Following up on Sandeep’s post about Alex Rodriguez’s alleged pitch-tipping, a game theorist is naturally led to ask a few questions.  How is a tipping ring sustainable?  If it is sustainable what is the efficient pitch-tipping scheme?  Finally, how would we spot it in the data?

A cooperative pitch-tipping arrangement would be difficult, but not impossible to support.  Just as with price-fixing and bid-rigging schemes, maintaining the collusive arrangement benefits the insiders as a group, but each individual member has an incentive to cheat on the deal if he can get away with it.  Ensuring compliance involves the implicit understanding that cheaters will be excluded from future benefits, or maybe even punished.

What would make this hard to enforce in the case of pitch-tipping is that it would be hard to detail exactly what compliance means and therefore hard to reach any firm understanding of what behavior would and would not be tolerated.  For example, if the game is not close but its still early innings is the deal on?  What if the star pitcher is on the mound, maybe a friend of one of the colluders?  Sometimes the shortstop might not be able to see the sign or he is not privy to an on-the-fly change in signs between the pitcher and catcher.  If he tips the wrong pitch by mistake, will he be punished?  If not, then he has an excuse to cheat on the deal.

These issues limit the scope of any pitch-tipping ring.  There must be clearly identifiable circumstances under which the deal is on.  Provided the colluders can reach an understanding of these bright-lines, they can enforce compliance.

There is not much to gain from pitch-tipping when the deal becomes active only in highly imbalanced games.  But the most efficient ring will make the most of it.  A deal between just two shortstops will benefit each only when their two teams meet.  A rare occurrence.  Each member of the group benefits if a shortstop from a heretofore unrepresented team is allowed in on the deal.  Increasing the value of the deal has the added benefit of making exclusion more costly and so helps enforcement.  So the most efficient ring will include the shortstop from every team. Another advantage of including a player from every team in the league is that it would make it harder to detect the pitch-tipping scheme in the data.  If instead some team was excluded then it would be possible to see in the data that A-Rod hit worse on average against that team, controlling for other factors.

But it should stop there.  There is no benefit to having a second player, say the second-baseman, from the same team on the deal.  While the second-baseman would benefit, he would add nothing new to the rest of the ring and would be one more potential cheater that would have to be monitored.

How could a ring be detected in data?  One test I already mentioned, but a sophisticated ring would avoid detection in that way.  Another test would be to compare the performance of the shortstops with the left-fielders.  But there is one smoking gun of any collusive deal:  the punishments.  As discussed above, when monitoring is not perfect, there will be times when it appears that a ring member has cheated and he will have to be punished.  In the data this will show up as a downgrade in that player’s performance in those scenarios where the ring is active.  And to distinguish this from a run-of-the-mill slump, one would look for downgrades in performance in the pitch-tipping scenarios (big lead by some team) which are not accompanied by downgrades in performance in the rest of the game (when it is close.)

The data are available.