(Single Winner ballots)
1. If it's a zero-info election, no information about winnability or likely frontrunners, then vote for all of the above-mean candidates.
2. Vote for the candidate for whom you'd vote in Plurality, and for everyone whom you like better.
3. Or vote for all the candidates whom you like better than what you expect from the election.
Number 2 is obvious. In Approval I, I showed why #1 maximizes utility expectationn by the mathematics introduced in that article. Here, I'll show why #3 maximizes utility by that same mathematics, and will then introduce, and show the utility expectation maximization of, some additional strategies, based on that same mathematics.
Remember that all this mathematics is optional. Strategy #2 above is obvious, without any mathematics. And stratgegy #3 clearly must improve your expectation.
But let me add one more: If it's zero-info, and you don't want to rate the candidates, and you're going only by your ranking of the candidates, then vote for the best half of the candidates. But of course if you can rank the candidates, you most likely can rate them, which makes it possible to use #1, above, and some additional ones to be described below.
Whenever I talk about tie probabilities here, I'm referring to the Pi or the Pij that I defined in Approval I. Strictly speaking, they're tie-or-neartie probabilities.
In Plurality, if you're sure that Smith & Jones will be the top 2 votegetters, then obviously you should vote for whichever of those 2 you prefer to the other. In Approval, vote for him and for everyone whom you like better. That could be called the "best frontrunner" strategy. It's the strategy that nearly everyone uses in Plurality.
There's a smooth, seamless gradation from the best-frontrunner strategy to the method that calculates the "strategic values" that I defined in Approval I. By that, I mean that we can add a little elaboration to "best frontrunner", and then a little more, and thereby make it as fancy as we want it to be, until we eventually have the Pij to put into the strategic value formula.
I'm going to show that gradation later here, but first I'd like to show why strategy #3 above maximizes utility expectation, by some reasonable approximations. Number 3 is the strategy of voting for all the candidates whom you like better than what you expect from the election. In other words, if you'd rather put candidate X in office instead of holding the election, then vote for candidate X in Approval, by this strategy.
I'll call that the "better than expectation" strategy. Here's why it maximizes utility expectation:
Say there are 4 candidates: a,b,c & d. From the strategy that I described yesterday for maximizing one's utility expectation, we should vote for candidate A if & only if:
Pab(Ua-Ub)+Pac(Ua-Uc)+Pad(Ua-Ud) > 0
As I was saying yesterday, it's a reasonable approximation to replace Pab with Pa*Pb, where Pa is the probability that if (before your vote is counted) there are 2 candidates tied for 1st, or the 2 top votegetters differ by only one vote, A is one of those 2 candidates.
So the inequality can be written:
PaPb(Ua-Ub)+PaPc(Ua-Ub)+PaPd(Ua-Ud) > 0
That can be written:
Pa[Pb(Ua-Ub)+Pc(Ua-Uc)+Pd(Ua-Ud)] > 0
We can drop the Pa, and have: Pb(Ua-Ub)+Pc(Ua-Uc)+Pd(Ua-Ud) > 0
That can be written:
Ua(Pb+Pc+Pd) - [PbUb+PcUc+PdUd] > 0
Let's say that Wi is the probability that i will win. Below, k is just some constant, one that's the same for all the Wi. If we assume that Pi is proportional to Wi, we can replace Pi with kWi.
The reason why we can approximately assume that Pi is proportional to Wi is that i is more likely to win than j is, then it's reasonable that, if there's a tie for 1st place, i is more likely to be in that tie than j is. As a rough approximation, we can suppose that if i is twice as likely to win than j is, we can guess that i is twice as likely as j to be in a tie for 1st if there is one.
So Pi = kWi
So Ua(Pb+Pc+Pd) becomes Ua(kWb+kWc+kWd), or kUa(Wb+Wc+Wd)
And that can be written:
(because if someone wins, the probability of it being a, b, c, or d adds up to 1)
So Ua(Pb+Pc+Pd) - [PbUb + PcUc + PdUd] > 0
kUa(1-Wa) - [kWbUb+kWcUc+kWdUd] > 0
dividing through by k, multiplying the left term out, and isolsting Ua:
Ua > WaUa+WbUb+WcUc+WdUd
The expression after the > sign is your expectation for the election--the probability of each candidate winning, multiplied by that candidate's utility, with all of those products summed to give your overall expectation for the election.
So, by our assumptions & approximations, the utility-maximizing strategy votes for the candidates who are better than the voter's expectation for the election.
In other words, if you vote for all the candidates whom you like better than what you expect from the election, all the candidates whom you'd rather put in office than hold the election, that will maximize your utility expectation, by a few reasonable approximations.
Furthermore, we can turn that around, and say that anyone who uses _any_ strategy that maximizes his utility expectation, is, whether intentionally or not, is voting for the candidates whom he likes better than his expectation for the election. That's because, by our reasonable assumptions, the strategy of voting for the candidates with positive strategic value is the only thing that will maximize utility expectation. Any different way to vote is sub-optimal. Therefore, however someone arrives at that way of voting that maximizes his utility expectation, he'll be voting for the candidates with positive strategic value (defined in Approval I). And that means that, by our reasonable approximations, he's also voting for all the candidates whom he prefers to what he expects from the election.
Why is that important? Because it means that Approval achieves this social optimization:
Approval maximizes the number of voters who consider the winner better than what they expected from the election. Approval maximizes the number of voters who'd have rather put the winner in office instead of holding the election.
In the zero-info case, we can also say that Approval maximizes the number of voters who consider the winner to be above-mean.
Though Approval has valuable guarantees related to getting rid of the lesser-of-2-evils problem (described by the criteria in the technical evaluation page), Approval has an additional, entirely separate advantage, because Approval maximizes the number of voters who'd have rather put the winner in office instead of holding the election.
Some feel that Approval doesn't quite equal Condorcet's ability to thoroughtly get rid of the lesser-of-2-evils problem, but, if you feel that you're getting a little less with Approval for that reason, then I suggest that the fairness of Approval's social optimization can outweigh that, meaning that there's a case for saying that this simplest and most modest of all single-winner reforms could be considered the best.
Approval brings a whole new benefit that other methods don't offer.
The more you find out about Approval, the better Approval is.
Now I'd like to answer a possible objection that someone could make, to the demonstration above:
One possible objection is that I've assumed that the win probabilities add up to 1, even though there could be ties. But when you guess the candidates' win probabilities, can you guess them so accurately that you can say that they're decisive win probabilities, not just probabilities of winning ultimately (maybe after a tiebreaker)? So it isn't so unreasonable for the Wi to add up to 1.
The demonstration above was written for 4 candidates. Obviously it could be written in a more general way that would cover all numbers of candidates, by such expressions as (W1U1 + W2U2...+Wn-1Un-1 +WnUn). I felt that writing it for 4 candidates instead of that more general form would be clearer.
In this article, Approval II, I've told why the better-than-expectation strategy maximizes utility expectation, and why that gives Approval its social optimization property. Approval III will discuss more Approval strategies, and show that they also are ways to maximize the voter's utility expectation.