# Condorcet Completion Methods that Inhibit Manipulation through Exploiting Knowledge of Electorate Preferences

## Abstract

**:**

## 1. Introduction

- (i)
- to elect a Condorcet winner (a candidate who could win pairwise against each opponent) if one exists according to the profile of true preferences of the electorate, and
- (ii)
- to be resistant to manipulation, with no lure to vote insincerely.

- (1)
- its ballot;
- (2)
- its procedure for aggregating the ballots to determine the winner; and
- (3)
- a set S of sincere voting strategies that specifies how all voters are to mark their ballots, given their true preferences (i.e., as a function of those preferences).

## 2. Secondary Assumptions, and Terminology

## 3. The Voting Systems

#### 3.1. The Ballots

#### 3.2. The Aggregation Procedures

#### 3.2.1. To Start: Ascribe Approval Votes within Each Candidate Triple

**Illustration for RR.**With four candidates, suppose that a ballot shows BDAC for the basic ranking and ABCD for the electorate ranking. Then, for that ballot, approval votes are ascribed for the ABC triple to both B and A under RR1, but only to B under RR2 and RR3; for the ABD triple to both B and D under RR1 and RR3, but only to B under RR2; for the ACD triple to both D and A under all three rules; and for the BCD triple to B alone under all three rules.

**Illustration for RA.**Suppose that a ballot shows a basic ranking of WXYZ for four candidates, with an approval vote for W only. Then an approval vote is ascribed to X (alone) in the XYZ triple. Ascribed approval votes go only to W in the other three triples.

#### 3.2.2. Steps in the Remainder of the Aggregation Procedure

**Step 1:**A triple will be called cyclical if the basic rankings (upon tabulation across all ballots) reveal that its three candidates exhibit a Condorcet vote cycle. Otherwise, it is noncyclical. Find which of the $\left(\begin{array}{c}m\\ 3\end{array}\right)$ triples are cyclical and which are noncyclical.

**Step 2:**For each of the $\left(\begin{array}{c}m\\ 3\end{array}\right)$ triples, tally the total number of ascribed approval votes (see Section 3.2.1) across all ballots for each candidate in the triple.

**Step 3:**In each of the $\left(\begin{array}{c}m\\ 3\end{array}\right)$ triples, rank the candidates 1, 2, 3. If a triple is cyclical, rank the candidates based on their numbers of approval votes within the triple. If a triple is noncyclical, ranks 1, 2, and 3 go (respectively) to the Condorcet vote winner, runner-up, and loser. (Note: Step 2 obtains approval-vote totals for both cyclical and noncyclical triples. The totals for the latter are not used at this stage but will sometimes be required later.)

**Steps 4{4}**(groups of j = 4 candidates),

**4{5}**(j = 5), ...,

**4{m}**(j = m)

**:**These step(s) (applicable only when m > 3) consist of ranking the candidates first in all $\left(\begin{array}{c}m\\ 4\end{array}\right)$ 4-tuples (quadruples) of candidates, then in all $\left(\begin{array}{c}m\\ 5\end{array}\right)$ 5-tuples, and so forth. To rank any of the $\left(\begin{array}{c}m\\ j\end{array}\right)$ j-tuples (j = 4, 5, ..., m), one first determines which candidate is to rank j-th (last), using the method given in Section 3.2.3 just below. Then the ranks of the remaining (j − 1) candidates are made the same as in the (j − 1)-tuple that contains them all.

**Step 5:**The candidate ranked first in the m‑tuple of all candidates (Step 4{m} if m > 3) is declared the winner of the election.

#### 3.2.3. Determining the Lowest-Ranked Candidate

_{1}caps i

_{2}, i

_{2}caps i

_{3}, ..., i

_{j}

_{−1}caps i

_{j}, and i

_{j}caps i

_{1}, where the candidates are numbered with j of the integers from 1 through m and (i

_{1}, i

_{2}, ..., i

_{j}) is some permutation of those j integers. (For j = 6, e.g., the set of six ordered quintuples in (i) and (ii) would have to be of the form ABCDE, BCDEF, CDEFA, DEFAB, EFABC, FABCD to reach operation (iii), so the loop would be A, B, C, D, E, F, and back to A.) Now consider the j triples that consist of three consecutive candidates in the loop, and call them contiguous triples. If (X, Y, Z) is a contiguous triple, with X capping Y and Y capping Z, call Y the central member of the triple. For each contiguous triple, find the number of approval votes won by its central member. (Capture this number regardless of whether the triple is noncyclical or cyclical.) A candidate Y will be called a lesser candidate if, in the contiguous triple of which Y is the central member, Y’s number of approval votes is less than half the total number of voters. Use Method M3 to choose L from among all lesser candidates, or from among all j candidates if there is no lesser candidate.

