### More post-season tournament design issues: MLS tiebreakers

Michael Giberson

For DC United fans, the MLS season is over.  While some fans contemplate coaching and roster changes, a few of us are still scratching our heads about the MLS tiebreaker rules and the complications presented by the final weekend of play (which had five teams angling for two remaining post-season positions. See here for an attempt to list them all, and an updated list.  See here for commentary.  Here a fan calls upon MLS to “stop the madness.”)

The combination of outcomes over the weekend put one of the five teams (New England, which improbably won its game) clearly over the others and one (Dallas, which lost its game) out of the running.  Three (Colorado, DC United, and Real Salt Lake) were tied for the final position and the tie breaker favored Real Salt Lake, which advanced to post-season play.

However, had New England lost, the situation gets interesting.  In this case, the tie breaking rules among Colorado, DC, and RSL would send Colorado and RSL into the playoffs.  However, if Dallas would have tied rather than lost its match, the tie breaking rules among Colorado, Dallas, DC, and RSL would have sent DC and RSL into the playoffs.

This interactive effect seems (at least to me) to violate an intuition about how these sorts of things should work.  Either DC was a better team than Colorado over the season or it was not, and whether Dallas won or lost against some another team in their final match should have little bearing on whether or not DC was better than Colorado during the season.

The intuition I’m talking about has been formalized in economic theory as the “independence of irrelevant alternatives” (IIA) principle. Formally:

If A is preferred to B out of the choice set {A,B}, then introducing a third alternative X, thus expanding the choice set to {A,B,X}, must not make B preferable to A.*

In this case: If Colorado is preferred to DC out of the set {Colorado, DC, RSL}, then introducing a fourth alternative Dallas, thus expanding the set to {Colorado, Dallas, DC, RSL}, should not make DC preferable to Colorado.  But the rules would have worked in just this way, had New England lost its final match.

Can this problem be fixed?  Why not the way that the professionals in Europe do it: first recourse in the event of a tie is to goal differential over the full season, then to total goals scored.

Sadly, such a rule would not have helped DC this season the way our defense gave up goals.  Is it too late to get Ryan Nelsen back at central defense?

*TECHNICAL NOTE: The formulation is stated in the simpler individual choice form, but the MLS tiebreaking rules may be seen more as a social choice mechanism.  Perhaps some form of Arrow’s impossibility theorem arises, meaning I’m unlikely to see a fully satisfying tiebreaking rule.  However, it does seem that goal differential avoids violating IIA.