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Addendum to our Theorem Regarding Preferential Delegation and Negative Voting Weight

by Jan Behrens, Berlin, March 28, 2015 other format: text version (UTF-8)

In our article on Preferential Delegation and Negative Voting Weight [PD], we have proven that a preferential delegation system with free choice of delegates may not fullfill the following 7 criteria at the same time:

While these 7 criteria make the proof easy to understand, it should be noted that if we define the absence of negative voting weight in a more general way, we could further reduce the number of conflicting properties to the following 5 properties:

The definition for the absence of negative voting weight, however, needs to be redefined as follows in this case:

“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, and if person B votes via delegation against a proposal that loses, then changing A's behavior to delegate to B instead of abstaining must not cause the previously losing proposal to win.”

As already explained in the original article, the property “Consistency” is implied by “Directionality”. Furthermore, the use of “Neutrality” isn't necessary until Case XXVI of our original proof. [Note: The original proof states on page 8 that Property 3 (Neutrality) is used implicitly until Case XXIV inclusive. This is not necessary though, because for each case, “x”, “y”, “z1”, etc. are variable.] We may therefore copy the findings regarding Case I through Case XXII from our previous proof and consider 6 new cases.

For Case I through Case XXII, see [PD]. Case XXIII through Case XXVIII will be (re)defined as follows.

Case I through Case XXII

See [PD] for Case I through XXII.

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”).

x ∈ {YES, NO, ∅}
y ∈ {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 “p(x,y)”. Voter K directly votes for “x”, or does not vote if x = ∅. Voter L 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 delegates to voter J. Voter I directly votes for “z_{3}”. Voter J directly votes for “p(x,y)”. Voter K directly votes for “x”, or does not vote if x = ∅. Voter L directly votes for “y”.
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 “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 delegates to voter J and votes via delegation for “p(x,y)”. Voter I directly votes for “z_{3}”. Voter J directly votes for “p(x,y)”. Voter K directly votes for “x”, or does not vote if x = ∅. Voter L directly votes for “y”. 7 votes for “p(x,y)”, 1 vote for “x”, 1 vote for “y”, 1 vote for “z_{1}”, 1 vote for “z_{2}”, 1 vote for “z_{3}”.
Considering Case XXII 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 primarily delegates to voter H, secondarily delegates to voter I, and votes via delegation for “p(x,y)”. Voter H delegates to voter J and votes via delegation for “p(x,y)”. Voter I directly votes for “z_{3}”. Voter J directly votes for “p(x,y)”. Voter K directly votes for “x”, or does not vote if x = ∅. Voter L directly votes for “y”. 7 votes for “p(x,y)”, 1 vote for “x”, 1 vote for “y”, 1 vote for “z_{1}”, 1 vote for “z_{2}”, 1 vote for “z_{3}”.
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.

x ∈ {YES, NO, ∅}
y ∈ {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 primarily delegates to voter K and secondarily delegates to voter L. Voter K directly votes for “x”, or does not vote if x = ∅. Voter L 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 delegates to voter J. Voter I directly votes for “z_{3}”. Voter J primarily delegates to voter K and secondarily delegates to voter L. Voter K directly votes for “x”, or does not vote if x = ∅. Voter L directly votes for “y”.
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(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 delegates to voter J and votes via delegation for “p(x,y)”. Voter I directly votes for “z_{3}”. Voter J primarily delegates to voter K, secondarily delegates to voter L, and votes via delegation for “p(x,y)”. Voter K directly votes for “x”, or does not vote if x = ∅. Voter L directly votes for “y”.
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(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 delegates to voter J and votes via delegation for “p(x,y)”. Voter I directly votes for “z_{3}”. Voter J primarily delegates to voter K, secondarily delegates to voter L, and votes via delegation for “p(x,y)”. Voter K directly votes for “x”, or does not vote if x = ∅. Voter L directly votes for “y”.
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 “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 delegates to voter J and votes via delegation for “p(x,y)”. Voter I directly votes for “z_{3}”. Voter J primarily delegates to voter K, secondarily delegates to voter L, and votes via delegation for “p(x,y)”. Voter K directly votes for “x”, or does not vote if x = ∅. Voter L directly votes for “y”.
Considering Case XXIII 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 delegates to voter J and votes via delegation for “p(x,y)”. Voter I directly votes for “z_{3}”. Voter J primarily delegates to voter K, secondarily delegates to voter L, and votes via delegation for “p(x,y)”. Voter K directly votes for “x”, or does not vote if x = ∅. Voter L directly votes for “y”.
Figure 24.3: Case XXIV (cont.)

