The Liquid Democracy Journal
on electronic participation, collective moderation, and voting systems
Issue 3
2015-01-23

Preferential Delegation and the Problem of Negative Voting Weight

by Jan Behrens & Björn Swierczek, Berlin, January 23, 2015 other format: text version (UTF-8)

In this article, we analyze all systems where each voter may freely choose to vote directly, or to delegate the decision to one or more persons of his or her free choice, or to abstain from voting (i.e. neither voting directly nor delegating to another person).

If two people are chosen as delegates to cast one's vote, then the delegating person must select one person as primary delegate, in which case the other person will be referred to as the secondary delegate. Accordingly, a preference list has to be provided by the voter in case of more than two delegates. If only one person is chosen as delegate, we also refer to that delegate as primary delegate for the remainder of this article.

We further assume that the reader is familiar with the general concept of vote delegation and the dualism of transferring voting weight and copying your delegate's vote. [PLF, p.23]

We expect such a system to fulfill at least the following 7 properties:

Property 1 (“Precedence”)

If a person A does not vote directly but has one delegate B, or two delegates B and C, where B is the primary and C is the secondary delegate, and none of A's delegates is either delegating to A, to each other, or to any other voter (i.e. if A's delegates are not delegating at all), then the following rules shall be fulfilled:

If the primary delegate B chooses to vote directly, then A votes (through delegation) as B does. If the primary delegate B doesn't vote directly and doesn't delegate, but the secondary delegate C votes directly, then A votes (through delegation) as C does.

For all other cases (e.g. when one of A's delegates is delegating further), no assumptions are made at this point.

Property 2 (“Anonymity”)

All voters are interchangeable with each other, as long as they behave in the same manner.

This property is also called “anonymity” in voting theory, [May, p.681] not to be confused with anonymous/secret voting. [PLF, p.148]

Property 3 (“Neutrality”)

All voting options are interchangable with each other, e.g. replacing all direct YES votes with direct NO votes while replacing all direct NO votes with direct YES votes will simply exchange their vote counts: the total number of votes for YES will become the total number of votes for NO, and the total number of votes for NO will become the total number of votes for YES. Thus, a tie will stay a tie, the previous outcome of YES as winner would change into NO being winner, and the previous outcome of NO as winner would change into YES being winner of the voting procedure if all direct YES votes are replaced with direct NO votes and vice versa. See also [May, p.682].

Property 4 (“Consistency”)

Unconnected subsets of the delegation graph can be considered separately (according to these 7 properties) and do not influence each other.

Property 5 (“Directionality”)

Influence of delegation is directional, i.e. if we split the electorate into two subsets R and S, and if none of the persons in S delegate to any person in R, then the behavior of the voters in subset S is independent of any voter in R. In particular: one person A delegating to another person B may affect how A's vote is used but must not change how B's vote is used, as long as there is no circular delegation path leading back to A.

Note: A delegation system fulfilling Property 5 always fulfills Property 4 as well. Therefore, Property 5 is a generalization of Property 4.

Property 6 (“Equality of Direct and Delegating Voters”)

Copying your delegates' votes according to Property 1 but acting as a directly voting person (instead of using the delegation system) doesn't change the outcome (i.e. the final vote counts) of the voting procedure. This rule only applies if the delegates whose votes are copied do not delegate futher. No assumptions are made otherwise (see also Property 1).

Fulfilling this property is particularly important to give all participants equal opportunities. Violating this property may cause some voters to have an advantage over other voters, depending on their social integration and/or technical abilities. [PLF, p.34-37]

Property 7 (“No Negative Voting Weight Through Delegation”)

If a person A doesn't vote directly and doesn't delegate to anyone, and if (in a binary yes/no-decision) a person B votes via delegation in favor of a proposal that wins, then changing A's behavior to delegate to B instead of abstaining (i.e. neither voting directly nor delegating) must not cause the previously winning proposal to lose.

Impossibility to fulfill all 7 properties

As we will show in the remainder of this article, it is impossible to fulfill all 7 properties under the given assumptions (e.g. freedom of choice regarding one's delegates). To prove this theorem, we will have a look at the following 26 cases.

For the remainder of this article, we define:

p(x,y) := x, if x ≠ ∅, otherwise y.

“∅” shall denote abstention from voting (i.e. neither voting directly nor through delegation). Primary delegation is depicted as an arrow, secondary delegation is depicted as a dashed arrow.

Note: In the following examples, Property 2 and Property 3 will be used implicitly until Case XXIV inclusive; the use of any other property will be explicitly noted in the text (and noted in the black arrows using a notation of “P1” for Property 1, and so on).

Case I

The first analyzed case consists of two voters: one voter B who directly casts a vote for option “x” (where x may be “YES” or “NO” in our example) and one voter A who delegates his or her decision to the other voter. Using Property 1, we can deduce that the delegating voter will also vote for “x” (via delegation).

x ∈ {YES, NO}

Voter A delegates to voter B.
Voter B directly votes for “x”.
Voter A delegates to voter B. Voter B directly votes for “x”.
Figure 1.1: Case I
Considering Property P1.
Voter A delegates to voter B and votes via delegation for “x”.
Voter B directly votes for “x”.
Considering Property P1. Voter A delegates to voter B and votes via delegation for “x”. Voter B directly votes for “x”.
Figure 1.2: Case I (cont.)

Case II

The second case consists of three voters: one voter B who either directly casts a vote for option “x” or abstains (i.e. doesn't vote and doesn't delegate), one voter C who directly casts a vote for option “y” (which may be equal to option “x” if voter B does not abstain), and one voter A who delegates his or her decision to the other two voters while selecting a preference in favor of voter B. Also here, we can use Property 1 to deduce how the delegating person's vote will be used. In this Case II, the delegating participant will vote for p(x,y) := [x, if x ≠ ∅, otherwise y] (whereas “∅” denotes abstention from voting).

x ∈ {YES, NO, ∅}
y ∈ {YES, NO}
Voter A primarily delegates to voter B and secondarily delegates to voter C.
Voter B directly votes for “x”, or does not vote if x = ∅.
Voter C directly votes for “y”.
Voter A primarily delegates to voter B and secondarily delegates to voter C. Voter B directly votes for “x”, or does not vote if x = ∅. Voter C directly votes for “y”.
Figure 2.1: Case II
Considering Property P1.
Voter A primarily delegates to voter B, secondarily delegates to voter C, and votes via delegation for “p(x,y)”.
Voter B directly votes for “x”, or does not vote if x = ∅.
Voter C directly votes for “y”.
Considering Property P1. Voter A primarily delegates to voter B, secondarily delegates to voter C, and votes via delegation for “p(x,y)”. Voter B directly votes for “x”, or does not vote if x = ∅. Voter C directly votes for “y”.
Figure 2.2: Case II (cont.)

Case III

We consider a new Case III that can be solved by using the previously solved Case I and applying the rules of Property 4 (“Consistency”).

x ∈ {YES, NO, ∅}
y ∈ {YES, NO}
Voter A delegates to voter B.
Voter B directly votes for “p(x,y)”.
Voter C directly votes for “x”, or does not vote if x = ∅.
Voter D directly votes for “y”.
Voter A delegates to voter B. Voter B directly votes for “p(x,y)”. Voter C directly votes for “x”, or does not vote if x = ∅. Voter D directly votes for “y”.
Figure 3.1: Case III
Considering Case I and Property 4.
Voter A delegates to voter B and votes via delegation for “p(x,y)”.
Voter B directly votes for “p(x,y)”.
Voter C directly votes for “x”, or does not vote if x = ∅.
Voter D directly votes for “y”.
2 votes for “p(x,y)”,
1 vote for “x”,
1 vote for “y”.
Considering Case I and Property 4. Voter A delegates to voter B and votes via delegation for “p(x,y)”. Voter B directly votes for “p(x,y)”. Voter C directly votes for “x”, or does not vote if x = ∅. Voter D directly votes for “y”. 2 votes for “p(x,y)”, 1 vote for “x”, 1 vote for “y”.
Figure 3.2: Case III (cont.)

Case IV

We consider a new Case IV that can be solved by first applying the rules of Property 5 (“Directivity”) to Case II in order to determine all votes but one, and then, due to Property 6 (“Equality of Direct and Delegating Voters”), using the vote counts determined in Case III to solve the last vote.

x ∈ {YES, NO, ∅}
y ∈ {YES, NO}
Voter A delegates to voter B.
Voter B primarily delegates to voter C and secondarily delegates to voter D.
Voter C directly votes for “x”, or does not vote if x = ∅.
Voter D directly votes for “y”.
Voter A delegates to voter B. Voter B primarily delegates to voter C and secondarily delegates to voter D. Voter C directly votes for “x”, or does not vote if x = ∅. Voter D directly votes for “y”.
Figure 4.1: Case IV
Considering Case II and Property 5.
Voter A delegates to voter B.
Voter B primarily delegates to voter C, secondarily delegates to voter D, and votes via delegation for “p(x,y)”.
Voter C directly votes for “x”, or does not vote if x = ∅.
Voter D directly votes for “y”.
Considering Case II and Property 5. Voter A delegates to voter B. Voter B primarily delegates to voter C, secondarily delegates to voter D, and votes via delegation for “p(x,y)”. Voter C directly votes for “x”, or does not vote if x = ∅. Voter D directly votes for “y”.
Figure 4.2: Case IV (cont.)
Considering Case III and Property 6.
Voter A delegates to voter B and votes via delegation for “p(x,y)”.
Voter B primarily delegates to voter C, secondarily delegates to voter D, and votes via delegation for “p(x,y)”.
Voter C directly votes for “x”, or does not vote if x = ∅.
Voter D directly votes for “y”.
Considering Case III and Property 6. Voter A delegates to voter B and votes via delegation for “p(x,y)”. Voter B primarily delegates to voter C, secondarily delegates to voter D, and votes via delegation for “p(x,y)”. Voter C directly votes for “x”, or does not vote if x = ∅. Voter D directly votes for “y”.
Figure 4.3: Case IV (cont.)

