The Two Envelopes Paradox

What’s going on

Someone puts money into two envelopes, and one of them holds exactly twice as much as the other. You get one at random and look inside: say $40. You may keep it, or swap it for the other one.

Here is a tempting argument:

• The other envelope holds either half of what you have, $20, or double, $80.
• Both seem equally likely, so on average it holds ½ × $20 + ½ × $80 = $50.
• That is 25% more than $40. So you should always switch.

But the argument never used the number 40: it says “switch” whatever you see. So you could switch without looking, and then the same argument says switch back, forever. And the holder of the other envelope can argue the same way, yet you cannot both gain.

Press Play 1,000 above. Each round, three players face the same envelopes: one always keeps, one always switches, one follows a rule we will come to. Keeping and switching both average about $76 a round. The 25% gain never shows up.

Where the argument breaks

The weak step is “both seem equally likely”. It cannot be true for every amount you might see.

In the scene the smaller amount is a whole number of dollars from $1 to $100. Press Why? to see what switching does, split by the amount in your envelope:

• If you see an odd amount, like $37, it cannot be double anything, so it is the smaller one. Switching wins.
• If you see an even amount up to $100, like $40, both cases really are equally likely. Here the 25% argument is right.
• If you see more than $100, you surely hold the bigger envelope. Switching loses half, and these are the biggest amounts in the game.

The small gains at small amounts and the large losses at large amounts cancel exactly. Over all rounds, switching gains nothing. That had to happen: before you open anything, the two envelopes are completely alike, so no blind rule can favour one of them.

Could someone fill the envelopes so that “half or double” is fifty-fifty for every amount? Then $1, $2, $4, $8 and so on, forever, would all need to be equally likely, and no list of infinitely many equal chances can add up to 1. Any real way of filling the envelopes has a top, and near the top switching hurts.

A rule that really helps

Looking inside is useful once you know something about how the amounts were chosen. In the scene the smaller amount is never above $100, so try: switch if you see $100 or less, keep otherwise.

• If you hold the smaller envelope, you always see $100 or less, so you switch and win.
• If you hold the bigger one, you keep it whenever it shows more than $100.

This rule ends with the bigger envelope three times out of four and averages about $94.63 a round, against $75.75 for keeping or switching blindly. Move the slider: no threshold beats $100.

Try it yourself

1. A different game

This time the host first puts $40 in your envelope. Then he tosses a coin: heads, he puts $80 in the other envelope; tails, $20. Should you switch?

Answer

Yes. Now "half or double, fifty-fifty" is really true, because a fair coin decides it after your amount is fixed. The other envelope averages $50, so switching gains $10 on average. The paradox came from treating the original game as if it were this one.

2. Read the number

In the scene’s game, with the smaller amount from $1 to $100, you open your envelope and see $150. Should you switch? What if you see $37?

Answer

With $150, keep: the smaller amount is at most $100, so $150 must be the bigger envelope. With $37, switch: an odd number cannot be twice a whole number, so $37 is the smaller one and the other holds $74.

3. Powers of two

Switch the scene to amounts $1, $2, $4 … $512 (the smaller amount is one of these ten, all equally likely). The best rule is “switch unless you see $1,024”. How often does it get the bigger envelope?

Answer

55% of the time. Half the time you hold the smaller envelope; you never see $1,024 then, so you switch and win. The other half you hold the bigger one, and the rule keeps it only if it shows $1,024, which happens one time in ten. So the chance is ½ + ½ × 1/10 = 0.55.

Go deeper

• In 1953 the Belgian mathematician Maurice Kraitchik told a version with two men comparing neckties: whoever has the nicer tie must give it to the other, and each man reasons that the bet favours him.
• There are ways of filling the envelopes where, whatever you see, switching gains on average. For example, make the smaller amount $2ⁿ with probability 2ⁿ ÷ 3ⁿ⁺¹. But then the average amount in an envelope is infinite, and averages stop behaving sensibly.
• Even if you know nothing about how the amounts were chosen, you can win more than half the time: pick a random threshold first, then switch only if your amount is below it. This trick is usually credited to the information theorist Thomas Cover.