I had the same initial reaction. I believe the logic of the proof is fine (it is similar to the Mazur swindle), basically because it it not operating on real numbers, but rather on mixtures of distributions.
The issue is more: why would you expect the dominance condition to hold in the first place? If you allow for unbounded utility functions, then you have to give it up anyway, for kind of trivial reasons. Consider two sequences Ai and Bi of gambles such that EA_i<EB_i and sum_i p_iEA_i and sum_i p_i EB_i both diverge. Does it follow that E(sum_i p_iA_i)< E(sum_i p_i B_i) ? Obviously not, since both quantities diverge. At best you can say <=. A bit more formally; in real analysis/measure theory one works with the so-called extended real numbers, in which the value “infinity” is assigned to any divergent sum, with this value assumed to be defined by the algebraic property x<=infinity for any x. In particular, there is no x in the extended real numbers such that infinity<x. So at least in standard axiomatizations of measure theory, you cannot expect the strict dominance condition to hold in complete generality; you will have to make some kind of exception for infinite values. Similar considerations apply to the Intermediate Mixtures assumption.
With surreals I might have transfinite quantities that can reliably compare every which way despite both members being beyond a finite bound. For “tame” entities all kinds of nice properties are easy to get/prove. The game of “how wild my entities can get while retaining a certain property” is a very different game. “These properties are impossible to get even for super-wild things” is even harder.
Mazur seems (atleast based on the wikipedia article) not to be a proof of certain things, so that warrants special interest whether the applicability conditions are met or not.
I had the same initial reaction. I believe the logic of the proof is fine (it is similar to the Mazur swindle), basically because it it not operating on real numbers, but rather on mixtures of distributions.
The issue is more: why would you expect the dominance condition to hold in the first place? If you allow for unbounded utility functions, then you have to give it up anyway, for kind of trivial reasons. Consider two sequences Ai and Bi of gambles such that EA_i<EB_i and sum_i p_iEA_i and sum_i p_i EB_i both diverge. Does it follow that E(sum_i p_iA_i)< E(sum_i p_i B_i) ? Obviously not, since both quantities diverge. At best you can say <=. A bit more formally; in real analysis/measure theory one works with the so-called extended real numbers, in which the value “infinity” is assigned to any divergent sum, with this value assumed to be defined by the algebraic property x<=infinity for any x. In particular, there is no x in the extended real numbers such that infinity<x. So at least in standard axiomatizations of measure theory, you cannot expect the strict dominance condition to hold in complete generality; you will have to make some kind of exception for infinite values. Similar considerations apply to the Intermediate Mixtures assumption.
With surreals I might have transfinite quantities that can reliably compare every which way despite both members being beyond a finite bound. For “tame” entities all kinds of nice properties are easy to get/prove. The game of “how wild my entities can get while retaining a certain property” is a very different game. “These properties are impossible to get even for super-wild things” is even harder.
Mazur seems (atleast based on the wikipedia article) not to be a proof of certain things, so that warrants special interest whether the applicability conditions are met or not.