Case XXV

We consider Case XXIV and set x=∅, y=YES, z1=YES, z2=NO, z3=NO to create a more specific Case XXV. The number of YES votes outnumbers the number of NO votes. Thus “YES” would 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”. 9 votes for “YES”, 2 votes for “NO”. 9 > 2 (“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”. 9 votes for “YES”, 2 votes for “NO”. 9 > 2 (“YES” would win).
Figure 25: Case XXV

Case XXVI

We create a 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 the requirement of the absence of negative voting weight through delegation, “YES” would need to win in Case XXVI (because it also wins in Case XXV).

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 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”.
Figure 26: Case XXVI

Case XXVII

We consider Case XXIV and set x=∅, y=NO, z1=NO, z2=YES, z3=YES to create a more specific Case XXVII. The number of NO votes outnumbers the number of YES votes. Thus “NO” would win here.

Voter A primarily delegates to voter B, secondarily delegates to voter C, and votes via delegation for “NO”. Voter B delegates to voter D and votes via delegation for “NO”. Voter C directly votes for “NO”. Voter D primarily delegates to voter E, secondarily delegates to voter F, and votes via delegation for “NO”. Voter E delegates to voter G and votes via delegation for “NO”. Voter F directly votes for “YES”. Voter G primarily delegates to voter H, secondarily delegates to voter I, and votes via delegation for “NO”. Voter H delegates to voter J and votes via delegation for “NO”. Voter I directly votes for “YES”. Voter J primarily delegates to voter K, secondarily delegates to voter L, and votes via delegation for “NO”. Voter K does not vote. Voter L directly votes for “NO”. 2 votes for “YES”, 9 votes for “NO”. 2 < 9 (“NO” would win).
Voter A primarily delegates to voter B, secondarily delegates to voter C, and votes via delegation for “NO”. Voter B delegates to voter D and votes via delegation for “NO”. Voter C directly votes for “NO”. Voter D primarily delegates to voter E, secondarily delegates to voter F, and votes via delegation for “NO”. Voter E delegates to voter G and votes via delegation for “NO”. Voter F directly votes for “YES”. Voter G primarily delegates to voter H, secondarily delegates to voter I, and votes via delegation for “NO”. Voter H delegates to voter J and votes via delegation for “NO”. Voter I directly votes for “YES”. Voter J primarily delegates to voter K, secondarily delegates to voter L, and votes via delegation for “NO”. Voter K does not vote. Voter L directly votes for “NO”. 2 votes for “YES”, 9 votes for “NO”. 2 < 9 (“NO” would win).
Figure 27: Case XXVII

Case XXVIII

We create a Case XXVIII equal to Case XXVII but with the sole difference that voter K (who was previously abstaining) delegates to voter A (who was previously voting for NO through delegation). According to the requirement of the absence of negative voting weight through delegation, “NO” would need to win in Case XXVIII (because it also wins in Case XXVII).

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 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”.
Figure 28: Case XXVIII

The property of “Anonymity”, however, forbids that that “YES” wins in Case XXVI and “NO” wins in case XXVIII. Therefore, the 5 properties are contradictory, quod erat demonstrandum.

[PD] Jan Behrens & Björn Swierczek: Preferential Delegation and the Problem of Negative Voting Weight. In “The Liquid Democracy Journal on electronic participation, collective moderation, and voting systems, Issue 3” (2015-01-23). ISSN 2198-9532. Published by Interaktive Demokratie e. V., available at http://www.liquid-democracy-journal.org/issue/3/ (referenced at: a b c)
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Published by: Interaktive Demokratie e. V., Berlin, Germany – Edited by: Jan Behrens, Axel Kistner, Andreas Nitsche, Björn Swierczek
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ISSN-L: 2198-9532 – ISSN print version: 2198-9532 – ISSN electronic version: 2199-1758