I was going to write a longer reply, but this seems to be the crux of the matter: Craig wrote: > The test [for the power of a vote] is clearly stated but it does not apply to > methods outside of the topic of preferential voting (i.e. checkbox methods, > e.g. the ASVB sub-vote Variant Block Vote voting method. I call it Approval). Now, previously you stated that Approval wasn't a preferential voting method, right? So the test doesn't apply to Approval, right? Great! So why exactly were we discussing whether Approval passes or fails the test? :-/ What have I missed? Something important clearly... Anyway, since you asked me some questions about the email I sent to you - here are some answers. -------------------------------- ORIGINAL TEXT > What ought be done is to have some of these weights be calculated. Well, first up we need to decide exactly what these weights are, hmm? You don't say yourself, so I'l trundle merrilly off with a definition of my own. -- begin definition Let C1, C2, ... Cn be the candidates. Let V be a voter, and B be his/her/its ballot paper. Let U1, U2, ... be V's normalised sincere utilities of the respective candidates. Let P1, P2, ... be the prior probabilities of the respective candidates winning. Let P1', P2', ... be the probabilities after the submssion of the ballot paper B. The Prior utility (U) of the election is the sum of Ux times Px over all x. The utility after the submission of B (U1) of the election is the sum of Ux times Px' over all x. U' - U is the effective weight of the voter V with ballot paper B. Find the ballot paper B which maximises U' - U. The effective weight of this ballot paper for V is the effective weight of V. Note that this ballot paper need not be sincere. The Sincere Effective Weight of V is the maximum effective weight available to V on any sincere ballot paper. -- end definition The justification is that the point of an election for a voter (ignoring 'sending a message' and such) is to get the best outcome from their perspective, and this 'bestness' is best measured from utilities, cos that's what they are. Every single election method in the world penalises voters who don't vote or who spoil their ballot papers or who vote for someone they hate, so we should concentrate on the effective weight of the *voter* not the *vote*. In an ideal world the sincere effective weight of any voter would be equal to the effective weight of that voter, and non-negative. In addition, the effective weights of each voter should be equal. Dictatorship or Random satisfies this, as all effective weights are zero. Random Ballot, Approval, Condorcet, IRV, etc, all fail utterly. In general, A>B=C voters will have no effective weight unless A is a contender for first place. -- Anyway, here's an example to see how the concept works in practice, for those who aren't keen on maths without examples. Skip otherwise. 100 voters have the following utilities, and represent the 'A' faction: 1.0 A 0.1 B 0.0 C zero or one voters (equiprobably) have the following utilities, and represent the 'B' faction: 1.0 B 0.0 A,C 100 voters have the following utilities, and represent the 'C' faction. 1.0 C 0.1 B 0.0 A -- Condorcet If we are a new member of the A faction, we get the values shown for our only possibly sincere vote: Initial: P(A) = 0, P(B) = 0.5, P(C) = 0, P(A,C) = 0.5, U = 0.3 After: P(A) = 0.5, P(B) = 0, P(C) = 0, P(A,B) = 0.5, U' = 0.775 So our effective weight is 0.475 A new member of the C faction has the same weight. If we are a new member of the B faction, we get the values shown for our only possible sincere vote: Initial: P(A) = 0, P(B) = 0.5, P(C) = 0, P(A,C) = 0.5, U = 0.5 After: P(A) = 0, P(B) = 1, P(C) = 0, U=1 So our effective weight is 0.5 Hence, the Condorcet effective weights are 0.475 and 0.5. -- Approval Presume that at least 60% of the A and C factions bullet vote. If we are a new member of the A faction, we get the values shown for an AB vote and an A vote (the two sincere votes available) Initial: P(A) = 0, P(B) = 0, P(C) = 0, P(A,C) = 1.0, U = 0.5 AB: P(A) = 1, P(B) = 0, P(C) = 0, U' = 1.0 A: P(A) = 1, P(B) = 0, P(C) = 0, U' = 1.0 So our effective weight is 0.5 A new member of the C faction has the same weight. If we are a new member of the B faction, we get the values shown for our only possible sincere vote (bullet vote for B): Initial: P(A) = 0, P(B) = 0, P(C) = 0, P(A,C) = 1.0, U = 0.5 After: P(A) = 0, P(B) = 0, P(C) = 0, P(A,C) = 1.0, U = 0.5 So our effective weight is zero. Hence the Approval effective weights are 0.5 and 0 -------------------------- QUESTIONS 1) what the "P" function is. Answer: I was abbreviating to save space. P(A) refers to "The probability that the candidate named 'A' will win the election". Similarly, P(A,B) refers to "The probability that the candidates named A and B will draw the election between them". It's not some kind of function. Sorry for the misunderstanding - I thought that this was standard notation. 2) derivation of the effective weight of 0.5 for the Approval example. Answer: The effective weight, as I defined it, is U' - U. U' = 1.0, U = 0.5, 1.0 - 0.5 = 0.5 Hence the effective weight is 0.5 3) derivation of U=0.5 for the Approval example. Answer: There is only one outcome in the initial case: a draw between A and C. This draw is resolved randomly. Therefore there is a 50% chance of electing A, with utility (for A voters) of 1.0, and a 50%, with a utility (for A voters) of 0.0 50% of 1.0 + 50% of 0.0 = 0.5 Therefore U=0.5 4) How the 60% figure affects things for the Approval example Answer: The requirement for greater than 60% bullet-voting is just to ensure that enough people bullet vote that B doesn't get elected. Given the utilities of A, B, C for the varying factions, this seemed a reasonable assumption to make. In practice I'd expect the proportion to be much, much higher than 60%. 5) What does the "if we are a new member of the B faction ... our sincere vote" sentence mean? Answer: If we are a member of the B faction, with the utilities given, then the only way we can sincerely vote in approval in the example given is to bullet vote for B. Any other vote would be insincere. 6) What's the definition of sincerity? My definition is: [subdefine] A greater insincere preference is a case where we prefer A to B, but express a preference on our ballot paper for B over A. A lesser insincere preference is a case where either we prefer A to B but do not express a preference, or we have no preference between them but we express one. [maindefine] A sincere vote is a vote which does not have any greater insincere preferences unless voting them is necessary to avoid other greater insincere paiwise preferences. In addition, a sincere vote does not have any lesser insincere preferences unless voting them is necessary to avoid other lesser insincere paiwise preferences Note that other people have other definitions - but I like mine. My definition of effective weight works with any definition of sincerity. Note that my defintion should not be applied to determine the sincerity of more complicated vote types than those used in Condorcet, Approval, Plurality - such as those used in Dyadic Approval or Average Ratings. 7) Invitation to prove that the values of P(...) are correct. Answer: That would be rather tedious, so I shall decline your invitation. However, for your benefit, I'll rewrite the stuff to not use the P() notation that you don't like. Initial Condorcet outcome - If there is one B voter, then B will win. Otherwise A and C will draw. [equiprobable] Condorcet outcome after an A vote - If there is one B voter, then A and B will draw. Otherwise A will win. [equiprobable] Condorcet outcome after a B vote - B will win. Condorcet outcome after an C vote - If there is one B voter, then C and B will draw. Otherwise C will win. [equiprobable] Initial Approval outcome - A and C draw Approval outcome after an A vote - A wins Approval outcome after a B vote - A and C draw Approval outcome after a C vote - C wins.
