## A fallacy with phones

Here’s a modern version of a problem I first encountered in one of Martin Gardner’s engrossing little books…

Two economics students, Zack and Yvonne, meet on their first day back after the winter break. Each of them is showing off a new 4G mobile phone that they were given as a Christmas present. Neither knows how much either phone cost, but it doesn’t take long for each of them to start worrying that the other’s phone might be better than theirs, and to become unhappy about this. (Economics students can be very competitive people.)

The obvious solution to their worries would be to look up the prices of both phones and compare them. Neither Yvonne nor Zack wants to do this, though, because each knows there’s a 50% chance that they’ll discover the other person’s phone is better than theirs.(Economics students don’t like to do anything unless they can expect to win. They’re willing to gamble, but the odds have to be on their side!)

A passing maths student, Xavier, suggests a way they can resolve the situation by means of a game, in which he will act as referee. If they both agree to play then Xavier will find out the prices of both phones, and tell both Zack and Yvonne whose phone is more expensive. The person who has the more expensive phone must then give their phone to the person who has the less expensive phone, while the person with the less expensive phone gets to keep theirs too.

Zack thinks about this for a moment, and reasons as follows. “Let’s say my phone is worth £Z. Now, there’s a 50% chance that Yvonne’s phone is worth more than mine, and a 50% chance that it’s worth less. So, with probability 50% I will gain something worth more than £Z, and with probability 50% I will lose something worth £Z. This means that my expected profit is

$\dfrac{1}{2}\left( \mbox{something more than}\ \pounds Z\right) - \dfrac{1}{2}\pounds Z$.

Since my expected profit is positive, it is to my advantage to play the game.”

Meanwhile, however, Yvonne is reasoning in exactly the same way, and she concludes that it’s to her advantage to play the game… So how can Zack and Yvonne both expect to profit from playing the game?

(DP)

PS. You might reasonably ask what Xavier gets out of the whole affair. That’s easy: he gets to see economics students looking puzzled. Mathematicians have some very simple pleasures in life!

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### 3 Responses to A fallacy with phones

1. A Thompson says:

Funny problem! I don’t know what the exact problem with it is, but it is a zero sum game so they don’t both profit.

2. A Thompson says:

I see that wikipedia has a huge article on this, calling it the “two envelopes problem”.

• strathmaths says:

I should have guessed somebody would see through the updated setting! In fact this is specifically a version of the “necktie paradox”, which I think was the original form of the more general “two envelopes problem” — but as even the thought of wearing a necktie makes me feel uncomfortable, I prefer this version…