Case V

We consider a new Case V that can be solved by using the previously solved Case II and applying the rules of Property 4 (“Consistency”).

x ∈ {YES, NO}
z ∈ {YES, NO}
Voter A primarily delegates to voter B and secondarily delegates to voter C.
Voter B directly votes for “x”.
Voter C directly votes for “z”.
Voter D directly votes for “x”.
Voter A primarily delegates to voter B and secondarily delegates to voter C. Voter B directly votes for “x”. Voter C directly votes for “z”. Voter D directly votes for “x”.
Figure 5.1: Case V
Considering Case II and Property 4.
Voter A primarily delegates to voter B, secondarily delegates to voter C, and votes via delegation for “p(x,z) = x”.
Voter B directly votes for “x”.
Voter C directly votes for “z”.
Voter D directly votes for “x”.
3 votes for “x”,
1 vote for “z”.
Considering Case II and Property 4. Voter A primarily delegates to voter B, secondarily delegates to voter C, and votes via delegation for “p(x,z) = x”. Voter B directly votes for “x”. Voter C directly votes for “z”. Voter D directly votes for “x”. 3 votes for “x”, 1 vote for “z”.
Figure 5.2: Case V (cont.)

Case VI

We consider a new Case VI that can be solved by first applying the rules of Property 5 (“Directivity”) to Case I in order to determine all votes but one, and then, due to Property 6 (“Equality of Direct and Delegating Voters”), using the vote counts determined in Case V to solve the last vote.

x ∈ {YES, NO}
z ∈ {YES, NO}
Voter A primarily delegates to voter B and secondarily delegates to voter C.
Voter B delegates to voter D.
Voter C directly votes for “z”.
Voter D directly votes for “x”.
Voter A primarily delegates to voter B and secondarily delegates to voter C. Voter B delegates to voter D. Voter C directly votes for “z”. Voter D directly votes for “x”.
Figure 6.1: Case VI
Considering Case I and Property 5.
Voter A primarily delegates to voter B and secondarily delegates to voter C.
Voter B delegates to voter D and votes via delegation for “x”.
Voter C directly votes for “z”.
Voter D directly votes for “x”.
Considering Case I and Property 5. Voter A primarily delegates to voter B and secondarily delegates to voter C. Voter B delegates to voter D and votes via delegation for “x”. Voter C directly votes for “z”. Voter D directly votes for “x”.
Figure 6.2: Case VI (cont.)
Considering Case V and Property 6.
Voter A primarily delegates to voter B, secondarily delegates to voter C, and votes via delegation for “x”.
Voter B delegates to voter D and votes via delegation for “x”.
Voter C directly votes for “z”.
Voter D directly votes for “x”.
Considering Case V and Property 6. Voter A primarily delegates to voter B, secondarily delegates to voter C, and votes via delegation for “x”. Voter B delegates to voter D and votes via delegation for “x”. Voter C directly votes for “z”. Voter D directly votes for “x”.
Figure 6.3: Case VI (cont.)

Case VII

We consider a new Case VII that can be solved by using the previously solved Case IV and applying the rules of Property 4 (“Consistency”).

x ∈ {YES, NO}
z ∈ {YES, NO}
Voter A delegates to voter B.
Voter B primarily delegates to voter C and secondarily delegates to voter D.
Voter C directly votes for “x”.
Voter D directly votes for “z”.
Voter E directly votes for “x”.
Voter A delegates to voter B. Voter B primarily delegates to voter C and secondarily delegates to voter D. Voter C directly votes for “x”. Voter D directly votes for “z”. Voter E directly votes for “x”.
Figure 7.1: Case VII
Considering Case IV and Property 4.
Voter A delegates to voter B and votes via delegation for “p(x,z) = x”.
Voter B primarily delegates to voter C, secondarily delegates to voter D, and votes via delegation for “p(x,z) = x”.
Voter C directly votes for “x”.
Voter D directly votes for “z”.
Voter E directly votes for “x”.
4 votes for “x”,
1 vote for “z”.
Considering Case IV and Property 4. Voter A delegates to voter B and votes via delegation for “p(x,z) = x”. Voter B primarily delegates to voter C, secondarily delegates to voter D, and votes via delegation for “p(x,z) = x”. Voter C directly votes for “x”. Voter D directly votes for “z”. Voter E directly votes for “x”. 4 votes for “x”, 1 vote for “z”.
Figure 7.2: Case VII (cont.)

Case VIII

We consider a new Case VIII that can be solved by first applying the rules of Property 5 (“Directivity”) to Case VI in order to determine all votes but one, and then, due to Property 6 (“Equality of Direct and Delegating Voters”), using the vote counts determined in Case VII to solve the last vote.

x ∈ {YES, NO}
z ∈ {YES, NO}
Voter A delegates to voter B.
Voter B primarily delegates to voter C and secondarily delegates to voter D.
Voter C delegates to voter E.
Voter D directly votes for “z”.
Voter E directly votes for “x”.
Voter A delegates to voter B. Voter B primarily delegates to voter C and secondarily delegates to voter D. Voter C delegates to voter E. Voter D directly votes for “z”. Voter E directly votes for “x”.
Figure 8.1: Case VIII
Considering Case VI and Property 5.
Voter A delegates to voter B.
Voter B primarily delegates to voter C, secondarily delegates to voter D, and votes via delegation for “x”.
Voter C delegates to voter E and votes via delegation for “x”.
Voter D directly votes for “z”.
Voter E directly votes for “x”.
Considering Case VI and Property 5. Voter A delegates to voter B. Voter B primarily delegates to voter C, secondarily delegates to voter D, and votes via delegation for “x”. Voter C delegates to voter E and votes via delegation for “x”. Voter D directly votes for “z”. Voter E directly votes for “x”.
Figure 8.2: Case VIII (cont.)
Considering Case VII and Property 6.
Voter A delegates to voter B and votes via delegation for “x”.
Voter B primarily delegates to voter C, secondarily delegates to voter D, and votes via delegation for “x”.
Voter C delegates to voter E and votes via delegation for “x”.
Voter D directly votes for “z”.
Voter E directly votes for “x”.
Considering Case VII and Property 6. Voter A delegates to voter B and votes via delegation for “x”. Voter B primarily delegates to voter C, secondarily delegates to voter D, and votes via delegation for “x”. Voter C delegates to voter E and votes via delegation for “x”. Voter D directly votes for “z”. Voter E directly votes for “x”.
Figure 8.3: Case VIII (cont.)

Case IX

We consider a new Case IX that can be solved by using the previously solved Case VI and applying the rules of Property 4 (“Consistency”).

x ∈ {YES, NO, ∅}
y ∈ {YES, NO}
z ∈ {YES, NO}
Voter A primarily delegates to voter B and secondarily delegates to voter C.
Voter B delegates to voter D.
Voter C directly votes for “z”.
Voter D directly votes for “p(x,y)”.
Voter E directly votes for “x”, or does not vote if x = ∅.
Voter F directly votes for “y”.
Voter A primarily delegates to voter B and secondarily delegates to voter C. Voter B delegates to voter D. Voter C directly votes for “z”. Voter D directly votes for “p(x,y)”. Voter E directly votes for “x”, or does not vote if x = ∅. Voter F directly votes for “y”.
Figure 9.1: Case IX
Considering Case VI and Property 4.
Voter A primarily delegates to voter B, secondarily delegates to voter C, and votes via delegation for “p(x,y)”.
Voter B delegates to voter D and votes via delegation for “p(x,y)”.
Voter C directly votes for “z”.
Voter D directly votes for “p(x,y)”.
Voter E directly votes for “x”, or does not vote if x = ∅.
Voter F directly votes for “y”.
3 votes for “p(x,y)”,
1 vote for “x”,
1 vote for “y”,
1 vote for “z”.
Considering Case VI and Property 4. Voter A primarily delegates to voter B, secondarily delegates to voter C, and votes via delegation for “p(x,y)”. Voter B delegates to voter D and votes via delegation for “p(x,y)”. Voter C directly votes for “z”. Voter D directly votes for “p(x,y)”. Voter E directly votes for “x”, or does not vote if x = ∅. Voter F directly votes for “y”. 3 votes for “p(x,y)”, 1 vote for “x”, 1 vote for “y”, 1 vote for “z”.
Figure 9.2: Case IX (cont.)

