Forest was the first to mention the Better-Than-Expectation strategy for Approval--the strategy whereby a voter votes for the candidates who are better than his/her expectation for the election, better than the value of the election. So the voter using that strategy votes for a candidate if that candidate is so good that s/he would rather have that candidate in office than hold the election.
One can come up with situations in which that isnīt optimal. But it maximizes oneīs utility expectation if certain approximations or assumptions are made. One usual assumption is that there are so many voters that oneīs own ballot wonīt change the probabilities significantly. By one approach, itīs also necessary to assume that the voters are so numerous that ties & near-ties will have only 2 members, and that Weberīs Pij = Wi*Wj, the product of the win-probabilities of i & j.
But, instead of the last 2 assumptions named in the previous paragraph, it would also be enough to assume that when your vote for a candidate increases his win-probability, it decreases everyone elseīs win-probability by a uniform factor.
Thatīs the approach that Russ used, except that he didnīt state that assumption.
Russ, donīt take any of this as criticism--Iīm just telling you so that youīll know.
In your derivation-description, you stated the goal "to keep the sum of all probabilities at unity without changing the probability ratios." But keeping the other candidates win-probabilities in the same ratios isnīt a goal of the derivation; itīs an assumption by which Better-Than-Expectation maximizes the voterīs utility expectation. Itīs important to state assumptions, and that particular assumption is really key to the derivation.
You said that delta Pj the increase in candidate jīs win-probability if the voter votes for j. But later you say that actually that increase is different from delta Pj. Preceding a quantiy by delta usually indicates a change in that quantity. You started out that way, but then, in your derivation, delta Pj no longer represented that change.
One could let delta Pj really always stand for the increase in jīs win-probability. That makes for a more direct derivation:
Let E = your expectation in the election if you donīt vote for j.
Let Pj = candidate jīs probability of winning if you donīt vote for j.
Let delta Pj = the amount by which jīs win-probability increases if you vote for j.
Let Uj be jīs utility for that you.
So, if the you vote for j, jīs win-probability will be Pj + delta Pj.
So, if the you vote for j, then jīs contribution to the expectation is (Pj + delta Pj)Uj.
When the you vote for j, whatīs the combined contribution of the other candidates to the expectation? Well, first, whatīs their contribution if you donīt vote for j? Itīs E minus jīs contribution if you donīt vote for j. Thatīs E - PjUj.
Now you want their combined contribution to the expectation when you make their wins less probable by voting for j. Aside from the fact that it turns out to accomplish our goal, itīs a reasonable simplifying assumption to assume that we reduce their win probabilities by a uniform factor.
Then, the combined contribution of the non-j candidates to your expectation is itīs initial value,
E - PjUj, mulitplied by the factor by which you reduce their win probabilities.
If you donīt vote for j, the probability that a non-j candidate will win is 1 - Pj.
If you vote for j, the probability that a non-j candidate will win is 1 - Pj - delta Pj. Thatīs one minus jīs new probability of winning.
So, multiply the initial non-j candidatesī expectation contribution, E - PjUj, by the ratio of the probabilities in he previous 2 paragraphs:
(E - PjUj) * ( (1 - Pj - delta Pj)/(1 - Pj)) Thatīs the j candidatesīnew contribution to your expectation.
So, add jīs new contribution to the expectation and that of the non-j candidates:
(Pj + delta Pj)Uj + (E - PjUj)(1 - Pj - delta Pj)/(1 - Pj)
We want the new expectation to be more than the initial expectation:
E < (Pj + delta Pj)Uj + (E - PjUj)(1 - Pj - delta Pj)/(1 - Pj)
Solve that for Uj. You get:
Uj > E
Because of the directness of that derivation, it doesnīt need summations to be written, and that makes it acceptable to people who wouldnīt want to look at something with summation notation.
Also, because delta Pj continues to mean the increase in Pj, this derivation is straightforward and direct.
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