[Election-Methods] [EM] Is "sincere" voting in Range suboptimal?

[Abd ul-Rahman Lomax]

Continuing my report on a study of voter-optimal Range Voting in a 
zero-knowledge candidate election with a sincere midrange candidate.

I realized that a way to study the expected utilities was to look 
only at situations where the voter's vote actually counts, that is, 
it makes a difference. To keep it simple, I ended up doing this by 
neglecting the very rare circumstance, in a large election, that 
there is a three-way near-tie. So I look at the three pairwise 
elections; we do not know which of these is the real one, we have no 
information about it.

If we did, then the optimal strategy would depend on which pairwise 
election was the one being resolved by the voter. And we know and 
acknowledge that where knowledge is sufficient, the voter's personal 
utility is enhanced by voting the extremes, i.e., Approval style. 
However, in some circumstances, where the risk is small, a voter may 
still consider that the normal overall improvement in social utility 
which results from voters voting sincerely makes it worthwhile.

There are three possible pairwise elections in the set, the AB pair, 
the AC pair, and the BC pair. In a large election, even if it were 
the case that many voters were voting strategically, the important 
pairwise contests, where the voter's vote can alter the outcome, show 
the same vote "dregs" as the possible voting pattern for a single 
voter. I modelled this by assigning a net vote of -3 to the candidate 
excluded from the pairwise election, thus our voter cannot bring this 
up to a tie and thus to a possible win; then the possible votes for 
the other two candidates were as before.

The results: expected utility for the "sincere" Range vote: 40. For 
the Approval style vote: 39. In the zero knowledge case, there is a 
slight improvement in utility by voting sincerely. How this is 
shifted when there is knowledge, I have not studied: I'd have to 
assign probabilities to the elections, I would use Range 3 instead of 
Range 2, or possibly even higher Range (but the vote patterns rapidly 
increase in number), but I assume that as the certainly of knowledge 
increases, the optimum Range Vote shifts toward Approval style, 
though details remain to be worked out.

Some critics have termed voting sincerely in Range, "altruistic." I'd 
strongly disagree. Giving up a small benefit, in a single 
transaction, in order to generate a larger social benefit is not 
altruism except when viewed in isolation. It looks like I'm going out 
of my way to help someone else. But I also know that if I treat 
people this way, they are more likely to treat me this way, and if 
everyone, or even a substantial fraction of people, so behave, *we 
all benefit.* This is the reason that actual human behavior in the 
Traveller's dilemma does not match the narrow calculation of 
immediate benefit that classic game theory would predict.

(In the Traveler's Dilemma, two travelers suffer loss of a valuable 
and identical item in their luggage, difficult to value. The airline 
luggage manager offers them a deal: they are asked to state the value 
of the item. If they both agree, that is what they get, up to $100. 
If they disagree, they both receive the lower of the stated values, 
except that the one with the lowest stated value gets a $2 bonus, 
subtracted from what is paid to the other.)

Contrary to classic game theory, I suggest (with others) that the 
optimum choice is $100, if the goal is to maximize return to the 
travellers, or the actual cost of the items, if overall SU is being 
maximized *including the airline.*

In the game theory analysis, there is a risk that the other traveller 
will go for maximum return, will be "greedy," or "selfish," as those 
who vote strategically were recently termed. Thus by stating $99 
instead of $100, the "selfish" traveller gains $1, costing the other 
traveller $3 below the $100 "bid."

So a common application of game theory would suggest that the payoff 
for stating $100 is lower than for stating $99. But then we can 
assume that the other traveler would realize this and so state $98, 
and the Nash equilibrium is actually at a bid of $2. Apparently, 
though, from experiment, most people are too smart to use the 
"rational" game theory approach and instead use a far more efficient 
and cooperative model, perhaps noticing that the maximum payoff to 
the set of players is for a bid of $100. By being "suckers" and 
taking the risk that the other player will bid $99, thus causing them 
to only get $97, they get a far better outcome than if they look for 
the best possible outcome for themselves personally.

Thus they risk the loss of $3 to a selfish player, and by being 
willing to take this risk, they generally maximize their gain, and if 
they suffer a loss due to the selfish manipulations of another, it is 
a small loss.

The scenarios where, allegedly, selfish voters exaggerate their 
preferences in order to maximize their personal outcome, and these 
selfish voters prevail, is one where the loss to the majority is 
small; voters who are considering the maximum total "payoff" do 
better, overall, than they would do, on average, if they themselves 
exaggerated.

To show this is complex unless one uses simulation, which is where 
IEVS comes in, Warren Smith's public-source voting simulator.

In any case, even if it were true that strategic voting seriously 
damaged Range, Range Voting would remain of interest as a voting 
method that maximized collective benefit where all voters are 
sincere. No doubt about it, strategic voting does damage, but it does 
not appear that it causes serious damage to sincere voters. And it is 
not a simple matter, aside from using a simulation, to quantify this; 
what too many writers have done is to note damage, fail to quantify 
it, and fail to balance it against benefit.

A voter who votes sincerely can certainly come to regret the vote; 
this is clear from the voting patterns I found in my study. However, 
this is not the norm (incidence 3/27) and is more than balanced by 
the votes where the voter would have regretted voting Approval style 
(incidence 5/27). In all cases the loss of utility was the same: 0.5, 
that is, 25%. For example, of the votes where the voter's vote can 
affect the outcome -- and these are, of course, the only ones where 
the voter's vote can properly be regretted -- there is only one of 
the possibilities where the outcome is the election of the least 
favorite, with a utility of zero. But, with an Approval style vote, 
it's a tie, thus the utility is 0.5. (Average of 0 and 1).

(Utilities are expressed in the example in the units of the Range 
votes; the range is 0-2, so 0 is the least favorable outcome and 2 is 
the most favorable.)

There is certainly a good possibility that I've erred in this analysis!