Talk (Work in Progress): Lennart Ackermans (MCMP)

Title:

Six versions of the Two-Envelope Problem

Abstract:

A discrete amount of money is put into an envelope, and twice that amount into another. The envelopes are shuffled, and subsequently labelled A and B. You are asked to pick one. After making the choice for envelope A, you are given the opportunity to switch. There's an argument (obviously mistaken) that you should: let x be the amount of money in envelope A. Then the amount in the other envelope must be either 2x or 1/2x, with both possibilities having a chance of 1/2. Thus, the expected value of sticking with envelope A is x while the expected value of switching is 1/2 * 2x + 1/2 *1/2x = 1.25x. It seems that you ought to switch! The challenge is to diagnose what is wrong with the argument.

Philosophical discussion of the Two-Envelope Problem has suffered from a lack of formal precision. This has led to confusion and miscommunication between philosophers and mathematicians. Richard Gill, a famous statistician, thinks philosophers are wasting their time on a problem that might be non-existent, writing: "[The Two-Envelope Paradox] is the kind of reason that formal probability theory was invented. Philosophers who work on [it] without knowing modern (elementary) probability are largely wasting their own time; at best they will reinvent the wheel." According to Gill, philosophers should give the argument a formally correct explication (if at all possible) before attempting to diagnose it.

Gill's challenge can be met. I discuss six versions of the paradoxical argument in a more formal manner. The mistakes in the arguments are revealed to be in one of three categories: (1) the argument makes a formal mistake such as an equivocation fallacy; (2) the argument uses an impoverished sample space; (3) the argument uses an invalid decision rule based on non-standard expected values. I review successful and unsuccessful existing diagnoses and discuss what kind of philosophical import the paradox does and does not have.