#### 3.2.4. Methods M1, M2, and M3

#### 3.2.5. A Full Ordering Sometimes

#### 3.3. The Sets of Base Strategies

## 4. Two Illustrations of How Manipulation Attempts Would Fail—Even Backfire

#### 4.1. An Illustration with Five Candidates

**Example 1.**

Group | G(ABCDE) | G(EDCAB) | G(BACDE) |

Number of voters | 35 | 25 | 40 |

RR1, RR2, RR3 vote | ABCDE/ABCDE | EDCAB/ABCDE | BCDEA/ABCDE |

RA1, RA2, RA3 vote | A|BCDE | EDCA|B | BCDE|A |

Noncyclical triples: | CDA, CEA, DEA, BCD, BCE, BDE, CDE |

Cyclical triples, RR1 and RR3: | C 65, A 60, B 40; D 65, A 60, B 40; E 65, A 60, B 40 |

Cyclical triples, RR2: | A 60, B 40, C 25; A 60, B 40, D 25; A 60, B 40, E 25 |

For RR1 and RR3: | CDAB (CAB, DAB, CDA, BCD; B is L) |

CEAB (CAB, EAB, CEA, BCE; B is L) | |

DEAB (DAB, EAB, DEA, BDE; B is L) | |

For RR2: | ABCD (ABC, ABD, CDA, BCD; D is L) |

ABCE (ABC, ABE, CEA, BCE; E is L) | |

ABDE (ABD, ABE, DEA, BDE; E is L) | |

For all three: | CDEA (CDA, CEA, DEA, CDE; A is L) |

BCDE (BCD, BCE, BDE, CDE; E is L) |

#### 4.2. An Illustration of How the Loop of Candidate Triples Works

**Example 2.**

Group | G(ABCD) | G(DCAB) | G(BACD) |

Number of voters | 35 | 25 | 40 |

RR vote | ABCD/ABCD | DCAB/ABCD | BCDA/ADBC |

Noncyclical triples: | CDA, BCD |

Cyclical triples: | A 60, B 40, C 25; D 65, A 60, B 40 |

## 5. Results on Non-Manipulability and Manipulability

#### 5.1. Non-Manipulability Proofs

**Lemma L1.**Let there be m = 3 candidates A, B, and C who are, respectively, the Condorcet preference winner, runner-up, and loser. Under any of the systems RR1, RR2, or RR3, let S1 denote the set of base strategies, which is the set of ballot strategies under which all voters mark their basic rankings according to their own true preferences, and mark their electorate rankings as A first, B second, and C third. Under the systems RA1, RA2, and RA3, let S2 denote the set of ballot strategies (again base strategies) under which all voters mark their basic rankings according to their own true preferences, and mark their approval votes in accord with Rules 1, 2, and 3, respectively. For any of the six systems, suppose that all preference groups other than G(BAC) follow S1 or S2 (as applicable) but that G(BAC) can deviate from S1 or S2. Then both of the following hold no matter what ballot strategy G(BAC) adopts:

**Remark.**With just three candidates there would be no occasion to tally approval votes unless cyclical vote majorities arise in the tally of the basic rankings. L1.1 is worded broadly, however, for use in later proofs where approval votes do have to be tallied within a subset of three out of m (> 3) candidates even when cyclical vote majorities do not occur in that subset.

**Proof of L1.1.**Regardless of whether an RR or RA system is in effect and regardless of which of the three rules for ascribing or marking approval votes applies, candidate A will receive approval votes from G(ABC), G(ACB), and G(CAB), and B will receive no approval votes from them. Because these three preference groups are the ones that prefer A to B, they make up more than half the electorate. G(BAC) can thus do nothing to forestall the conclusion stated in L1.1.

**Proof of L1.2.**If the ballot strategy of G(BAC) causes cyclical vote majorities to occur, then A will get more approval votes than B by virtue of L1.1 and will thus place above B. If not, then A will still place above B because the basic rankings on more than half the ballots—those from G(ABC), G(ACB), and G(CAB)—will show A above B.

**Proposition 1.**Let there be m = 3 candidates with the same conditions and notation as in the first three sentences of Lemma L1. (Among other things, the domain for admissible true-preference profiles is thus F0, or F1, with m = 3.) Then each of the systems RR1, RR2, RR3, RA1, RA2, and RA3 is non-manipulable. That is, no preference group G, by deviating from S1 or S2 when no one else does, can adopt a ballot strategy that will dethrone A (the Condorcet preference winner) as the election winner in favor of a candidate whom G prefers to A.

**Proof**(similar to that of Proposition 3 in Po0, p. 106). For each of the six preference groups, one needs to show that that group can gain nothing by deviating from S1 or S2 (whichever one is applicable), given that the other five groups all adhere to S1 or S2. The following statements apply to all six systems. G(ABC) or G(ACB) voters cannot gain by deviating, because their most preferred candidate (A) already wins. Because G(BCA) or G(CBA) voters are already marking A last in their basic rankings under S1 or S2, they can adopt no ballot strategy that would prevent A from being the Condorcet vote winner. Because G(CAB) voters are already marking C first in their basic rankings under S1 or S2, they can do nothing to stop C from being the Condorcet vote loser; thus, they can achieve nothing better than the election of A, their second preference. Finally, G(BAC) voters can likewise achieve nothing better than the election of their second preference (A), because, by L1.2, they cannot prevent B from placing below A regardless of whether they do or do not bring about cyclical vote majorities.

**Remark.**Brams and Sanver [6] showed that, under approval voting with voting assumed to be sincere, there exists with respect to a Condorcet preference winner a (rather stringent) specific set of strategies, the “critical strategy profile,” under which no coalition of preference groups can dislodge that winner in favor of a candidate whom everyone in the coalition prefers. That is, this is a strong Nash equilibrium (for any m), not just a Nash equilibrium pertaining to manipulation by a single preference group. The present paper concentrates on systems that inhibit manipulation by an individual preference group (rather than by a coalition of voters with differing preference rankings, which would entail far greater complexity). One might ask, though, whether the various Nash equilibria that our systems enjoy are also strong Nash equilibria. For m = 3 candidates, an argument a bit more general than the one in the proof of Proposition 1 shows that the answer is yes for RR1 and RA1. But for the other four systems, the answer is no even for m = 3, as can be shown by suitable minor modification of Example 3 of Po0.

