<div dir="ltr"><div dir="ltr"><div class="gmail_default" style="font-family:trebuchet ms,sans-serif;font-size:small"><br></div></div><br><div class="gmail_quote"><div dir="ltr" class="gmail_attr">On Wed, Jan 12, 2022 at 5:01 AM Kristofer Munsterhjelm <<a href="mailto:km_elmet@t-online.de">km_elmet@t-online.de</a>> wrote:<br></div><blockquote class="gmail_quote" style="margin:0px 0px 0px 0.8ex;border-left:1px solid rgb(204,204,204);padding-left:1ex">Yeah, you would think so, but in practice it seems to work pretty well.<br>
Perhaps this is an indication that if an election is susceptible to<br>
strategy, the strategy usually is not very exotic?<br></blockquote><div><br></div><div><br></div><div><div class="gmail_default" style="font-family:"trebuchet ms",sans-serif;font-size:small">That certainly seems to be the case. Yesterday I also added a "simple strategy" check where the ballot puts c_k at the top, w_A at the bottom, and the other candidates are randomized. So in my loop I first check whether that strategy works, and if it doesn't, I apply JGA's method, and I just keep track of that. I noticed that the simple strategy always worked, but I paid no heed because at the time I was doing tests with C=3 and the impartial model. But today I implemented the spatial model and ran several tests with more candidates and it really looks like the simple strategy works *almost* every time that JGA's method works:</div><div class="gmail_default" style="font-family:"trebuchet ms",sans-serif;font-size:small"><br></div><div class="gmail_default" style="font-family:"trebuchet ms",sans-serif;font-size:small">Spatial model + Benham<br>V=29, C=3 --> 0.1233-0.1365 (95% c.i.), simple=1.00, majority=0.73<br>V=29, C=4 --> 0.2811-0.2989 (95% c.i.), simple=0.97, majority=0.52<br>V=29, C=5 --> 0.4186-0.4384 (95% c.i.), simple=0.94, majority=0.37<br></div><div class="gmail_default" style="font-family:"trebuchet ms",sans-serif;font-size:small">V=29, C=6 --> 0.5388-0.5575 (95% c.i.), simple=0.92, majority=0.26<br></div><br></div><div><div class="gmail_default" style="font-family:"trebuchet ms",sans-serif;font-size:small">So the 'simple' value is the fraction out of all the JGA-susceptible elections, where the simple strategy worked. Incidentally, the 'majority' column is the fraction of the JGA-resilient elections where the election was resilient because there was a majority winner. At some point I want to do the non-JGA version of these tests, where each voter in the coalition has a different ballot. It would be interesting to measure susceptibility in terms of how often the strategy is simple vs complicated. The simplest strategy that a group of conspirators can have is "put c_k on top, w_A at the bottom, and rank the rest honestly". So I want to test that strategy.</div><br></div><div><br></div><div> </div><blockquote class="gmail_quote" style="margin:0px 0px 0px 0.8ex;border-left:1px solid rgb(204,204,204);padding-left:1ex">I have noticed a dropoff with >10 candidates, though.<br>
<br>
Other (obvious?) optimizations include:<br>
- If there's a majority winner and the method passes the majority<br>
criterion, give up directly: the election is unmanipulable.<br>
- Trying the naive exaggeration strategy (if A was the winner, make B>A<br>
voters vote B first and A last); requires just one check per other<br>
candidate.<br></blockquote><div><br></div><div><div class="gmail_default" style="font-family:"trebuchet ms",sans-serif;font-size:small">Yeah. Those are the two I came up with. The numbers above give you an indication of how often they'll speed up the program (i.e. quite often, actually).</div></div><div><br></div><div> </div><blockquote class="gmail_quote" style="margin:0px 0px 0px 0.8ex;border-left:1px solid rgb(204,204,204);padding-left:1ex">In case it's useful, I can also mention that I'm using the Jeffreys<br>
interval for binomial c.i. to avoid problems with the Gaussian when the<br>
expected value is very close to 0 or 1. Though reading the Wikipedia<br>
article, it seems that the consensus is that the Wilson interval is better.<br></blockquote><div><br></div><div><div class="gmail_default" style="font-family:"trebuchet ms",sans-serif;font-size:small">Yeah, I don't trust those a whole lot because I'm never quite sure what the assumptions are or whether they apply. I compute my c.i. with a bootstrap. It's easy to implement. In the main loop instead of incrementing SS and SF, I use an array to keep track of the boolean result (i.e. success vs failure). From that array I can compute SS and SF, but more importantly, I can do 500 bootstrap resamples (i.e. sampling with replacement) and basically bruteforce the c.i. with no assumptions made about the distribution. This is expensive compared to an analytic approach, but in this context, "expensive" means that it adds 0.025s to the runtime.</div></div><div class="gmail_default" style="font-family:"trebuchet ms",sans-serif;font-size:small"><br></div><div class="gmail_default" style="font-family:"trebuchet ms",sans-serif;font-size:small">Cheers,</div></div>-- <br><div dir="ltr" class="gmail_signature"><div dir="ltr"><div><div dir="ltr"><div dir="ltr"><div dir="ltr"><div dir="ltr"><div dir="ltr"><div dir="ltr"><div dir="ltr"><font face="trebuchet ms, sans-serif">Dr. Daniel Carrera</font></div><div dir="ltr"><font face="trebuchet ms, sans-serif">Postdoctoral Research Associate</font></div><div><font face="trebuchet ms, sans-serif">Iowa State University</font></div></div></div></div></div></div></div></div></div></div></div>