Consider the following game: At any time t∈[0,1], you may say “stop!”, in which case you’ll get the lottery that resolves to an outcome you value at (0,0,…) with probability t, and to an outcome you value at (1+t,0,…) with probability 1−t. If you don’t say “stop!” in that time period, we set t=1.
Let’s say at every instant in [0,1] you can decide to either say “stop!” or to wait a little longer. (A dubious assumption, as it lets you make infinitely many decisions in finite time.) Then you’ll naturally wait until t=1 and get a payoff of (0,0,…). It would have been better for you to say “stop!” at t=0, in which case you’d get (1,0,…).
You can similarly argue that it’s irrational for your utility to be discontinuous in the amount of wine in your glass: Otherwise you’ll let the waiter fill up your glass and then be disappointed the instant it’s full.
Consider the following game: At any time t∈[0,1], you may say “stop!”, in which case you’ll get the lottery that resolves to an outcome you value at (0,0,…) with probability t, and to an outcome you value at (1+t,0,…) with probability 1−t. If you don’t say “stop!” in that time period, we set t=1.
Let’s say at every instant in [0,1] you can decide to either say “stop!” or to wait a little longer. (A dubious assumption, as it lets you make infinitely many decisions in finite time.) Then you’ll naturally wait until t=1 and get a payoff of (0,0,…). It would have been better for you to say “stop!” at t=0, in which case you’d get (1,0,…).
You can similarly argue that it’s irrational for your utility to be discontinuous in the amount of wine in your glass: Otherwise you’ll let the waiter fill up your glass and then be disappointed the instant it’s full.
Why would you wait until t=1? It seems like at any time t the expected payoff will be (1−t2,0,…), which is strictly decreasing with t.
Oh you’re right, I was confused.
I’ve no idea if this example has appeared anywhere else. I’m not sure how seriously to take it.