#### 5.2. Counterexamples that Show Manipulability

**Example 3.**

Group | G(CADB) | G(DABC) | G(BACD) |

Number of voters | 35 | 25 | 40 |

RR2 or RR3 vote | CADB/ACDB | DABC/ADBC | BCDA/BCDA |

Noncyclical triples: | CDA, DAB |

Cyclical triples: | B 40, C 35, A 25; B 40, C 35, D 25 |

**Example 4.**

Group | G(ABCD) | G(DCBA) | G(DABC) | G(DBAC) | G(CABD) |

Number of voters | 43 | 40 | 4 | 4 | 9 |

RA2 or RA3 vote | A|BCD | DCB|A | DA|BC | DB|AC | C|DAB |

Noncyclical triples: | ABC, DAB |

Cyclical triples: | C 49, D 48, A 47; C 49, D 48, B 47 |

## 6. RA* (from Po0)versus RA

**Example 5.**

Group | G(ABCD) | G(BACD) | G(DABC) | G(CABD) |

Number of voters | 7 | 27 | 29 | 37 |

RA2, RA3, RA*2, or RA*3 vote | A|BCD | B|ACD | DA|BC | C|DBA |

Noncyclical triples: | BAC, DBA |

Cyclical triples: | A 63, C 37, D 29; C 37, B 34, D 29 |

## 7. Simplification of the Aggregation Procedures

## 8. Circumventing the Minor Assumptions

_{g}(g = 1, ..., n) be independent random variables each drawn from the uniform (rectangular) distribution with range from 0 to 1. Now suppose that each voter g receives a weight of (1 + u

_{g}/n) instead of 1. Such slightly differing weights for the voters automatically produce a random tie‑breaking mechanism. It ensures that vote tallies and preference tallies will both satisfy the assumptions of no ties between candidates and no totals exactly equal to n/2. Moreover, as one can easily show, the differential weighting will give the same results as the equal weighting when there is no assumption failure under equal weighting.

## 9. Are RR1 and RA1 Still Best if the Information Postulate Is Relaxed?

**Example 6.**

Group | G(ABC) | G(BAC) | G(CAB) | G(CBA) |

Number of voters | ||||

Actual | 40 | 45 | 9 | 6 |

Believed to be | 40 | 45 | 11 | 4 |

RR1, RR2, or RR3 vote | ACB/ACB | BAC/ABC | CAB/ABC | CBA/ABC |

RA1 vote | A|CB | BA|C | CA|B | CB|A |

RA2 or RA3 vote | A|CB | B|AC | CA|B | CB|A |

## 10. General Remarks

**Remark 10.1.**Although goals (i) and (ii) stated at the start of the paper are not the only ones that one might want to examine, there is probably broad but not universal agreement that (i) and (ii) are both highly desirable. The selection of the Condorcet preference winner (i), including even the selection of the majority winner in the simple special case of a two-way race, may not maximize utility (in some form) in situations where preference intensities are distributed asymmetrically. Such situations, though, may be uncommon or at least not be easy to identify or remedy; and utility maximization is not the goal here anyway.

**Remark 10.2.**Instead of candidate, a broader term such as option could have been used throughout the paper. Options could refer not only to candidates in an election but also (e.g.) to rival legislative proposals. All of our results for candidates would hold also for other options.

**Remark 10.3.**In a system with an ordered agenda, voters vote on successive pairs of options. Other multi-stage systems also exist. We confine our attention, though, to voting systems that select a winner in a single stage.

**Remark 10.4.**Although the Nash equilibria in our systems are relatively natural, understandable, and straightforward, there is, of course, no claim that any of them are unique. Other Nash equilibria would not generally have the same desirable properties as ours.

**Remark 10.5.**Computation for our systems may become prohibitive, when there is no Condorcet vote winner, if m (the number of candidates) is too large. (But a big value of n (number of voters) should cause little trouble.) Typical (e.g., single-digit) values of m should be manageable. Moreover, computational burden can be greatly reduced when the shortcut described in Section 7 is applicable. Although the main calculations then involve just m* candidates (where 3 ≤ m* ≤ m), the value of m* cannot be known before the election.

^{m}− 1 − m − $\left(\begin{array}{c}m\\ 2\end{array}\right)$ < 2

^{m}, a number that would seem to be small enough to preclude prohibitive computational time so long as m is below 10 or even slightly greater. Larger values of m would probably be rare in practice anyway.

**Remark 10.6.**The concept of unilateral deviation or strategic voting by a homogeneous bloc, rather than a single voter, is not unique to this paper. Other works, such as Niou [14] and Brams and Sanver [6], as well as Po0, have also used it.

**Remark 10.7.**Note that we do not require true (or reported) preferences to be single-peaked. One would hardly expect voter preferences for m > 2 candidates in most usual elections to be strictly single-peaked. (On the other hand, if voters are choosing not from m candidates, but rather from (e.g.) numerical values along a continuum that represent money to be spent or taxed, then single-peaked preferences may be reasonable to assume. For such a case, methods far simpler than ours (e.g., asking voters to report just their first-preference value and then declaring the median to be the winner, as suggested long ago by Francis Galton) could be applied. See (e.g.) Black [15] (Chapter IV and p. 188) and Balinski and Laraki [16] (Section 5.2).)

