What matters is not how well your ballot reflects your preferences, but how well the outcome matches the preferences of the electorate. IRV can have very strange results where when a candidate starts to get more support it actually hurts them (http://zesty.ca/voting/sim/) and if voters/campaigns are strategic you can get worse outcomes than first past the post (http://electionscience.github.io/vse-sim/VSE/)
I guess the thing I’m questioning is how well does Approval actually reflect the preferences of the electorate. Let’s use Melbourne as an example: it was a safe Labor seat, but is now controlled by the Greens (there a useful summary of the seat’s history at https://www.abc.net.au/news/federal-election-2016/guide/melb/ ). Under Approval, Liberal voters (who know their candidate isn’t going to win, and want to deny Labor the seat) would vote for the Greens over Labor (e.g. approve Liberal and Greens); Greens voters would only vote Greens (they definitely won’t vote Liberal); Labor vote Labor (as adding the Greens to the their vote only weakens their candidate). If the Greens win, Liberal voters can switch to Labor next election (causing more division); or if Labor wins can continue forcing Labor to spend resources on the Melbourne electorate—in both cases the result isn’t reflecting what people actually want, rather the number of people following how-to-vote cards (are how-to-vote cards a thing in the US/outside Australia?). IRV has the same problem, except due to the problem of choosing whether to approve a candidate or not (vs giving all preferences), people are more likely to follow the how-to-vote card from the appropriate party (which is going to be governed by strategic voting, meaning you’re getting more false information about the elecorate’s preferences). It’s also much harder to vote against someone under Approval, such as happened against Stephen Conroy ( https://www.pcworld.idg.com.au/article/355744/what_would_it_take_unseat_conroy_/ has some information, but I recall there were websites devoted to providing information about how to vote against Conroy — this would be much easier now since there were changes in the senate ballot in 2016).
What matters is not how well your ballot reflects your preferences, but how well the outcome matches the preferences of the electorate. IRV can have very strange results where when a candidate starts to get more support it actually hurts them (http://zesty.ca/voting/sim/) and if voters/campaigns are strategic you can get worse outcomes than first past the post (http://electionscience.github.io/vse-sim/VSE/)
I guess the thing I’m questioning is how well does Approval actually reflect the preferences of the electorate. Let’s use Melbourne as an example: it was a safe Labor seat, but is now controlled by the Greens (there a useful summary of the seat’s history at https://www.abc.net.au/news/federal-election-2016/guide/melb/ ). Under Approval, Liberal voters (who know their candidate isn’t going to win, and want to deny Labor the seat) would vote for the Greens over Labor (e.g. approve Liberal and Greens); Greens voters would only vote Greens (they definitely won’t vote Liberal); Labor vote Labor (as adding the Greens to the their vote only weakens their candidate). If the Greens win, Liberal voters can switch to Labor next election (causing more division); or if Labor wins can continue forcing Labor to spend resources on the Melbourne electorate—in both cases the result isn’t reflecting what people actually want, rather the number of people following how-to-vote cards (are how-to-vote cards a thing in the US/outside Australia?). IRV has the same problem, except due to the problem of choosing whether to approve a candidate or not (vs giving all preferences), people are more likely to follow the how-to-vote card from the appropriate party (which is going to be governed by strategic voting, meaning you’re getting more false information about the elecorate’s preferences). It’s also much harder to vote against someone under Approval, such as happened against Stephen Conroy ( https://www.pcworld.idg.com.au/article/355744/what_would_it_take_unseat_conroy_/ has some information, but I recall there were websites devoted to providing information about how to vote against Conroy — this would be much easier now since there were changes in the senate ballot in 2016).