A jointPolitics andEconomics series
Social choice andelectoral systems
Social choice Mechanism design Comparative politics Comparison List (By country)
Single-winner methods Single vote –plurality methods First preference plurality (FPP) Two-round (US:Jungle primary) Partisan primary Instant-runoff UK: Alternative vote (AV) US: Ranked-choice (RCV) Party block voting Plurality block voting Condorcet methods Condorcet-IRV Round-robin voting Minimax Kemeny Schulze Ranked pairs Maximal lottery Positional voting Plurality (el.IRV) Borda count (el.Baldwin,Mdn.Bucklin) Antiplurality (el.Coombs) Cardinal voting Score voting Approval voting Majority judgment STAR voting
Proportional representation Party-list Apportionment Highest averages Largest remainders National remnant Biproportional List type Closed list Open list Panachage List-free PR Localized list Quota-remainder methods Single transferable vote (Hare,Droop) Schulze STV CPO-STV Quota Borda Approval-based committees Thiele's method Phragmen's method Expanding approvals rule Method of equal shares Fractional social choice Direct representation Interactive representation Liquid democracy Fractional approval voting Maximal lottery Random ballot Semi-proportional representation Cumulative SNTV Limited voting
Mixed systems By results of combination Mixed-member majoritarian Mixed-member proportional By mechanism of combination Non-compensatory Parallel (superposition) Coexistence Conditional Fusion (majority bonus) Compensatory Seat linkage system UK:'AMS' NZ:'MMP' Vote linkage system Negative vote transfer Mixed ballot Supermixed systems Dual-member proportional Rural–urban proportional Majority jackpot By ballot type Single vote Double simultaneous vote Dual-vote
Paradoxes and pathologies Spoiler effects Spoiler effect Cloning paradox Frustrated majorities paradox Center squeeze Pathological response Perverse response Best-is-worst paradox No-show paradox Multiple districts paradox Strategic voting Lesser evil voting Exaggeration Truncation Turkey-raising Wasted vote Paradoxes ofmajority rule Tyranny of the majority Discursive dilemma Conflicting majorities paradox
Social and collective choice Impossibility theorems Arrow's theorem Majority impossibility Moulin's impossibility theorem McKelvey–Schofield chaos theorem Gibbard's theorem Positive results Median voter theorem Condorcet's jury theorem May's theorem Condorcet dominance theorems Harsanyi's utilitarian theorem VCG mechanism Quadratic voting
Politics portal Economics portal Mathematics portal
v t e

Avoting system satisfiesjoin-consistency (also called thereinforcement criterion) if combining two sets of votes, both electingA overB, always results in a combined electorate that ranksA overB.^[1] It is a stronger form of theparticipation criterion. Systems that fail the consistency criterion (such asinstant-runoff voting orCondorcet methods) are susceptible to themultiple-district paradox, apathological behavior where a candidate can win an election without carrying even a single precinct.^[1] Conversely, it can be seen as allowing for a particularly egregious kind ofgerrymander: it is possible to draw boundaries in such a way that a candidate who wins the overall election fails to carry even a singleelectoral district.^[1]

Rules susceptible to the multiple-districts paradox include allCondorcet methods^[2] andinstant-runoff (or ranked-choice) voting. Rules that are not susceptible to it include allpositional voting rules (such asfirst-preference plurality and theBorda count) as well asscore voting andapproval voting.

There are three variants of join-consistency:

1. Winner-consistency—if two districts elect the same winnerA,A also wins in a new district formed by combining the two.
2. Ranking-consistency—if two districts rank a set of candidates exactly the same way, then the combined district returns the same ranking of all candidates.
3. Grading-consistency—if two districts assign a candidate the sameoverall grade, then the combined district assigns that candidate the same grade.

A voting system is winner-consistent if and only if it is a point-summing method; in other words, it must be apositional voting system orscore voting (includingapproval voting).^[3][2]

As shown below for theKemeny rule andmajority judgment, these three variants do not always agree with each other (which contrasts with most other voting criteria). Kemeny is the only ranking-consistent Condorcet method, and no Condorcet method can be winner-consistent.^[2]

This example shows that Copeland's method violates the consistency criterion. Assume five candidates A, B, C, D and E with 27 voters with the following preferences:

Now, the set of all voters is divided into two groups at the bold line. The voters over the line are the first group of voters; the others are the second group of voters.

