The eternal Kevin Bryan writes to me:

Consider an NFL team down 15 who scores very late in the game, as happened twice this weekend. Everybody kicks the extra point in that situation instead of going for two, and is then down 8.  But there is no conceivable “value of information” model that can account for this – you are just delaying the resolution of uncertainty (since you will go for two after the next touchdown).  Strange indeed.

Let me restate his puzzle.  If you are in a contest and success requires costly effort, you want to know the return on effort in order to make the most informed decision.  In the situation he describes if you go for the 2-pointer after the first touchdown you will learn something about the return on future effort.  If you make the 2 points you will know that another touchdown could win the game.  If you fail you will know that you are better off saving your effort (avoiding the risk of injury, getting backups some playing time, etc.)

If instead you kick the extra point and wait until a second touchdown before going for two there is a chance that all that effort is wasted.  Avoiding that wasted effort is the value of information.

The upshot is that a decision-maker always wants information to be revealed as soon as possible.  But in football there is a separation between management and labor.  The coach calls the plays but the players spend the effort.  The coach internalizes some but not all of the players’ cost of effort. This can make the value of information negative.

Suppose that both the coach and the players want maximum effort whenever the probability of winning is above some threshold, and no effort when its below.  Because the coach internalizes less of the cost of effort, his threshold is lower.  That is, if the probability of winning falls into the intermediate range below the players’ threshold and above the coach’s threshold, the coach still wants effort from them but the players give up.  Finally, suppose that after the first touchdown the probability of winning is above both thresholds.

Then the coach will optimally choose to delay the resolution of uncertainty.  Because going for two is either going to move the probability up or down.  Moving it up has no effect since the players are already giving maximum effort.  Moving it down runs the risk of it landing in that intermediate area where the players and coach have conflicting incentives.  Instead by taking the extra point the coach gets maximum effort for sure.

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