Case X

We consider a new Case X that can be solved by first applying the rules of Property 5 (“Directivity”) to Case IV in order to determine all votes but one, and then, due to Property 6 (“Equality of Direct and Delegating Voters”), using the vote counts determined in Case IX to solve the last vote.

x ∈ {YES, NO, ∅}
y ∈ {YES, NO}
z ∈ {YES, NO}
Voter A primarily delegates to voter B and secondarily delegates to voter C.
Voter B delegates to voter D.
Voter C directly votes for “z”.
Voter D primarily delegates to voter E and secondarily delegates to voter F.
Voter E directly votes for “x”, or does not vote if x = ∅.
Voter F directly votes for “y”.
Voter A primarily delegates to voter B and secondarily delegates to voter C. Voter B delegates to voter D. Voter C directly votes for “z”. Voter D primarily delegates to voter E and secondarily delegates to voter F. Voter E directly votes for “x”, or does not vote if x = ∅. Voter F directly votes for “y”.
Figure 10.1: Case X
Considering Case IV and Property 5.
Voter A primarily delegates to voter B and secondarily delegates to voter C.
Voter B delegates to voter D and votes via delegation for “p(x,y)”.
Voter C directly votes for “z”.
Voter D primarily delegates to voter E, secondarily delegates to voter F, and votes via delegation for “p(x,y)”.
Voter E directly votes for “x”, or does not vote if x = ∅.
Voter F directly votes for “y”.
Considering Case IV and Property 5. Voter A primarily delegates to voter B and secondarily delegates to voter C. Voter B delegates to voter D and votes via delegation for “p(x,y)”. Voter C directly votes for “z”. Voter D primarily delegates to voter E, secondarily delegates to voter F, and votes via delegation for “p(x,y)”. Voter E directly votes for “x”, or does not vote if x = ∅. Voter F directly votes for “y”.
Figure 10.2: Case X (cont.)
Considering Case IX and Property 6.
Voter A primarily delegates to voter B, secondarily delegates to voter C, and votes via delegation for “p(x,y)”.
Voter B delegates to voter D and votes via delegation for “p(x,y)”.
Voter C directly votes for “z”.
Voter D primarily delegates to voter E, secondarily delegates to voter F, and votes via delegation for “p(x,y)”.
Voter E directly votes for “x”, or does not vote if x = ∅.
Voter F directly votes for “y”.
Considering Case IX and Property 6. Voter A primarily delegates to voter B, secondarily delegates to voter C, and votes via delegation for “p(x,y)”. Voter B delegates to voter D and votes via delegation for “p(x,y)”. Voter C directly votes for “z”. Voter D primarily delegates to voter E, secondarily delegates to voter F, and votes via delegation for “p(x,y)”. Voter E directly votes for “x”, or does not vote if x = ∅. Voter F directly votes for “y”.
Figure 10.3: Case X (cont.)

Case XI

We consider a new Case XI that can be solved by using the previously solved Case VIII and applying the rules of Property 4 (“Consistency”).

x ∈ {YES, NO, ∅}
y ∈ {YES, NO}
z ∈ {YES, NO}
Voter A delegates to voter B.
Voter B primarily delegates to voter C and secondarily delegates to voter D.
Voter C delegates to voter E.
Voter D directly votes for “z”.
Voter E directly votes for “p(x,y)”.
Voter F directly votes for “x”, or does not vote if x = ∅.
Voter G directly votes for “y”.
Voter A delegates to voter B. Voter B primarily delegates to voter C and secondarily delegates to voter D. Voter C delegates to voter E. Voter D directly votes for “z”. Voter E directly votes for “p(x,y)”. Voter F directly votes for “x”, or does not vote if x = ∅. Voter G directly votes for “y”.
Figure 11.1: Case XI
Considering Case VIII and Property 4.
Voter A delegates to voter B and votes via delegation for “p(x,y)”.
Voter B primarily delegates to voter C, secondarily delegates to voter D, and votes via delegation for “p(x,y)”.
Voter C delegates to voter E and votes via delegation for “p(x,y)”.
Voter D directly votes for “z”.
Voter E directly votes for “p(x,y)”.
Voter F directly votes for “x”, or does not vote if x = ∅.
Voter G directly votes for “y”.
4 votes for “p(x,y)”,
1 vote for “x”,
1 vote for “y”,
1 vote for “z”.
Considering Case VIII and Property 4. Voter A delegates to voter B and votes via delegation for “p(x,y)”. Voter B primarily delegates to voter C, secondarily delegates to voter D, and votes via delegation for “p(x,y)”. Voter C delegates to voter E and votes via delegation for “p(x,y)”. Voter D directly votes for “z”. Voter E directly votes for “p(x,y)”. Voter F directly votes for “x”, or does not vote if x = ∅. Voter G directly votes for “y”. 4 votes for “p(x,y)”, 1 vote for “x”, 1 vote for “y”, 1 vote for “z”.
Figure 11.2: Case XI (cont.)

Case XII

We consider a new Case XII that can be solved by first applying the rules of Property 5 (“Directivity”) to Case X in order to determine all votes but one, and then, due to Property 6 (“Equality of Direct and Delegating Voters”), using the vote counts determined in Case XI to solve the last vote.

x ∈ {YES, NO, ∅}
y ∈ {YES, NO}
z ∈ {YES, NO}
Voter A delegates to voter B.
Voter B primarily delegates to voter C and secondarily delegates to voter D.
Voter C delegates to voter E.
Voter D directly votes for “z”.
Voter E primarily delegates to voter F and secondarily delegates to voter G.
Voter F directly votes for “x”, or does not vote if x = ∅.
Voter G directly votes for “y”.
Voter A delegates to voter B. Voter B primarily delegates to voter C and secondarily delegates to voter D. Voter C delegates to voter E. Voter D directly votes for “z”. Voter E primarily delegates to voter F and secondarily delegates to voter G. Voter F directly votes for “x”, or does not vote if x = ∅. Voter G directly votes for “y”.
Figure 12.1: Case XII
Considering Case X and Property 5.
Voter A delegates to voter B.
Voter B primarily delegates to voter C, secondarily delegates to voter D, and votes via delegation for “p(x,y)”.
Voter C delegates to voter E and votes via delegation for “p(x,y)”.
Voter D directly votes for “z”.
Voter E primarily delegates to voter F, secondarily delegates to voter G, and votes via delegation for “p(x,y)”.
Voter F directly votes for “x”, or does not vote if x = ∅.
Voter G directly votes for “y”.
Considering Case X and Property 5. Voter A delegates to voter B. Voter B primarily delegates to voter C, secondarily delegates to voter D, and votes via delegation for “p(x,y)”. Voter C delegates to voter E and votes via delegation for “p(x,y)”. Voter D directly votes for “z”. Voter E primarily delegates to voter F, secondarily delegates to voter G, and votes via delegation for “p(x,y)”. Voter F directly votes for “x”, or does not vote if x = ∅. Voter G directly votes for “y”.
Figure 12.2: Case XII (cont.)
Considering Case XI and Property 6.
Voter A delegates to voter B and votes via delegation for “p(x,y)”.
Voter B primarily delegates to voter C, secondarily delegates to voter D, and votes via delegation for “p(x,y)”.
Voter C delegates to voter E and votes via delegation for “p(x,y)”.
Voter D directly votes for “z”.
Voter E primarily delegates to voter F, secondarily delegates to voter G, and votes via delegation for “p(x,y)”.
Voter F directly votes for “x”, or does not vote if x = ∅.
Voter G directly votes for “y”.
Considering Case XI and Property 6. Voter A delegates to voter B and votes via delegation for “p(x,y)”. Voter B primarily delegates to voter C, secondarily delegates to voter D, and votes via delegation for “p(x,y)”. Voter C delegates to voter E and votes via delegation for “p(x,y)”. Voter D directly votes for “z”. Voter E primarily delegates to voter F, secondarily delegates to voter G, and votes via delegation for “p(x,y)”. Voter F directly votes for “x”, or does not vote if x = ∅. Voter G directly votes for “y”.
Figure 12.3: Case XII (cont.)

Case XIII

We consider a new Case XIII that can be solved by using the previously solved Case X and applying the rules of Property 4 (“Consistency”).

x ∈ {YES, NO}
z1 ∈ {YES, NO}
z2 ∈ {YES, NO}
Voter A primarily delegates to voter B and secondarily delegates to voter C.
Voter B delegates to voter D.
Voter C directly votes for “z_{1}”.
Voter D primarily delegates to voter E and secondarily delegates to voter F.
Voter E directly votes for “x”.
Voter F directly votes for “z_{2}”.
Voter G directly votes for “x”.
Voter A primarily delegates to voter B and secondarily delegates to voter C. Voter B delegates to voter D. Voter C directly votes for “z_{1}”. Voter D primarily delegates to voter E and secondarily delegates to voter F. Voter E directly votes for “x”. Voter F directly votes for “z_{2}”. Voter G directly votes for “x”.
Figure 13.1: Case XIII
Considering Case X and Property 4.
Voter A primarily delegates to voter B, secondarily delegates to voter C, and votes via delegation for “p(x,z_{2}) = x”.
Voter B delegates to voter D and votes via delegation for “p(x,z_{2}) = x”.
Voter C directly votes for “z_{1}”.
Voter D primarily delegates to voter E, secondarily delegates to voter F, and votes via delegation for “p(x,z_{2}) = x”.
Voter E directly votes for “x”.
Voter F directly votes for “z_{2}”.
Voter G directly votes for “x”.
5 votes for “x”,
1 vote for “z_{1}”,
1 vote for “z_{2}”.
Considering Case X and Property 4. Voter A primarily delegates to voter B, secondarily delegates to voter C, and votes via delegation for “p(x,z_{2}) = x”. Voter B delegates to voter D and votes via delegation for “p(x,z_{2}) = x”. Voter C directly votes for “z_{1}”. Voter D primarily delegates to voter E, secondarily delegates to voter F, and votes via delegation for “p(x,z_{2}) = x”. Voter E directly votes for “x”. Voter F directly votes for “z_{2}”. Voter G directly votes for “x”. 5 votes for “x”, 1 vote for “z_{1}”, 1 vote for “z_{2}”.
Figure 13.2: Case XIII (cont.)

