The main thrust is that either the utility of Heaven isn’t the best possible thing, or it is the best possible thing and a mixed strategy of betting on heaven with probability p and betting on nothing with probability 1-p also gives infinite utility, for positive p. Thus, if Heaven is the best possible thing, Pascal’s Wager doesn’t rule out mixed strategies.
If someone could check my math here—I don’t think surreal numbers let you assign a utility to the St. Petersburg paradox. The expected utility received at each step is 1, so the total utility is 1 + 1 + 1 + … . Suppose that sum is X. Then X + 1 = X. This is not true for any surreal number, right?
Thanks for the pointer to a free version of Hajek’s “Waging War on Pascal’s Wager” paper. One of his alternative formulations uses surreal numbers for utilities, much to my surprise.
The main thrust is that either the utility of Heaven isn’t the best possible thing, or it is the best possible thing and a mixed strategy of betting on heaven with probability p and betting on nothing with probability 1-p also gives infinite utility, for positive p. Thus, if Heaven is the best possible thing, Pascal’s Wager doesn’t rule out mixed strategies.
If someone could check my math here—I don’t think surreal numbers let you assign a utility to the St. Petersburg paradox. The expected utility received at each step is 1, so the total utility is 1 + 1 + 1 + … . Suppose that sum is X. Then X + 1 = X. This is not true for any surreal number, right?