**Remark 10.8.**The value of our systems is greater the lower the frequency of Condorcet preference cycles. Empirically, such cycles seem to be uncommon (see, e.g., Section 10 of Po0).

**Remark 10.9.**Positing knowledge of other voters’ preferences, as this paper does, is not a novel concept and has a long tradition. The idea played a basic role (e.g.) in Farquharson’s [17] (p. 38) classic development of “sophisticated voting”; in related schemes of McKelvey and Niemi [18]; and in the articles of Niemi and Frank [19], Eckel and Holt [20], and Felsenthal, Maoz, and Rapoport [21]. Both in those works and in the voting systems proposed here, each voter’s knowledge of the preference profile of the electorate is the key to bringing about improvements in the voting process. More recent articles that assume voters’ knowledge of other voters’ preferences include Niou [14], Brams and Sanver [6], and Peress [7], as well as Po0. Any posited knowledge of electorate preferences is used only for determining voter strategies and, obviously, could not properly be used in any aggregation procedure.

**Remark 10.10.**How might voters obtain the needed information about the preference profile of the electorate? For various ideas and possibilities, see (e.g.) Section 9 of Po0. The information sources mentioned there include past election results, discussions in the media, and interactions with other voters. Especially, though, they include results from a certain type of public-opinion poll (Condorcet polling, covered in Potthoff [22]), which could apply in somewhat the same fashion as described in the work of McKelvey and Ordeshook [23] on rational-expectations equilibrium. More speculatively, they also include political stock markets.

**Remark 10.11.**To the extent that our systems work as intended, the discerned electorate preferences do not often play a role in the eventual aggregation, thus somewhat limiting their import. Paradoxically, their purpose is, in essence, to prevent them from being needed, or at least from needing to be used. Their function is to deter manipulation. If that deterrence succeeds, then manipulated vote cycles are not created, and only the voters’ own reported preference rankings will generally play a role in the aggregation.

**Remark 10.12.**The voting systems of this paper (and of Po0) can be seen as falling under the wide umbrella of mechanism design (e.g., Moore [24]; Maskin [25]; Jackson [26]; Maskin and Sjöström [27]), sometimes referred to also as design of game forms or as reverse game theory. Mechanism design does cover voting systems but deals especially with economic problems. Typically, it does not involve preference groups of differing sizes, as in our work, nor use anything like our unusual ballots. Its results are often highly theoretical and abstract, and limited in their practicality (e.g., Moore [24] (pp. 209 ff.))—unlike the systems of the present paper, which, though not as suited as those of Po0 to real-world elections, are nonetheless workable.

**Remark 10.13.**Ranking of m candidates for either preferences or votes (Section 2) is more involved the larger m is, although any ranked voting system becomes more burdensome for voters as m increases. The points in Section 8 mitigate the complexities, though.

**Remark 10.14.**The RA ballot can be simply designed so as to require only a single mark beyond what an ordinary ranked ballot requests (cf. Po0, and the displays in the examples above).

**Remark 10.15.**Section 3.3 did not define the set S for any case where no Condorcet preference winner is discerned (although for definiteness it could have done so). For some other cases, Section 3.3 could allow leeway for base strategies without affecting non-manipulability proofs but, for definiteness, it still defined S. But any indefiniteness concerning S does not prevent the use of any RR or RA system in an actual election, since the aggregation procedure is independent of S and is thus always well defined (though non-manipulability may fail in some cases).

**Remark 10.16.**Although Examples 3 and 4 in Section 5.2 prove that manipulation is indeed mathematically possible, the situations may be so complex that, practically speaking, voters may be unable to discover and carry out the successful manipulation strategies. Difficulty in understanding the role of the candidate triples and of the ascription of approval votes, along with uncertain discernment of electorate profiles, may deter manipulation efforts.

**Remark 10.17.**Although we have pointed out some contrasts among the six systems of this paper, we have attempted no assessments to compare RR2 versus RR3 or RA2 versus RA3.

**Remark 10.18.**Unfortunately, any voting system whose ballot asks for voters’ candidate rankings and nothing more is doomed to be manipulable even in the rather extreme simple case where the domain of true-preference profiles is restricted to those that are single‑peaked. This is shown by the following example, which resembles that of Blin and Satterthwaite [30] but is more general (it applies to any Condorcet completion method, not just Borda as in the earlier article) and is also like that of Penn, Patty, and Gailmard [31] (pp. 447–448) though the perspective differs:

**Example 7.**Suppose that 100 voters each report rankings of three candidates X, Y, and Z with the result that there are 36 ballots marked XYZ (X first, Y second, Z third), 33 marked YZX, and 31 with ZXY. Thus, there is no Condorcet vote winner. Consider the following three voter preference profiles (each of which is consistent with single-peaked preferences:

Profile | Number of Voters Whose True Preference Order Is | CondorcetPreference Winner | |||||

XYZ | XZY | YXZ | YZX | ZXY | ZYX | ||

1 | 36 | 0 | 33 | 0 | 31 | 0 | X |

2 | 36 | 0 | 0 | 33 | 0 | 31 | Y |

3 | 0 | 36 | 0 | 33 | 31 | 0 | Z |

**Remark 10.19.**The Impossibility Theorem of Arrow [33] and different results on ease of manipulation (e.g., Gibbard [34]; Satterthwaite [35]; Gärdenfors [36]) are not encouraging to those who try to find good voting systems. Nonetheless, as our work is intended to convey, the outlook is not as bleak as one might suppose, at least for situations where cycles in voter preferences are absent or of limited impact.

