| So, first, you're right to be skeptical. The basic claim is this: "Expected value (a mathematical object belonging to the very rigorous field of probability theory) is often used by economists in a non-rigorous way." The covert implication is something along the lines of, economists do this deliberately to gain the benefit of rigor that the empirical performance of economic theory hasn't earned. You should definitely interpret that first as a political argument, motivated by a metaphysical argument, motivated by a mathematical argument. And you're right to hear some alarm bells, because that's certainly a controversial claim. But it's not an outlandish claim. The field of psycho-economics is basically the study of the many ways that people observably, empirically don't make economic decisions on the basis of EV maximization, so what more need be said? And there are plenty of other metaphysical arguments that you probably accept in principle. Pascal's Wager, and the converse Pascal's Mugging, clearly show that EV maximization at least has to "break down at high energy levels", to borrow a metaphor. So, this thought experiment is one of those, but it's actually a pretty elegant one: In a series of two coin flips you have four outcomes, so your expected value is a quarter of (0.6 x 0.6) + (0.6 x 1.5) + (1.5 x 0.6) + (1.5 x 1.5) = about 4.4 over four = about 110% of your stake. So this is a winning gamble, so EV maximization predicts you will take it. However: more than half of that EV, (1.5 x 1.5)/4 = 56% out of that 110%, comes from the relatively rare event in which you win two coin flips in a row. The other three outcomes have to share the remaining 54% EV between them. It should make sense by now that if you lose, and play again, you can expect to lose again, despite still having a positive EV on the toss. So the essential question the author is asking, the thing that makes this metaphysically controversial but not mathematically so, is this: Assuming there's no mechanism to actually share the EV, how much should those three out of four parallel-universe versions of you who individually lose, and collectively have to come up with the cash to pay out 2x winnings, care about the one out of four of you who's taking those 2x winnings home? And the political question is: Does more than 1/4 of you still think EV is a good basis for decision making absent a mechanism to actually share the EV? |
Too late to edit but I feel like I should highlight this more, because it looks like the place something would be swept under the rug:
Yes, the individual who wins two flips in a row is still ahead even if they lose the next two flips. If they win the first four, they can afford to lose three more.
And if they win the first five, they can afford to lose… still only three. Five to four no longer breaks even.
That’s the thing that, if it doesn’t seem intuitive, is a meaningful insight. “This +EV dynamics is not ergodic” means:
In the limit, a) the individual who wins every single coin flip can afford to pay the losses of everyone else and profit, and b) no other individual breaks even.