Fun with St. Petersburg

A generous patron makes an offer to you. You are to pick out a positive integer n and you will get 2ⁿ units of value. You have the ability to pick out any positive integer at no cost to yourself (maybe you can engage in a supertask and name long numbers really fast).

You think about naming a million, but then a billion would pay so much better, and a billion and two is four times better! You agonize. And then you have a brilliant idea. You will randomize by choosing positive integer n with probability 2⁻ⁿ (say, by flipping a coin until you get heads and counting how many flips that took). Your expected payoff will be

• (1/2)(2) + (1/4)(4) + (1/8)(8) + ... = ∞.

That beats any specific number you could choose. So you go for it.

And, poof, you get 4. Regrets! You don’t want to stick to what the random choice gave you, as you’ll “only” get 2⁴ = 16 units of value. Disappointing! So you try again. You choose another positive integer. Now it is, mirabile dictu, a billion and two. But you think: 2^1000000002 may be a lot, but infinity is more, and if you randomly choose another number, your expected payoff is ∞. So you randomly choose again. And whatever you get, you are dissatisfied.

Fun with St Petersburg

Consider any game, like St Petersburg where the expected payoff is infinite but the prizes are guaranteed to be finite. For instance, a number x is picked uniformly at random in the interval from 0 to 1 not inclusive, and your prize is 1/x.

Suppose you and I independently play this game, and we find our winnings. Now I go up to you and say: “Hey, I’ve got a deal for you: you give me your winnings plus a million dollars, and then you’ll toss a hundred coins, and if they’re all heads, you’ll get one percent of what I won.” That’s a deal you can’t rationally refuse (assuming I’m dead-set against your negotiating a better one). For the payoff for refusing is the finite winnings you have. The payoff for accepting is −1000000 + 2⁻¹⁰⁰⋅0.01⋅(+∞) = +∞.

Wow!

Now let’s play doubles! There are two teams: (i) I and Garibaldi, and (ii) you and Delenn. The members of each team don’t get to talk to each other during the game, but after the game each team evenly splits its winnings. This is what happens. The house calculates two payoffs using independent runs of our St Petersburg style game, w₁ and w₂. I am in a room with you; Garibaldi is in a room with Delenn. I and Delenn are each given w₁; you and Garibaldi are each given w₂. Now, by pre-arrangement with Garibaldi, I offer you the deal above: You give me a million, and then toss a hundred coins, and then you get one percent of my winnings if they’re all heads. You certainly accept. And Garibaldi offers exactly the same deal to Delenn, and she accepts. What’s the result? Well, the vast majority of the time, the Pruss and Garibaldi team ends up with all the winnings (w₁ + w₂ + w₁ + w₂ = 2w₁ + 2w₂), plus two million, and the you and Delenn team end up out two million. But about once in 2¹⁰⁰ runs, the Pruss and Garibaldi team ends up with 1.99w₁ + 1.99w₂, plus two million, while you and Delenn end up with 0.01w₁ + 0.01w₂ − 2000000.

And, alas, I don’t see a way to use Causal Finitism to solve this paradox.