<div dir="ltr"><div><div><div><div>In our last post we showed that direct approval voting for parties does not yield a satisfactory answer to question two, i.e. there are strategic incentives against directly approving the moderate party, even when many of the voters consider it almost as good as their favorite extreme party.<br>
<br></div>The solution to this problem comes from Jobst Heitzig, in an idea which he came up with in the context of lottery methods, but which applies equally well in this PR context. Let's call the idea "Conditional Approval."<br>
<br></div>The idea is to give the voters a way of saying "I'll cooperate if they will cooperate, and I'll defect if they defect." In other words, "Put me down for approving M (the moderate compromise party) as long as there are 74 other voters who are willing to do the same, otherwise no."<br>
<br></div>Now suppose that all 75 voters in the middle two factions indicate this conditional approval of M on their conditional approval ballots (along with unconditional approval of their favorites). I claim that this position is a strategic equilibrium, because this position makes impossible the pesky unilateral defections we had to deal with in our direct approval approach. If any of the 75 voters from the two middle factions defect, then the deal is off, and the payoffs jump from the (cooperate, cooperate) corner of the (old) payoff matrix to the (defect, defect) corner, resulting in a loss for both of the middle factions.<br>
<br></div><div>Here's a recap of how to combine our answers to questions one and two into a PR method that will tend to give seats to moderate compromise parties when they exist:<br><br></div><div>First the voters submit conditional approval ballots which may give unconditional approval to some parties and conditional approval to others. Your conditional approval is indicated by specifying the minimum percentage of other voters that would have to also give conditional approval to the party in question before it could have your approval.<br>
<br></div><div>From these conditional approval ballots, standard approval ballots are constructed respecting the conditions specified by the voters, in each case awarding the maximum approval consistent with the specifications.<br>
<br></div><div>Then these standard approval ballots are used as input to Martin Harper's method of assigning votes to parties: the vote of an approval ballot is awarded to the party (among those approved on that ballot) which is approved on the greatest number of other ballots.<br>
<br></div><div>Seats are assigned to parties in proportion to the number of votes awarded them.<br><br></div><div>That's it!<br><br></div><div>Jobst has worked out all of the nice properties of this method in the context of lotteries, which (except for the fractional seat problem) should carry over mutatis mutandis to this PR setting.<br>
<br></div><div>I think this method should be given a chance in the aforementioned deeply divided nations. (However, if the likes of Obama and Kissinger are favored for Nobel Peace prizes, then Harper, Heitzig, and Simmons should not get their hopes up.)<br>
<br></div><div>Thoughts? Questions?<br></div><div><div><div><div><div><div><div><div class="gmail_extra"><br></div><div class="gmail_extra">Forest<br></div><div class="gmail_extra"><br><div class="gmail_quote">On Wed, Jun 25, 2014 at 4:29 PM, Forest Simmons <span dir="ltr"><<a href="mailto:fsimmons@pcc.edu" target="_blank">fsimmons@pcc.edu</a>></span> wrote:<br>
<blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex"><div dir="ltr"><div><div><div><div><div><div><div><div><div>Now let's take up question 2 which, in view of our tentative answer to question one, can be reformulated in the following way:<br>
<br></div>Given honest benefit expectations<div class=""><br>
<br>10 A(100)<br>
30 A(100)>M(80)<br>
45 B(100)>M(80)<br>
15 B(100)<br>
<br></div></div>from the parties A, B, and M,<br><br></div>how to elicit the approvals that we needed as input to our answer to question one?<br><br></div>The input approvals for our Martin Harper inspired method were<div class="">
