Editing Approval voting (section)

=== Strategy with cardinal utilities ===

Voting strategy under approval is guided by two competing features of approval. On the one hand, approval fails the [[later-no-harm criterion]], so voting for a candidate can cause that candidate to win instead of a candidate more preferred by that voter. On the other hand, approval satisfies the [[monotonicity criterion]], so not voting for a candidate can never help that candidate win, but can cause that candidate to lose to a less preferred candidate. Either way, the voter can risk getting a less preferred election winner. A voter can balance the risk-benefit trade-offs by considering the voter's cardinal utilities, particularly via the [[von Neumann–Morgenstern utility theorem]], and the probabilities of how others vote.

A [[Tactical voting#Myerson-Weber strategy|rational voter model]] described by [[Roger Myerson|Myerson]] and Weber specifies an approval strategy that votes for those candidates that have a positive prospective rating.<ref name=":2">{{Cite journal | doi = 10.2307/2938959 |last1=Myerson |first1=R. |last2=Weber |first2=R. J. | year = 1993 | title = A theory of Voting Equilibria | jstor = 2938959| journal = American Political Science Review | volume = 87 | issue = 1| pages = 102–114 |url=http://www.kellogg.northwestern.edu/research/math/papers/782.pdf |archive-url=https://ghostarchive.org/archive/20221009/http://www.kellogg.northwestern.edu/research/math/papers/782.pdf |archive-date=October 9, 2022 |url-status=live | hdl = 10419/221141 |s2cid=143828854 | hdl-access = free }}</ref> This strategy is optimal in the sense that it maximizes the voter's [[Utility#Expected utility|expected utility]], subject to the constraints of the model and provided the number of other voters is sufficiently large.

An optimal approval vote always votes for the most preferred candidate and not for the least preferred candidate, which is a [[dominant strategy]]. An optimal vote can require supporting one candidate and not voting for a more preferred candidate if there 4 candidates or more, e.g. the third and fourth choices are correlated to gain or lose decisive votes together; however, such situations are inherently unstable, suggesting such strategy should be rare.<ref>{{Cite journal | last1=Dutta |first1=B |last2=De Sinopoli |first2=F. | last3=Laslier |first3= J.-F.| year = 2006 | title = Approval voting: three examples | journal = International Journal of Game Theory | volume = 35 | pages = 27–38 |doi=10.1007/s00182-006-0053-2 |s2cid=801286 | citeseerx=10.1.1.365.8090 }}</ref>

Other strategies are also available and coincide with the optimal strategy in special situations. For example:
* Vote for the candidates that have above average utility. This strategy coincides with the optimal strategy if the voter thinks that all pairwise ties are equally likely.{{sfn|Brams|Fishburn|1983|p=85}}
* Vote for any candidate that is more preferred than the expected winner and also vote for the expected winner if the expected winner is more preferred than the expected runner-up. This strategy coincides with the optimal strategy if there are three or fewer candidates or if the pivot probability for a tie between the expected winner and expected runner-up is sufficiently large compared to the other pivot probabilities. This strategy, if used by all voters, implies at equilibrium the election of the Condorcet winner whenever it exists.<ref name=":3">{{Cite journal  | last1=Laslier |first1= J.-F.| year = 2009 | title = The Leader rule: a model of strategic approval voting in a large electorate | journal = Journal of Theoretical Politics | volume = 21 | issue=1 | pages = 113–136 |doi= 10.1177/0951629808097286|s2cid= 153790214}}</ref>
*Vote for the most preferred candidate only. This strategy coincides with the optimal strategy when the best candidate is either much better than all others (i.e. is the only one with a positive expected value).{{sfn|Brams|Fishburn|1983|p=74, 81}}
*If all voters are rational and cast a strategically optimal vote based on a common knowledge of how all other voters vote except for small-probability, statistically independent errors, then the winner will be the Condorcet winner, if one exists.<ref name=":4">Laslier, J.-F. (2006) [http://halshs.archives-ouvertes.fr/docs/00/12/17/51/PDF/stratapproval4.pdf "Strategic approval voting in a large electorate,"] ''IDEP Working Papers'' No. 405 (Marseille, France: Institut D'Economie Publique)</ref>