**Remark 10.20.**It may not be widely recognized that the Arrow results and manipulation results just mentioned do not apply if information is available that is not derivable from the true‑preference profile. That is because the results follow from premises that assume availability of nothing other than the profile. They thus do not apply to approval voting nor, in fact, to any of the voting schemes covered here or in Po0. Of course, the unfavorable conclusions of those earlier works may well still hold under much broader premises (it would be highly optimistic to think otherwise), but different proofs would be required. Still, the lack of full applicability of the results may suggest that exploration of unusual voting systems can lead to unexpected benefits.

## 11. Conclusions

## Acknowledgments

## Appendixes

## A. Non-Manipulability of RR1, RR2, and RR3 When There Are No Preference Cycles

**Theorem T1(m).**Let U

_{m}be a set of m (> 2) candidates among whom the profile shows no Condorcet preference cycles. Let A

_{m}and B

_{m}then denote the candidates in U

_{m}who are the Condorcet preference winner and runner‑up, respectively. Let G be any preference group. For any of the election systems RR1, RR2, or RR3, let S1 denote the set of base strategies, under which all voters simply mark their basic rankings according to their own true preferences, and mark their electorate rankings according to the Condorcet preference ranking for the electorate as a whole (i.e., A

_{m}first, B

_{m}second, and so on). If all preference groups other than G follow S1 but G can deviate from S1, then all of the following hold for each of RR1, RR2, and RR3:

_{m}as its first preference, the members of G can adopt no ballot strategy that will cause A

_{m}to place last in U

_{m}.

_{m}and A

_{m}as its first and second preferences, respectively, its members can adopt no ballot strategy that will prevent A

_{m}from placing above B

_{m}in U

_{m}.

_{m}as its second preference and does not have B

_{m}as its first preference, the members of G can adopt no ballot strategy that will cause B

_{m}to place last in U

_{m}.

**Proof.**The proof of Theorem T1(m) is by mathematical induction on m, and thus has two main steps. The first one is to prove T1(3). The m = 3 candidates will be denoted by A (=A

_{3}), B (=B

_{3}), and C (Condorcet preference loser). There are, of course, 3! = 6 preference groups G.

**Proof of T1.1(3).**T1.1(3) applies to four of these six preference groups. Because G(BCA) and G(CBA) are already marking their basic rankings with A last, neither one can switch to a ballot strategy that will stop A from placing first. Because G(CAB) voters are already marking C first, they can do nothing to pull C ahead of either A or B in the two-way comparisons with each, so C must remain last. By L1.2 (see Section 5.1), G(BAC) voters cannot stop A from placing above B, regardless of whether the election system is RR1, RR2, or RR3.

**Proof of T1.2(3).**G(BAC) is the only applicable preference group. Again, the result follows from L1.2.

**Proof of T1.3(3).**The only applicable preference group is G(CAB). This group is unable to stop C from placing last, and so B cannot place last.

_{m}

_{+1}as third preference or lower.

_{m}

_{+1}as second preference, and B

_{m}

_{+1}is not first preference.

_{m}

_{+1}as second preference and B

_{m}

_{+1}as first preference.

**Proof of T1.1.1(m + 1) given T1(m).**G cannot have A

_{m}

_{+1}higher than second preference in any of the m U

_{m}sets that result from removing a candidate other than A

_{m}

_{+1}from U

_{m}

_{+1}. Thus, by T1.1(m) with A

_{m}

_{+1}identified therein as A

_{m}, it follows that A

_{m}

_{+1}cannot place last in any of these m U

_{m}sets, regardless of G’s ballot strategy. It is therefore not possible for A

_{m}

_{+1}to place last in U

_{m}

_{+1}, because to do so A

_{m}

_{+1}would have to place last in at least one of the m U

_{m}sets to which the candidate belongs.

**Proof of T1.1.2(m + 1) given T1(m).**In the U

_{m}set obtained by removing A

_{m}

_{+1}from U

_{m}

_{+1}, no ballot strategy adopted by G can cause B

_{m}

_{+1}to place last, by virtue of T1.1(m) with B

_{m}

_{+1}identified therein as A

_{m}. Similarly, in the U

_{m}set with B

_{m}

_{+1}removed from U

_{m}

_{+1}, G cannot cause A

_{m}

_{+1}to place last, by virtue of T1.1(m) with A

_{m}

_{+1}identified therein as A

_{m}. In any U

_{m}set obtained by removing any candidate other than A

_{m}

_{+1}, B

_{m}

_{+1}, or G’s first preference from U

_{m}

_{+1}, A

_{m}

_{+1}cannot place last, thanks to T1.1(m) with A

_{m}

_{+1}identified therein as A

_{m}, and neither can B

_{m}

_{+1}place last, owing to T1.3(m) with A

_{m}

_{+1}and B

_{m}

_{+1}identified therein as A

_{m}and B

_{m}, respectively. Thus, neither A

_{m}

_{+1}nor B

_{m}

_{+1}can place last in any of the (m + 1) possible U

_{m}sets except perhaps the one from which G’s first-preference candidate is removed. This means that some candidate other than A

_{m}

_{+1}or B

_{m}

_{+1}has to place last in at least two of the U

_{m}sets, which precludes any possibility for either A

_{m}

_{+1}or B

_{m}

_{+1}to place last in U

_{m}

_{+1}.