Case XIV

We consider a new Case XIV that can be solved by first applying the rules of Property 5 (“Directivity”) to Case VIII in order to determine all votes but one, and then, due to Property 6 (“Equality of Direct and Delegating Voters”), using the vote counts determined in Case XIII to solve the last vote.

x ∈ {YES, NO}
z1 ∈ {YES, NO}
z2 ∈ {YES, NO}
Voter A primarily delegates to voter B and secondarily delegates to voter C.
Voter B delegates to voter D.
Voter C directly votes for “z_{1}”.
Voter D primarily delegates to voter E and secondarily delegates to voter F.
Voter E delegates to voter G.
Voter F directly votes for “z_{2}”.
Voter G directly votes for “x”.
Voter A primarily delegates to voter B and secondarily delegates to voter C. Voter B delegates to voter D. Voter C directly votes for “z_{1}”. Voter D primarily delegates to voter E and secondarily delegates to voter F. Voter E delegates to voter G. Voter F directly votes for “z_{2}”. Voter G directly votes for “x”.
Figure 14.1: Case XIV
Considering Case VIII and Property 5.
Voter A primarily delegates to voter B and secondarily delegates to voter C.
Voter B delegates to voter D and votes via delegation for “x”.
Voter C directly votes for “z_{1}”.
Voter D primarily delegates to voter E, secondarily delegates to voter F, and votes via delegation for “x”.
Voter E delegates to voter G and votes via delegation for “x”.
Voter F directly votes for “z_{2}”.
Voter G directly votes for “x”.
Considering Case VIII and Property 5. Voter A primarily delegates to voter B and secondarily delegates to voter C. Voter B delegates to voter D and votes via delegation for “x”. Voter C directly votes for “z_{1}”. Voter D primarily delegates to voter E, secondarily delegates to voter F, and votes via delegation for “x”. Voter E delegates to voter G and votes via delegation for “x”. Voter F directly votes for “z_{2}”. Voter G directly votes for “x”.
Figure 14.2: Case XIV (cont.)
Considering Case XIII and Property 6.
Voter A primarily delegates to voter B, secondarily delegates to voter C, and votes via delegation for “x”.
Voter B delegates to voter D and votes via delegation for “x”.
Voter C directly votes for “z_{1}”.
Voter D primarily delegates to voter E, secondarily delegates to voter F, and votes via delegation for “x”.
Voter E delegates to voter G and votes via delegation for “x”.
Voter F directly votes for “z_{2}”.
Voter G directly votes for “x”.
Considering Case XIII and Property 6. Voter A primarily delegates to voter B, secondarily delegates to voter C, and votes via delegation for “x”. Voter B delegates to voter D and votes via delegation for “x”. Voter C directly votes for “z_{1}”. Voter D primarily delegates to voter E, secondarily delegates to voter F, and votes via delegation for “x”. Voter E delegates to voter G and votes via delegation for “x”. Voter F directly votes for “z_{2}”. Voter G directly votes for “x”.
Figure 14.3: Case XIV (cont.)

Case XV

We consider a new Case XV that can be solved by using the previously solved Case XII and applying the rules of Property 4 (“Consistency”).

x ∈ {YES, NO}
z1 ∈ {YES, NO}
z2 ∈ {YES, NO}
Voter A delegates to voter B.
Voter B primarily delegates to voter C and secondarily delegates to voter D.
Voter C delegates to voter E.
Voter D directly votes for “z_{1}”.
Voter E primarily delegates to voter F and secondarily delegates to voter G.
Voter F directly votes for “x”.
Voter G directly votes for “z_{2}”.
Voter H directly votes for “x”.
Voter A delegates to voter B. Voter B primarily delegates to voter C and secondarily delegates to voter D. Voter C delegates to voter E. Voter D directly votes for “z_{1}”. Voter E primarily delegates to voter F and secondarily delegates to voter G. Voter F directly votes for “x”. Voter G directly votes for “z_{2}”. Voter H directly votes for “x”.
Figure 15.1: Case XV
Considering Case XII and Property 4.
Voter A delegates to voter B and votes via delegation for “p(x,z_{2}) = x”.
Voter B primarily delegates to voter C, secondarily delegates to voter D, and votes via delegation for “p(x,z_{2}) = x”.
Voter C delegates to voter E and votes via delegation for “p(x,z_{2}) = x”.
Voter D directly votes for “z_{1}”.
Voter E primarily delegates to voter F, secondarily delegates to voter G, and votes via delegation for “p(x,z_{2}) = x”.
Voter F directly votes for “x”.
Voter G directly votes for “z_{2}”.
Voter H directly votes for “x”.
6 votes for “x”,
1 vote for “z_{1}”,
1 vote for “z_{2}”.
Considering Case XII and Property 4. Voter A delegates to voter B and votes via delegation for “p(x,z_{2}) = x”. Voter B primarily delegates to voter C, secondarily delegates to voter D, and votes via delegation for “p(x,z_{2}) = x”. Voter C delegates to voter E and votes via delegation for “p(x,z_{2}) = x”. Voter D directly votes for “z_{1}”. Voter E primarily delegates to voter F, secondarily delegates to voter G, and votes via delegation for “p(x,z_{2}) = x”. Voter F directly votes for “x”. Voter G directly votes for “z_{2}”. Voter H directly votes for “x”. 6 votes for “x”, 1 vote for “z_{1}”, 1 vote for “z_{2}”.
Figure 15.2: Case XV (cont.)

Case XVI

We consider a new Case XVI that can be solved by first applying the rules of Property 5 (“Directivity”) to Case XIV in order to determine all votes but one, and then, due to Property 6 (“Equality of Direct and Delegating Voters”), using the vote counts determined in Case XV to solve the last vote.

x ∈ {YES, NO}
z1 ∈ {YES, NO}
z2 ∈ {YES, NO}
Voter A delegates to voter B.
Voter B primarily delegates to voter C and secondarily delegates to voter D.
Voter C delegates to voter E.
Voter D directly votes for “z_{1}”.
Voter E primarily delegates to voter F and secondarily delegates to voter G.
Voter F delegates to voter H.
Voter G directly votes for “z_{2}”.
Voter H directly votes for “x”.
Voter A delegates to voter B. Voter B primarily delegates to voter C and secondarily delegates to voter D. Voter C delegates to voter E. Voter D directly votes for “z_{1}”. Voter E primarily delegates to voter F and secondarily delegates to voter G. Voter F delegates to voter H. Voter G directly votes for “z_{2}”. Voter H directly votes for “x”.
Figure 16.1: Case XVI
Considering Case XIV and Property 5.
Voter A delegates to voter B.
Voter B primarily delegates to voter C, secondarily delegates to voter D, and votes via delegation for “x”.
Voter C delegates to voter E and votes via delegation for “x”.
Voter D directly votes for “z_{1}”.
Voter E primarily delegates to voter F, secondarily delegates to voter G, and votes via delegation for “x”.
Voter F delegates to voter H and votes via delegation for “x”.
Voter G directly votes for “z_{2}”.
Voter H directly votes for “x”.
Considering Case XIV and Property 5. Voter A delegates to voter B. Voter B primarily delegates to voter C, secondarily delegates to voter D, and votes via delegation for “x”. Voter C delegates to voter E and votes via delegation for “x”. Voter D directly votes for “z_{1}”. Voter E primarily delegates to voter F, secondarily delegates to voter G, and votes via delegation for “x”. Voter F delegates to voter H and votes via delegation for “x”. Voter G directly votes for “z_{2}”. Voter H directly votes for “x”.
Figure 16.2: Case XVI (cont.)
Considering Case XV and Property 6.
Voter A delegates to voter B and votes via delegation for “x”.
Voter B primarily delegates to voter C, secondarily delegates to voter D, and votes via delegation for “x”.
Voter C delegates to voter E and votes via delegation for “x”.
Voter D directly votes for “z_{1}”.
Voter E primarily delegates to voter F, secondarily delegates to voter G, and votes via delegation for “x”.
Voter F delegates to voter H and votes via delegation for “x”.
Voter G directly votes for “z_{2}”.
Voter H directly votes for “x”.
Considering Case XV and Property 6. Voter A delegates to voter B and votes via delegation for “x”. Voter B primarily delegates to voter C, secondarily delegates to voter D, and votes via delegation for “x”. Voter C delegates to voter E and votes via delegation for “x”. Voter D directly votes for “z_{1}”. Voter E primarily delegates to voter F, secondarily delegates to voter G, and votes via delegation for “x”. Voter F delegates to voter H and votes via delegation for “x”. Voter G directly votes for “z_{2}”. Voter H directly votes for “x”.
Figure 16.3: Case XVI (cont.)