<br><br>10 A<br>
30 A, M<br>
45 B, M<br>
15 B<br>
<br></div></div>There is a pretty good case that these would be sincere approvals, so why not just instruct the voters to list the parties that they approve of?<br><br></div>The trouble is that under direct approval voting the above approvals would not form a Nash equilibrium. In particular the third faction would have a great incentive to defect from this position.<br>
<br></div>to see this let's compare the expected benefit of the third faction members before and after unilateral defection :<br><br></div>Before: 10*0+30*80+45*80+15*100 = 7500<br></div>After: 10*0+30*80+45*100+15*100 = 8400<br>
<div><div><div><div><div><div><div><div><div><br></div><div>Their expectation difference is 45*(100-80)=900<br><br></div><div>Here's the full payoff matrix for the respective middle factions:<br><br></div><div> defect cooperate<br>
<br></div><div>defect .............(4000, 6000)...............(7600, 5100)<br><br></div><div>cooperate.........(3400, 8400)..............(7000, 7500)<br></div><div><br></div><div>In other words, we are dealing with a game of "prisoner's dilemma" which has the greatest total payoff when neither player defects, but the only equilibrium is the position of both players defecting, which has the worst total payoff.<br>
<br></div><div>In summary, our analysis has revealed two things: (1) It shows that direct approval voting gives strong game theoretic incentives for disapproving M in the two middle factions. (2) It shows that best over-all expectation is the case in which the two middle factions do approve M. This second point is the "dispassionate" response that I promised to Toby's question about the desirability of M getting the full middle 75 percent.<br>
<br></div><div>In the next installment I will explainn how to solve this dilemma.<br><br></div><div>Forest<br></div><div><br></div><div><div><div><div><div><div><div class="gmail_extra"><br><br><div class="gmail_quote"><br>
<blockquote class="gmail_quote" style="margin:0px 0px 0px 0.8ex;border-left:1px solid rgb(204,204,204);padding-left:1ex">
Date: Wed, 25 Jun 2014 15:15:44 -0700<br>
From: Forest Simmons <<a href="mailto:fsimmons@pcc.edu" target="_blank">fsimmons@pcc.edu</a>><br>
To: Toby Pereira <<a href="mailto:tdp201b@yahoo.co.uk" target="_blank">tdp201b@yahoo.co.uk</a>><br>
Cc: EM <<a href="mailto:election-methods@lists.electorama.com" target="_blank">election-methods@lists.electorama.com</a>><br>
Subject: Re: [EM] PR for ethnically polarized electorates<br>
Message-ID:<br>
<CAP29oneWAh9dhh3YoOtcJ=<a href="mailto:1Qbv2cCM6RLLbeyn8VTYeNGuzW_A@mail.gmail.com" target="_blank">1Qbv2cCM6RLLbeyn8VTYeNGuzW_A@mail.gmail.com</a>><br>
Content-Type: text/plain; charset="utf-8"<div><div class="h5"><br>
<br>
Thanks to Juho and Toby for their insights.<br>
<br>
It is true, as they suggest, that question 2 is the harder one.<br>
<br>
The simplest answer to question one that I know of is based on an idea that<br>
Martin Harper came up with 12 years ago as a way of showing that ordinary<br>
Approval satisfies "one voter one vote" in the same strict sense that IRV<br>
does (through vote transfer):<br>
<br>
First list the candidates in order of most approval to least approval.<br>
Then on each ballot transfer the entire support of the voter to the highest<br>
candidate on the list that is approved on the ballot. In other words, the<br>
voter's one and only vote is for the candidate she approves that is most<br>
approved by other voters. As Martin pointed out, this assignment of votes<br>
still elects the ordinary Approval winner in the single winner case. (Half<br>
a dozen years later Jobst pointed out that this same idea can be used to<br>
assign probabilities in a single winner lottery method.)<br>
<br>
I am now pointing out that Martin Harper's vote transfer scheme is a simple<br>
way of designing a PR method (based on approval ballots) that solves<br>
problem one. In the given example let us assume that the truncations are<br>
