A Post about Envelopes
Submitted by John C on April 6, 2009 - 18:34
I promised a few weeks ago that Iâ€™d make another post about â€œseemingly paradoxical mattersâ€, and Iâ€™m finally making good on that promise.1 Be forewarned, however, that this post doesnâ€™t really have anything to do with anything else. Itâ€™s just a cool paradox (or â€œpuzzleâ€ or â€œproblemâ€ or whatever other â€˜pâ€™-word you want to call it) I learned from my roommate.
The paradox hinges on the notion of expected value, so first weâ€™d better understand what that is. Letâ€™s say I flip a coin, and if it lands heads I give you a dollar. But if it lands tails, you have to give me a dollar. Then your expected value from this game is $0 overallâ€”in any given round youâ€™ll either gain or lose a dollar, but in the long run youâ€™ll break even. Your average expected gain from any single round is $0.
Letâ€™s change the numbers a bit. Now I give you $2 if the coin lands heads, and you give me $1 if itâ€™s tails. Assuming Iâ€™m not using a weighted coin, this sounds like a pretty sweet deal for you, right? In any given round you have an equal chance of winning $2 or losing $1. Over many rounds you should average a net gain of 50 cents per round (.5 * 2 + .5 * -1). So we say that the expected value of flipping the coin is $.50.
In general, you can calculate the expected value of any random variable by multiplying the probability of some outcome by the value of that outcome, then summing these over all possible outcomes. For example, the expected value of rolling a die would be (1/6) * (1 + 2 + 3 + 4 + 5 + 6) = 3.5, since every value from 1 to 6 has an equal probability of coming up.
Now that we have this idea of expected value, we can also define what it means to play a game rationally. Quite simply, a rational player is someone who always makes the choice that maximizes their expected value from the game.
Alright. With those preliminaries out of the way, letâ€™s get to the good stuff. Hereâ€™s the game: I offer you two envelopes, both of which contain checks written out to you. One of them is worth twice as much as the other. With no other information, you pick one of the envelopes and get to keep whateverâ€™s inside. Not a bad game for you, eh?
So letâ€™s say youâ€™ve selected one of the envelopes. But before you can open it, I interrupt you:
â€œHold up! If youâ€™re having regrets, Iâ€™ll let you switch to the other envelope, no strings attached.â€
â€œWell thatâ€™s kind of a dumb offer, Greg. If I had really wanted the other envelope, I would have chosen it in the first place. What do you take me for? Some kind of flippity-floppity flip-flopper?â€
â€œOf course not! Iâ€™d never dream of insinuating such a thing. But hereâ€™s the deal: I just so happen to know that the envelope in your hand contains a check for $10. (I know this because I put the check in there myself.) Which means that this other envelopeâ€”the one Iâ€™m holding right hereâ€”has either $5 or $20 in it. Itâ€™s a 50-50 chance either way, right?â€
â€œIâ€¦ guess? Sure.â€
â€œThat means the expected value of my envelope is .5 * $5 + .5 * $20 = $12.50. As a rational individual, you should obviously switch envelopes, since the expected value from switching is higher than what youâ€™ve got now.â€
â€œBut Greg, that is stupid. If I had chosen that other envelope in the first place, you could have gone through the exact same song and dance, convincing me to switch to this one.â€
â€œSo why would I switch?â€
â€œBecause your expected value is higher, and youâ€™re rational.â€
â€œBut I donâ€™t want to switch.â€
â€œBut you should.â€
â€œBut I like this envelope.â€
â€œSo are you some kind of irrational crazy-person, then? Is that what you are?â€
â€œYou know what? Screw this.â€
And then you punch me in the face and snatch the other envelope from my hand. As you run away, you open the other envelope to find a $5 check inside.
â€œThat sneaky snake! That snakey sneak! I knew he was full of baloney.â€
Butâ€¦. was I? Was there really anything wrong with my expected value argument?
1 You thought I forgot, didnâ€™t you? I didnâ€™t forget! I never forget.