Strategic voting Approval voting

1 strategic voting

1.1 overview
1.2 sincere voting

1.2.1 examples


1.3 sincere strategy ordinal preferences

1.3.1 dichotomous preferences
1.3.2 approval threshold


1.4 strategy cardinal utilities
1.5 strategy examples
1.6 dichotomous cutoff

strategic voting

overview

approval voting vulnerable bullet voting , compromising, while immune push-over , burying.

bullet voting occurs when voter approves candidate instead of both , b reason voting b can cause lose.

compromising occurs when voter approves additional candidate otherwise considered unacceptable voter prevent worse alternative winning.

strategic approval voting differs ranked choice voting methods voters might reverse preference order of 2 options. strategic approval voting, more 2 options, involves voter changing approval threshold. voter decides options give same rating, despite having strict preference order between them.

sincere voting

approval voting experts describe sincere votes ... directly reflect true preferences of voter, i.e., not report preferences falsely. give specific definition of sincere approval vote in terms of voter s ordinal preferences being vote that, if votes 1 candidate, votes more preferred candidate. definition allows sincere vote treat strictly preferred candidates same, ensuring every voter has @ least 1 sincere vote. definition allows sincere vote treat equally preferred candidates differently. when there 2 or more candidates, every voter has @ least 3 sincere approval votes choose from. 2 of sincere approval votes not distinguish between of candidates: vote none of candidates , vote of candidates. when there 3 or more candidates, every voter has more 1 sincere approval vote distinguishes between candidates.

examples

based on definition above, if there 4 candidates, a, b, c, , d, , voter has strict preference order, preferring b c d, following voter s possible sincere approval votes:

vote a, b, c, , d
vote a, b, , c
vote , b
vote a
vote no candidates

if voter instead equally prefers b , c, while still preferred candidate , d least preferred candidate, of above votes sincere , following combination sincere vote:

vote , c

the decision between above ballots equivalent deciding arbitrary approval cutoff. candidates preferred cutoff approved, candidates less preferred not approved, , candidates equal cutoff may approved or not arbitrarily.

sincere strategy ordinal preferences

a sincere voter multiple options voting sincerely still has choose sincere vote use. voting strategy way make choice, in case strategic approval voting includes sincere voting, rather being alternative it. differs other voting systems typically have unique sincere vote voter.

when there 3 or more candidates, winner of approval voting election can change, depending on sincere votes used. in cases, approval voting can sincerely elect 1 of candidates, including condorcet winner , condorcet loser, without voter preferences changing. extent electing condorcet winner , not electing condorcet loser considered desirable outcomes voting system, approval voting can considered vulnerable sincere, strategic voting. in 1 sense, conditions can happen robust , not isolated cases. on other hand, variety of possible outcomes has been portrayed virtue of approval voting, representing flexibility , responsiveness of approval voting, not voter ordinal preferences, cardinal utilities well.

dichotomous preferences

approval voting avoids issue of multiple sincere votes in special cases when voters have dichotomous preferences. voter dichotomous preferences, approval voting strategy-proof (also known strategy-free). when voters have dichotomous preferences , vote sincere, strategy-proof vote, approval voting guaranteed elect condorcet winner, if 1 exists. however, having dichotomous preferences when there 3 or more candidates not typical. unlikely situation voters have dichotomous preferences when there more few voters.

having dichotomous preferences means voter has bi-level preferences candidates. of candidates divided 2 groups such voter indifferent between 2 candidates in same group , candidate in top-level group preferred candidate in bottom-level group. voter has strict preferences between 3 candidates—prefers b , b c—does not have dichotomous preferences.

being strategy-proof voter means there unique way voter vote strategically best way vote, regardless of how others vote. in approval voting, strategy-proof vote, if exists, sincere vote.

approval threshold

another way deal multiple sincere votes augment ordinal preference model approval or acceptance threshold. approval threshold divides of candidates 2 sets, voter approves of , voter not approve of. voter can approve of more 1 candidate , still prefer 1 approved candidate approved candidate. acceptance thresholds similar. such threshold, voter votes every candidate meets or exceeds threshold.

with threshold voting, still possible not elect condorcet winner , instead elect condorcet loser when both exist. however, according steven brams, represents strength rather weakness of approval voting. without providing specifics, argues pragmatic judgements of voters candidates acceptable should take precedence on condorcet criterion , other social choice criteria.

