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]

Sorry drb, I only really dealt with part of your answer. i totally agree that everyone has different needs and attitudes to risk etc. In that sense it does matter whether they see the amount in the envelope.

What I was commenting on and challenging was the idea that there is an equally likely chance of the other envelope containing twice as much or half as much. If someone calculated expected values on the assumption of a 50/50 chance of doubling or having then they'd always swap. So whether they see the amount or not doesn't affect the calculation. they'd still swap. But then they'd swap again. And again... and that's the paradox.

[dr b]

Ah, but people do not deal in expected values. Yes, having seen the amount in the first envelope you'd say the expected value in the other envelope is .5(4800)+.5(1200) = 3000, so why not swap? But individuals will discount this expected value for the risk involved, so the "certainty equivalent" (more econ-speak) of the second envelope is less than 3000. Depending on your attitudes towards risk, this discount can be more or less than 600, so the decision to swap or not remains an individual one.

[Old_Geezer]

They wouldn't swop again and again, not on the same pair of envelopes anyway, as two consecutive swops with the same envelopes are linked. It isn't a new 'double or half' situation, it's a 'stay as you are or go back to where you were' situation on the second swop.

Now a different pair of envelopes, that's a different matter, that's a new choice, and yes, swop again.

Hmmm taking into consideration personal circumstances isn't really playing the game as far as I see it. Just complicating things with human frailties.

[Old_Geezer]

Come on someone, do the spreadsheet, I want to know if I'm right or wrong, and why ;-)

[factor-fiction]

You wanted a spreadsheet O_G.
This spreadsheet proves nothing O_G but it may help show what's happening. Let's play 20 rounds. To make it fair I've put an equal number of Large and Small values in envelopes 1 and 2.

round Env1 Env2 stick swap

1 600 1200 600 1200
2 500 1000 500 1000
3 1200 600 1200 600
4 500 1000 500 1000
5 400 200 400 200
6 400 800 400 800
7 800 400 800 400
8 250 500 250 500
9 500 250 500 250
10 4000 2000 4000 2000
11 3000 6000 3000 6000
12 1000 2000 1000 2000
13 2000 4000 2000 4000
14 400 800 400 800
15 800 400 800 400
16 200 400 200 400
17 1000 500 1000 500
18 6000 3000 6000 3000
19 2000 1000 2000 1000
20 1000 500 1000 500
TOTAL 26550 26550

The total gained from sticking or swapping is the same in the long run.

Let's look at Round 1. We didn't know this until afterwards but the time the two envelopes had £600 and £1200 in. If we'd picked the £600 one first and swapped we'd have gained £600 by swapping. If we'd picked the £1200 one first we'd have lost £600 by swapping.

If the first pair of envelopes had £X and £2X in then by swapping you'd either gain X or lose X. So why bother swopping.

This of course is just based on maths- it ignores the issues someone called 'human frailties'

[Old_Geezer]

Thanks. I'm going to have a good think about that, it intrigues me.

[Old_Geezer]

Ah I managed to get to a working spredsheet and got a different answer.

Column 1 = "Round", column 2 = "Envelope 1", column 3 = "Envelope 2"
A RAND function determines the value of Envelope 2 (which you don't get to see.) Sum for always stick with 1 and always stick with 2 shown at the end.

ROUND,Env 1,Env 2
1,2400,1200
2,2400,1200
3,2400,1200
4,2400,4800
5,2400,4800
6,2400,1200
7,2400,1200
8,2400,1200
9,2400,1200
10,2400,4800
11,2400,1200
12,2400,1200
13,2400,4800
14,2400,1200
15,2400,1200
16,2400,1200
17,2400,1200
18,2400,4800
19,2400,1200
20,2400,1200
21,2400,4800
22,2400,4800
23,2400,4800
24,2400,1200
25,2400,1200
26,2400,4800
27,2400,4800
28,2400,4800
29,2400,4800
30,2400,4800
31,2400,4800
32,2400,4800
33,2400,1200
34,2400,1200
35,2400,1200
36,2400,1200
37,2400,4800
38,2400,4800
39,2400,1200
40,2400,1200
41,2400,4800
42,2400,1200
43,2400,4800
44,2400,4800
45,2400,4800
46,2400,1200
47,2400,4800
48,2400,1200
49,2400,1200
50,2400,1200
51,2400,4800
52,2400,1200
53,2400,4800
54,2400,1200
55,2400,4800
56,2400,1200
57,2400,1200
58,2400,1200
59,2400,4800
60,2400,1200
61,2400,1200
62,2400,4800
63,2400,4800
64,2400,1200
65,2400,4800
66,2400,1200
67,2400,1200
68,2400,1200
69,2400,1200
70,2400,4800
71,2400,4800
72,2400,4800
73,2400,4800
74,2400,1200
75,2400,4800
76,2400,4800
77,2400,1200
78,2400,4800
79,2400,4800
80,2400,1200
81,2400,4800
82,2400,1200
83,2400,1200
84,2400,1200
85,2400,4800
86,2400,4800
87,2400,1200
88,2400,4800
89,2400,1200
90,2400,4800
91,2400,4800
92,2400,4800
93,2400,1200
94,2400,1200
95,2400,4800
96,2400,1200
97,2400,4800
98,2400,1200
99,2400,1200
100,2400,4800
,240000,289200

