Continuity: Consider the scenario where each of the [LMN] bets refer to one (guaranteed) outcome, which we’ll also call L, M and N for simplicity.
Let U(L) = 0, U(M) = 1, U(N) = 10**100
For a simple EU maximizer, you can then satisfy continuity by picking p=(1-1/10**100). A PESTI agent, OTOH, may just discard a (1-p) of 1/10**100, which leaves no other options to satisfy it.
The 10**100 value is chosen without loss of generality. For PESTI agents that still track probabilities of this magnitude, increase it until they don’t.
Independence:
Set p to a number small enough that it’s Small Enough To Ignore. At that point, the terms for getting L and M by that probability become zero, and you get equality between both sides.
Continuity and independence.
Continuity: Consider the scenario where each of the [LMN] bets refer to one (guaranteed) outcome, which we’ll also call L, M and N for simplicity.
Let U(L) = 0, U(M) = 1, U(N) = 10**100
For a simple EU maximizer, you can then satisfy continuity by picking p=(1-1/10**100). A PESTI agent, OTOH, may just discard a (1-p) of 1/10**100, which leaves no other options to satisfy it.
The 10**100 value is chosen without loss of generality. For PESTI agents that still track probabilities of this magnitude, increase it until they don’t.
Independence: Set p to a number small enough that it’s Small Enough To Ignore. At that point, the terms for getting L and M by that probability become zero, and you get equality between both sides.