Game Theory and Volleyball

I just saw this write up about John Nash at Five Thirty Eight, and I'm still trying to formulate my thoughts on game theory as it applies to volleyball. I saw A Beautiful Mind, but I never really took that much time looking into game theory. In some small way, it is part of my thought process as I think about offensive and defensive strategies. I have looked a little at set distribution in the past here and here. I like my setters to spread the hitting opportunities around. I especially like this when it is clear that one particular player is going to get most of the points on offense.

In the Five Thirty Eight article there is the specific mention of pass/run decisions in football, and penalty kick direction in the other football. For volleyball there are a few decisions that can be made on offense, and there are two different overarching conditions that govern what decisions can be made. There is the condition of a perfect pass where all offensive options are possible, and the condition where the team is out of system. Even with those two conditions it isn't a simple binary one or the other. There is a continuum of perfect pass with all available hitters an option to a bad pass where there might only be one possible hitting option available.

In the perfect pass, in system situation, there can be 4 hitting options (more if there isn't a libero on our side of the net, or if the setter is on the front row). A first tempo hitting option to the middle should force the opposing middle to make a decision to block the middle, or to block one of the other three available hitters, but it should be impossible to attempt to block both the middle and another hitter. This is an ideal situation where the middle blocker must decide between 4 hitting options. With a front row setter and an opposite hitting from the back row, a wing blocker faces a similar decision whether to attempt to block the setter, or attempting to block the opposite. Here good setting distribution (if I'm getting this right, this is Nash Equilibrium) gives the offense a huge advantage.

Taking it down to the most simple out of system situation where the middle is not a hitting option, there are still three hitting options. The best situation for the offense is three hitters. With Nash Equilibrium neither team has an inherent advantage. There are three hitters and three blockers and the middle doesn't have the same decision of blocking the first tempo or the second tempo. The middle might even be able to see where the set is going and not have to make any decision at all. In this situation it is only a matter of which side executes the skills of the game better, the hitter or the blocker. In many of these situations there is often an obvious setting decision that will be made because of where the setter must be to set and the direction the setter is moving. At this point I don't think there is any possibility for equilibrium.  There are no decisions.

The take home message from this is to train our teams to a) extend the area of the court that the setter can set a first tempo set, and b) give the setters opportunities to set away from the obvious out of system hitters. Extending the area for first tempo hits keeps the defense in a position where the middle has to make a decision, and our team still has the four hitting options. Setting to the non obvious hitters keeps the game in a place where the defense still must make decisions. The offense is always going to be making the first decision. Extending first tempo offense and forcing the defense to make decisions when they aren't expecting to will keep the game in a place where there can be an equilibrium strategy, and keep the probability of success in our favor. Beyond that I have to spend more time looking into game theory as it applies to volleyball.