Pepsi Billion Dollar Challange

[sexypanda]

Ok, so I saw the Pepsi Billion Dollar Challange on TV yesterday and thought it was by far one of the worst game shows I've ever seen. Firstly, it was a game that required absolutely no skill; it was pretty much the lottery. They first randomly chose 200 people to play the game and they all picked 6 digit numbers. A monkey then picked a 6 digit number which was the "winning number." They then chose the 7 people whose numbers were closest to the monkey's and they played a game of "chicken." The winner of this game of chicken won a million dollars.

Here they eliminated the contestants one by one by who had the number least close to the monkey's. They gave the contestants an opportunity to quit and take a set amount of money. The problem was that this amount was sooo small that it was a wildly -EV play to do. Here's how it broke down.

7 contestants
They give you $10,000 if you quit
(1/7) chance of winning $1,000,000
Therefore EV is $142,857
That makes this play -$132,857

6 Contestants
They give you $20,000 if you quit
(1/6) Chance of winning $1,000,000
Therefore EV is $166,666
That makes this play -$146,666

5 Contestants
They give you $30,000 if you quit
(1/5) Chance of winning $1,000,000
Therefore EV is $200,000
That makes this play -$170,000

4 Contestants
They give you $40,000 if you quit
(1/4) Chance of winning $1,000,000
Therefore EV is $250,000
That makes this play -$210,000

3 Contestants
They give you $50,000 if you quit
(1/3) Chance of winning $1,000,000
Therefore EV is $333,333
That makes this play -$283,333

2 Contestants
They give you $100,000 if you quit
(1/2) Chance of winning $1,000,000
Therefore EV is $500,000
That makes this play -$400,000

Ok, so my question is, is did anyone else notice this? No contestant even thought about quitting. Would this game be better if it wasn't so lopsided and if the guaranteed cash prizes made the plays closer to 0 EV rather than being SO negative? Any thoughts?

[Ghazban]

I didn't actually see this but the purely +EV play (i.e. staying in) isn't necessarily the best for an average person. I know many people that would quit at the final two and take the $100K rather than take a 50/50 shot at a million just because they can do so much that's important to them with $100K that having that much guaranteed is much more useful than possibly having a million (with the risk of getting nothing).

Think of it this way-- if someone offered you a trillion dollars or you could take a coinflip for 5 trillion dollars (getting nothing if you lose), why would you take the flip? 5 trillion dollars doesn't have all that much more utility than 1 trillion as you can pretty much do anything you want either way and the 1 trillion would be risk free.

[sexypanda]

About the $100,00 play. At this point you have a 50% chance of winning $1,000,000, therefore EV is $500,000. But also, if your number exactly matches the monkey's you get $1,000,000,000. Since you have one of the two closest numbers your chances are probably at worst 1/100 to get the billion. So at this point your EV is (1/2) (1,000,000) + (1/2)(1/100)(1,000,000,000) (50% to get this million plus 50% to try for the billion times 1% to get the billion) = 500,000+5,000,000 = 5,500,000. Including the chance for winning a billion this is unbelievably -EV!!

[Lost Wages]

Listen to the other guy. Yes, it's -EV that's clear, but this is a choice that you only get to make once in a lifetime, not as many times as you want. So, it is not an EV decision it's a utility of money decision. Making a high variance +EV choice isn't always correct. This comes up in tournament situations all the time. You might pass on calling with all of your chips early on even if you think that you are a slight favorite.

Suppose that you go on the show flat broke and get down to the $100,000 play. Now suppose that your kid needs a heart transplant that costs $100,000 and she will die if she doesn't get it. What is the correct decision?

Lost Wages

[mostsmooth]

i was wondering how the decided who was the closer of the contestants. i dont know if anybody noticed, but the guys number started with a 7 while the monkeys number started with a 1 (or vice versa). so the guy was 600000 off. surely there had to be other people closer than that margin? or by closest did they mean the most matching numbers?

[thylacine]

It's still not even close. $1,000,000 has much more than twice the utility of $100,000 for anyone in the western world. You are just not being realistic with this example.

[Lost Wages]

$1,000,000 has much more than twice the utility of $100,000 for anyone in the western world.

And $100,000 has infinitely more utility than $0.

Lost Wages

[thylacine]

[ QUOTE ]
And $100,000 has infinitely more utility than $0.

[/ QUOTE ]

No it doesn't. You can do lots of things for $0.

For me, $100k would give me some extra security, whereas $1M would totally transform my life. I would take this gamble in an instant.

[Lost Wages]

I would take this gamble in an instant.

Ah, me too. But that doesn't mean that it is the correct decision for everyone.

Lost Wages

[SossMan]

[ QUOTE ]
[ QUOTE ]
And $100,000 has infinitely more utility than $0.

[/ QUOTE ]

No it doesn't. You can do lots of things for $0.

For me, $100k would give me some extra security, whereas $1M would totally transform my life. I would take this gamble in an instant.

