#### Preferential Delegation and the Problem of Negative Voting Weight
#### by Jan Behrens & Björn Swierczek, Berlin, January 23, 2015


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}

Figure 1.1 “Case I”:

    Voter A delegates to voter B.
    Voter B directly votes for “x”.

End of Figure 1.1.

Figure 1.2 “Case I (cont.)”:

    Considering Property P1.
    Voter A delegates to voter B and votes via delegation for “x”.
    Voter B directly votes for “x”.

End of Figure 1.2.


== 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}

Figure 2.1 “Case II”:

    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”.

End of Figure 2.1.

Figure 2.2 “Case II (cont.)”:

    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”.

End of Figure 2.2.


== 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}

Figure 3.1 “Case III”:

    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”.

End of Figure 3.1.

Figure 3.2 “Case III (cont.)”:

    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”.

End of Figure 3.2.


== 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}

Figure 4.1 “Case IV”:

    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”.

End of Figure 4.1.

Figure 4.2 “Case IV (cont.)”:

    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”.

End of Figure 4.2.

Figure 4.3 “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”.

End of Figure 4.3.


== 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}

Figure 5.1 “Case V”:

    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”.

End of Figure 5.1.

Figure 5.2 “Case V (cont.)”:

    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”.

End of Figure 5.2.


== 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}

Figure 6.1 “Case VI”:

    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”.

End of Figure 6.1.

Figure 6.2 “Case VI (cont.)”:

    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”.

End of Figure 6.2.

Figure 6.3 “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”.

End of Figure 6.3.


== 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}

Figure 7.1 “Case VII”:

    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”.

End of Figure 7.1.

Figure 7.2 “Case VII (cont.)”:

    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”.

End of Figure 7.2.


== 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}

Figure 8.1 “Case VIII”:

    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”.

End of Figure 8.1.

Figure 8.2 “Case VIII (cont.)”:

    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”.

End of Figure 8.2.

Figure 8.3 “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”.

End of Figure 8.3.


== 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}

Figure 9.1 “Case IX”:

    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”.

End of Figure 9.1.

Figure 9.2 “Case IX (cont.)”:

    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”.

End of Figure 9.2.


== 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}

Figure 10.1 “Case 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 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”.

End of Figure 10.1.

Figure 10.2 “Case X (cont.)”:

    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”.

End of Figure 10.2.

Figure 10.3 “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”.

End of Figure 10.3.


== 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}

Figure 11.1 “Case XI”:

    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”.

End of Figure 11.1.

Figure 11.2 “Case XI (cont.)”:

    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”.

End of Figure 11.2.


== 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}

Figure 12.1 “Case XII”:

    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”.

End of Figure 12.1.

Figure 12.2 “Case XII (cont.)”:

    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”.

End of Figure 12.2.

Figure 12.3 “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”.

End of Figure 12.3.


== 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}

Figure 13.1 “Case XIII”:

    Voter A primarily delegates to voter B and secondarily delegates to voter C.
    Voter B delegates to voter D.
    Voter C directly votes for “z1”.
    Voter D primarily delegates to voter E and secondarily delegates to voter F.
    Voter E directly votes for “x”.
    Voter F directly votes for “z2”.
    Voter G directly votes for “x”.

End of Figure 13.1.

Figure 13.2 “Case XIII (cont.)”:

    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,z2) = x”.
    Voter B delegates to voter D and votes via delegation for “p(x,z2) = x”.
    Voter C directly votes for “z1”.
    Voter D primarily delegates to voter E, secondarily delegates to voter F, and votes via delegation for “p(x,z2) = x”.
    Voter E directly votes for “x”.
    Voter F directly votes for “z2”.
    Voter G directly votes for “x”.

    5 votes for “x”,
    1 vote for “z1”,
    1 vote for “z2”.

End of Figure 13.2.


== 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}

Figure 14.1 “Case XIV”:

    Voter A primarily delegates to voter B and secondarily delegates to voter C.
    Voter B delegates to voter D.
    Voter C directly votes for “z1”.
    Voter D primarily delegates to voter E and secondarily delegates to voter F.
    Voter E delegates to voter G.
    Voter F directly votes for “z2”.
    Voter G directly votes for “x”.

