The two envelopes problem

[factor-fiction]

The Deal or No Deal thread reminded me of two tricky probability/choice problems. One is the Monty Hall goat and cars problem which has been covered on here several times before so I'll leave that one.The other is the Two Envelopes problem:

Suppose you are in a Game Show. The host has 2 identical looking envelopes The host then tells you that one envelope contains twice as much as the other but neither of you knows which is which.

He asks you to choose one envelope and says you can open it and he will then offer you the chance to 'swap' or 'stick'.

You pick an envelope and open it to find it contains £2400. You think
"Mmm. The other envelope contains either £4800 or £1200. Should I swap or stick?"

What would you do?

[factor-fiction]

Hi jno- isn't it a "double or halving " bet rather than "double or nothing"?

[Postdog]

I'd stick, simply because it is money for nothing and why be greedy

[Jonathan-Joe]

Where you're allowed to see the amount in the envelope, before you open it, ask yourself "What amount would I expect the Game Show producers to set as top prize?" If the amount in the envelope exceeds your guess, or comes close, accept that amount. If it is signifcantly lower go for the swap.

[jno]

true. Same answer, though - and as you'll have noted it has nothing directly to do with probablilities, just with how much money is offered and how much I've got and how much I'm happy with.

[jno]

sorry, that answer was to factor30.

[Old_Geezer]

Are these the same gameshow producers that produced 'bullseye', or 'who wants to be a millionaire' ?

[Prudie]

More likely 'Take Your Pick' or 'Double Your Money' OG!

[ludwig]

I think if you feed the problem into a computer it will tell you that the logical option is to swap, but as we're humans other factors come into play such as how wealthy you are.

I mean if you're loaded, you'll just swap because you couldn't really give a monkeys about the possibility of losing £1200.
But if £2400 could really do something useful for you like buy the dream holiday you want, and £1200 wouldn't cover it, you're probably wiser to stick with the £2400.

[factor-fiction]

Thanks all.

I partly agree with you- Jonathon Joe. I tried to cover that point in my thread at the bottom of the last page when I said:
"I suppose the question of seeing the amount or not seeing it could become relevant if you knew there was an lower and upper limit on the values- for example if it was known that the amounts ranged from £1 to £1048576 and you picked one of the extremes you'd be able to work out for sure what the other one was."

Thanks jno- yes, different people have different attitudes to risk some prefer a bird in the bush etc, and for some people the beneficial value of money is not always proportional to the monetary value. Mr A may be desperate for £10000 and would take it rather than risking getting a smaller amount, whereas Mr B might already be well off and feel £10000 won't make much difference to him whereas £20000 would give him the flashy new car he wants.

I suppose that's the reason some people do the lottery (they won't miss a few pounds but would like a fortune) and some don't (they value every pound they have).

[Old_Geezer]

Some do anyway factor30 foolishly thinking that the lotteries are the only chance they have for a better life. Spending more than they can afford.

Ah Prudie, probably so. :-)

[Old_Geezer]

Are there 2 lives in the second envelope ?

[Davethedog]

Its not a problem, its a philisophical position on greed or decisiveness.

The simple answer is whatever you decide is right. The only quandary is will you be regretful if you end up with the smallest amount.

Unless there are any missing clues you can't make a balanced decision.

[factor-fiction]

Sorry ludwig, your answer wasn't there when I started typing mine.

The official answer I believe is that there is no logical reason to swap in this game. In simple terms you had a 50/50 chance of picking the larger amount. You are still none-the wiser after picking your envelope. There is still a 50/50 chance that you have the larger amount.

You can tie yourself in knots (and I and others have done-see the Wikipedia link) by trying to calculate probabilities and considering the likely distribution .

Thanks for some excellent and interesting contributions. These show that appears to be a simple choice/probability problem can be interpreted in different ways and might explain what appear to be irrational choices in game shows and in life. Even mathematicians can make big mistakes with probability and there have been serious misuses in court rooms for example.

[factor-fiction]

"Get a life", Groupie?
Do you mean I we should ask questions like this? http://www.theanswerb.../Question1073742.html

[Old_Geezer]

Indeed, there is still a 50/50 chance that you have the larger amount. But that is only half the story. The other half is to compare what can be gained from what can be lost. I have to say that I'm most unimpressed with the official answer.

[dr b]

"The official answer I believe is that there is no logical reason to swap in this game." I disagree. There's a big difference between the wiki problem and this one. In the Wiki problem you can't open the envelope, whereas here you can. So the problem is actually, "Do you prefer 2400, or a 50/50 chance at 4800/1200?" As noted by others, the choice depends on your starting wealth and your attitudes towards risk. So some people will have a logical (for them) reason to swap, and other will have a logical (for them) reason to hold.

[factor-fiction]

I'm not clear why it matters, dr b? If knowing then amount causes you to swap on that basis then you'd swap whatever the amount wouldn't you? Or would a certain amount cause you not to swap. If you'd always swap whatever the amount then why bother opening the envelope first?

[dr b]

No, you would not always swap. A person with no money might prefer the sure thing to the chance of having less. A millionaire might feel the amounts involved are trivial and therefore say, what the heck, let's gamble.

Two people might have the same wealth, but one is predisposed to taking risks (so he'd swap), the other is more conservative (so he'd hold).

(These are examples of relative and absolute risk aversion, in economist-speak.)

Someone might be disposed to swap given the amounts in the problem as stated, but add two zeroes to everything and the sure 120,000 now looks good.

The "right" answer will be an individual choice for all.

[dr b]

sorry, 'sure 240,000'.