reliable indicators of disapproval. Then the approval ballots are<br>
<br>
10 A<br>
30 A, M<br>
45 B, M<br>
15 B<br>
<br>
The approval order is M>B>A<br>
<br>
The first faction ballots all count for A. The last faction ballots all<br>
count for B, and the other 75 ballots all count for M, yielding the desired<br>
quotas of 10, 15, and 75 respectively.<br>
<br>
Toby asks the question of why this M heavy proportion is so desirable.<br>
<br>
One answer is that in these polarized countries (the ones that inspired<br>
this thread in the first place) the fewer extremists in power the better.<br>
But in my next post, the one addressing question two, I will give a more<br>
dispassionate answer to that question.<br>
<br>
Forest<br>
<br>
<br>
<br>
On Wed, Jun 25, 2014 at 12:23 PM, Toby Pereira <<a href="mailto:tdp201b@yahoo.co.uk" target="_blank">tdp201b@yahoo.co.uk</a>> wrote:<br>
<br>
> At first glance it seems that 10, 15 and 75 for A, B, M respectively seems<br>
> a little optimistic from a voting system. It's not just that party list PR<br>
> would shut out M - I can't see any system calling itself PR could award the<br>
> seats in those proportions. Something like reweighted range voting or the<br>
> score PR system I detailed a couple of weeks ago would stop M being shut<br>
> out with honest voting, but they would go nowhere near as far as you are<br>
> suggesting.<br>
><br>
> Regarding voter honesty, it may be difficult to ensure it anyway with a<br>
> normal score-based PR method, but I can't see how you could get it to work<br>
> given that you would want the middle two factions' support for A and B to<br>
> be effectively ignored. To be clear, 10, 15, 75 are the proportions<br>
> you'd expect if the 75 people who gave a positive score to M completely<br>
> lost all their support for A/B and raised M to from 80 to 100.<br>
><br></div></div>
> *From:* Forest Simmons <<a href="mailto:fsimmons@pcc.edu" target="_blank">fsimmons@pcc.edu</a>><br>
> *To:* EM <<a href="mailto:election-methods@lists.electorama.com" target="_blank">election-methods@lists.electorama.com</a>><br>
> *Sent:* Wednesday, 25 June 2014, 1:21<br>
> *Subject:* [EM] PR for ethnically polarized electorates<div><div class="h5"><br>
><br>
> In Rwanda it was the Hutu and the Tutsi tribal division. In Iraq the<br>
> Sunni, Shia, and Kurds. In the former Yugoslavia it was the Serbs Croats<br>
> and Bosnians. There are similar divisions today in the Ukraine, Israel,<br>
> Syria, Bolivia, etc.<br>
><br>
> What do they have in common? A need for electing a representative body<br>
> that has as many moderates and as much consensus as possible so that<br>
> minorities are not so desperate for separation, i.e. to prevent the scourge<br>
> of Balkanization that seems to be spreading like a plague.<br>
><br>
> Suppose that there are two extreme groups A and B supported by two<br>
> individual ethnicities, as well as a more moderate group M with preferences<br>
> like<br>
><br>
> 10 A(100)<br>
> 30 A(100)>M(80)<br>
> 45 B(100)>M(80)<br>
> 15 B(100)<br>
><br>
> (The numbers in parentheses represent voter expectations of relative<br>
> benefits.)<br>
><br>
> In ordinary party list PR methods the parliament would be formed by 40<br>
> representatives from A and 60 representatives from B. The moderate party<br>
> would be shut out entirely.<br>
><br>
> Here are my questions:<br>
><br>
> 1. What method(s) would take this information and elect a parliament with<br>
> respective party strengths of 10, 15, and 75 for A, B, and M?<br>
><br>
> 2. What election method could possibly get the two middle factions to<br>
> honestly convey this information via their ballots? In other words, how to<br>
> keep the two middle factions from defecting from their common interest?<br>
><br>
> Forest<br>
><br>
><br>
> ----<br></div></div></blockquote></div><br></div></div></div></div></div></div></div></div></div></div></div></div></div></div></div></div>
</blockquote></div><br></div></div></div></div></div></div></div></div></div>