End of Figure 14.1.

Figure 14.2 “Case XIV (cont.)”:

    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 “z1”.
    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 “z2”.
    Voter G directly votes for “x”.

End of Figure 14.2.

Figure 14.3 “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 “z1”.
    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 “z2”.
    Voter G directly votes for “x”.

End of Figure 14.3.


== 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}

Figure 15.1 “Case XV”:

    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 “z1”.
    Voter E primarily delegates to voter F and secondarily delegates to voter G.
    Voter F directly votes for “x”.
    Voter G directly votes for “z2”.
    Voter H directly votes for “x”.

End of Figure 15.1.

Figure 15.2 “Case XV (cont.)”:

    Considering Case XII and Property 4.
    Voter A delegates to voter B and votes via delegation for “p(x,z2) = x”.
    Voter B primarily delegates to voter C, secondarily delegates to voter D, and votes via delegation for “p(x,z2) = x”.
    Voter C delegates to voter E and votes via delegation for “p(x,z2) = x”.
    Voter D directly votes for “z1”.
    Voter E primarily delegates to voter F, secondarily delegates to voter G, and votes via delegation for “p(x,z2) = x”.
    Voter F directly votes for “x”.
    Voter G directly votes for “z2”.
    Voter H directly votes for “x”.

    6 votes for “x”,
    1 vote for “z1”,
    1 vote for “z2”.

End of Figure 15.2.


== 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}

Figure 16.1 “Case XVI”:

    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 “z1”.
    Voter E primarily delegates to voter F and secondarily delegates to voter G.
    Voter F delegates to voter H.
    Voter G directly votes for “z2”.
    Voter H directly votes for “x”.

End of Figure 16.1.

Figure 16.2 “Case XVI (cont.)”:

    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 “z1”.
    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 “z2”.
    Voter H directly votes for “x”.

End of Figure 16.2.

Figure 16.3 “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 “z1”.
    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 “z2”.
    Voter H directly votes for “x”.

End of Figure 16.3.


== 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}

Figure 17.1 “Case XVII”:

    Voter A primarily delegates to voter B and secondarily delegates to voter C.
    Voter B delegates to voter D.
    Voter C directly votes for “z1”.
    Voter D primarily delegates to voter E and secondarily delegates to voter F.
    Voter E delegates to voter G.
    Voter F directly votes for “z2”.
    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”.

End of Figure 17.1.

Figure 17.2 “Case XVII (cont.)”:

    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 “z1”.
    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 “z2”.
    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 “z1”,
    1 vote for “z2”.

End of Figure 17.2.


== 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}

Figure 18.1 “Case XVIII”:

    Voter A primarily delegates to voter B and secondarily delegates to voter C.
    Voter B delegates to voter D.
    Voter C directly votes for “z1”.
    Voter D primarily delegates to voter E and secondarily delegates to voter F.
    Voter E delegates to voter G.
    Voter F directly votes for “z2”.
    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”.

End of Figure 18.1.

Figure 18.2 “Case XVIII (cont.)”:

    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 “z1”.
    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 “z2”.
    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”.

End of Figure 18.2.

Figure 18.3 “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 “z1”.
    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 “z2”.
    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”.

End of Figure 18.3.


== 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}

Figure 19.1 “Case XIX”:

    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 “z1”.
    Voter E primarily delegates to voter F and secondarily delegates to voter G.
    Voter F delegates to voter H.
    Voter G directly votes for “z2”.
    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”.

End of Figure 19.1.

Figure 19.2 “Case XIX (cont.)”:

    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 “z1”.
    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 “z2”.
    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 “z1”,
    1 vote for “z2”.

End of Figure 19.2.


== 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}

Figure 20.1 “Case XX”:

    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 “z1”.
    Voter E primarily delegates to voter F and secondarily delegates to voter G.
    Voter F delegates to voter H.
    Voter G directly votes for “z2”.
    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”.

End of Figure 20.1.