So I'm better off by £49,200 if I swop each time (judged over 100 attempts). Or an average of £492 a shot.

[factor-fiction]

Thanks. I'll have a look at your data later.

[joggerjayne]

Isn't this an illustration not of the law of probabilities, but of the law of big numbers?

Mathematically, you will gain, on average, by swapping.

But that does not mean that the "average" outcome will apply on this occasion.

But if you had a chance to enter 1,000 times, you should swap every time and you would, in the long run, be better off.

It's like Blackjack.

You know what the statistical correct decision is for any hand of yours, against any hand of the Dealer. Making the correct decision might still backfire. But if you continue making the correct decision through the whole six decks of cards in the shoe then you will, in the long run, be better off.

[factor-fiction]

I have copied it into Excel. I see that what you have done is started with £2400 as your starting envelope.
Unfortunately I keep getting zero when I try to sum your final column in Excel. Have you generated this column by either halving or doubling on a random basis or have you ensured that half the time it's £1200 and half the time it's £4800?

[factor-fiction]

The theory is that, unlike the Monty Hall problem where switching is better, in the 2 envelope game there is no advantage in the long run in switching. However it's fair to say that many eminent mathematicians have tied themselves in knots over both problems and have disagreed on the answers.
I'm off out now but I'll try to explain it later. In the meantime I wonder whether Buenchico has a view on this. Like Chris and me, this puzzle has been around for alongtme

[Old_Geezer]

Unsure why you get zero. Yes it is random and in fairness there just so happened to turn up a slight bias in the example shown, but refreshing the spreadsheet a few times brought up a 50/50 split and again the benefit of swopping is seen.

But in fairness it needs not 100 goes. 2 will do so long as 1 is higher amount in the envelope you looked in, and 1 is higher amount in envelope you don't know.

Stick 2400 + 2400 = 4800 whereas Swop 4800 + 1200 = 6000, Swop wins.

I've now had a pint at the local so I am thinking much more clearly. And it occurs to me that earlier I agreed that knowing the amount in the first envelope ought not change anything. Instinctively it does not seem to be useful information. But now I wonder if that is so. It has allowed me to anchor the problem and enter figures to prove the different outcomes. Maybe that has changed things in a way I do not yet understand.

Ah yes I always have £2,400 in the starting envelope because that is what the original question defined. And that brings me back to the idea that knowing this may make a difference somehow.

[Jonathan-Joe]

Your spreadsheet calculations assume ONE group of people, some of whom have either £ 1,200 / £ 2,400 envelopes, others have £ 2,400 / £ 4,800.

But look at this from the point of view of the person providing the money (forget that it's a game show - perhaps its a team testing a theory).

They prepare pairs of envelopes, with one holding £ 1,200, the other £ 2,400. One envelope of each pair, chosen at random, is handed to each of a number of people who act independently of each other; about half will receive an envelope with £ 1,200, the rest £ 2,400.

Some people will not swap, their average reward is (1200 + 2400)/2 = £ 1,800.

Of the rest, about half swap £ 1,200 for £ 2,400, the other half swap £ 2,400 for £ 1,200. Net result is that these people have an average reward (1,200 + 2,400)/2 = £ 1,800; the same as the first group.

So no advantage in swapping.

This is why I think knowledge of the first amount only helps if it gives a clue to the amount potentially available.

[factor-fiction]

Thanks for all your input to this. It just confirms that probability and decision making is a very difficult areas and even what appears to be a simple problem can cause great minds (as we have on here) to scratch their heads and disagree. As was mentioned, before many authors have written about this problem at great length.

I'm pretty certain I know the answer but I'll tie myself in knots trying to explain so I'll leave it there for now unless anyone wants to pick me up on it again or has any further thoughts after mulling it over for a few days.

Good night all and thanks. (I was hoping Buenchico would see the thread though- Prudie saw it but wisely sidestepped it!)

[ludwig]

// 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. //

I get that - there's a 50:50 chance of Loss vs Gain, but the amount you stand to gain is greater than the amount you stand to lose, which makes it a logical option to take the 50:50 chance and swap.