strategy cardinal utilities

voting strategy under approval guided 2 competing features of approval voting. on 1 hand, approval voting fails later-no-harm criterion, voting candidate can cause candidate win instead of more preferred candidate. on other hand, approval voting satisfies monotonicity criterion, not voting candidate can never candidate win, can cause candidate lose less preferred candidate. either way, voter can risk getting less preferred election winner. voter can balance risk-benefit trade-offs considering voter s cardinal utilities, particularly via von neumann–morgenstern utility theorem, , probabilities of how others vote.

a rational voter model described myerson , weber specifies approval voting strategy votes candidates have positive prospective rating. strategy optimal in sense maximizes voter s expected utility, subject constraints of model , provided number of other voters sufficiently large.

an optimal approval vote votes preferred candidate , not least preferred candidate. however, optimal vote can require voting candidate , not voting more preferred candidate if there 4 candidates or more.

other strategies available , coincide optimal strategy in special situations. example:

vote candidates have above average utility. strategy coincides optimal strategy if voter thinks pairwise ties equally likely
vote candidate more preferred expected winner , vote expected winner if expected winner more preferred expected runner-up. strategy coincides optimal strategy if there 3 or fewer candidates or if pivot probability tie between expected winner , expected runner-up sufficiently large compared other pivot probabilities.
vote preferred candidate only. strategy coincides optimal strategy when there 1 candidate positive prospective rating.

another strategy vote top half of candidates, candidates have above-median utility. when voter thinks others balancing votes randomly , evenly, strategy maximizes voter s power or efficacy, meaning maximizes probability voter make difference in deciding candidate wins.

optimal strategic approval voting fails satisfy condorcet criterion , can elect condorcet loser. strategic approval voting can guarantee electing condorcet winner in special circumstances. example, if voters rational , cast strategically optimal vote based on common knowledge of how other voters vote except small-probability, statistically independent errors in recording votes, winner condorcet winner, if 1 exists.

strategy examples

in example election described here, assume voters in each faction share following von neumann–morgenstern utilities, fitted interval between 0 , 100. utilities consistent rankings given earlier , reflect strong preference each faction has choosing city, compared weaker preferences other factors such distance other cities.

using these utilities, voters choose optimal strategic votes based on think various pivot probabilities pairwise ties. in each of scenarios summarized below, voters share common set of pivot probabilities.

in first scenario, voters choose votes based on assumption pairwise ties equally likely. result, vote candidate above-average utility. voters vote first choice. knoxville faction votes second choice, chattanooga. result, winner memphis, condorcet loser, chattanooga coming in second place.

in second scenario, of voters expect memphis winner, chattanooga runner-up, , pivot probability memphis-chattanooga tie larger pivot probabilities of other pair-wise ties. result, each voter votes candidate prefer more leading candidate, , vote leading candidate if prefer candidate more expected runner-up. each remaining scenario follows similar pattern of expectations , voting strategies.

in second scenario, there three-way tie first place. happens because expected winner, memphis, condorcet loser , ranked last voter did not rank first.

only in last scenario actual winner , runner-up match expected winner , runner-up. result, can considered stable strategic voting scenario. in language of game theory, equilibrium. in scenario, winner condorcet winner.

dichotomous cutoff

as voting method cardinal rather ordinal, possible model voters in way not simplify ordinal method. modelling voters dichotomous cutoff assumes voter has immovable approval cutoff, while having meaningful cardinal preferences. means rather voting top 3 candidates, or candidates above average approval (which may result in vote changing if 1 candidate drops out, resulting in system not satisfy iia), instead vote candidates above approval cutoff have decided. cutoff not change, regardless of , how many candidates running, when available alternatives either above or below cutoff, voter votes or none of candidates, despite preferring on others. while extreme appears unrealistic, reflects reality in way many voters become disenfranchised , apathetic if see no candidates approve of. in way, there evidence suggest many voters may have internal cutoff, , not vote top 3, or above average candidates, although not entirely immovable.

for example, - in scenario, voters voting candidates approval above 50% (bold signifies voters voted candidate):

c wins 65% of voters approval, beating b 60%, d 40% , 35%

if voters threshold receiving vote candidate has above average approval, or vote 2 approved of candidates, not dichotomous cutoff, can change if candidates drop out. on other hand, if voters threshold receiving vote fixed (say 50%), dichotomous cutoff, , satisfies iia shown below:

b wins 60%, beating c 55% , d 40%

with dichotomous cutoff, c still wins.

b wins 70%, beating c , 65%

with dichotomous cutoff, c still wins.