Dear Jobst, Your ingenious use of coins was very inspiring to me. It encouraged me to come up with a dice throwing realization of our benchmark function f(x) = 1/(5-4x) in another solution of our challenge problem.
Also, since our goal is mutually beneficial cooperation, let me define two ballots to be "friends" of each other iff they co-approve one or more candidates. First, I give the method without the benefit of the dice: 1. Draw a ballot at random. Let Y be its favorite, let Z be the most approved of its approved candidates, and let x be the percentage of ballots that are friends with this one. 2. Elect Z with probability f(x), else Y. Now here's the dice rolling version: 1. Draw a ballot at random. Let Y be its favorite, and let Z be the most approved of its approved candidates. 2. Roll a die until some number k other than six shows on top. If k = 1, then elect Z, else ... 3. Draw a new ballot at random. If this new ballot is a friend of the first ballot, go back to step 2, else ... 4. Elect Y. This method, like yours, guarantees a probability proportional to faction size for those factions that choose to bullet, yet it gently encourages friendship. What do you think? Forest ----- Original Message ----- From: Jobst Heitzig Date: Friday, July 4, 2008 9:45 am Subject: Re: [Election-Methods] Challenge Problem To: [EMAIL PROTECTED] Cc: election-methods@lists.electorama.com > Hi again. > > There is still another slight improvement which might be useful > in > practice: Instead of using the function 1/(5-4x), use the function > (1 + 3x + 3x^7 + x^8) / 8. > This is only slightly smaller than 1/(5-4x) and has the same > value of 1 > and slope of 4 for x=1. Therefore, it still encourages unanimous > cooperation in our benchmark situation > 50: A(1) > C(gamma) > B(0) > 50: B(1) > C(gamma) > A(0) > whenever gamma > (1+1/(1+(slope at x=1)))/2 = 0.6, just as the > other > methods did. > > The advantage of using (1 + 3x + 3x^7 + x^8) / 8 is that then > there is a > procedure in which you don't need any calculator or random > number > generator, only three coins: ---- Election-Methods mailing list - see http://electorama.com/em for list info