I am confused about something. You write that a preference ordering L⪯M is geometrically rational ifGU∼PEO∼LU(O)≤GU∼PEO∼MU(O).
This is compared to VNM rationality which favours L⪯M if and only if EO∼LU(O)≤EO∼MU(O).
Why, in the the definition of geometric rationality, do we have both the geometric average and the arithmetic average? Why not just say “an ordering is geometrically rational if it favours L⪯M if and only if GO∼LU(O)≤GO∼MU(O) ” ?
As I understand it, this is what Kelly betting does. It doesn’t favour lotteries over either outcome, but it does reject the VNM continuity axiom, rather than the independence axiom.
I am confused about something. You write that a preference ordering L⪯M is geometrically rational ifGU∼PEO∼LU(O)≤GU∼PEO∼MU(O).
This is compared to VNM rationality which favours L⪯M if and only if EO∼LU(O)≤EO∼MU(O).
Why, in the the definition of geometric rationality, do we have both the geometric average and the arithmetic average? Why not just say “an ordering is geometrically rational if it favours L⪯M if and only if GO∼LU(O)≤GO∼MU(O) ” ?
As I understand it, this is what Kelly betting does. It doesn’t favour lotteries over either outcome, but it does reject the VNM continuity axiom, rather than the independence axiom.