No they do not convey equivalent information, because the information content isn't just the number of states, it's the encoding mechanism too. For instance su

No they do not convey equivalent information, because the information content isn't just the number of states, it's the encoding mechanism too. For instance suppose you encode the position of missile strikes on some land area in a 16x16 grid, meaning 8 bits of information. Another person uses only six bits of information, encoding the missile strike position as degrees and distance from the center point of the grid. Then it turns out the latter method is more accurate because the missiles tend to always strike pretty close to the center of the grid. The specific patterns of voter preference that emerge in a realistic scenario are counterintuitive and can end up being more accurately measured by the kind of expression approval voting allows. You can get people who have radically different intensity of preference but the same order of preference so that they look identical to a ranked voting method but appear different to approval voting based on their thresholds. In addition, you have tactical behavior, as well as the efficiency of how the voting method uses the information on the ballots. When you combine all of these factors together, approval voting tends to perform better than ranked voting methods. https://www.electionscience.org/library/expressiveness-in-approval-vs-ranked-ballots/ https://www.electionscience.org/library/expressiveness-in-ap...