Hallo, it has been proven by Moulin that the Condorcet criterion and the participation criterion are incompatible:    Herve Moulin, "Condorcet's principle implies    the no show paradox", Journal of Economic Theory,    volume 45, number 1, pages 53-64, 1988,    DOI: 10.1016/0022-0531(88)90253-0 Here is a short version of Moulin's proof: http://lists.electorama.com/pipermail/election-methods-electorama.com/2003-October/011042.html Markus Schulze

 Herve Moulin, "Condorcet's principle implies
 the no show paradox", Journal of Economic Theory,
 volume 45, number 1, pages 53-64, 1988,
 DOI: 10.1016/0022-0531(88)90253-0

Hi Markus, Thank you for sharing this. I believed the Wikipedia article that Maximal Lotteries did not violate the participation criterion, but this simple proof does demonstrate that all Condorcet methods, even non-deterministic ones, will violate the participation criterion. That is good to know. Thanks again for sharing your knowledge. -Dan On Tue, Jun 24, 2025, 8:27 AM Markus Schulze via Election-Methods < election-methods@lists.electorama.com> wrote: > Hallo, > > it has been proven by Moulin that the Condorcet > criterion and the participation criterion are > incompatible: > > Herve Moulin, "Condorcet's principle implies > the no show paradox", Journal of Economic Theory, > volume 45, number 1, pages 53-64, 1988, > DOI: 10.1016/0022-0531(88)90253-0 > > Here is a short version of Moulin's proof: > > > http://lists.electorama.com/pipermail/election-methods-electorama.com/2003-October/011042.html > > Markus Schulze > > ---- > Election-Methods mailing list - see https://electorama.com/em for list > info >

 Herve Moulin, "Condorcet's principle implies
 the no show paradox", Journal of Economic Theory,
 volume 45, number 1, pages 53-64, 1988,
 DOI: 10.1016/0022-0531(88)90253-0

Markus—different generalizations/definitions of no-show (equivalent in the deterministic case) yield different results in when you allow lotteries. I'd have to double-check which is satisfied for Maximal Lotteries, but the most common are either: 1. Turning out to vote will always yield a better lottery than not turning out, or 2. Turning out to vote will probably improve the outcome for you, i.e. if you do a random draw from the winning lottery if you do vs. don't turn out to vote, you will prefer the random draw from the one where you turn out more often than vice-versa. On Tue, Jun 24, 2025 at 5:27 AM Markus Schulze via Election-Methods < election-methods@lists.electorama.com> wrote: > Hallo, > > it has been proven by Moulin that the Condorcet > criterion and the participation criterion are > incompatible: > > Herve Moulin, "Condorcet's principle implies > the no show paradox", Journal of Economic Theory, > volume 45, number 1, pages 53-64, 1988, > DOI: 10.1016/0022-0531(88)90253-0 > > Here is a short version of Moulin's proof: > > > http://lists.electorama.com/pipermail/election-methods-electorama.com/2003-October/011042.html > > Markus Schulze > > ---- > Election-Methods mailing list - see https://electorama.com/em for list > info >

 Herve Moulin, "Condorcet's principle implies
 the no show paradox", Journal of Economic Theory,
 volume 45, number 1, pages 53-64, 1988,
 DOI: 10.1016/0022-0531(88)90253-0

I have always seen the participation criterion defined as 'Adding votes that rank candidate A over B will not make it more likely for candidate B to win over candidate A'. The proof that Markus shared appears to establish that any Condorcet method will fail this criterion. It is not dependent on how the lotteries are defined over the Smith set, because it hinges on the fact that when a Condorcet winner exists, their probability of winning is 100%. It constructs an election with a cyclic tie between 4 candidates. Then, it shows that the tie can be broken to create a Condorcet winner by adding votes that rank that Condorcet winner second, thus moving those votes' first place candidate from a non-zero probability of winning to a zero probability of winning. On Wed, Jun 25, 2025 at 12:57 PM Closed Limelike Curves via Election-Methods <election-methods@lists.electorama.com> wrote: > Markus—different generalizations/definitions of no-show (equivalent in the > deterministic case) yield different results in when you allow lotteries. > I'd have to double-check which is satisfied for Maximal Lotteries, but the > most common are either: > 1. Turning out to vote will always yield a better lottery than not turning > out, or > 2. Turning out to vote will probably improve the outcome for you, i.e. if > you do a random draw from the winning lottery if you do vs. don't turn out > to vote, you will prefer the random draw from the one where you turn out > more often than vice-versa. > > On Tue, Jun 24, 2025 at 5:27 AM Markus Schulze via Election-Methods < > election-methods@lists.electorama.com> wrote: > >> Hallo, >> >> it has been proven by Moulin that the Condorcet >> criterion and the participation criterion are >> incompatible: >> >> Herve Moulin, "Condorcet's principle implies >> the no show paradox", Journal of Economic Theory, >> volume 45, number 1, pages 53-64, 1988, >> DOI: 10.1016/0022-0531(88)90253-0 >> >> Here is a short version of Moulin's proof: >> >> >> http://lists.electorama.com/pipermail/election-methods-electorama.com/2003-October/011042.html >> >> Markus Schulze >> >> ---- >> Election-Methods mailing list - see https://electorama.com/em for list >> info >> > ---- > Election-Methods mailing list - see https://electorama.com/em for list > info >