Case XVII

We consider a new Case XVII that can be solved by using the previously solved Case XIV and applying the rules of Property 4 (“Consistency”).

x ∈ {YES, NO, ∅}
y ∈ {YES, NO}
z1 ∈ {YES, NO}
z2 ∈ {YES, NO}
Voter A primarily delegates to voter B and secondarily delegates to voter C.
Voter B delegates to voter D.
Voter C directly votes for “z_{1}”.
Voter D primarily delegates to voter E and secondarily delegates to voter F.
Voter E delegates to voter G.
Voter F directly votes for “z_{2}”.
Voter G directly votes for “p(x,y)”.
Voter H directly votes for “x”, or does not vote if x = ∅.
Voter I directly votes for “y”.
Voter A primarily delegates to voter B and secondarily delegates to voter C. Voter B delegates to voter D. Voter C directly votes for “z_{1}”. Voter D primarily delegates to voter E and secondarily delegates to voter F. Voter E delegates to voter G. Voter F directly votes for “z_{2}”. Voter G directly votes for “p(x,y)”. Voter H directly votes for “x”, or does not vote if x = ∅. Voter I directly votes for “y”.
Figure 17.1: Case XVII
Considering Case XIV and Property 4.
Voter A primarily delegates to voter B, secondarily delegates to voter C, and votes via delegation for “p(x,y)”.
Voter B delegates to voter D and votes via delegation for “p(x,y)”.
Voter C directly votes for “z_{1}”.
Voter D primarily delegates to voter E, secondarily delegates to voter F, and votes via delegation for “p(x,y)”.
Voter E delegates to voter G and votes via delegation for “p(x,y)”.
Voter F directly votes for “z_{2}”.
Voter G directly votes for “p(x,y)”.
Voter H directly votes for “x”, or does not vote if x = ∅.
Voter I directly votes for “y”.
5 votes for “p(x,y)”,
1 vote for “x”,
1 vote for “y”,
1 vote for “z_{1}”,
1 vote for “z_{2}”.
Considering Case XIV and Property 4. Voter A primarily delegates to voter B, secondarily delegates to voter C, and votes via delegation for “p(x,y)”. Voter B delegates to voter D and votes via delegation for “p(x,y)”. Voter C directly votes for “z_{1}”. Voter D primarily delegates to voter E, secondarily delegates to voter F, and votes via delegation for “p(x,y)”. Voter E delegates to voter G and votes via delegation for “p(x,y)”. Voter F directly votes for “z_{2}”. Voter G directly votes for “p(x,y)”. Voter H directly votes for “x”, or does not vote if x = ∅. Voter I directly votes for “y”. 5 votes for “p(x,y)”, 1 vote for “x”, 1 vote for “y”, 1 vote for “z_{1}”, 1 vote for “z_{2}”.
Figure 17.2: Case XVII (cont.)

Case XVIII

We consider a new Case XVIII that can be solved by first applying the rules of Property 5 (“Directivity”) to Case XII in order to determine all votes but one, and then, due to Property 6 (“Equality of Direct and Delegating Voters”), using the vote counts determined in Case XVII to solve the last vote.

x ∈ {YES, NO, ∅}
y ∈ {YES, NO}
z1 ∈ {YES, NO}
z2 ∈ {YES, NO}
Voter A primarily delegates to voter B and secondarily delegates to voter C.
Voter B delegates to voter D.
Voter C directly votes for “z_{1}”.
Voter D primarily delegates to voter E and secondarily delegates to voter F.
Voter E delegates to voter G.
Voter F directly votes for “z_{2}”.
Voter G primarily delegates to voter H and secondarily delegates to voter I.
Voter H directly votes for “x”, or does not vote if x = ∅.
Voter I directly votes for “y”.
Voter A primarily delegates to voter B and secondarily delegates to voter C. Voter B delegates to voter D. Voter C directly votes for “z_{1}”. Voter D primarily delegates to voter E and secondarily delegates to voter F. Voter E delegates to voter G. Voter F directly votes for “z_{2}”. Voter G primarily delegates to voter H and secondarily delegates to voter I. Voter H directly votes for “x”, or does not vote if x = ∅. Voter I directly votes for “y”.
Figure 18.1: Case XVIII
Considering Case XII and Property 5.
Voter A primarily delegates to voter B and secondarily delegates to voter C.
Voter B delegates to voter D and votes via delegation for “p(x,y)”.
Voter C directly votes for “z_{1}”.
Voter D primarily delegates to voter E, secondarily delegates to voter F, and votes via delegation for “p(x,y)”.
Voter E delegates to voter G and votes via delegation for “p(x,y)”.
Voter F directly votes for “z_{2}”.
Voter G primarily delegates to voter H, secondarily delegates to voter I, and votes via delegation for “p(x,y)”.
Voter H directly votes for “x”, or does not vote if x = ∅.
Voter I directly votes for “y”.
Considering Case XII and Property 5. Voter A primarily delegates to voter B and secondarily delegates to voter C. Voter B delegates to voter D and votes via delegation for “p(x,y)”. Voter C directly votes for “z_{1}”. Voter D primarily delegates to voter E, secondarily delegates to voter F, and votes via delegation for “p(x,y)”. Voter E delegates to voter G and votes via delegation for “p(x,y)”. Voter F directly votes for “z_{2}”. Voter G primarily delegates to voter H, secondarily delegates to voter I, and votes via delegation for “p(x,y)”. Voter H directly votes for “x”, or does not vote if x = ∅. Voter I directly votes for “y”.
Figure 18.2: Case XVIII (cont.)
Considering Case XVII and Property 6.
Voter A primarily delegates to voter B, secondarily delegates to voter C, and votes via delegation for “p(x,y)”.
Voter B delegates to voter D and votes via delegation for “p(x,y)”.
Voter C directly votes for “z_{1}”.
Voter D primarily delegates to voter E, secondarily delegates to voter F, and votes via delegation for “p(x,y)”.
Voter E delegates to voter G and votes via delegation for “p(x,y)”.
Voter F directly votes for “z_{2}”.
Voter G primarily delegates to voter H, secondarily delegates to voter I, and votes via delegation for “p(x,y)”.
Voter H directly votes for “x”, or does not vote if x = ∅.
Voter I directly votes for “y”.
Considering Case XVII and Property 6. Voter A primarily delegates to voter B, secondarily delegates to voter C, and votes via delegation for “p(x,y)”. Voter B delegates to voter D and votes via delegation for “p(x,y)”. Voter C directly votes for “z_{1}”. Voter D primarily delegates to voter E, secondarily delegates to voter F, and votes via delegation for “p(x,y)”. Voter E delegates to voter G and votes via delegation for “p(x,y)”. Voter F directly votes for “z_{2}”. Voter G primarily delegates to voter H, secondarily delegates to voter I, and votes via delegation for “p(x,y)”. Voter H directly votes for “x”, or does not vote if x = ∅. Voter I directly votes for “y”.
Figure 18.3: Case XVIII (cont.)

Case XIX

We consider a new Case XIX that can be solved by using the previously solved Case XVI and applying the rules of Property 4 (“Consistency”).

x ∈ {YES, NO, ∅}
y ∈ {YES, NO}
z1 ∈ {YES, NO}
z2 ∈ {YES, NO}
Voter A delegates to voter B.
Voter B primarily delegates to voter C and secondarily delegates to voter D.
Voter C delegates to voter E.
Voter D directly votes for “z_{1}”.
Voter E primarily delegates to voter F and secondarily delegates to voter G.
Voter F delegates to voter H.
Voter G directly votes for “z_{2}”.
Voter H directly votes for “p(x,y)”.
Voter I directly votes for “x”, or does not vote if x = ∅.
Voter J directly votes for “y”.
Voter A delegates to voter B. Voter B primarily delegates to voter C and secondarily delegates to voter D. Voter C delegates to voter E. Voter D directly votes for “z_{1}”. Voter E primarily delegates to voter F and secondarily delegates to voter G. Voter F delegates to voter H. Voter G directly votes for “z_{2}”. Voter H directly votes for “p(x,y)”. Voter I directly votes for “x”, or does not vote if x = ∅. Voter J directly votes for “y”.
Figure 19.1: Case XIX
Considering Case XVI and Property 4.
Voter A delegates to voter B and votes via delegation for “p(x,y)”.
Voter B primarily delegates to voter C, secondarily delegates to voter D, and votes via delegation for “p(x,y)”.
Voter C delegates to voter E and votes via delegation for “p(x,y)”.
Voter D directly votes for “z_{1}”.
Voter E primarily delegates to voter F, secondarily delegates to voter G, and votes via delegation for “p(x,y)”.
Voter F delegates to voter H and votes via delegation for “p(x,y)”.
Voter G directly votes for “z_{2}”.
Voter H directly votes for “p(x,y)”.
Voter I directly votes for “x”, or does not vote if x = ∅.
Voter J directly votes for “y”.
6 votes for “p(x,y)”,
1 vote for “x”,
1 vote for “y”,
1 vote for “z_{1}”,
1 vote for “z_{2}”.
Considering Case XVI and Property 4. Voter A delegates to voter B and votes via delegation for “p(x,y)”. Voter B primarily delegates to voter C, secondarily delegates to voter D, and votes via delegation for “p(x,y)”. Voter C delegates to voter E and votes via delegation for “p(x,y)”. Voter D directly votes for “z_{1}”. Voter E primarily delegates to voter F, secondarily delegates to voter G, and votes via delegation for “p(x,y)”. Voter F delegates to voter H and votes via delegation for “p(x,y)”. Voter G directly votes for “z_{2}”. Voter H directly votes for “p(x,y)”. Voter I directly votes for “x”, or does not vote if x = ∅. Voter J directly votes for “y”. 6 votes for “p(x,y)”, 1 vote for “x”, 1 vote for “y”, 1 vote for “z_{1}”, 1 vote for “z_{2}”.
Figure 19.2: Case XIX (cont.)

