Limited Predictive Ability

[BMAD]

Say you are granted the ability to know the outcome of two consecutive rounds in a 8-team elimination tournament.  Each round that you win, the amount you bet is doubled while losing any bets when you pick the wrong team.  Every round you take all of the money in your possession and split it evenly over each team that you believe will win. Is it in your best interest to use your ability of foresight in the first two rounds, middle two rounds, last two rounds or does it not matter?

[bubbled]

A few clarifications. It seems like you are expecting there to be 4 rounds? In that case, there needs to be 16 teams. I'm also assuming that each game is 50-50 and you have no insight into the outcome?

[bonanova]

@bubbled, I would interpret it this way:

Round 1: Four games => 4 winners.
Round 2: Two games => 2 winners.
Round 3: One game => 1 winner.

Seven games in all, and there are no "middle two" rounds.

If that's right, my take is

It doesn't matter.

Your expectation is the same for every round: break even. For each round that you know the outcomes, your expectation (certainty) is to double your money.

So suppose you start with $4. Then for example if you know the outcomes for Rounds 1 and 2, you will take $16 into the final round, where you'll win 1/2 the time, ending up with either $32 or $0. Your expected return is 2x2x1. If you knew the last two rounds it would be 1x2x2.

This of course uses an assumption implied by the wording of the OP although it's not stated explicitly. You make the bet on each round AFTER the completion of the previous round.

UNLESS I am misinterpreting what "doubling" your money means. I'm assuming this describes an even-money bet. i.e. I bet 1 unit of currency. Winning means I now have 2 units. Losing means I now have 0 units.
 

[bubbled]

@bubbled, I would interpret it this way:

Round 1: Four games => 4 winners.
Round 2: Two games => 2 winners.
Round 3: One game => 1 winner.

Seven games in all, and there are no "middle two" rounds.

If that's right, my take is

Hidden Content

I agree. But

I would use my power on the last two rounds as that will make my probability  of losing it all 1/16. Same expected profit, $12, but less volatility.

[BMAD]

A few clarifications. It seems like you are expecting there to be 4 rounds? In that case, there needs to be 16 teams. I'm also assuming that each game is 50-50 and you have no insight into the outcome?

@bubbled, I would interpret it this way:

Round 1: Four games => 4 winners.
Round 2: Two games => 2 winners.
Round 3: One game => 1 winner.

Seven games in all, and there are no "middle two" rounds.

If that's right, my take is

Hidden Content

i wanted 4 rounds so my apologies. There should be 16 teams.  Each game has a 50-50 chance of being picked by you in that you have no knowledge of who you should pick though one team may or may not be better than the other team.

[bonanova]

OK, and I have a suspicion that my answer is too easy and there's more to it.

[bubbled]

OK, and I have a suspicion that my answer is too easy and there's more to it.

Nope:

I think you were right. It doesn't matter how many rounds there are. Your total EV will never change on the rounds where you have to guess. So your finishing money will always be S * (2^R) where S = starting money and R = number of rounds you can know the results.

Let's say you start with $8, in this 16 team tournament and your magical power is during...

first two rounds: 8 (start) -> 16 (after round 1) -> 32 -> 32 -> 32

middle two rounds: 8 -> 8 -> 16 - > 32 - > 32

last two rounds: 8 -> 8 -> 8-> 16 -> 32

However, I would definitely use the power on the last two rounds to minimize chances of ruin.

[BMAD]

OK, and I have a suspicion that my answer is too easy and there's more to it.

Nope:

Hidden Content

my suspicion is that given that most people would pick a particular round to use their ability that there is in fact a particular advantage even if the expected payouts are the same. Maybe game theory should be utilized here over strict statistics??

[bubbled]

OK, and I have a suspicion that my answer is too easy and there's more to it.

Nope:

Hidden Content

my suspicion is that given that most people would pick a particular round to use their ability that there is in fact a particular advantage even if the expected payouts are the same. Maybe game theory should be utilized here over strict statistics??

This feels very subjective. I simulated the event using the three different options 1M times each. Here's what I get with graphs. Seems like it totally depends on your appetite for risk:

16 teams, $8 starting bankroll.

Magic power last two rounds (my choice): mean 32.006668

My choice has nothing to do with the fact that the mean happened to be a tiny bit better. Instead, I would always choose the strategy that reduces the chance of ruin by 7-1 (versus middle two) and 9-1 (versus first two). 

Magic power middle two rounds: mean  31.980512

 

Magic power first two rounds: mean 31.993984

 

 

Edited November 26, 2015 by bubbled
Images didn't load first time

[bubbled]

OK, and I have a suspicion that my answer is too easy and there's more to it.

Nope:

Hidden Content

my suspicion is that given that most people would pick a particular round to use their ability that there is in fact a particular advantage even if the expected payouts are the same. Maybe game theory should be utilized here over strict statistics??

This feels very subjective. I simulated the event using the three different options 1M times each. Here's what I get with graphs. Seems like it totally depends on your appetite for risk:

Hidden Content

 

One follow-up:

The only way I could imagine using game theory or human psychology to decide, is if my goal was to beat a theoretical opponent who is granted the same ability. In that case, I maximize my chances of "beating" him by using my ability on the last two rounds. If he chooses the same thing, we have an equal chance of beating each other. But, if he chooses either of the other two options, I am more likely than not to finish ahead of him.

[BMAD]

OK, and I have a suspicion that my answer is too easy and there's more to it.

Nope:

Hidden Content

my suspicion is that given that most people would pick a particular round to use their ability that there is in fact a particular advantage even if the expected payouts are the same. Maybe game theory should be utilized here over strict statistics??

This feels very subjective. I simulated the event using the three different options 1M times each. Here's what I get with graphs. Seems like it totally depends on your appetite for risk:

Hidden Content

 

One follow-up:

Hidden Content

would this not imply that the first two rounds use for the magic power are worst.  Imagine that you are trying to beat your best score, your opponent is yourself. An opponent with the exact same power. Trying to do the best in this case, according to your previous post would include using this power at the end. Hence there is an advantage.

Edited November 26, 2015 by BMAD

[bubbled]

OK, and I have a suspicion that my answer is too easy and there's more to it.

Nope:

Hidden Content

my suspicion is that given that most people would pick a particular round to use their ability that there is in fact a particular advantage even if the expected payouts are the same. Maybe game theory should be utilized here over strict statistics??

This feels very subjective. I simulated the event using the three different options 1M times each. Here's what I get with graphs. Seems like it totally depends on your appetite for risk:

Hidden Content

 

One follow-up:

Hidden Content

would this not imply that the first two rounds use for the magic power are worst.  Imagine that you are trying to beat your best score, your opponent is yourself. An opponent with the exact same power. Trying to do the best in this case, according to your previous post would include using this power at the end. Hence there is an advantage.

OK, I agree, but it's really really thin. First of all, I might be a cowboy and want to maximize my chances of turning 8 into 128. In that case, I'd use the power on first two rounds. And who's to say that's wrong? if you put clear parameters in place, like "minimize chance of ruin," I'm totally with you.

Also, if you offered me $.50 to use my power first two rounds, I'd be foolish not to.

In the case where we are taking about $8 at risk, you can't make a definitive choice and say it's "right." If we say a player has to risk all the money they have in the world, now I'm feeling like it's clear that last two is right.