Approval Percentages

[Clay Shentrup]

I'd like to know from Warren, what were the average approvals per ballot as a function of the number of candidates, at both ends of the honesty/strategy spectrum?

[Warren D Smith]

half-approved.


--
Warren D. Smith
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[Brian Langstraat]

Warren,

In Approval Voting, approving about 50% of the candidates maximizes an individual voter's influence on an election, since approving 0% and 100% have no influence on the election.

Is there a mathematical proof or other research that proves that approving a specific percentage of the candidates maximizes an individual voter's influence on an election?

Is there a difference between this specific percentage for elections with an even or odd number of candidates?

[Clay Shentrup]

On Thursday, January 12, 2017 at 10:26:48 AM UTC-8, Warren D. Smith (CRV cofounder, http://RangeVoting.org) wrote:

half-approved.

Because you're randomly picking the frontrunners? What about for "honest" voters just responding to average utilities? Why would that be exactly half?

[Warren D Smith]

In some versions of my sims, such as random-normals as utilities,
the strategic and honest approval fractions both are on average 50%,
and you can basically see that because of "reflection symmetry."

In other versions of the sims in unsymmetric scenarios, not necessarily
so.

In certain simplified models, similarly, 50% is the best
way to approval vote, maximizing your "power." E.g. that discriminates
between the most candidate-pairs. However, it is not so
simple in fancier models.

Here were some experiments comparing some voting strategies:
http://www.rangevoting.org/RVstrat3.html

[Brian Langstraat]

After reading http://www.rangevoting.org/RVstrat3.html, I was confusing "mean-based thresholding" with what could be called "middle-based thresholding" with definitions:

Mean-based thresholding. The voter gives max to every candidate at least as good as the average value of all candidates, and gives min to the others.

Middle-based thresholding. The voter gives max to every candidate at least as good as the middle value of all candidates, and gives min to the others.

[The middle would be the same as the median, but fixes a flaw with medians.

In cases of multiple median value candidates, then the remaining approval votes among the median candidates would be randomly selected.]

In certain simplified models, similarly, 50% is the best 

way to approval vote, maximizing your "power."

Middle-based thresholding forces a 50% (+1 for odd number of candidates) approval vote to maximize voter power, while mean-based thresholding could result in a single (bullet) approval vote.

Example:

Candidate   Sincere   Mean (0.32)   Middle (0.3)

A                1.0           1                  1

B                0.3           0                  1

C                0.3           0                  1

D                0.0           0                  0

E                0.0           0                  0

I assume that "mean-based thresholding" would be better than "middle-based thresholding" per the proof for the theorem:

Mean-based thresholding is optimal range-voting strategy in the limit of a large number of other voters, each random independent full-range.

I think that the expected utility of "mean-based thresholding" and "middle-based thresholding" would be very similar since the mean and middle would usually result in the same approvals.  Any difference in approvals would likely approach the same expected utility after many trials.

Have there been any simulations comparing "mean-based thresholding" to "middle-based thresholding" (or "median-based thresholding without equal sincerity values")?

Is voter utility more effective than voter power?

[William Waugh]

Given Approval Voting, I don't see why a voter would want to give up the strategic advantage of simulating Range Voting using probability.

On Thursday, April 20, 2017 at 12:35:28 PM UTC-4, Brian Langstraat wrote https://groups.google.com/d/msg/electionscience/J_auapPG-DI/qbdDFNpaAgAJ

[Brian Langstraat]

William,

I think that simulations in http://www.rangevoting.org/RVstrat3.html were for a sincere/honest voter using various voting strategies without knowledge of how other voters could vote.

I was not considering strategies that involved knowledge of how other voters could vote.

I will try to breakdown your reply, since it is quite dense.

Given Approval Voting, I don't see why a voter would want to give up the strategic advantage of simulating Range Voting using probability.

You are assuming that there is at least one highly strategic voter.