In the following the Copeland winner for the first group of voters is determined.

The results would be tabulated as follows:

Result: With the votes of the first group of voters, A can defeat three of the four opponents, whereas no other candidate wins against more than two opponents. Thus,A is elected Copeland winner by the first group of voters.

Now, the Copeland winner for the second group of voters is determined.

The results would be tabulated as follows:

Result: Taking only the votes of the second group in account, again, A can defeat three of the four opponents, whereas no other candidate wins against more than two opponents. Thus,A is elected Copeland winner by the second group of voters.

Finally, the Copeland winner of the complete set of voters is determined.

The results would be tabulated as follows:

Result: C is the Condorcet winner, thus Copeland choosesC as winner.

This example shows that Instant-runoff voting violates the consistency criterion. Assume three candidates A, B and C and 23 voters with the following preferences:

Now, the set of all voters is divided into two groups at the bold line. The voters over the line are the first group of voters; the others are the second group of voters.

In the following the instant-runoff winner for the first group of voters is determined.

B has only 2 votes and is eliminated first. Its votes are transferred to A. Now, A has 6 votes and wins against C with 4 votes.

Result:A wins against C, after B has been eliminated.

Now, the instant-runoff winner for the second group of voters is determined.

C has the fewest votes, a count of 3, and is eliminated. A benefits from that, gathering all the votes from C. Now, with 7 votes A wins against B with 6 votes.

Result:A wins against B, after C has been eliminated.

Finally, the instant runoff winner of the complete set of voters is determined.

C has the fewest first preferences and so is eliminated first, its votes are split: 4 are transferred to B and 3 to A. Thus, B wins with 12 votes against 11 votes of A.

Result:B wins against A, after C is eliminated.

A is the instant-runoff winner within the first group of voters and also within the second group of voters. However, both groups combined elect B as the instant-runoff winner. Thus, instant-runoff voting fails the consistency criterion.

This example shows that the Kemeny method violates the consistency criterion. Assume three candidates A, B and C and 38 voters with the following preferences:

Now, the set of all voters is divided into two groups at the bold line. The voters over the line are the first group of voters; the others are the second group of voters.

In the following the Kemeny winner for the first group of voters is determined.

The Kemeny method arranges the pairwise comparison counts in the following tally table:

The ranking scores of all possible rankings are:

Result: The ranking A > B > C has the highest ranking score. Thus,A wins ahead of B and C.

Now, the Kemeny winner for the second group of voters is determined.

The Kemeny method arranges the pairwise comparison counts in the following tally table:

The ranking scores of all possible rankings are:

Result: The ranking A > C > B has the highest ranking score. Hence,A wins ahead of C and B.

Finally, the Kemeny winner of the complete set of voters is determined.

The Kemeny method arranges the pairwise comparison counts in the following tally table:

The ranking scores of all possible rankings are:

Result: The ranking B > A > C has the highest ranking score. So,B wins ahead of A and C.

A is the Kemeny winner within the first group of voters and also within the second group of voters. However, both groups combined elect B as the Kemeny winner. Thus, the Kemeny method fails the reinforcement criterion.

The Kemeny method satisfies ranking consistency; that is, if the electorate is divided arbitrarily into two parts and separate elections in each part result in the same ranking being selected, an election of the entire electorate also selects that ranking. In fact, it is the onlyCondorcet method that satisfies ranking consistency.

The Kemeny score of a ranking${\displaystyle \mathcal{R}}$ is computed by summing up the number of pairwise comparisons on each ballot that match the ranking${\displaystyle \mathcal{R}}$. Thus, the Kemeny score${\displaystyle s_V(\mathcal{R})}$ for an electorate${\displaystyle V}$ can be computed by separating the electorate into disjoint subsets${\displaystyle V=V_1\cup V_2}$ (with${\displaystyle V_1\cap V_2=\mathrm{\varnothing }}$), computing the Kemeny scores for these subsets and adding it up:

${\displaystyle \text{(I)}{s}_V(\mathcal{R})=s_{V_1}(\mathcal{R})+s_{V_2}(\mathcal{R})}$.

