One way players might play a game is by learning over time till they reach a best response to strategies they have observed in the past.  If learning converges, then a natural  hypothesis due to Fudenberg and Levine , is that it settles on a self-confirming equilibrium:

Self-Confirming Equilibrium (SCE) is a relaxation of Nash equilibrium: Each player chooses a best response to his beliefs and his beliefs are “correct” on the path of play.  But different players may have different beliefs over strategies off the path of play and may believe that players’ actions are correlated.  Nash equilibrium (NE) requires that players’ beliefs are also correct off the path of play, that all players have the same beliefs over off the path play and that players’ strategies are independent.  As the definition of Nash equilibrium puts extra constraints on beliefs, the set of Nash equilibria of a game cannot be larger than the set of self-confirming equilibria.

There is no reason why learning based on past play should tell us anything about off path play.  So SCE is a more natural prediction for the outcome of learning than NE.  Finally, we come to college football!

The University of Oregon football team has been pursuing an innovative “off the path” strategy:

“Oregon plays so fast that it is not uncommon for it to snap the ball 7 seconds into the 40-second play clock, long before defenses are accustomed to being set. That is so quick that opponents have no ability to substitute between plays, and fans at home do not have time to run to the fridge.”

Opposing teams on defense are just not used to playing  against this strategy and have not developed a best-response.   So far they have come up with an import from soccer, the old fake an injury strategy.  This has yielded great moments like the YouTube video above.

I am trying to relate this football scenario to SCE.  SCE does not incorporate experimentation which is what the Oregon Ducks are trying so this is immediately inconsistent with SCE.  But set that aside – even without experimentation, is the status quo of slower snaps and best responses to them an SCE?  I think it is and that it is even consistent with NE.

Even in a SCE of the two player sequential move game of football, the offense has to hypothesize what the defense would do if the offense plays fast.  Given their conjecture about the defense’s play if the offense plays fast, it is better for the offense to play slow rather than play fast.  Their conjecture about the defense’s play to fast snaps does not have to be at a best response for the defense as this node is unreached.  And the defense plays a best response to what they observe – slow play by the offense. So both players are at a best response and the offense’s conjecture about the defense play off the path of play can be taken to be “correct” as neither SCE nor NE put restrictions on the defense being at a best response off the path of play.

In other words, in two player games, a SCE is automatically a NE.  From diagonalizing Fudenberg and Levine, it seems this that this is true if you rule out correlated strategies (but I am administering an exam as I write this so I cannot concentrate!). If I am right, the football example is consistent with SCE and hence NE. (In three (or more) player games, there can be a substantive difference between SCE and NE as different players can have different conjectures on off path play in SCE but not NE and this can turn out to be important.)

But the football experience is not necessarily a Subgame Perfect Equilibrium. This adds the requirement of sequential rationality to Nash equilibrium:  Each player’s strategy at all decision nodes, even those off the path of play, has to be a best response to his beliefs and beliefs have to be correct etc.   So, it may be that football teams on offense have been assuming there is some devastating loss to playing fast.  First, it is simply hard to play fast and perhaps they thought it was easy to defend fast snaps.  But since this was never really tested, no-one really knew it for a fact.

Now the Oregon Ducks are experimenting and their opponents are trying to find a best response.  So far they have come up with faking injuries.  Eventually they will find a best response.  Then and only then will the teams learn whether it is better for the offense to have fast snaps or slow snaps.  And then they will play subgame perfect equilibrium: the offense may switch back to slow snaps if the best response to fast snaps if sufficiently devastating.

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