**Proof of T1.1.3(m + 1) given T1(m).**In any of the (m − 1) sets U

_{m}obtained by removing a candidate other than A

_{m}

_{+1}or B

_{m}

_{+1}from U

_{m}

_{+1}, A

_{m}

_{+1}must place above B

_{m}

_{+1}, no matter what ballot strategy G uses. This follows from T1.2(m) with A

_{m}

_{+1}and B

_{m}

_{+1}identified therein as A

_{m}and B

_{m}, respectively. Thus, among the (m + 1) sets U

_{m}obtainable by dropping one candidate from U

_{m}

_{+1}, A

_{m}

_{+1}can place last only in the U

_{m}set that results when B

_{m}

_{+1}is the candidate removed. In fact, A

_{m}

_{+1}must place last in that U

_{m}set if there is to be any possibility that A

_{m}

_{+1}will place last in U

_{m}

_{+1}.

_{m}

_{+1}is to place last in U

_{m}

_{+1}, the number of the U

_{m}sets in which B

_{m}

_{+1}places last cannot exceed 1 (since then B

_{m}

_{+1}would have more last places than A

_{m}

_{+1}) and cannot be 0 (since then some other candidate would have more last places than A

_{m}

_{+1}). Thus, B

_{m}

_{+1}must place last in exactly one of the (m + 1) sets U

_{m}to keep alive the possibility of A

_{m}

_{+1}placing last in U

_{m}

_{+1}.

_{m}

_{+1}must place above B

_{m}

_{+1}in all sets U

_{m}that include both of these candidates, A

_{m}

_{+1}can place second from last in only one of these sets—the one in which B

_{m}

_{+1}places last. Arguing as before, one finds that A

_{m}

_{+1}must place next to last in this set, and also that B

_{m}

_{+1}must place second from last in exactly one set U

_{m}(if A

_{m}

_{+1}is to have any chance of placing last in U

_{m}

_{+1}). Continued application of this argument leads to the conclusion that the (m + 1) sets U

_{m}have to consist of one set with B

_{m}

_{+1}first and A

_{m}

_{+1}omitted, one with A

_{m}

_{+1}first and B

_{m}

_{+1}second, one with A

_{m}

_{+1}second and B

_{m}

_{+1}third, ..., one with A

_{m}

_{+1}third from last and B

_{m}

_{+1}second from last, one with A

_{m}

_{+1}second from last and B

_{m}

_{+1}last, and one with A

_{m}

_{+1}last and B

_{m}

_{+1}omitted.

_{m}

_{+1}is a capping candidate, and A

_{m}

_{+1}caps B

_{m}

_{+1}. Because A

_{m}

_{+1}is a capping candidate, A

_{m}

_{+1}cannot place last in U

_{m}

_{+1}unless all (m + 1) candidates are capping candidates. To complete the proof that A

_{m}

_{+1}cannot place last in U

_{m}

_{+1}, it remains to deal with the case where every candidate is a capping candidate.

_{m}

_{+1}is the central member will also contain A

_{m}

_{+1}. By L1.1, the number of approval votes for B

_{m}

_{+1}in this triple will be less than half the electorate (regardless of whether the election system is RR1, RR2, or RR3). Thus, B

_{m}

_{+1}is a lesser candidate. The contiguous triple of which A

_{m}

_{+1}is the central member will also contain B

_{m}

_{+1}. By L1.1, the number of approval votes for A

_{m}

_{+1}in this triple will be more than half the electorate. Thus, A

_{m}

_{+1}is not a lesser candidate. (Remember that approval votes are tallied for the central member of each contiguous triple even if the triple is noncyclical.)

_{m}

_{+1}), and because A

_{m}

_{+1}is not a lesser candidate, the last‑place candidate in U

_{m}

_{+1}will be chosen from candidates who do not include A

_{m}

_{+1}. This eliminates the only remaining avenue by which A

_{m}

_{+1}could place last in U

_{m}

_{+1}.

**Proof of T1.2(m + 1) given T1(m).**By T1.1(m + 1) (which was just proved), A

_{m}

_{+1}cannot place last in U

_{m}

_{+1}, regardless of what ballot strategy G uses. If B

_{m}

_{+1}places last in U

_{m}

_{+1}, the conclusion of T1.2(m + 1) is immediate. Thus, one needs to consider only the case where some candidate other than A

_{m}

_{+1}or B

_{m}

_{+1}places last in U

_{m}

_{+1}. The proof of T1.2(m + 1) is completed by applying T1.2(m), with A

_{m}

_{+1}and B

_{m}

_{+1}identified therein as A

_{m}and B

_{m}, respectively, to the U

_{m}set that is obtained by removing the last‑place candidate from U

_{m}

_{+1}.

**Proof of T1.3(m + 1) given T1(m).**See proof of T1.1.2(m + 1) given T1(m), above.

**Proposition 2.**Let there be m candidates with the same conditions and notation as in Theorem T1. (Thus, the domain for admissible true-preference profiles is F0.) Then each of the systems RR1, RR2, and RR3 is non-manipulable. That is, no preference group G, by deviating from S1 when no one else does, can adopt a ballot strategy that will dethrone A

_{m}(the Condorcet preference winner) as the election winner in favor of a candidate whom G prefers to A

_{m}.

