lower than I would think. But, I thought my results were skewing high, so I'm not sure.

I just ran your scenario in my (comparatively rudimentary) excel:

Entrants: 50
ROI: 10%
Buy-In: $120
Pot: $5000 (same ratio as before)

I get that optimal strategy to max EV is to enter 10 times.

1 entry only: 10% of $120 = $12
2 entries: $22.69
3 entries: $32.08
4 entries: $40.16
5 entries: $46.94
6 entries: $52.41
7 entries: $56.57
8 entries: $59.43
9 entries: $60.98
10 entries: $61.22
-----
11 entries: $60.16 (decreasing now)

I don't know that the above is "right", just reporting my output.

Would it be simple to re-run your sim with the entire pool at equal talent level? I'm curious as to how much of a difference that assumption makes? That is how my calcs are being done.

Appreciate your feedback.

What are you doing in Excel? You assume all your opponents are equally talented (to use my term)? Then you use a binomial formula to calculate expected number of wins for each of your opponents?

But what do you do for Hero's matches? How do you determine who wins (or expected wins)? Is it log5 method?

Then you use Excel's solver to find the Hero "talent" level to find where expected ROI is around 10% for a single entry?

Oh, I guess you can just skip to the last step in Excel since that is all that matters under those assumptions.

Seems interesting.

I started typing an explanation, but it would probably be far easier for you if I just share my spreadsheet via google docs? If that works, please let me know your email address and I will send a link. You can PM me if you prefer.

Thank you!