Case XX

We consider a new Case XX that can be solved by first applying the rules of Property 5 (“Directivity”) to Case XVIII in order to determine all votes but one, and then, due to Property 6 (“Equality of Direct and Delegating Voters”), using the vote counts determined in Case XIX to solve the last vote.

x ∈ {YES, NO, ∅}
y ∈ {YES, NO}
z1 ∈ {YES, NO}
z2 ∈ {YES, NO}
Voter A delegates to voter B.
Voter B primarily delegates to voter C and secondarily delegates to voter D.
Voter C delegates to voter E.
Voter D directly votes for “z_{1}”.
Voter E primarily delegates to voter F and secondarily delegates to voter G.
Voter F delegates to voter H.
Voter G directly votes for “z_{2}”.
Voter H primarily delegates to voter I and secondarily delegates to voter J.
Voter I directly votes for “x”, or does not vote if x = ∅.
Voter J directly votes for “y”.
Voter A delegates to voter B. Voter B primarily delegates to voter C and secondarily delegates to voter D. Voter C delegates to voter E. Voter D directly votes for “z_{1}”. Voter E primarily delegates to voter F and secondarily delegates to voter G. Voter F delegates to voter H. Voter G directly votes for “z_{2}”. Voter H primarily delegates to voter I and secondarily delegates to voter J. Voter I directly votes for “x”, or does not vote if x = ∅. Voter J directly votes for “y”.
Figure 20.1: Case XX
Considering Case XVIII and Property 5.
Voter A delegates to voter B.
Voter B primarily delegates to voter C, secondarily delegates to voter D, and votes via delegation for “p(x,y)”.
Voter C delegates to voter E and votes via delegation for “p(x,y)”.
Voter D directly votes for “z_{1}”.
Voter E primarily delegates to voter F, secondarily delegates to voter G, and votes via delegation for “p(x,y)”.
Voter F delegates to voter H and votes via delegation for “p(x,y)”.
Voter G directly votes for “z_{2}”.
Voter H primarily delegates to voter I, secondarily delegates to voter J, and votes via delegation for “p(x,y)”.
Voter I directly votes for “x”, or does not vote if x = ∅.
Voter J directly votes for “y”.
Considering Case XVIII and Property 5. Voter A delegates to voter B. Voter B primarily delegates to voter C, secondarily delegates to voter D, and votes via delegation for “p(x,y)”. Voter C delegates to voter E and votes via delegation for “p(x,y)”. Voter D directly votes for “z_{1}”. Voter E primarily delegates to voter F, secondarily delegates to voter G, and votes via delegation for “p(x,y)”. Voter F delegates to voter H and votes via delegation for “p(x,y)”. Voter G directly votes for “z_{2}”. Voter H primarily delegates to voter I, secondarily delegates to voter J, and votes via delegation for “p(x,y)”. Voter I directly votes for “x”, or does not vote if x = ∅. Voter J directly votes for “y”.
Figure 20.2: Case XX (cont.)
Considering Case XIX and Property 6.
Voter A delegates to voter B and votes via delegation for “p(x,y)”.
Voter B primarily delegates to voter C, secondarily delegates to voter D, and votes via delegation for “p(x,y)”.
Voter C delegates to voter E and votes via delegation for “p(x,y)”.
Voter D directly votes for “z_{1}”.
Voter E primarily delegates to voter F, secondarily delegates to voter G, and votes via delegation for “p(x,y)”.
Voter F delegates to voter H and votes via delegation for “p(x,y)”.
Voter G directly votes for “z_{2}”.
Voter H primarily delegates to voter I, secondarily delegates to voter J, and votes via delegation for “p(x,y)”.
Voter I directly votes for “x”, or does not vote if x = ∅.
Voter J directly votes for “y”.
Considering Case XIX and Property 6. Voter A delegates to voter B and votes via delegation for “p(x,y)”. Voter B primarily delegates to voter C, secondarily delegates to voter D, and votes via delegation for “p(x,y)”. Voter C delegates to voter E and votes via delegation for “p(x,y)”. Voter D directly votes for “z_{1}”. Voter E primarily delegates to voter F, secondarily delegates to voter G, and votes via delegation for “p(x,y)”. Voter F delegates to voter H and votes via delegation for “p(x,y)”. Voter G directly votes for “z_{2}”. Voter H primarily delegates to voter I, secondarily delegates to voter J, and votes via delegation for “p(x,y)”. Voter I directly votes for “x”, or does not vote if x = ∅. Voter J directly votes for “y”.
Figure 20.3: Case XX (cont.)

Case XXI

We consider a new Case XXI that can be solved by using the previously solved Case XVIII and applying the rules of Property 4 (“Consistency”).

x ∈ {YES, NO}
z1 ∈ {YES, NO}
z2 ∈ {YES, NO}
z3 ∈ {YES, NO}
Voter A primarily delegates to voter B and secondarily delegates to voter C.
Voter B delegates to voter D.
Voter C directly votes for “z_{1}”.
Voter D primarily delegates to voter E and secondarily delegates to voter F.
Voter E delegates to voter G.
Voter F directly votes for “z_{2}”.
Voter G primarily delegates to voter H and secondarily delegates to voter I.
Voter H directly votes for “x”.
Voter I directly votes for “z_{3}”.
Voter J directly votes for “x”.
Voter A primarily delegates to voter B and secondarily delegates to voter C. Voter B delegates to voter D. Voter C directly votes for “z_{1}”. Voter D primarily delegates to voter E and secondarily delegates to voter F. Voter E delegates to voter G. Voter F directly votes for “z_{2}”. Voter G primarily delegates to voter H and secondarily delegates to voter I. Voter H directly votes for “x”. Voter I directly votes for “z_{3}”. Voter J directly votes for “x”.
Figure 21.1: Case XXI
Considering Case XVIII and Property 4.
Voter A primarily delegates to voter B, secondarily delegates to voter C, and votes via delegation for “p(x,z_{3}) = x”.
Voter B delegates to voter D and votes via delegation for “p(x,z_{3}) = x”.
Voter C directly votes for “z_{1}”.
Voter D primarily delegates to voter E, secondarily delegates to voter F, and votes via delegation for “p(x,z_{3}) = x”.
Voter E delegates to voter G and votes via delegation for “p(x,z_{3}) = x”.
Voter F directly votes for “z_{2}”.
Voter G primarily delegates to voter H, secondarily delegates to voter I, and votes via delegation for “p(x,z_{3}) = x”.
Voter H directly votes for “x”.
Voter I directly votes for “z_{3}”.
Voter J directly votes for “x”.
7 votes for “x”,
1 vote for “z_{1}”,
1 vote for “z_{2}”,
1 vote for “z_{3}”.
Considering Case XVIII and Property 4. Voter A primarily delegates to voter B, secondarily delegates to voter C, and votes via delegation for “p(x,z_{3}) = x”. Voter B delegates to voter D and votes via delegation for “p(x,z_{3}) = x”. Voter C directly votes for “z_{1}”. Voter D primarily delegates to voter E, secondarily delegates to voter F, and votes via delegation for “p(x,z_{3}) = x”. Voter E delegates to voter G and votes via delegation for “p(x,z_{3}) = x”. Voter F directly votes for “z_{2}”. Voter G primarily delegates to voter H, secondarily delegates to voter I, and votes via delegation for “p(x,z_{3}) = x”. Voter H directly votes for “x”. Voter I directly votes for “z_{3}”. Voter J directly votes for “x”. 7 votes for “x”, 1 vote for “z_{1}”, 1 vote for “z_{2}”, 1 vote for “z_{3}”.
Figure 21.2: Case XXI (cont.)