**Proof.**G cannot gain by deviating from S1 if G has A

_{m}as its first preference. With this case disregarded, T1.1(m) implies that G can adopt no ballot strategy that will cause A

_{m}to place last in U

_{m}. Therefore some other candidate will have to place at the bottom of U

_{m}. Let U

_{m}

_{−1}denote the set obtained by removing that candidate from U

_{m}. If A

_{m}is G’s top preference within U

_{m}

_{−1}, then G can achieve nothing better than the election of A

_{m}. If not, then apply T1.1(m − 1) to U

_{m}

_{−1}, with A

_{m}identified therein as A

_{m}

_{−1}, to conclude that G can do nothing to cause A

_{m}to place last in U

_{m}

_{−1}. Let U

_{m}

_{−2}denote the set obtained by removing from U

_{m}

_{−1}the candidate who does place last in U

_{m}

_{−1}. If A

_{m}is G’s first preference in U

_{m}

_{−2}, then, as before, the conclusion is proved. If not, then apply T1.1(m − 2). Continue in this way until either the conclusion is proved because A

_{m}is G’s top preference among remaining (non‑excluded) candidates or else a candidate (other than A

_{m}) is removed from U

_{4}to produce U

_{3}. In the latter case, apply Proposition 1 to show that G can do nothing in U

_{3}to dethrone A

_{m}in favor of another candidate in U

_{3}whom G prefers to A

_{m}.

## B. Non-Manipulability of RR1 and RA1 When There Is a Condorcet Preference Winner

**Lemma L2.**Let there be a set of three candidates A, B, and C who are, respectively, the Condorcet preference winner, runner‑up, and loser. Under the system RR1, let S1* denote the set of ballot strategies under which all voters mark their basic rankings according to their own true preferences, and mark their electorate rankings either as A first, B second, and C third or as A first, C second, and B third. Under RA1, let S2* denote the set of ballot strategies under which all voters mark their basic rankings according to their own true preferences, and mark their approval votes in accord with Rule 1 (that is, their approval votes go only to A if A is their first preference, or to their first two preferences otherwise). Let G* refer to either G(BAC) or G(CAB) (but not to both simultaneously), and let G** refer to the other one. For either RR1 or RA1, suppose that all preference groups other than G* follow S1* or S2* (respectively) but that G* can deviate from S1* or S2*. Then both of the following will hold no matter what ballot strategy G* adopts:

**Proof of L2.1.**Under either RR1 or RA1 (and under S1* or S2*, respectively), G(ABC) and G(ACB) provide approval votes only to A, and G** gives approval votes to its top two preferences. Thus, G(ABC), G(ACB), and G** provide approval votes to A but not to G*’s most preferred candidate. Those are the three groups that prefer A to G*’s top preference (B or C) and thus constitute more than half the electorate. L2.1 then follows.

**Proof of L2.2.**The argument is like that in the proof of L1.2.

**Lemma L3.**Let there be m = 3 candidates with the same conditions and notation as in the first three sentences of Lemma L2. Then each of the systems RR1 and RA1 is non-manipulable. That is, no preference group G, by deviating from S1* or S2* when no one else does, can adopt a ballot strategy that will dethrone A (the Condorcet preference winner) as the election winner in favor of a candidate whom G prefers to A.

**Proof for RA1.**For RA1 the proof is already covered by Proposition 1, because S2* in Lemma L3 and Lemma L2 is the same as S2 in Proposition 1 and Lemma L1.

**Proof for RR1.**Because S1* here and S1 earlier are not likewise the same, for RR1 the proof of Proposition 1 has to be modified slightly. Again the proof considers each of the six preference groups separately. The argument is the same as before for the five preference groups other than G(BAC). Then, from L2.2 (where G* is G(BAC)) it follows that G(BAC) cannot stop A from placing above B, and thus can bring about nothing better than the election of A.

**Theorem T2(m).**Let U

_{m}be a set of m (> 2) candidates that includes a Condorcet preference winner, to be denoted by A

_{m}, but otherwise may have Condorcet preference cycle(s). Let G be any preference group. For the election system RR1, let S1* denote any set of ballot strategies under which all voters simply mark their basic rankings according to their own true preferences, and mark their electorate rankings with A

_{m}ranked first but with the other candidates in any order. For RA1, let S2* denote the set of ballot strategies under which all voters mark their basic rankings according to their own true preferences, and mark their approval votes in accord with Rule 1 (which means that their approval votes go to A and to any candidate preferred to A unless A is their lowest preference, in which case all candidates except A receive their approval votes). For either RR1 or RA1, if all preference groups other than G follow S1* or S2* (respectively) but G can deviate from S1* or S2*, then the following will hold:

_{m}as its first preference, the members of G can adopt no ballot strategy that will cause A

_{m}to place last in U

_{m}.

_{m}as its second preference and some other candidate W

_{m}as its first preference, its members can adopt no ballot strategy that will prevent A

_{m}from placing above W

_{m}in U

_{m}.

**Proof.**The proof of Theorem T2(m) is much like that of T1(m). It uses mathematical induction again and starts by proving T2(3). Then it proves T2(m + 1) given T2(m). For RR1 and RA1 the arguments are essentially the same.

**Proof of T2.1(3).**With m = 3 (though not when m > 3), there can be no Condorcet preference cycle, because it is given that A

_{3}(hereafter A) is the Condorcet preference winner. The premises of Lemma L2 are thus satisfied. Then T2.1(3) follows from L2.2 if G is G(BAC) or G(CAB), where B and C denote (respectively) the Condorcet preference runner-up and loser within the set of three candidates. For the other two applicable preference groups, G(BCA) and G(CBA), the proof of T2.1(3) is the same as that for T1.1(3).