Now, consider an election with electorate${\displaystyle V}$. The premise of reinforcement is to divide the electorate arbitrarily into two parts${\displaystyle V=V_1\cup V_2}$, and in each part the same ranking${\displaystyle \mathcal{R}}$ is selected. This means, that the Kemeny score for the ranking${\displaystyle \mathcal{R}}$ in each electorate is bigger than for every other ranking${\displaystyle {\mathcal{R}}^{\prime }}$:

Now, it has to be shown, that the Kemeny score of the ranking${\displaystyle \mathcal{R}}$ in the entire electorate is bigger than the Kemeny score of every other ranking${\displaystyle {\mathcal{R}}^{\prime }}$:

${\displaystyle s_V(\mathcal{R})\stackrel{(I)}{=}{s}_{V_1}(\mathcal{R})+s_{V_2}(\mathcal{R})\stackrel{(II)}{>}{s}_{V_1}({\mathcal{R}}^{\prime })+s_{V_2}(\mathcal{R})\stackrel{(III)}{>}{s}_{V_1}({\mathcal{R}}^{\prime })+s_{V_2}({\mathcal{R}}^{\prime })\stackrel{(I)}{=}{s}_V({\mathcal{R}}^{\prime })q.e.d.}$

Thus, the Kemeny method is consistent with respect to complete rankings.

This example shows that majority judgment violates reinforcement. Assume two candidates A and B and 10 voters with the following ratings:

Now, the set of all voters is divided into two groups at the bold line. The voters over the line are the first group of voters; the others are the second group of voters.

In the following the majority judgment winner for the first group of voters is determined.

The sorted ratings would be as follows:

Result: With the votes of the first group of voters, A has the median rating of "Excellent" and B has the median rating of "Fair". Thus,A is elected majority judgment winner by the first group of voters.

Now, the majority judgment winner for the second group of voters is determined.

The sorted ratings would be as follows:

Result: Taking only the votes of the second group in account, A has the median rating of "Fair" and B the median rating of "Poor". Thus,A is elected majority judgment winner by the second group of voters.

Finally, the majority judgment winner of the complete set of voters is determined.

The sorted ratings would be as follows:

The median ratings for A and B are both "Fair". Since there is a tie, "Fair" ratings are removed from both, until their medians become different. After removing 20% "Fair" ratings from the votes of each, the sorted ratings are now:

Result: Now, the median rating of A is "Poor" and the median rating of B is "Fair". Thus,B is elected majority judgment winner.

A is the majority judgment winner within the first group of voters and also within the second group of voters. However, both groups combined elect B as the Majority Judgment winner. Thus, Majority Judgment fails the consistency criterion.

This example shows that theranked pairs method violates the consistency criterion. Assume three candidates A, B and C with 39 voters with the following preferences:

Now, the set of all voters is divided into two groups at the bold line. The voters over the line are the first group of voters; the others are the second group of voters.

In the following the ranked pairs winner for the first group of voters is determined.

The results would be tabulated as follows:

The sorted list of victories would be:

Result: B > C and A > B are locked in first (and C > A can't be locked in after that), so the full ranking is A > B > C. Thus,A is elected ranked pairs winner by the first group of voters.

Now, the ranked pairs winner for the second group of voters is determined.

The results would be tabulated as follows:

The sorted list of victories would be:

Result: Taking only the votes of the second group in account, A > C and C > B are locked in first (and B > A can't be locked in after that), so the full ranking is A > C > B. Thus,A is elected ranked pairs winner by the second group of voters.

Finally, the ranked pairs winner of the complete set of voters is determined.

The results would be tabulated as follows:

The sorted list of victories would be:

Result: Now, all three pairs (A > C, B > C and B > A) can be locked in without a cycle. The full ranking is B > A > C. Thus, ranked pairs choosesB as winner, which is the Condorcet winner, due to the lack of a cycle.

A is the ranked pairs winner within the first group of voters and also within the second group of voters. However, both groups combined elect B as the ranked pairs winner. Thus, the ranked pairs method fails the consistency criterion.

1. ^John H Smith, "Aggregation of preferences with variable electorate",Econometrica, Vol. 41 (1973), pp. 1027–1041.
2. ^D. R. Woodall, "Properties of preferential election rules",Voting matters, Issue 3 (December 1994), pp. 8–15.
3. ^H. P. Young, "Social Choice Scoring Functions",SIAM Journal on Applied Mathematics Vol. 28, No. 4 (1975), pp. 824–838.

• Electoral system criteria

• Articles with short description
• Short description matches Wikidata