Case XXII

We consider a new Case XXII that can be solved by first applying the rules of Property 5 (“Directivity”) to Case XVI in order to determine all votes but one, and then, due to Property 6 (“Equality of Direct and Delegating Voters”), using the vote counts determined in Case XXI to solve the last vote.

x ∈ {YES, NO}
z1 ∈ {YES, NO}
z2 ∈ {YES, NO}
z3 ∈ {YES, NO}
Voter A primarily delegates to voter B and secondarily delegates to voter C.
Voter B delegates to voter D.
Voter C directly votes for “z_{1}”.
Voter D primarily delegates to voter E and secondarily delegates to voter F.
Voter E delegates to voter G.
Voter F directly votes for “z_{2}”.
Voter G primarily delegates to voter H and secondarily delegates to voter I.
Voter H delegates to voter J.
Voter I directly votes for “z_{3}”.
Voter J directly votes for “x”.
Voter A primarily delegates to voter B and secondarily delegates to voter C. Voter B delegates to voter D. Voter C directly votes for “z_{1}”. Voter D primarily delegates to voter E and secondarily delegates to voter F. Voter E delegates to voter G. Voter F directly votes for “z_{2}”. Voter G primarily delegates to voter H and secondarily delegates to voter I. Voter H delegates to voter J. Voter I directly votes for “z_{3}”. Voter J directly votes for “x”.
Figure 22.1: Case XXII
Considering Case XVI and Property 5.
Voter A primarily delegates to voter B and secondarily delegates to voter C.
Voter B delegates to voter D and votes via delegation for “x”.
Voter C directly votes for “z_{1}”.
Voter D primarily delegates to voter E, secondarily delegates to voter F, and votes via delegation for “x”.
Voter E delegates to voter G and votes via delegation for “x”.
Voter F directly votes for “z_{2}”.
Voter G primarily delegates to voter H, secondarily delegates to voter I, and votes via delegation for “x”.
Voter H delegates to voter J and votes via delegation for “x”.
Voter I directly votes for “z_{3}”.
Voter J directly votes for “x”.
Considering Case XVI and Property 5. Voter A primarily delegates to voter B and secondarily delegates to voter C. Voter B delegates to voter D and votes via delegation for “x”. Voter C directly votes for “z_{1}”. Voter D primarily delegates to voter E, secondarily delegates to voter F, and votes via delegation for “x”. Voter E delegates to voter G and votes via delegation for “x”. Voter F directly votes for “z_{2}”. Voter G primarily delegates to voter H, secondarily delegates to voter I, and votes via delegation for “x”. Voter H delegates to voter J and votes via delegation for “x”. Voter I directly votes for “z_{3}”. Voter J directly votes for “x”.
Figure 22.2: Case XXII (cont.)
Considering Case XXI and Property 6.
Voter A primarily delegates to voter B, secondarily delegates to voter C, and votes via delegation for “x”.
Voter B delegates to voter D and votes via delegation for “x”.
Voter C directly votes for “z_{1}”.
Voter D primarily delegates to voter E, secondarily delegates to voter F, and votes via delegation for “x”.
Voter E delegates to voter G and votes via delegation for “x”.
Voter F directly votes for “z_{2}”.
Voter G primarily delegates to voter H, secondarily delegates to voter I, and votes via delegation for “x”.
Voter H delegates to voter J and votes via delegation for “x”.
Voter I directly votes for “z_{3}”.
Voter J directly votes for “x”.
Considering Case XXI and Property 6. Voter A primarily delegates to voter B, secondarily delegates to voter C, and votes via delegation for “x”. Voter B delegates to voter D and votes via delegation for “x”. Voter C directly votes for “z_{1}”. Voter D primarily delegates to voter E, secondarily delegates to voter F, and votes via delegation for “x”. Voter E delegates to voter G and votes via delegation for “x”. Voter F directly votes for “z_{2}”. Voter G primarily delegates to voter H, secondarily delegates to voter I, and votes via delegation for “x”. Voter H delegates to voter J and votes via delegation for “x”. Voter I directly votes for “z_{3}”. Voter J directly votes for “x”.
Figure 22.3: Case XXII (cont.)

Case XXIII

We consider a new Case XXIII that can be solved by using the previously solved Case XXII and applying the rules of Property 4 (“Consistency”).

Voter A primarily delegates to voter B and secondarily delegates to voter C.
Voter B delegates to voter D.
Voter C directly votes for “YES”.
Voter D primarily delegates to voter E and secondarily delegates to voter F.
Voter E delegates to voter G.
Voter F directly votes for “NO”.
Voter G primarily delegates to voter H and secondarily delegates to voter I.
Voter H delegates to voter J.
Voter I directly votes for “NO”.
Voter J directly votes for “YES”.
Voter K does not vote.
Voter L directly votes for “YES”.
Voter A primarily delegates to voter B and secondarily delegates to voter C. Voter B delegates to voter D. Voter C directly votes for “YES”. Voter D primarily delegates to voter E and secondarily delegates to voter F. Voter E delegates to voter G. Voter F directly votes for “NO”. Voter G primarily delegates to voter H and secondarily delegates to voter I. Voter H delegates to voter J. Voter I directly votes for “NO”. Voter J directly votes for “YES”. Voter K does not vote. Voter L directly votes for “YES”.
Figure 23.1: Case XXIII
Considering Case XXII and Property 4.
Voter A primarily delegates to voter B, secondarily delegates to voter C, and votes via delegation for “YES”.
Voter B delegates to voter D and votes via delegation for “YES”.
Voter C directly votes for “YES”.
Voter D primarily delegates to voter E, secondarily delegates to voter F, and votes via delegation for “YES”.
Voter E delegates to voter G and votes via delegation for “YES”.
Voter F directly votes for “NO”.
Voter G primarily delegates to voter H, secondarily delegates to voter I, and votes via delegation for “YES”.
Voter H delegates to voter J and votes via delegation for “YES”.
Voter I directly votes for “NO”.
Voter J directly votes for “YES”.
Voter K does not vote.
Voter L directly votes for “YES”.
9 votes for “YES”,
2 votes for “NO”.
Considering Case XXII and Property 4. Voter A primarily delegates to voter B, secondarily delegates to voter C, and votes via delegation for “YES”. Voter B delegates to voter D and votes via delegation for “YES”. Voter C directly votes for “YES”. Voter D primarily delegates to voter E, secondarily delegates to voter F, and votes via delegation for “YES”. Voter E delegates to voter G and votes via delegation for “YES”. Voter F directly votes for “NO”. Voter G primarily delegates to voter H, secondarily delegates to voter I, and votes via delegation for “YES”. Voter H delegates to voter J and votes via delegation for “YES”. Voter I directly votes for “NO”. Voter J directly votes for “YES”. Voter K does not vote. Voter L directly votes for “YES”. 9 votes for “YES”, 2 votes for “NO”.
Figure 23.2: Case XXIII (cont.)

Case XXIV

We consider a new Case XXIV that can be solved by first applying the rules of Property 5 (“Directivity”) to Case XX in order to determine all votes but one, and then, due to Property 6 (“Equality of Direct and Delegating Voters”), using the vote counts determined in Case XXIII to solve the last vote.

Voter A primarily delegates to voter B and secondarily delegates to voter C.
Voter B delegates to voter D.
Voter C directly votes for “YES”.
Voter D primarily delegates to voter E and secondarily delegates to voter F.
Voter E delegates to voter G.
Voter F directly votes for “NO”.
Voter G primarily delegates to voter H and secondarily delegates to voter I.
Voter H delegates to voter J.
Voter I directly votes for “NO”.
Voter J primarily delegates to voter K and secondarily delegates to voter L.
Voter K does not vote.
Voter L directly votes for “YES”.
Voter A primarily delegates to voter B and secondarily delegates to voter C. Voter B delegates to voter D. Voter C directly votes for “YES”. Voter D primarily delegates to voter E and secondarily delegates to voter F. Voter E delegates to voter G. Voter F directly votes for “NO”. Voter G primarily delegates to voter H and secondarily delegates to voter I. Voter H delegates to voter J. Voter I directly votes for “NO”. Voter J primarily delegates to voter K and secondarily delegates to voter L. Voter K does not vote. Voter L directly votes for “YES”.
Figure 24.1: Case XXIV
Considering Case XX and Property 5.
Voter A primarily delegates to voter B and secondarily delegates to voter C.
Voter B delegates to voter D and votes via delegation for “p(∅,YES) = YES”.
Voter C directly votes for “YES”.
Voter D primarily delegates to voter E, secondarily delegates to voter F, and votes via delegation for “p(∅,YES) = YES”.
Voter E delegates to voter G and votes via delegation for “p(∅,YES) = YES”.
Voter F directly votes for “NO”.
Voter G primarily delegates to voter H, secondarily delegates to voter I, and votes via delegation for “p(∅,YES) = YES”.
Voter H delegates to voter J and votes via delegation for “p(∅,YES) = YES”.
Voter I directly votes for “NO”.
Voter J primarily delegates to voter K, secondarily delegates to voter L, and votes via delegation for “p(∅,YES) = YES”.
Voter K does not vote.
Voter L directly votes for “YES”.
Considering Case XX and Property 5. Voter A primarily delegates to voter B and secondarily delegates to voter C. Voter B delegates to voter D and votes via delegation for “p(∅,YES) = YES”. Voter C directly votes for “YES”. Voter D primarily delegates to voter E, secondarily delegates to voter F, and votes via delegation for “p(∅,YES) = YES”. Voter E delegates to voter G and votes via delegation for “p(∅,YES) = YES”. Voter F directly votes for “NO”. Voter G primarily delegates to voter H, secondarily delegates to voter I, and votes via delegation for “p(∅,YES) = YES”. Voter H delegates to voter J and votes via delegation for “p(∅,YES) = YES”. Voter I directly votes for “NO”. Voter J primarily delegates to voter K, secondarily delegates to voter L, and votes via delegation for “p(∅,YES) = YES”. Voter K does not vote. Voter L directly votes for “YES”.
Figure 24.2: Case XXIV (cont.)
Considering Case XXIII and Property 6.
Voter A primarily delegates to voter B, secondarily delegates to voter C, and votes via delegation for “YES”.
Voter B delegates to voter D and votes via delegation for “YES”.
Voter C directly votes for “YES”.
Voter D primarily delegates to voter E, secondarily delegates to voter F, and votes via delegation for “YES”.
Voter E delegates to voter G and votes via delegation for “YES”.
Voter F directly votes for “NO”.
Voter G primarily delegates to voter H, secondarily delegates to voter I, and votes via delegation for “YES”.
Voter H delegates to voter J and votes via delegation for “YES”.
Voter I directly votes for “NO”.
Voter J primarily delegates to voter K, secondarily delegates to voter L, and votes via delegation for “YES”.
Voter K does not vote.
Voter L directly votes for “YES”.
Considering Case XXIII and Property 6. Voter A primarily delegates to voter B, secondarily delegates to voter C, and votes via delegation for “YES”. Voter B delegates to voter D and votes via delegation for “YES”. Voter C directly votes for “YES”. Voter D primarily delegates to voter E, secondarily delegates to voter F, and votes via delegation for “YES”. Voter E delegates to voter G and votes via delegation for “YES”. Voter F directly votes for “NO”. Voter G primarily delegates to voter H, secondarily delegates to voter I, and votes via delegation for “YES”. Voter H delegates to voter J and votes via delegation for “YES”. Voter I directly votes for “NO”. Voter J primarily delegates to voter K, secondarily delegates to voter L, and votes via delegation for “YES”. Voter K does not vote. Voter L directly votes for “YES”.
Figure 24.3: Case XXIV (cont.)

