Here's the new text on the SODA page<http://wiki.electorama.com/wiki/Simple_Optionally-Delegated_Approval#Criteria_Compliance>relating to the Condorcet criterion:
It fails the Condorcet criterion<http://wiki.electorama.com/wiki/Condorcet_criterion>, although the majority Condorcet winner over the ranking-augmented ballots is the unique strong, subgame-perfect equilibrium winner. That is to say that, the method would in fact pass the *majority* Condorcet winner criterion, assuming the following: - *Candidates are honest* in their pre-election rankings. This could be because they are innately unwilling to be dishonest, because they are unable to calculate a useful dishonest strategy, or, most likely, because they fear dishonesty would lose them delegated votes. That is, voters who disagreed with the dishonest rankings might vote approval-style instead of delegating, and voters who perceived the rankings as dishonest might thereby value the candidate less. - *Candidates are rationally strategic* in assigning their delegated vote. Since the assignments are sequential, game theory states that there is always a subgame-perfect Nash equilibrium, which is always unique except in some cases of tied preferences. - *Voters* are able to use the system to *express all relevant preferences*. That is to say, all voters fall into one of two groups: those who agree with their favored candidate's declared preference order and thus can fully express that by delegating their vote; or those who disagree with their favored candidate's preferences, but are aware of who the Condorcet winner is, and are able to use the approval-style ballot to express their preference between the CW and all second-place candidates. "Second place" means the Smith set if the Condorcet winner were removed from the election; thus, for this assumption to hold, each voter must prefer the CW to all members of this second-place Smith set or vice versa. That's obviously always true if there is a single second-place CW. The three assumptions above would probably not strictly hold true in a real-life election, but they usually would be close enough to ensure that the system does elect the CW. SODA does even better than this if there are only 3 candidates, or if the Condorcet winner goes first in the delegation assignment order, or if there are 4 candidates and the CW goes second. In any of those circumstances, under the assumptions above, it passes the *Condorcet* criterion, not just the majority Condorcet criterion. The important difference between the Condorcet criterion (beats all others pairwise) and the majority Condorcet criterion (beats all others pairwise by a strict majority) is that the former is clone-proof while the latter is not. Thus, with few enough strong candidates, SODA also passes the independence of clones criterion<http://wiki.electorama.com/wiki/index.php?title=Independence_of_clones_criterion&action=edit&redlink=1> . Note that, although the circumstances where SODA passes the Condorcet criterion are hemmed in by assumptions, when it does pass, it does so in a perfectly strategy-proof sense. That is *not* true of any actual Condorcet system (that is, any system which universally passes the Condorcet criterion). Therefore, for rationally-strategic voters who believe that the above assumptions are likely to hold, *SODA may in fact pass the Condorcet criterion more often than a Condorcet system*.
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