A highly strategic voter wants the election results to maximize their expected value (sum of each probability multiplied by each candidate value).

The probabilities would be based on knowledge of how other voters could vote such as polling statistics.

The candidate value of each candidate would be based on their utility to the voter.

Scores within a Range would be given to each candidate to optimize the voter's strategic advantage (expected value).

Approval votes would be derived in a strategic way from the range votes.

Is this a decent interpretation?

How should each step be optimized?

Would the typical voter be able to be this highly strategic?

If every voter was highly strategic, then would the final result be worse than honest voters (http://www.rangevoting.org/BR52002bw.png)?

[William Waugh]

I intersperse replies below.

On Monday, April 24, 2017 at 12:31:32 PM UTC-4, Brian Langstraat wrote:

William,

I think that simulations in http://www.rangevoting.org/RVstrat3.html were for a sincere/honest voter using various voting strategies without knowledge of how other voters could vote.

The "without knowledge" part is correct, as in that writing, Warren D. Smith, Ph.D. says he reports on modeling of a "zero-info" election. That's what info he is referring to. He models several strategies, only one of which is honest.
 

I was not considering strategies that involved knowledge of how other voters could vote.

Well, that's pretty important.

I will try to breakdown your reply, since it is quite dense.

[You quote me accurately:] 

Given Approval Voting, I don't see why a voter would want to give up the strategic advantage of simulating Range Voting using probability.

You are assuming that there is at least one highly strategic voter.

I assume that after party strategists become familiar with how a voting system works, the bulk of the voters will act as highly strategic.
 

A highly strategic voter wants the election results to maximize their expected value (sum of each probability multiplied by each candidate value).

Good definition.
 

The probabilities would be based on knowledge of how other voters could vote such as polling statistics.

Yes.
 

The candidate value of each candidate would be based on their utility to the voter.

Yes.
 

Scores within a Range would be given to each candidate to optimize the voter's strategic advantage (expected value).

Yes. I think this involves exaggerating the score given to a compromise candidate, if the voter thinks her favorite candidate is unpopular, and if the voter has a preference between the "evil" candidates whose campaigns have the big-money support. If I expect 1% of the other voters to support my candidate, and if I care about the evils, I think I should score the lesser evil at 1% from the high end of the scale, relative to the width of the whole scale.
 

Approval votes would be derived in a strategic way from the range votes.

Yes. Map the range linearly to [0, 1]. Treat each candidate's score on that range as a probability. Approve that candidate with that probability. 

Is this a decent interpretation?

Yes, it's exactly what I meant.
 

How should each step be optimized?

Stated above.
 

Would the typical voter be able to be this highly strategic?

I think that in the long run under any given system of rewards and punishments, people tend to behave according to the actual relations of power. That is why capitalism makes people behave as "greedy". Greed, as a personal character flaw, is not the problem; the system is the problem. Similarly, I suggest that after a given voting system has been in place for a time, people find and act on the power relations. This will lead them to a strategy that will serve their interests quite well as compared to alternative strategies such as the "honest" strategy. I was tempted to say it will lead them to the strategy that will serve their interests best, but as pointed out in this forum, there is a proof in Game Theory that there isn't necessarily a single strategy that will do the optimal thing. The game may be somewhat like rock-paper-scissors. However, I suspect that with Range Voting, there are strategies that are so effective that they'll be found and fixed on by the voters. Some might not understand the reasoning for using the particular strategies, but they might trust party strategists to tell them how to vote.
 

If every voter was highly strategic, then would the final result be worse than honest voters (http://www.rangevoting.org/BR52002bw.png)?

It is my opinion that this question is not worth investigating, on the grounds that honest voting is just a theoretical construct that never will get realized in practice when public office is involved (it can happen among friends deciding what restaurant to go to). People want to exercise what political power they can, because they care about public policy. A plus of Range Voting is that it gives everyone the same amount of power, which is not true of for example the choose-one plurality system (called First Past the Post for whatever crazy reasons, but there it is).