[/ QUOTE ]

screw that, i'd take the 100k. And i understand EV.

[Ghazban]

I'd take the 100K, too. With that kind of capital, I could do an awful lot that I can't do right now. Having an extra 100K guaranteed is worth more to me than a 50/50 shot at a million (plus an even tinier shot at a billion). If I could have multiple appearances on the show, it would be the complete opposite.

[Cerril]

Not infinitely more... it has $100,000 worth more. If I get $0 tomorrow, or I get $100,000 tomorrow, I won't be infinitely better off. In fact, I'll still have to move out of my house in a couple months. With $1,000,000, on the other hand, even after taxes the house would be no problem and I wouldn't even have to work for a few years (at least!)

[Greeksquared]

This all depends on your utility function. If you have a square root utility function then
sqrt(1000000)=1000
sqrt(100000)=316

.5(1000)=500>316 so you would take the chance.

If your utility function is the natural log function then
log(1000000)=13.8
log(100000)=11.5

.5*13.8<11.5 so therefore you take the 100000

[pzhon]

[ QUOTE ]
This all depends on your utility function... If your utility function is the natural log function then
log(1000000)=13.8
log(100000)=11.5
.5*13.8<11.5 so therefore you take the 100000

[/ QUOTE ]
No. The correct comparison in expected utility theory are utility(x+10^5) vs. (utility(x) + utility(x+10^6))/2. You are assuming that the utility of winning 0 is 0, but log(0) = -infinity.

If x=0, and your utility function is log, you would give up a 99.9% chance of $1 billion for a sure penny. If that seems absurd, then either this is an implausible utility function, or it is implausible that your initial bankroll is 0.

Expected utility theory is contradicted by the Allais paradox: People often prefer A to B, but a 10% chance of B (and 90% nothing) to a 10% chance of A (and 90% nothing). For example, this is true if A is a sure $1 million and B is a 90% chance at 1.5 million.

Would you prefer a sure million, or a 90% chance at $1.5 million? If you choose the gamble, 10% of the the time you greatly regret your decision.

Would you prefer a 10% chance of $1 million, or a 9% chance at $1.5 million? If you go for the larger prize and lose, you can console yourself that you would probably have lost anyway.

Expected utility theory says the choices are the same. You will either choose A both times or B both times.

[tpir90036]

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[SossMan]

log($100,000 is a sh!tload of money) = "whoo hoo!!"
log($1,000,000 *50% chance of "goddammmit!!!") = "i have no balls".

clearly, i prefer the utility of choice 1.

[pzhon]

My gambling bankroll is not huge. In late rounds of backgammon tournaments where I will be playing for tens of thousands of dollars, I look for ways to hedge. If I face a weak opponent in a late round, it is unlikely that my opponent will make a fair offer.

The right thing to do may be to hedge with third parties. Some of my friends are happy to put a few grand on a wager with a 5% edge. I'm not, so I try to hedge with them. If you can't stomach a hugely profitable gamble, sell shares of yourself or hedge with friends until you will be happy to go for it.

[ QUOTE ]
2 Contestants
They give you $100,000 if you quit
(1/2) Chance of winning $1,000,000
Therefore EV is $500,000
That makes this play -$400,000

[/ QUOTE ]
I presume the rules prevented the contestants from saying to each other, "$100k save, ok everyone? Sign this," or in the last two, "I'll pay you $450k to quit. That way we each get $550k."

One possible deal with a third party would be to say ahead of time that if you get into this situation, you bet $400k that you will lose, giving 2:1 odds. If you win, you net $600k = $1m-$400k. If you lose, you net $200k, still much better than if you took the $100k for quitting. You could also sell a percentage of your action at earlier stages so losing would not bother you as much.

[junkmail3]

There is one question though.

Let's be extreme. Out of 200 people everyone 'picked' 000000 - 000200. (I don't know if you can repeat your numbers, but why wouldn't you be able to)

Now the monkey picks 846295.

We're left with 7 players:
000194 - 000200.

If I'm 000200 ... I'm in the whole way, because the chance of the monkey picking a number 000000 - 000194 (best case) is absurd compared to him picking one I'm closer to (plus I can get higher in between numbers.)

So, I would look at my number and ride it out if I had 000200 and it was the highest number in the crowd.

[sthief09]

the key word here is variance. sometimes I'll log onto Party Poker, check my buddy list, and see some of my favorite maniacs from my days at 5/10, and really want to get in the game. However, I'm not bankrolled for 5/10, and I can't afford the variance. Clearly playing in a juicy 5/10 game is more profitable than a random 3/6 table, and thus has a higher EV, but I can't afford to take the chance.

for me, there's a lot I could do with $100,000. Sure, there's more that I can do with $1,000,000 but I don't have enough of a living bankroll, ie enough money in the bank, to take that kind of risk.

If I was Bill Gates and had a huge living bankroll, or if I was bankrolled for 5/10, then I'd take the shot in the high variance game and go for $1,000,000, or play in the juicy game.