Fictional Prop Bet

[djames]

Please settle an argument. I believe this is a straight-forward application of EV, but I'm having trouble convincing others to accept this game.

Game:
10 player tournament, top 3 get paid, 1st > 2nd > 3rd.

Assume:
1) you have an unlimited bankroll
2) this game will be offered to you forever
3) you have a positive ROI for this game, and always will, where ROI = prize / buyin - 1 as usual.

Bet:
Someone offers to bet $X that you'll be negative after 10 tourneys.

I claim you should take this every day. I have a positive ROI, so I'm expected to be ahead after 10. I say screw variance because I have an unlimited bankroll and can always just play again.

Why can't I convince people of this? Can you?

[SonnyJay]

Say you have an ITM of 40% and a ROI of 20%.

If they did this bet for one tournament. Would you do it? Of course not, you're taking even money for a bet where you're 40%.

Looking at more than this, it would depend on your distribution of 1st, 2nd, 3rd, and 4th-10th. I don't have the time to do the math at the moment (though I'd appreciate it if someone did it), but it depends on how you achieve your X% ROI. If you do it solely through 3rds (ITM distibution of 0% 1st, 0% 2nd, 100% 3rd), it's a better bet for you than if you do it solely through 1sts (100%-0%-0%). This is because your ITM percentage is higher in the 1st example than in the second one, and thus there is less variability in the number of winning sessions.

In the original thread someone gave the good example of a lottery where a $1 ticket will give you a 1 in 1,000,000 chance of winning $10,000,000. This is definitely a +EV bet for you, but if the same bet was made (will you be ahead after 10 tickets) it is -EV for you because you'll hit very rarely.

Say that you make the 10 ticket bet 100,000 times. On average, you'll hit once and win and lose the other 99,999 times. If you bet $1 each time on this, you'd lose $99,998 on the bet. You seem to be confusing this with the actual bet itself ($1 tickets on 1 in 1,000,000 chance that pays $10,000,000), which is very +EV.

-SonnyJay

[djames]

I accept this 10 SnG bet with anyone that would care to offer it to me.

I can't accept the lotto bet itself, since I do not have infinite funds.

I would never accept the lotto prop bet, since the variance in outcomes is gigantic. That is, over any stretch of 10 tourneys, I expect to win $90, but nearly always lose $10.

Not sure who is confusing what at this point, but again, I'd like to speak with anyone offering the SnG prop bet!

[SonnyJay]

[ QUOTE ]
Bet:
Someone offers to bet $X that you'll be negative after 10 tourneys.

I claim you should take this every day. I have a positive ROI, so I'm expected to be ahead after 10. I say screw variance because I have an unlimited bankroll and can always just play again.

[/ QUOTE ]

This seems to be where you're confused. Think of it as though he's betting you even money if you're up after one tournament. This is always a bad bet for you unless your ITM is higher than 50%. It's not an issue of "riding out variance", as any individual tournament leaves you at less than 50% to finish ahead.

This doesn't mean that the actual game isn't profitable for you, it just means that you win less frequently but win larger amounts when you do hit. The issue is that in this bet it doesn't matter whether you finish up $1 or up $1,000,000 in your tournaments, you still finish with the same result in the bet.

I'm not saying that the bet is inherently -EV, depending on your stats it may very well be the case that it's a good bet. However, just because you're +EV in the game doesn't mean that this bet that you propose is +EV.

-SonnyJay

[djames]

Construct me a finish distribution equating to an ROI of say 15% which has so much variance that it makes the prop bet -EV. Maybe it exists, maybe it doesn't.

[ciaran]

I think with a "normal" distribution of finishes you're going to be +$ in any given 10 SNG run more than 50% of the time, which would make this bet +EV. This based on messing around with breakeven ROI/ITM numbers in the ROI simulator.

There are ITM distributions (like all 1sts or all 1st/3rds) which, combined with a low enough ROI, would appear to make it possible for the odds on a given 10 SNG stretch being +$ less than 1-1 even with an overall +ROI player, but I don't think the distributions required are very realistic, and the ROI required is awfully low.

FWIW, I tried to work out the proper way to correctly solve this problem in Excel, but couldn't do it. Seems like it wouldn't be that difficult, but I quit doing higher-than-simple-algebra math without computer help about 20 years ago, so someone else will have to work out the exact solution.

[SonnyJay]

I don't have the time at the moment to do this, and at 15% ROI I would guess that most (all?) distributions would result in this being a positive bet. I was responding to your assertion that, given a positive ROI, this bet is +EV. I was arguing on the statistical validity of the assertion, not necessarily that your bet is -EV. I was trying to show that your thought process was flawed (particularly the assertion that you're just "riding out variance") since it is, after all, the probability forum. If you had a 15% ROI I would think that you should take the bet.

-SonnyJay

[djames]

Fair enough. Minimizing ITM while retaining a positive ROI clearly is what would be necessary to get a negative expected prop bet. Realistic finish distributions are probably rare for a truly positive ROI player, but I yield that some might theoretically exist. Care to stab at a near-realistic set?

So yes, I'm mistaken that all positive ROI players should take such a bet. I still would certainly accept [img]/images/graemlins/smile.gif[/img]

[ciaran]

[ QUOTE ]
Care to stab at a near-realistic set?

[/ QUOTE ]

Not that it's realistic, but a player with a positive ROI with all 1st place finishes should have a -EV return on the bet for low enough ROIs.

This becomes a BINOMDIST problem in Excel, so I can solve for that case with my limited math. Given a 1st place finish of 22.1% (which in a 10-handed Party SNG yields a plus 0.45% ROI for 10% rake games) you'll win 2 or fewer SNGs in 10 (so a -$ run of 10) ~61% of the time. You have to have a 1st place finish of roughly 25.85%(assuming no 2nd/3rd) to get to break even, which translates to a 17.5% ROI.

Hmm, I'm starting to wonder about how high an ROI you need generally to make the bet +EV (assuming I haven't done something stupid above). Someone with better math skills than me should actually work out the answer.

[djames]

Realistic finish distributions will certainly have 2nds & 3rds though. I guess I only care about those. Plus, the challenge was made from one great player to another, so I'm willing to further restrict to %'es where 1 > 2 > 3, and 1 + 2 + 3 between 39% and 42% since the bet would probably take place at a low buyin.