Convergence

In this section we study the convergence of the restricted iterative voting process associated with our proposed manipulation strategies.

Theorem 4.3.1 An iterative process defined using second-chance (SC) converges for every (deterministic) voting rule F and turn function τ.

Proof. The proof of this statement is straightforward from our definitions. The iteration process starts at the truthful profile P0, and each agent is allowed to switch the top candidate

with the one in second position only once. We stress the fact that in this chapter we consider only deterministic voting rules, i.e., no randomised procedure is used in their definition, therefore when no individual changes their preferences anymore the result of the voting rule remain invariant and the process converges. 

Theorem 4.3.2 An iterative process defined using best-upgrade (BU) converges for every turn function τ if F is a PSR, the Copeland rule or the Maximin rule.

4.3 Restricted Iterative Voting 67

Proof. The winner of an election using a PSR, Copeland or Maximin is defined as the candidate maximizing a certain score (or with maximal score and highest rank in the tie-breaking order). Since the maximal score of a candidate is bounded, it is sufficient to show that the score of the winner increases at every iteration step (or, in case the score remains constant that the position of the winner in the tie-breaking order increases) to show that the iterative process converges. Let us start with PSR. Recall that the score of a candidate c under a PSR is ∑_isiwhere siis the score given by the position of c in ballot

Pi. Using BU, the manipulator moves to the top a candidate which lies above the current

winner c. Thus, the position – and hence the score – of c remains unchanged, and the new winner must have a strictly higher score (or a better position in the tie-breaking order) than the previous one. Since the maximal score of a candidate is bounded by n times the maximal score that can be given, the process eventually stop. The case of Copeland and Maximin can be solved in a similar fashion: it is sufficient to observe that the relative position of the current winner c with all other candidates (and thus also its score) remain unchanged when ballots are manipulated using BU. Thus, the Copeland score and the Maximin score of a new winner must be higher than that of c (or the new winner must be placed higher in the tie-breaking order). 

This proof generalizes to show the convergence of iterative processes using BU for any voting rule where BU does not change the score of the winner and which outputs as winners those candidates maximizing a notion of score.

For the case of STV, we observed experimentally that its iteration always terminates using the fair turn function. However, as shown in the example in Figure 4.2, convergence of STV is not guaranteed if the turn function used is sequential. For space constraints we are omitting the preference symbol> or P between candidates, reading preference from left to right. Let the tie-breaking rule be e>Cd>Cc>Cb>Ca, and let voters manipulate

using BU. The initial truthful profile is the one on the top left corner.

A closer look at the proofs of Theorem 4.3.1 and Theorem 4.3.2 suggests a bound on the number of iterations of an iterative process defined with our manipulation strategies. Since SC can be applied only once, the iteration process associated to it stops after at most|V | steps for every voting rule. The case of BU is slightly more complex. At every step of the iteration the score of the winner must increase (or she should be placed higher in the tie-breaking order). Thus, in the worst case different winners will touch all possible scores and for each score we will climb up the tie-breaking order until we reach the highest score on the highest tie-breaking position. Since the maximal score of the winner is bounded by a polynomial, so is the number of steps. These observations are summed up in the following statement.

v1: daebc v2: bedac v3: baedc v4: cbeda v5: cebda v6: adbec Winner = d v1: daebc v2: ebdac v3: baedc v4: cbeda v5: cebda v6: adbec Winner = e v1: adebc v2: ebdac v3: baedc v4: cbeda v5: cebda v6: adbec Winner = a v1: adebc v2: bedac v3: baedc v4: cbeda v5: cebda v6: adbec Winner = b v2 v1 v2 v1

Figure 4.2: STV with sequential turn function does not converge.

Theorem 4.3.3 An iterative voting process defined using SC will terminate after at most

O(|V |) steps. An iterative voting process defined using BU for a PSR, Copeland, or Maximin, converges after at most O(s × |C|) steps, where s is the maximal score that a candidate can receive in an election.

We conclude by showing that the veto rule does not iterate using our proposed manipulation strategies, and therefore will not be included in our experimental analysis.

Theorem 4.3.4 If|C| > 3 the iterative voting processes defined using SC or BU with the veto rule does not iterate, i.e., no agent has incentives to manipulate in the truthful state.

Proof. Recall that the veto rule gives one point to all candidates but the one ranked last in the individual ranking. It is then straightforward to observe that SC cannot change the outcome of the veto rule if there are more than 3 candidates, since the swap occurs only in the top part of the individual preference orders. The same holds for BU: since the candidates that can be upgraded must lie above the current winner of the election, even if the winner is in last position the upgrade will not change the score of any of the candidates, as the veto rule gives one point to all candidates but the last. 

4.3 Restricted Iterative Voting 69