I think I must be missing something. As the number of traders increases, each trader can be less risk averse as their personal wealth is now a much smaller fraction of the whole, and this changes their strategy. In what way are these individuals now not EU-maximizing?
That example with traders was to show that in the limit these non EU-maximizers actually become EU-maximizers, now with linear utility instead of logaritmic. And in other sections I tried to demonstrate that they are not EU-maximizers for a finite number of agents.
First, in the expression for their utility based on the outcome distribution, you integrate something of the formf(x1,x2)p(x1)p(x2)dx, a quadratic form, instead of f(x)p(x)dx as you do to compute expected utility. By itself it doesn’t prove that there is no utility function, because there might be some easy cases like ∫(x1+x2)p(x1)p(x2)dx1dx2=∫x1p(x1)dx1+∫x2p(x2)dx2, and I didn’t rigorously proof that this utility function can’t be split, though it feels very unlikely to me that something can be done with such non-linearity.
Second, in the example about Independence axiom we have U(0.5A+0.5B)≠0.5U(A)+0.5U(B), which should have been equal if U was equivalent to expectation of some utility function.
I think I must be missing something. As the number of traders increases, each trader can be less risk averse as their personal wealth is now a much smaller fraction of the whole, and this changes their strategy. In what way are these individuals now not EU-maximizing?
That example with traders was to show that in the limit these non EU-maximizers actually become EU-maximizers, now with linear utility instead of logaritmic. And in other sections I tried to demonstrate that they are not EU-maximizers for a finite number of agents.
First, in the expression for their utility based on the outcome distribution, you integrate something of the formf(x1,x2)p(x1)p(x2)dx, a quadratic form, instead of f(x)p(x)dx as you do to compute expected utility. By itself it doesn’t prove that there is no utility function, because there might be some easy cases like ∫(x1+x2)p(x1)p(x2)dx1dx2=∫x1p(x1)dx1+∫x2p(x2)dx2, and I didn’t rigorously proof that this utility function can’t be split, though it feels very unlikely to me that something can be done with such non-linearity.
Second, in the example about Independence axiom we have U(0.5A+0.5B)≠0.5U(A)+0.5U(B), which should have been equal if U was equivalent to expectation of some utility function.