Case XXV

We copy the delegation graph from Case XXIV and add a single NO vote (using Property 4). Despite adding a NO vote, the number of YES votes still outnumbers the number of NO votes. Thus “YES” would still win here.

Voter A primarily delegates to voter B, secondarily delegates to voter C, and votes via delegation for “YES”.
Voter B delegates to voter D and votes via delegation for “YES”.
Voter C directly votes for “YES”.
Voter D primarily delegates to voter E, secondarily delegates to voter F, and votes via delegation for “YES”.
Voter E delegates to voter G and votes via delegation for “YES”.
Voter F directly votes for “NO”.
Voter G primarily delegates to voter H, secondarily delegates to voter I, and votes via delegation for “YES”.
Voter H delegates to voter J and votes via delegation for “YES”.
Voter I directly votes for “NO”.
Voter J primarily delegates to voter K, secondarily delegates to voter L, and votes via delegation for “YES”.
Voter K does not vote.
Voter L directly votes for “YES”.
Voter M directly votes for “NO”.
9 votes for “YES”,
3 votes for “NO”.
9 > 3
(“YES” would win).
Voter A primarily delegates to voter B, secondarily delegates to voter C, and votes via delegation for “YES”. Voter B delegates to voter D and votes via delegation for “YES”. Voter C directly votes for “YES”. Voter D primarily delegates to voter E, secondarily delegates to voter F, and votes via delegation for “YES”. Voter E delegates to voter G and votes via delegation for “YES”. Voter F directly votes for “NO”. Voter G primarily delegates to voter H, secondarily delegates to voter I, and votes via delegation for “YES”. Voter H delegates to voter J and votes via delegation for “YES”. Voter I directly votes for “NO”. Voter J primarily delegates to voter K, secondarily delegates to voter L, and votes via delegation for “YES”. Voter K does not vote. Voter L directly votes for “YES”. Voter M directly votes for “NO”. 9 votes for “YES”, 3 votes for “NO”. 9 > 3 (“YES” would win).
Figure 25: Case XXV

Case XXVI

We create a final Case XXVI equal to Case XXV but with the sole difference that voter K (who was previously abstaining) delegates to voter A (who was previously voting for YES through delegation). According to Property 7, “YES” would need to win in Case XXVI (because it also wins in Case XXV). However, due to symmetry of the circular structure in Case XXVI (using Property 4, Property 3, and Property 2 to transform the circular structure), we can show that (because of voter M) there must be more “NO” votes than “YES” votes, which, in turn, means that Property 7 is contradictory to the previously defined properties, quod erat demonstrandum.

Voter A primarily delegates to voter B, secondarily delegates to voter C.
Voter B delegates to voter D.
Voter C directly votes for “YES”.
Voter D primarily delegates to voter E, secondarily delegates to voter F.
Voter E delegates to voter G.
Voter F directly votes for “NO”.
Voter G primarily delegates to voter H, secondarily delegates to voter I.
Voter H delegates to voter J.
Voter I directly votes for “NO”.
Voter J primarily delegates to voter K, secondarily delegates to voter L.
Voter K delegates to voter A.
Voter L directly votes for “YES”.
Voter M directly votes for “NO”.
‘a’ votes for “YES”,
‘b+1’ votes for “NO”.
Voter A primarily delegates to voter B, secondarily delegates to voter C. Voter B delegates to voter D. Voter C directly votes for “YES”. Voter D primarily delegates to voter E, secondarily delegates to voter F. Voter E delegates to voter G. Voter F directly votes for “NO”. Voter G primarily delegates to voter H, secondarily delegates to voter I. Voter H delegates to voter J. Voter I directly votes for “NO”. Voter J primarily delegates to voter K, secondarily delegates to voter L. Voter K delegates to voter A. Voter L directly votes for “YES”. Voter M directly votes for “NO”. ‘a’ votes for “YES”, ‘b+1’ votes for “NO”.
Figure 26.1: Case XXVI
Considering Property 4 and Property 3.
Voter A primarily delegates to voter B, secondarily delegates to voter C.
Voter B delegates to voter D.
Voter C directly votes for “NO”.
Voter D primarily delegates to voter E, secondarily delegates to voter F.
Voter E delegates to voter G.
Voter F directly votes for “YES”.
Voter G primarily delegates to voter H, secondarily delegates to voter I.
Voter H delegates to voter J.
Voter I directly votes for “YES”.
Voter J primarily delegates to voter K, secondarily delegates to voter L.
Voter K delegates to voter A.
Voter L directly votes for “NO”.
Voter M directly votes for “NO”.
‘b’ votes for “YES”,
‘a+1’ votes for “NO”.
Considering Property 4 and Property 3. Voter A primarily delegates to voter B, secondarily delegates to voter C. Voter B delegates to voter D. Voter C directly votes for “NO”. Voter D primarily delegates to voter E, secondarily delegates to voter F. Voter E delegates to voter G. Voter F directly votes for “YES”. Voter G primarily delegates to voter H, secondarily delegates to voter I. Voter H delegates to voter J. Voter I directly votes for “YES”. Voter J primarily delegates to voter K, secondarily delegates to voter L. Voter K delegates to voter A. Voter L directly votes for “NO”. Voter M directly votes for “NO”. ‘b’ votes for “YES”, ‘a+1’ votes for “NO”.
Figure 26.2: Case XXVI (cont.)
Considering Property 2.
Voter G primarily delegates to voter H, secondarily delegates to voter I.
Voter H delegates to voter J.
Voter I directly votes for “NO”.
Voter J primarily delegates to voter K, secondarily delegates to voter L.
Voter K delegates to voter A.
Voter L directly votes for “YES”.
Voter A primarily delegates to voter B, secondarily delegates to voter C.
Voter B delegates to voter D.
Voter C directly votes for “YES”.
Voter D primarily delegates to voter E, secondarily delegates to voter F.
Voter E delegates to voter G.
Voter F directly votes for “NO”.
Voter M directly votes for “NO”.
‘a’ votes for “YES”,
‘b+1’ votes for “NO”,
and also
‘b’ votes for “YES”,
‘a+1’ votes for “NO”.
a = b !
(NO would win).
Considering Property 2. Voter G primarily delegates to voter H, secondarily delegates to voter I. Voter H delegates to voter J. Voter I directly votes for “NO”. Voter J primarily delegates to voter K, secondarily delegates to voter L. Voter K delegates to voter A. Voter L directly votes for “YES”. Voter A primarily delegates to voter B, secondarily delegates to voter C. Voter B delegates to voter D. Voter C directly votes for “YES”. Voter D primarily delegates to voter E, secondarily delegates to voter F. Voter E delegates to voter G. Voter F directly votes for “NO”. Voter M directly votes for “NO”. ‘a’ votes for “YES”, ‘b+1’ votes for “NO”, and also ‘b’ votes for “YES”, ‘a+1’ votes for “NO”. a = b ! (NO would win).
Figure 26.3: Case XXVI (cont.)
[PLF] Behrens, Kistner, Nitsche, Swierczek: “The Principles of LiquidFeedback”. ISBN 978-3-00-044795-2. Published January 2014 by Interaktive Demokratie e. V., available at http://principles.liquidfeedback.org/ (referenced at: a b c)
[May] Kenneth O. May: A Set of Independent Necessary and Sufficient Conditions for Simple Majority Decision. In “Econometrica, Vol. 20, No. 4” (October 1952), pp. 680–684. Published by the Econometric Society (Wiley-Blackwell). (referenced at: a b)