The basic approval voting game

We survey results about Approval Voting obtained within the standard framework of game theory. Restricting the set of strategies to undominated and sincere ballots does not help to predict Approval Voting outcomes, which is also the case under strategic equilibrium concepts such as Nash equilibrium and its usual refinements. Strong Nash equilibrium in general does not exist but predicts the election of a Condorcet winner when one exists

Multi winner Approval Voting: An Apportionment Approach

We extend approval voting so as to elect multiple candidates, who may be either individuals or members of a political party, in rough proportion to their approval in the electorate. We analyze two divisor methods of apportionment, first proposed by Jefferson and Webster, that iteratively depreciate the approval votes of voters who have one or more of their approved candidates already elected. We compare the usual sequential version of these methods with a nonsequential version, which is computationally complex but feasible for many elections. Whereas Webster apportionments tend to be more representative of the electorate than those of Jefferson, the latter, whose equally spaced vote thresholds for winning seats duplicate those of cumulative voting in 2-party elections, is even-handed or balanced

Multi winner Approval Voting: An Apportionment Approach

We extend approval voting so as to elect multiple candidates, who may be either individuals or members of a political party, in rough proportion to their approval in the electorate. We analyze two divisor methods of apportionment, first proposed by Jefferson and Webster, that iteratively depreciate the approval votes of voters who have one or more of their approved candidates already elected. We compare the usual sequential version of these methods with a nonsequential version, which is computationally complex but feasible for many elections. Whereas Webster apportionments tend to be more representative of the electorate than those of Jefferson, the latter, whose equally spaced vote thresholds for winning seats duplicate those of cumulative voting in 2-party elections, is even-handed or balanced

Jugement majoritaire vs. vote majoritaire

Cet article met en évidence les lacunes du scrutin majoritaire à un ou à deux tours en théorie et en pratique, se reposant sur l'histoire récente des élections présidentielles françaises et de résultats expérimentaux. Et il propose un remède : plus d'information doit être demandée aux électeurs et le sens de la " majorité " doit être reformulé

The Paradox of Grading Systems

We distinguish between (i) voting systems in which voters can rank candidates and (ii) those in which they can grade candidates, such as approval voting, in which voters can give two grades—approve (1) or not approve (0)—to candidates. While two grades rule out a discrepancy between the average-grade winners, who receive the highest average grade, and the superior-grade winners, who receive more superior grades in pairwise comparisons (akin to Condorcet winners), more than two grades allow it. We call this discrepancy between the two kinds of winners the paradox of grading systems, which we illustrate with several examples and whose probability we estimate for sincere and strategic voters through a Monte Carlo simulation. We discuss the tradeoff between (i) allowing more than two grades, but risking the paradox, and (ii) precluding the paradox, but restricting voters to two grades

The Paradox of Grading Systems

We distinguish between (i) voting systems in which voters can rank candidates and (ii) those in which they can grade candidates, such as approval voting, in which voters can give two grades—approve (1) or not approve (0)—to candidates. While two grades rule out a discrepancy between the average-grade winners, who receive the highest average grade, and the superior-grade winners, who receive more superior grades in pairwise comparisons (akin to Condorcet winners), more than two grades allow it. We call this discrepancy between the two kinds of winners the paradox of grading systems, which we illustrate with several examples and whose probability we estimate for sincere and strategic voters through a Monte Carlo simulation. We discuss the tradeoff between (i) allowing more than two grades, but risking the paradox, and (ii) precluding the paradox, but restricting voters to two grades

Satisfaction approval voting

We propose a new voting system, satisfaction approval voting (SAV), for multiwinner elections, in which voters can approve of as many candidates or as many parties as they like. However, the winners are not those who receive the most votes, as under approval voting (AV), but those who maximize the sum of the satisfaction scores of all voters, where a voter’s satisfaction score is the fraction of his or her approved candidates who are elected. SAV may give a different outcome from A--in fact, SAV and AV outcomes may be disjoint—but SAV generally chooses candidates representing more diverse interests than does AV (this is demonstrated empirically in the case of a recent election of the Game Theory Society). A decision-theoretic analysis shows that all strategies except approving of a least-preferred candidate are undominated, so voters will often find it optimal to approve of more than one candidate. In party-list systems, SAV apportions seats to parties according to the Jefferson/d’Hondt method with a quota constraint, which favors large parties and gives an incentive to smaller parties to coordinate their policies and forge alliances, even before an election, that reflect their supporters’ coalitional preferences.multiwinner election; voting system; approval ballot; proportional representation; apportonment

The Excess Method: A Multiwinner Approval Voting Procedure to Allocate Wasted Votes

In using approval voting to elect multiple winners to a committee or council, it is desirable that excess votes—approvals beyond those that a candidate needs to win a seat—not be wasted. The excess method does this by sequentially allocating excess votes to a voter’s as-yet-unelected approved candidates, based on the Jefferson method of apportionment. It is monotonic—approving of a candidate never hurts and may help him or her get elected—computationally easy, and less manipulable than related methods. In parliamentary systems with party lists, the excess method is equivalent to the Jefferson method and thus ensures the approximate proportional representation of political parties. As a method for achieving proportional representation (PR) on a committee or council, we compare it to other PR methods proposed by Hare, Andrae, and Droop for preferential voting systems, and by Phragmén for approval voting. Because voters can vote for multiple candidates or parties, the excess method is likely to abet coalitions that cross ideological and party lines and to foster greater consensus in voting bodies

The Excess Method: A Multiwinner Approval Voting Procedure to Allocate Wasted Votes

In using approval voting to elect multiple winners to a committee or council, it is desirable that excess votes—approvals beyond those that a candidate needs to win a seat—not be wasted. The excess method does this by sequentially allocating excess votes to a voter’s as-yet-unelected approved candidates, based on the Jefferson method of apportionment. It is monotonic—approving of a candidate never hurts and may help him or her get elected—computationally easy, and less manipulable than related methods. In parliamentary systems with party lists, the excess method is equivalent to the Jefferson method and thus ensures the approximate proportional representation of political parties. As a method for achieving proportional representation (PR) on a committee or council, we compare it to other PR methods proposed by Hare, Andrae, and Droop for preferential voting systems, and by Phragmén for approval voting. Because voters can vote for multiple candidates or parties, the excess method is likely to abet coalitions that cross ideological and party lines and to foster greater consensus in voting bodies