To make this more obvious, imagine that the other envelope contains either £10 million or £10 - I know what I'd do - I'd take a 50:50 chance and try for the 10 million. It'd be crazy not to.

But as I said, when you get down to smaller more realistic differences, the human element comes into play and overrides the cold logic.

[ludwig]

I posted the above without looking at all the inbetween stuff.

The best way to think of it is if you had to repeat the process 100 times.
I'll play you and I'll swap every time, whereas you'll stick with £2400 every time.

After 100 tries you'll have 100 X 2400 = £240,000

After 100 tries I'll have (50 x 1200) + (50 x 4800) = £300,000

We don't need a spreadsheet - we can assume from the law of averages that half the time I'll get the higher amount, and half the time I'll get the lower, but I'll still end up with more money.

--

This obviously assumes certain things such as the envelopes are all independent and they only contain a combination of 1200 and 2400 or 4800 and 2400, and we assume that we both pick the one with 2400 in as a starting position each time.

[Old_Geezer]

I believe there's an issue in that the original question is not quite right.

When one mentions a figure and then say the other envelope has either half or double that figure then the situation changes from when no amount is mentioned.

When you know you have £2400 in your first envelope, and the other envelope must hold either £1200 or £4800 then we are considering 2 possible scenarios, not 1. We may have a situation where £3600 is 'on the table', or we may have a situation where £7200 is 'on the table',

Consequently this skews the result and we get:

Attempt 1. £2400 in envelope 1, £1200 in envelope 2. Stick = £2400, Swop = £1200
Attempt 2. £2400 is still in envelope 1, £4800 in envelope 2. Stick = £2400, Swop = £4800

For the fair trial of 2 goes we get Stick = £4800, Swop = £6000, and Swop wins.

However for amounts hidden the 2 amounts have to be selected in advance (either £1200 & £2400 or £2400 and £4800, it doesn't matter but let's go with the former).

This time, because the amount is not known, we can get either £1200 or £2400 in the first envelope:

Attempt 1. £2400 in envelope 1, £1200 in envelope 2. Stick = £2400, Swop = £1200
Attempt 2. £1200 in envelope 1, £2400 in envelope 2. Stick = £1200, Swop = £2400

For the fair trial of 2 goes we get Stick = £3600, Swop = £3600, and it's a draw.

Perhaps you could start a new thread and we could all say it doesn't matter ? :-D

[factor-fiction]

That's why it's known as the two envelope Paradox, Ludwig. The more you think about it the more complicated our brain makes it. I agree with the first paragraph but start to disagree on your second. Basically all you know at the outset is that one envelope contains £X and the other contains £2X. You have a 50/50chance of picking the higher one first.

You then say that when considering a swap " the amount you stand to gain is greater than the amount you stand to lose, which makes it a logical option to take the 50:50 chance and swap.
But is that true?
You either started with £X or £2X, but don't know which you have. If it turned out that you'd started with the £X then you would gain £X (i.e. double your money) by swapping. If instead you'd started with the £2X envelope and swapped you'd be left with the smaller £X, so you'd have lost £X by swapping.

The flaw (which is the trap I fell into when i first looked at this) is to say I have an envelope with £Y in, therefore there is a 50% chance the other contains £2Y (so I'd gain £Y by swapping) and a 50% chance that the other contains £Y/2, so I'd lose half of £Y by swapping; therefore it's better to swap as the potential gain exceeds the potential loss. But that isn't the choice. There are only 2 envelopes not 3. What's in the other envelope has already been determined before the game started. That's the bit that is difficult to explain, and I know I'm not able to explain it very well.

Let's stand back from the problem and think of it as a simple choice. We are offered two envelopes- all we know is one contains more than the other. It doesn't really matter which we pick does it? If you choose one and then have the chance to change it, is there really any reason to swap? Whatever you do you always have a 50/50 chance of getting the bigger amount.

[Old_Geezer]

You say the flaw is to say I have an envelope with £Y in, therefore there is a 50% chance the other contains £2Y (so I'd gain £Y by swapping) and a 50% chance that the other contains £Y/2, so I'd lose half of £Y by swapping. But I believe that has been defined by the question. And the spreadsheet confirms the benefit so it's not a flaw in that case. In fairness were that penultimate paragraph not there then one may come up with a different answer.

[Old_Geezer]

Standing back from the problem then yes I agree there is no difference, but that is because you are looking at a single transaction and a benefit is only seen as an emergent quality from a number of runs. Emergance isn't applicable to a single transaction where there is a simple 50:50 chance of losing or gaining regardless the amounts involved.

But over many runs the criteria for the offer makes a difference to whether it pays to swop or not. If the offer involves £7200 in two envelopes when you swop and win, but just £3600 when you swop and lose, over a long period the swop will be beneficial.