User:Silvio Gesell/Super-generalized cumulative voting

Super-generalized cumulative voting (SGC) is a further generalization of Generalized cumulative voting (GC) that includes ordinal and multi-round systems. I define a specific system by its cost exponent (¢), the power to which the points (as a fraction of MAX+1/∞, MAX being the highest possible score when the lowest is normalized to 0) given to a candidate is raised to obtain its cost to the voter (as a fraction of the budget); and its speed ($), the number of candidates eliminated per round. I include ordinal systems through the use of an infinitesimal cost exponent and the re-imagination of a rank as a number of points greater than the rank below by an infinite factor. The table below summarizes:

	Cost exponent (¢)	Speed ($)
Plurality (FPTP)	1/∞	N-1
Contingent vote	1/∞	N-2
Instant-runoff voting (IRV)	1/∞	1
Cumulative voting	1	N-1
Distributed voting (DV)	1	1
Quadratic voting (QV)	2	N-1
Score (SV)	∞	N-1
Score then automatic runoff (STAR)	∞	N-2
Cardinal Baldwin (CB)	∞	1

N signifies the number of candidates. A system's bias is inversely related to ¢, the above systems (with non-infinite ¢) being extreme-biased and their duals being center-biased (and having opposite optimal strategies), with speed exacerbating bias. I regard Approval voting (AV) as SV with the useless intermediate scores conveniently removed, and I make no distinction between systems that are effectively the same in the absence of blanks (e.g. AV and Disapproval voting).

Strategy

I take off from the Meyerson-Weber strategy, my first step being to derive a special strategy for single-round SGC. This entails the introduction of the concept of the prospective rating (R) of a set of candidates (C):

Failed to parse (syntax error): {\displaystyle R_C=\frac{\sum_{i\in C}R_i}{\sqrt[¢]{|C|}}}

Optimal strategy is to distribute all points equally among the set of candidates with the highest R. From this, we can derive the familiar Meyerson-Weber FPTP and Approval/Score strategies as well as optimal strategy for Cumulative, Quadratic or any other single-round specialization of SGC. In particular, we find that any ¢≤1 system (i.e. FPTP, Cumulative or anything in between) is strategically identical to FPTP, and its dual is strategically identical to Anti-plurality voting.

Optimal two-round strategy is the same, except that we rank the candidates after bifurcating them as above. So strategic Contingent is to rank the candidate with highest R (the lone member of the set with highest R) first and the rest honestly. And strategic STAR is to make 2 honest-ordered series of candidates, one at the top of the range (consisting of the candidates with positive R) and one at the bottom, with no more than 1 point separating any two adjacent candidates in a series. In the zero information scenario, optimal strategy is approximated by applying this process to each series:

1. Separate all adjacent candidates with unequal utilities by 1 point and set a variable x to 0.
2. Go through each pair of unequal adjacent candidates (i,j) in ascending order of |S| (absolute value of surplus utility, i.e. utility net of the weighted average, with weights proportional to win probabilities, which in the zero information scenario is equal to the unweighted average), applying the following steps:
   1. Add $MAX*{\frac {|u_{i}-u_{j}|}{u_{max}-u_{min}}}$ to x (u_max and u_min are the highest and lowest utilities).
   2. If x<.5, bring all candidates (in the series) with |S|≤min(|S|_i,|S|_j) 1 point closer to the appropriate extreme (MAX for S>0, 0 for S<0); otherwise, subtract 1 from x.

Optimal sequential elimination strategy requires the introduction of new variables:

v_maxC=the points to be voted for the highest rated in C in the first round.

v_minC=the points to be voted for the lowest rated in C in the first round.

v_iC=the points to be voted for candidate i in a round whose candidates are C:

Failed to parse (syntax error): {\displaystyle v_{iC}=\frac{v_i-v_{min}}{(v_{max}-v_{min})\sqrt[¢]{\sum_{j\in C}\frac{v_j-v_{min}}{v_{max}-v_{min}}}}}

p_CD=the probability that C would be the survivors of a round whose candidates were D

p_ijC=the probability that i and j would near-tie for the last safe rank in a round whose candidates were C.

p_C=the probability that the round with |C| candidates consists of C:

$p_{C}=\sum p_{D}p_{CD}$ or, with no information, $p_{C}={\frac {|C|!(N-|C|)!}{N!}}$

u_C=the voter's gain in utility if the round with |C| candidates consists of C:

$\forall |C|=1,u_{C}=u_{i\in C};\forall |C|>1,u_{C}=\sum _{D\subset C}p_{DC}u_{D}$; or, with no information, $u_{C}=\sum _{i\in C}{\frac {u_{i}}{|C|}}$

R_iC=the prospective rating of i in a round whose candidates are C:

$R_{iC}=\sum _{j\in C}p_{ijC}(u_{C\backslash j}-u_{C\backslash i})$ or, with no information, $R_{iC}={\frac {2u_{i}-2u_{C}}{(|C|-1)^{2}}}$

The gain in expected utility for a given vote, becomes:

