Dear Kristofer, would the constant relative risk function be of any help for Approval voting?
F=( s(1)^(r-1)+...+s(n)^(r-1) ) / (r-1). s(i) is the number of approved council members that are elected, where 1<=i<=n, n is the number of voters r is a coefficient of risk aversion, which determines the rate at which marginal utility of the voter declines with the number of council members awarded to the voter. The constant relative risk aversion function (CRRA) is a special case of Arrow-Pratts relative risk aversion function http://en.wikipedia.org/wiki/Constant_relative_risk_aversion#Relative_risk_aversion. Other names of this function is: isoelastic utility function, CES. Why this function: Each of the elected council member represents one unit of a consumption good. If a voter approves of a candidate and this candidate is elected, then the voter gets one unit, if the voter approves of two elected council members, then he/she gets two units and so on. The marginal utility of the next unit is defined by the function F above given a value on r. Thus we want to distribute S units of this good, where S is the number of council members in such a way that happiness or utility (i.e. F) is maximized among the voters, given a value on r. The selection of r determines the behavior of F: If r=1, then F becomes the Bernoulli-Nash social welfare function: log(s(1))+...+log(s(n)) in the limit. As log(0) is minus infinity, this function requires that each voter gets at least one council member, thus it over-represents minorities and insures, that "everyone has their representative" in the council. This function is useless, if all bullet vote for themselves. If r=0, then F becomes the standard utilitarian function used to calculate Bayesian regret: s(1)+...+s(n), which just counts the number of approvals and is indifferent about the distribution (i.e. if we have two voters and two seats and four candicates a b c d, and the ballots (approve a b) and (approve c d), then all elected councils have the same value of F). If r<0, then F favors winner-takes-it-all block voting. As r goes to minus infinity, it favors dictatorial council appointments: max(s(i), 1<=i<=n) As r goes to infinity then it becomes min(s(i), 1<=i<=n) If r is around 0.5, then it seems to prefer droop quota proportionality, at least for the case of two seats and five voters. Maybe there is a value of r or a function to determine the value of r. Usage for Satisfaction Approval voting (SAV): The function F can also be used to construct election systems, provided that the utility can be measured (which is the case in Approval voting and in the simulations at: http://munsterhjelm.no/km/elections/multiwinner_tradeoffs/). The task for SAV is to find a suitable value of r, for instance which finds proportional representations meeting the droop quota and optimizes utility. I don't know if there is one, but values of around 1/2 could be a good point to start (i.e. F=(sqrt(s(i))+...+sqrt(s(n)))*2. Other uses: It seems that F can be used both as a proportionality index and as a majoritan preference index for suitable values of r. F can be used to explain why there are different voting systems - they simply have different values of r, i.e. they have different utility functions. For a suitable value of r, block-voting has higher utility and is thus "better" than proportional representation, like STV. F can also be used to measure which voting systems are the "best" and to measure the "distance" or "similarity" between voting systems against some benchmark values of r. It would be cool to see how the chart at http://munsterhjelm.no/km/elections/multiwinner_tradeoffs/ would look like using F with different values of r at the axis. For instance, if r=0, then we would get Bayesian regret. F with r=0.5 (or some other value of r between 0<r<1) could maybe be used in the chart instead of the Sainte-Lague proportionality index (SLI). Different proportionality indexes like F with r=0.5 vs SLI could be plotted against each other. Maybe F with r=0.5 could be shown to be related to some of the proportionality indexes in http://www.mcdougall.org.uk/VM/ISSUE20/I20P4.PDF. Maybe the index could be generalized for condorcet methods, but I don't know how. Best regards Peter Zborník On Sun, May 23, 2010 at 11:03 AM, Kristofer Munsterhjelm < [email protected]> wrote: > [email protected] wrote: > >> Satisfaction Approval Voting - A Better Proportional Representation >> Electoral Method >> >> One way to generalize Proportional Approval voting to range ballots is by >> finding the most natural smooth extension of the function f that takes >> each >> natural number n to the sum >> >> f(n) = 1 + 1/2 + ... + 1/n. >> >> It turns out that we can extend f(n) to all positive real values of n via >> the >> integral >> >> Integral from zero to one of (1-t^n)/(1-t) with respect to t >> >> For PAV generalized to range ballots, first normalize the ratings to be >> between >> zero and one. >> > > As might be obvious by my messages, I find Sainte-Lague of interest. What > would the integral be for the corresponding "generalized divisor" > C/(n+C)? > > If C is 1, we have D'Hondt. If C is 0.5, we have Sainte-Lague. > > > Then for each proposed coalition C of k candidates (assuming there are to >> be k >> winners) and each range ballot r, let g(r,C) be f(S) where S is the sum of >> the >> ratings (according to r) of the alternatives in the coalition C. Elect >> the >> coalition C with the greatest sum of g(r,C) over the range ballots r. >> > > It would also be possible to do a sequential version, as with PAV. > > > One should take as many nominations of winning coalitions as anybody wants >> to >> submit along with the results of SAV, sequential PAV, STV, and any other >> multiwinner method that can be computed from range style (including >> approval) >> ballots, and see which one of them has the highest generalized PAV score. >> > > Or for that matter, determine its proportionality and BR as by my program. > I haven't implemented Approval strategy into it yet, but the generator does > rate each candidate (not just rank them), so the Range version could work. > > How would you calculate the harmonic number for fractional values in a > program? Perhaps the expansion: > > ln(n) + gamma + 1/2n^-1 - 1/12n^-2 + 1/120n^-4 > > would be good enough, at least for test purposes. > > ---- > Election-Methods mailing list - see http://electorama.com/em for list info >
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