Mathematical Proofs: Notation and Formats

Algebraic Notation: Upper Case Characters

• A, B, C & D = Names of example candidates or parties. [Party candidates are differentiated by subscript.]
• CI = Consensus Index (where 0 ≤ CI ≤ 1).
• F (as subscript) = Forward slate ordering of clone candidates; from 1 to K.
• K = Number of clone candidates within the clone set (K = N - M).
• M = Number of non-clone candidates competing against the clone set (M = N - K).
• N = Number of candidates / alternatives / preferences.
• P = Preference.
• P₁ = First (highest) preference.
• P_n = nth preference.
• P_N = Last (lowest) preference.
• PI = Polarization Index (where 0 ≤ PI ≤ 1).
• R = Reciprocal of the common ratio (1/r).
• R (as subscript) = Reverse slate ordering of clone candidates; from K to 1.
• S = Number of candidate sets / parties.
• Σ = Sum of all the normalized preference weightings in a vector or conjugate vector (when calculating CI or PI).
• T = Tally for a candidate or party.
• T_A = Tally for candidate A or party A.
• V = Number of voters casting a valid ballot.
• V_A = Number of voters casting a first preference for candidate A.
• V_n = Number of voters casting an nth preference for a given candidate.
• W = Number of winners / seats.

Algebraic Notation: Lower Case Characters

• ∝ = Constant (common offset).
• a = Weighting of first preference (w₁ = a).
• a,b,c = Weighting co-ordinates (for three-preference maps only).
• c = Y-axis intercept (for a straight line only).
• d = Common difference (= w_n - w_n+1) as used in the Borda Count.
• δ = Common difference (where d → 0).
• k = Constant (general).
• m = Majority threshold (where 1/2 ≤ m ≤ 1) for guaranteed victory.
• m = Gradient (for a straight line only).
• n = Rank position of preference on ballot (where 1 ≤ n ≤ N).
• η = Relative (per-unit) rank position of preference on ballot (where 0 ≤ η ≤ 1).
• p = Proportion of votes supporting named party as first preference (P₁); where p + q = 1.
• q = Proportion of votes not supporting named party as first preference (P₁); where p + q = 1.
• r = Common ratio (= w_n+1/w_n) as used in GV/CHPV.
• s = Seat share (where 0 ≤ s ≤ 1).
• t = Tally share (where 0 ≤ t ≤ 1).
• t = Average tally per voter (= T/V for three-candidate maps only).
• u_n = Weighting of nth preference in a conjugate vector.
• v = Given number of valid votes.
• v_n = Share of votes casting nth preferences for a given candidate (where 0 ≤ v_n ≤ 1).
• w_n = Weighting of nth preference in a vector.
• x,y = Cartesian co-ordinates.
• x, y & z = Variables (general).

Election Profile Format

The general format is QUANTITY: BALLOT TYPE (where candidate preferences P₁ to P_N are listed in strict rank order). Four illustrative examples follow and definitions are given below them.

• V/2: C,A,D,B.
• p: A,B,C,D,~~~~~~.
• q: B,C,A,********.
• v: D,B,...........

• Quantity = A proportion, fraction or integer number of votes for the specified ballot type.
• Semi-colon (:) = Separator between quantity and rank order listing of candidates.
• Comma (,) = Separator between adjacent candidates / alternatives / preferences.
• Full stop (.) = Terminator after last preference (P_N).
• Tildes (~~~~~) = Sequence continues (other candidates in strict implied rank order).
• Asterisks (*****) = No sequence (all unnamed candidates in any rank order).
• Repeated full stops (.....) = Truncated sequence (unnamed candidates not awarded any preference).

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