Doubling up early MORE than doubles the ROI of a winning player

[jcm4ccc]

This article got me to thinking: To flip or not to flip

Imagine a hypothetical coin-flip tournament with 1024 players. The prize structure is as follows:

Buy-in: $100 + $10

1st place: (25%) $25,600
2nd place: (14%) $14,336
3rd place & 4th place (8%): $8192
5th place – 8th place: (4%): $4096
9th place – 16th place (2%): $2048
17th place – 32nd place (.05%): $512
33rd place – 64th place (.001565%): $160

In this hypothetical tournament, it takes coin flips to win. Winning 10 coin flips in a row gets first place; 9 coin flips in a row gets 2nd place, etc etc etc.

Let’s say that you win coin flips 55.47% of the time (rather than 50% of the time). Your expected winnings in this tournament will be $110.21. In other words, your ROI will be 100%. You spent $110, and you expect to earn a profit of $110.

Now, let’s say that you win that first coin flip. Now half the field is gone, and it only takes 9 coin flips to get first place, 8 coin flips to get 2nd place. What is your ROI now? You may expect that your ROI is a little less than twice 100%, perhaps something like 190%. This would conform with the notion that doubling up early does not double your ROI for a winning player.

But in fact, winning that first coin flip increases your expected profit to $286.68, and increases your ROI to 260%. This seems to indicate that doubling up early more than doubles your ROI as a winning player.

Here’s why your ROI more than doubles. If you win coin flips 55.47% of the time, your expected winnings in the tournament are $220.21. Your expected profit is $110.21.

When you win that first coin flip, your expected winnings are now $396.68. Your expected winnings increase by less than twice as much. Your expected profit, though, is $286, which is more than twice as much.

I ran some other figures, and here is what I came up with:

If your ROI is -75% or below, than doubling up early does NOT double your ROI.
If your ROI is 220% or higher, than doubling up early does NOT double your ROI.


Maybe there is some flaw here in my reasoning, and maybe expected winnings is really the important thing, not ROI. I’ll leave it to others to tear apart.

[AtticusFinch]

[ QUOTE ]

Maybe there is some flaw here in my reasoning, and maybe expected winnings is really the important thing, not ROI. I’ll leave it to others to tear apart.

[/ QUOTE ]

The first flaw I can see is that everyone participates and is all-in on every flip. Your ROI has improved not just from your gain in chips, but from the elimination of half the field as well. (Which just adds to the notion that there are other factors in play besides stack size).

That being said, I believe there are cases where a double more than doubles a good player's $EV in a tourney, but it depends on your relative stack size and skill level. I'm still working on math to support this.

[nath]

That article is already being discussed here.

[LearnedfromTV]

[ QUOTE ]

Maybe there is some flaw here in my reasoning, and maybe expected winnings is really the important thing, not ROI. I’ll leave it to others to tear apart.

[/ QUOTE ]

The flaw is: The tournaments in which you win the flip is offset by the tournaments in which you lose the flip.

That the per tournament ROI more than doubles after winning the flip isn't what you should use to decide whether to take the flip. The expected ROI pre-flip is what matters, and that is dependent on a probability weighted EV calculation.

386*0.55+0.45*0 < 210*100