Editing Approval voting (section)

=== Strategy examples ===
{{more citations needed section|date=June 2019}}In the example election described [[Condorcet method#Example: Voting on the location of Tennessee's capital|here]], assume that the voters in each faction share the following [[Von Neumann–Morgenstern utility theorem|von Neumann–Morgenstern utilities]], fitted to the interval between 0 and 100. The utilities are consistent with the rankings given earlier and reflect a strong preference each faction has for choosing its city, compared to weaker preferences for other factors such as the distance to the other cities.<!-- This seems like a very weird choice of utilities. Wouldn't it be better to make the disutility equal to the distance between the two cities? -->

{| class="wikitable" style="text-align: center"
|+Voter utilities for each candidate city
|-
! scope="col" rowspan="2" | Fraction of voters <small>(living close to)</small>
! scope="colgroup" colspan="4" | Candidates
! scope="col" rowspan=2 | Average
|-
! scope="col" style="width:6em;"| Memphis
! scope="col" style="width:6em;"| Nashville
! scope="col" style="width:6em;"| Chattanooga
! scope="col" style="width:6em;"| Knoxville
|-
| style="text-align:left;"| Memphis (42%)
| 100 || 15 || 10 || 0 || 31.25
|-
| style="text-align:left;"| Nashville (26%)
| 0 || 100 || 20 || 15 || 33.75
|-
| style="text-align:left;"| Chattanooga (15%)
| 0 || 15 || 100 || 35 || 37.5
|-
| style="text-align:left;"| Knoxville (17%)
| 0 || 15 || 40 || 100 || 38.75
|}

Using these utilities, voters choose their optimal strategic votes based on what they think the various pivot probabilities are for pairwise ties. In each of the scenarios summarized below, all voters share a common set of pivot probabilities.

{| class="wikitable" style="text-align: center"
|+Approval Voting results for scenarios using optimal strategic voting
|-
! scope="col" rowspan="2" | Strategy scenario
! scope="col" rowspan="2" | Winner
! scope="col" rowspan="2" | Runner-up
! scope="colgroup" colspan="4" | Candidate vote totals
|-
! scope="col" style="width:6em" | Memphis
! scope="col" style="width:6em" | Nashville
! scope="col" style="width:6em" | Chattanooga
! scope="col" style="width:6em" | Knoxville
|-
| style="text-align: left" | Zero-info
| Memphis || Chattanooga || 42 || 26 || 32 || 17
|-
| style="text-align: left" | Memphis leading Chattanooga
| colspan="2" | Three-way tie || 42 || 58 || 58 || 58
|-
| style="text-align: left" | Chattanooga leading Knoxville
| Chattanooga || Nashville || 42 || 68 || 83 || 17
|-
| style="text-align: left" | Chattanooga leading Nashville
| Nashville || Memphis || 42 || 68 || 32 || 17
|-
| style="text-align: left" | Nashville leading Memphis
| Nashville || Memphis || 42 || 58 || 32 || 32
|}

In the first scenario, voters all choose their votes based on the assumption that all pairwise ties are equally likely. As a result, they vote for any candidate with an above-average utility. Most voters vote for only their first choice. Only the Knoxville faction also votes for its second choice, Chattanooga. As a result, the winner is Memphis, the Condorcet loser, with Chattanooga coming in second place. In this scenario, the winner has minority approval (more voters disapproved than approved) and all the others had even less support, reflecting the position that no choice gave an above-average utility to a majority of voters.

In the second scenario, all of the voters expect that Memphis is the likely winner, that Chattanooga is the likely runner-up, and that the pivot probability for a Memphis-Chattanooga tie is much larger than the pivot probabilities of any other pair-wise ties. As a result, each voter votes for any candidate they prefer more than the leading candidate, and also vote for the leading candidate if they prefer that candidate more than the expected runner-up. Each remaining scenario follows a similar pattern of expectations and voting strategies.

In the second scenario, there is a three-way tie for first place. This happens because the expected winner, Memphis, was the Condorcet loser and was also ranked last by any voter that did not rank it first.

Only in the last scenario does the actual winner and runner-up match the expected winner and runner-up. As a result, this can be considered a stable strategic voting scenario. In the language of [[game theory]], this is an "equilibrium." In this scenario, the winner is also the Condorcet winner.