**Proof of T2.2(3).**G(BAC) and G(CAB) are the only applicable preference groups. T2.2(3) follows from L2.2.

_{m}

_{+1}as third preference or lower.

_{m}

_{+1}as second preference.

**Proof of T2.1.1(m + 1) given T2(m).**The proof follows the same wording as in the proof of T1.1.1(m + 1) given T1(m) (see Appendix A), except that “T1.1(m)” therein is to be replaced by “T2.1(m).”

**Proof of T2.1.2(m + 1) given T2(m).**Let W

_{m}

_{+1}denote the first‑preference candidate of G within U

_{m}

_{+1}. Then the proof follows the same wording as the proof of T1.1.3(m + 1) given T1(m), except that “B

_{m}

_{+1}” therein is to be replaced (wherever “B

_{m}

_{+1}” appears) by “W

_{m}

_{+1},” “B

_{m}” is to be replaced by “W

_{m},” “T1.2(m)” by “T2.2(m),” and “L1.1” by “L2.1.” In addition, the reference to “RR1, RR2, or RR3” is to be changed to “RR1 or RA1.”

**Proof of T2.2(m + 1) given T2(m).**The proof follows the same wording as the proof of T1.2(m + 1) given T1(m), except that “B

_{m}

_{+1}” therein is to be replaced by “W

_{m}

_{+1}, “B

_{m}” by “W

_{m},” “T1.1(m + 1)” by “T2.1(m + 1),” “T1.2(m + 1)” by “T2.2(m + 1),” and “T1.2(m)” by “T2.2(m).”

**Proposition 3.**Let there be m candidates with the same conditions and notation as in Theorem T2. (In particular, the domain for admissible true-preference profiles is F1.) Then each of the systems RR1 and RA1 is non-manipulable. That is, no preference group G, by deviating from S1* (for RR1) or S2* (for RA1) when no one else does, can adopt a ballot strategy that will dethrone A

_{m}(the Condorcet preference winner) as the election winner in favor of a candidate whom G prefers to A

_{m}.

**Proof.**For either RR1 or RA1, the proof has the same wording as the proof of Proposition 2, except that “S1” therein is to be changed to “S1* or S2*,” “T1.1(m)” to “T2.1(m),” “T1.1(m − 1)” to “T2.1(m − 1),” “T1.1(m − 2)” to “T2.1(m − 2),” and “Proposition 1” to “Lemma L3.”

## C. Validity of Restricting Condorcet Completion Method to the Top Vote Cycle

**Proposition 4.**Consider any of the four aggregation procedures (for RR1, RR2, or RR3, or for the RA ballot). Suppose that there is no Condorcet vote winner and that the top vote cycle consists of m* (3 ≤ m* ≤ m) of the m candidates. Call any candidate in the top vote cycle a B candidate, and call any other candidate a W candidate. Then both of the following hold:

**Proof.**The same proof of Proposition 4 applies to all four of the aggregation procedures.

**Proof of P4.1(j).**The proof is by mathematical induction on j (for any fixed m). We first prove P4.1(3), and then prove P4.1(j + 1) given P4.1(j).

**Proof of P4.1(3).**The result is trivially true if the three candidates are either all B’s or all W’s. If there are two B’s and one W or vice versa, then the triple must be noncyclical and no W can place above a B; this is because no W is rated above a B in the basic rankings.

**Proof of P4.1(j + 1) given P4.1(j).**Let U

_{j}

_{+1}denote any set of (j + 1) of the m candidates. If there are no W’s in U

_{j}

_{+1}, then P4.1(j + 1) is trivially true. If U

_{j}

_{+1}has exactly one W, then, by P4.1(j), that W places at the bottom of all but one of the (j + 1) subsets of U

_{j}

_{+1}that are of size j, and thus places at the bottom of U

_{j}

_{+1}. If U

_{j}

_{+1}has two or more W’s, then P4.1(j) implies that all (j + 1) of the subsets must have a W placing at the bottom. Thus, a W must place at the bottom of U

_{j}

_{+1}. Because the remaining candidates are in the same order as in the subset that contains them all, no W can place above a B in that subset (again by P4.1(j)) nor (therefore) in U

_{j}

_{+1}itself.

**Proof of P4.2.**By P4.1(m), the m* B candidates will place at the top of the m candidates when the Condorcet completion method is applied to all m candidates. Moreover, by virtue of the aggregation procedure, the ordering of these m* candidates at the top will be the same as in the subset that consists of them alone. This is, of course, the same ordering that is obtained upon applying the Condorcet completion method to just those m* candidates.

## Conflicts of Interest

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**MDPI and ACS Style**

Potthoff, R.F.
Condorcet Completion Methods that Inhibit Manipulation through Exploiting Knowledge of Electorate Preferences. *Games* **2014**, *5*, 204-233.
https://doi.org/10.3390/g5040204

**AMA Style**

Potthoff RF.
Condorcet Completion Methods that Inhibit Manipulation through Exploiting Knowledge of Electorate Preferences. *Games*. 2014; 5(4):204-233.
https://doi.org/10.3390/g5040204

**Chicago/Turabian Style**

Potthoff, Richard F.
2014. "Condorcet Completion Methods that Inhibit Manipulation through Exploiting Knowledge of Electorate Preferences" *Games* 5, no. 4: 204-233.
https://doi.org/10.3390/g5040204