I’m increasingly coming to the conclusion that formal Smith compliance may be much less informative about strategic behavior than I previously assumed. A Smith-compliant method will, of course, continue to elect from the Smith set of the ballots actually cast. But strategic burial can change that reported Smith set. So a method can remain perfectly Smith-compliant while nevertheless failing to elect a candidate who would have been the Condorcet winner (or more generally Smith compliant) under sincere preferences.

Put differently, there seems to be an important distinction between formal Smith compliance and strategic preservation of the sincere Smith set.

I’ve been exploring this through a large number of LLM-assisted simulations of adaptive strategic behavior. In the simulations, strategic blocs are allowed to alter their ballots freely and adapt in response to the strategies of other blocs. These are obviously exploratory tests and should be taken with a grain of salt, as I may be out of my element.

In any case, conditioning specifically on elections in which a unique sincere Condorcet winner exists, I’ve found that even fairly sophisticated two-round mechanisms designed specifically to counter burial do not seem to improve the probability that the sincere CW actually wins by very much. The best variants I’ve tested have been around 72% under adaptive strategy.

What surprised me is that this is very close to the performance of much simpler one-shot methods under the same strategic conditions. IRV, for example, elected the sincere Condorcet winner about 70% of the time. Stable Voting and Ranked Pairs were roughly in the high-60% range, and differences between the better-performing methods were often pretty small.

So I don’t mean that IRV is somehow “more Condorcet compliant” than Ranked Pairs or Stable Voting. Obviously it isn’t in the formal criterion sense. Rather, it seems that formal Condorcet/Smith compliance did not translate into dramatically greater preservation of the sincere Condorcet winner once strategic behavior was introduced.

This seems important, because we often talk about Smith compliance as though it provides strong protection against electing the “wrong” candidate. But the guarantee of Smith compliance applies only to the Smith set induced by the submitted ballots. If strategic voting substantially changes the pairwise structure, that guarantee can diverge from the thing we may actually care about: whether a candidate who would beat everyone else under sincere preferences wins the election.

That has made me wonder whether formal Smith compliance is less useful as a practical discriminator between methods than I had thought. Maybe a more relevant question is: how difficult, risky, or dependent on coordination is it for strategic voting to displace the sincere Condorcet winner? And that’s a different property from formal Smith compliance itself.

In fact, a formally non-Smith-compliant method might preserve the sincere CW about as often as a formally Smith-compliant method under strategic conditions, since a Smith-compliant method might remain perfectly compliant with its reported ballots even after burial has manipulated the pairwise structure away from sincerity.

I’m still testing this, and I’m not an expert in this area, so I wouldn’t put too much weight on the values yet. But the general pattern has been pretty consistent: once sufficiently adaptive strategic behavior is allowed, a number of very different methods seem to converge toward fairly similar rates of sincere-CW preservation—namely, in this case, ~65-75%.

What do you think of this? Personally, it makes me more interested in IRV, since it consistently outperformed many other methods under strategic pressure. I haven’t looked into the sincere bottom-Smith situation yet, but that’s another thing to consider.

At the same time, it makes me feel that prioritizing some criteria that are incompatible with formal Smith may be a wise decision, such as participation.

Shout out to this post: https://www.votingtheory.org/forum/topic/620/score-voting-is-king-condorcet-not-so-much/2