$G(p,v,u)=\sum _{i=1}^{N}\sum _{C\ni i}p_{C}v_{iC}R_{iC}$

Inducing pivot probabilities from overall win probabilities greatly simplifies the problem:

p_i=the probability that i will win, which gives us R_i=p_iS_i and lets us induce $u_{C}={\frac {\sum _{i\in C}p_{i}u_{i}}{\sum _{i\in C}p_{i}}}$

p_iC=the probability that i would lose a round whose candidates were C:

$p_{iC}={\frac {1}{p_{i}\sum _{j\in C}{\frac {1}{p_{j}}}}}$, from which we can induce $p_{ijC}\propto p_{iC}p_{jC}$ and $p_{ijC}\propto {\frac {1}{p_{i}p_{j}(\sum _{k\in C}{\frac {1}{p_{k}}})^{2}}}$

Thus, given only overall win probabilities, optimal IRV strategy is just nested optimal plurality strategy: repeatedly give the highest remaining rank to the remaining candidate with the highest R in a round consisting of only the remaining candidates. This reduces to honesty in the zero information scenario and, in the perfect information scenario, giving the highest remaining rank to one's favorite of the two with highest win probability among the remaining candidates.

Optimal cardinal sequential elimination strategy is more complex, but some rules carry over from one- and two-round strategy:

Optimal DV, like any biased system's optimal strategy, has a point distribution that is more skewed than the R distribution. But the points aren't always given to a single candidate (as in Cumulative) or even distributed equally among a set (as in GC). And yet, optimal DV is, like strategic voting in any biased system, strongly dishonest; it is sometimes optimal to give A a lower score than B<A.

Optimal CB is less centrophobic than optimal STAR (which is only marginally less centrophobic than optimal SV); for instance, it is sometimes optimal to leave a gap larger than 1 point between adjacent candidates in a series. However, neither optimal CB nor optimal STAR are always at least as honest as optimal SV; both sometimes round up strengths of preference an honest voter would round down to 0. Another difference is that optimal CB (unlike optimal SV and optimal STAR) does not use the average candidate as the threshold (between candidates scored above the mid-range and candidates scored below the mid-range) in the zero-information scenario; the threshold is relatively sensitive to outliers in CB.

The table below summarizes:

Method	Zero information strategy	General strategy	Perfect information strategy
FPTP	Honest	Highest R	Best frontrunner
Contingent vote	Honest	Highest R then honest	Best frontrunner then honest
IRV	Honest	Nested highest R	Nested best frontrunner
Cumulative voting	Bullet best	Bullet highest R	Bullet best frontrunner
DV	Max $\sum _{i=1}^{N}\sum _{C\ni i}{\frac {2(u_{i}-u_{C})|C|!(N-|C|)!(v_{i}-v_{minC})}{N!(|C|-1)^{2}(v_{maxC}-v_{minC})\sum _{j\in C}{\frac {v_{j}-v_{minC}}{v_{maxC}-v_{minC}}}}}$	Max $\sum _{i=1}^{N}\sum _{C\ni i}p_{C}v_{iC}R_{iC}$	Bullet best frontrunner
QV	Equally among highest ${\frac {\sum _{i\in C}u_{i}-{\bar {u}}}{\sqrt {|C|}}}$	Equally among highest ${\frac {\sum _{i\in C}R_{i}}{\sqrt {|C|}}}$	Bullet best frontrunner
SV	Min-max, ${\bar {u}}$ threshold	Min-max, R=0 threshold	Min-max, frontrunner threshold
STAR	Min-max, ${\bar {u}}$ threshold, 0-1-pt gaps	Min-max, R=0 threshold, 0-1-pt gaps	Min-max, frontrunner threshold
CB	Max $\sum _{i=1}^{N}\sum _{C\ni i}{\frac {2(u_{i}-u_{C})|C|!(N-|C|)!(v_{i}-v_{minC})}{N!(|C|-1)^{2}(v_{maxC}-v_{minC})}}$	Centrophobic scoring	Min-max, frontrunner threshold
GC	Equally among highest Failed to parse (syntax error): {\displaystyle \frac{\sum_{i\in C}u_i-\bar{u}}{\sqrt[¢]{|C|}}}	Equally among highest Failed to parse (syntax error): {\displaystyle \frac{\sum_{i\in C}R_i}{\sqrt[¢]{|C|}}}	Bullet best frontrunner
SGC	Max Failed to parse (syntax error): {\displaystyle \textstyle \sum_{i=1}^N \displaystyle\sum_{C\ni i}\frac{2(u_i-u_C)|C|!(N-|C|)!(v_i-v_{minC})}{N!(|C|-1)^2(v_{maxC}-v_{minC}) \sqrt[¢]{\sum_{j \in C}\frac{v_j-v_{minC}}{v_{maxC}-v_{minC}}}} }	Max $\sum _{i=1}^{N}\sum _{C\ni i}p_{C}v_{iC}R_{iC}$	Bullet best frontrunner