If one uses surreals for chances then one can provide the required chance to make continuity happen.
There might be multiple ways to map a lexiographic weight to 1 + w + w^2 + w^3… but I would expect that simiarly that real functions can be scaled for no effect using different transfinites would just be a matter of consistency. Ie whether you map (1,2,0) to 1+2w or 1+ 2w^2 is a choice that can be made freely if it is just sticked to.
Then you can have a 1/w=e where 0<e<1 which can function as the p to make the lottery mix exactly ambipreferable.
If one uses surreals for chances then one can provide the required chance to make continuity happen.
There might be multiple ways to map a lexiographic weight to 1 + w + w^2 + w^3… but I would expect that simiarly that real functions can be scaled for no effect using different transfinites would just be a matter of consistency. Ie whether you map (1,2,0) to 1+2w or 1+ 2w^2 is a choice that can be made freely if it is just sticked to.
Then you can have a 1/w=e where 0<e<1 which can function as the p to make the lottery mix exactly ambipreferable.