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 fulfill 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.
See [PD] for Case I through XXII.
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, ∅}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, ∅}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.
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).
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.
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).
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.