 Herve Moulin, "Condorcet's principle implies
 the no show paradox", Journal of Economic Theory,
 volume 45, number 1, pages 53-64, 1988,
 DOI: 10.1016/0022-0531(88)90253-0

Yes, by that definition the Maximal Lottery fails. The intuition for why it's a much smaller issue here is that when the voter adds their ballot, they are transferring probability from their top-ranked candidate to their second-ranked candidate. However, their third, 4th, etc. ranked candidates also lose probability in the winning lottery to their 2nd, which cancels out, such that the voter is still happier with the result than they would have been if they had not voted. Another intuition is that the algorithm is switches from being forced to compromise between several irreconcilable alternatives by randomizing and over to being able to choose a single "consensus pick". On Wed, Jun 25, 2025 at 11:09 AM Daniel Kirslis <dankirslis@gmail.com> wrote: > I have always seen the participation criterion defined as 'Adding votes > that rank candidate A over B will not make it more likely for candidate B > to win over candidate A'. The proof that Markus shared appears to establish > that any Condorcet method will fail this criterion. It is not dependent on > how the lotteries are defined over the Smith set, because it hinges on the > fact that when a Condorcet winner exists, their probability of winning is > 100%. It constructs an election with a cyclic tie between 4 candidates. > Then, it shows that the tie can be broken to create a Condorcet winner by > adding votes that rank that Condorcet winner second, thus moving those > votes' first place candidate from a non-zero probability of winning to a > zero probability of winning. > > On Wed, Jun 25, 2025 at 12:57 PM Closed Limelike Curves via > Election-Methods <election-methods@lists.electorama.com> wrote: > >> Markus—different generalizations/definitions of no-show (equivalent in >> the deterministic case) yield different results in when you allow >> lotteries. I'd have to double-check which is satisfied for Maximal >> Lotteries, but the most common are either: >> 1. Turning out to vote will always yield a better lottery than not >> turning out, or >> 2. Turning out to vote will probably improve the outcome for you, i.e. if >> you do a random draw from the winning lottery if you do vs. don't turn out >> to vote, you will prefer the random draw from the one where you turn out >> more often than vice-versa. >> >> On Tue, Jun 24, 2025 at 5:27 AM Markus Schulze via Election-Methods < >> election-methods@lists.electorama.com> wrote: >> >>> Hallo, >>> >>> it has been proven by Moulin that the Condorcet >>> criterion and the participation criterion are >>> incompatible: >>> >>> Herve Moulin, "Condorcet's principle implies >>> the no show paradox", Journal of Economic Theory, >>> volume 45, number 1, pages 53-64, 1988, >>> DOI: 10.1016/0022-0531(88)90253-0 >>> >>> Here is a short version of Moulin's proof: >>> >>> >>> http://lists.electorama.com/pipermail/election-methods-electorama.com/2003-October/011042.html >>> >>> Markus Schulze >>> >>> ---- >>> Election-Methods mailing list - see https://electorama.com/em for list >>> info >>> >> ---- >> Election-Methods mailing list - see https://electorama.com/em for list >> info >> >

On 2025-06-25 14:55, Daniel Kirslis via Election-Methods wrote: > Hi Markus, > > Thank you for sharing this. I believed the Wikipedia article that > Maximal Lotteries did not violate the participation criterion, but this > simple proof does demonstrate that all Condorcet methods, even > non-deterministic ones, will violate the participation criterion. Wikipedia articles are sometimes wrong and often incomplete. I have some knowledge of the reasons - ask me privately if you want to know :-) -km