In the previous article in this series, Approval Strategy: Summary and Mathematical Basis, we discussed four Approval strategies:
- Acceptable/Unacceptable: If there are unacceptable candidates who could win, then vote for all the acceptables but for none of the unacceptables.
- Best-Frontrunner: Vote for whichever of the likely top two candidates you prefer, and also for every candidate you like more.
- Better-Than-Expectation: (for when it isn't obvious who the top two candidates are likely to be) Vote for each candidate whom you consider so good that you'd rather have him/her in office instead of holding the election.
- If no information is available on the candidates' chances of winning: Vote for the above-mean candidates.
Strategies 1 and 2 are obvious. In Approval Strategy I, we justified strategy 4 in terms of Weber's mathematical strategy. In this article, we'll justify strategy 3 in terms of the same mathematics. We'll also show that strategy 4 is a special case of strategy 3 for the case of a zero-info election.
Better-Than-Expectation is plausible without any mathematics, but we will demonstrate its optimality in terms of Weber's mathematical strategy.
When we discuss tie probabilities, we're referring to the Pi or the Pij that we defined in Approval I. Strictly speaking, they're tie/neartie probabilities.
A demonstration with four candidates will help explain why the Better-Than-Expectation strategy maximizes your utility expectation, with a few reasonable approximations. We could develop this more generally for any number of candidates, but we feel it will be clearer to a larger audience for the special case of four candidates.
Let the four candidates be designated A, B, C and D. From the strategy that we described in the previous article for maximizing one's utility expectation, we should vote for candidate A if and only if
Pab(Ua-Ub) + Pac(Ua-Uc) + Pad(Ua-Ud) > 0
As before, we approximate Pab with Pa*Pb, where Pa is the probability that (counting all ballots but your own) if there are two candidates tied for first, or if the top two votegetters differ by only one vote, A is one of those two candidates. So the inequality becomes
PaPb(Ua-Ub) + PaPc(Ua-Ub) + PaPd(Ua-Ud) > 0
This can be rearranged to
Pa[Pb(Ua-Ub) + Pc(Ua-Uc) + Pd(Ua-Ud)] > 0
We can factor out the Pa, and have
Pb(Ua-Ub) + Pc(Ua-Uc) + Pd(Ua-Ud) > 0
or, equivalently,
Ua(Pb+Pc+Pd) - [PbUb + PcUc + PdUd] > 0
Let Wi be the probability that i will win. Below, k is just some constant, one that's the same for each Wi. If we assume that Pi is proportional to Wi, we can replace Pi with kWi.
We can assume that Pi is approximately proportional to Wi because if i is more likely than j to win, then in the event of a tie for first place, i is more likely than j to be in that tie. 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 first if there is one. Hence
Pi = kWi
So Ua(Pb+Pc+Pd) becomes Ua(kWb+kWc+kWd), or kUa(Wb+Wc+Wd). But since Wb+Wc+Wd = 1-Wa, this can be simplifie
kUa(1-Wa)
So the expression
Ua(Pb+Pc+Pd) - [PbUb + PcUc + PdUd] > 0
becomes
kUa(1-Wa) - [kWbUb + kWcUc + kWdUd] > 0
By factoring out k, expanding the left term, and isolating the Ua term, we obtain
Ua > WaUa + WbUb + WcUc + WdUd
The right side 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, given our assumptions and approximations, the utility-maximizing strategy is to vote for the candidates who are better than the voter's overall 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, that will maximize your utility expectation.
Furthermore, we can turn that around and say that anyone who uses any strategy that maximizes his utility expectation is, whether intentionally or not, voting for the candidates 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 Strategy I). And that means that, by our reasonable approximations, he's also voting for all the candidates he prefers to what he expects from the election.
In the case of a zero-info election, your expectation from the election is the mean of all the candidates. So our derivation justifies strategy 4, the above-mean strategy for the zero-info case.
This result is important because it means that Approval Voting achieves the following social optimization. Approval maximizes the number of voters who consider the winner better than what they expected from the election. That is, Approval maximizes the number of voters who would rather put the winner in office instead of holding the election.
Some feel that Approval doesn't quite match Condorcet's ability to thoroughly eliminate the "lesser of two evils" problem, because Approval doesn't meet SFC, GSFC or SDSC. Likewise, Condorcet, in simulations, maximizes average social utility better than Approval does. However, Approval meets FBC, along with WDSC, and it can achieve the social optimization described above. Note that IRV meets none of the defensive strategy criteria, and it does significantly worse than Approval is social utility simulations.
Approval Voting is a better system than it seems at first to most people.
In this article, we've explained why the better-than-expectation strategy maximizes utility expectation, and why that gives Approval its social optimization property. Approval III will elaborate the Best-Frontrunner strategy.