Figure 20.2 “Case XX (cont.)”:

    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 “z1”.
    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 “z2”.
    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”.

End of Figure 20.2.

Figure 20.3 “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 “z1”.
    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 “z2”.
    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”.

End of Figure 20.3.


== 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}

Figure 21.1 “Case XXI”:

    Voter A primarily delegates to voter B and secondarily delegates to voter C.
    Voter B delegates to voter D.
    Voter C directly votes for “z1”.
    Voter D primarily delegates to voter E and secondarily delegates to voter F.
    Voter E delegates to voter G.
    Voter F directly votes for “z2”.
    Voter G primarily delegates to voter H and secondarily delegates to voter I.
    Voter H directly votes for “x”.
    Voter I directly votes for “z3”.
    Voter J directly votes for “x”.

End of Figure 21.1.

Figure 21.2 “Case XXI (cont.)”:

    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,z3) = x”.
    Voter B delegates to voter D and votes via delegation for “p(x,z3) = x”.
    Voter C directly votes for “z1”.
    Voter D primarily delegates to voter E, secondarily delegates to voter F, and votes via delegation for “p(x,z3) = x”.
    Voter E delegates to voter G and votes via delegation for “p(x,z3) = x”.
    Voter F directly votes for “z2”.
    Voter G primarily delegates to voter H, secondarily delegates to voter I, and votes via delegation for “p(x,z3) = x”.
    Voter H directly votes for “x”.
    Voter I directly votes for “z3”.
    Voter J directly votes for “x”.

    7 votes for “x”,
    1 vote for “z1”,
    1 vote for “z2”,
    1 vote for “z3”.

End of Figure 21.2.


== 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}

Figure 22.1 “Case XXII”:

    Voter A primarily delegates to voter B and secondarily delegates to voter C.
    Voter B delegates to voter D.
    Voter C directly votes for “z1”.
    Voter D primarily delegates to voter E and secondarily delegates to voter F.
    Voter E delegates to voter G.
    Voter F directly votes for “z2”.
    Voter G primarily delegates to voter H and secondarily delegates to voter I.
    Voter H delegates to voter J.
    Voter I directly votes for “z3”.
    Voter J directly votes for “x”.

End of Figure 22.1.

Figure 22.2 “Case XXII (cont.)”:

    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 “z1”.
    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 “z2”.
    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 “z3”.
    Voter J directly votes for “x”.

End of Figure 22.2.

Figure 22.3 “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 “z1”.
    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 “z2”.
    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 “z3”.
    Voter J directly votes for “x”.

End of Figure 22.3.


== 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”).

Figure 23.1 “Case XXIII”:

    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”.

End of Figure 23.1.

Figure 23.2 “Case XXIII (cont.)”:

    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”.

End of Figure 23.2.


== 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.

Figure 24.1 “Case XXIV”:

    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”.

End of Figure 24.1.

Figure 24.2 “Case XXIV (cont.)”:

    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”.

End of Figure 24.2.

Figure 24.3 “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”.

End of Figure 24.3.


== 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.

Figure 25 “Case XXV”:

    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).

End of Figure 25.


== 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.

Figure 26.1 “Case XXVI”:

    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”.

End of Figure 26.1.

Figure 26.2 “Case XXVI (cont.)”:

    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”.

End of Figure 26.2.

Figure 26.3 “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).

End of Figure 26.3.


References:
[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/
[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).

    
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This file is part of:

The Liquid Democracy Journal
on electronic participation, collective moderation, and voting systems
Issue 3, Berlin 2015-01-23 (electronic version, rev2)

The full issue is available at: http://www.liquid-democracy-journal.org/issue/3/

All rights reserved
Copyright © 2015 Interaktive Demokratie e. V., Berlin, Germany
http://www.interaktive-demokratie.org/

Published by: Interaktive Demokratie e. V., Berlin, Germany
Edited by: Jan Behrens, Axel Kistner, Andreas Nitsche, Björn Swierczek

Imprint and contact: http://www.liquid-democracy-journal.org/imprint
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ISSN-L: 2198-9532
ISSN print version: 2198-9532
ISSN electronic version: 2199-1758

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