To box kick or not to box kick

On 3/11/2011 Wales played Australia in a rugby test at the Millennium Stadium in Cardiff. I watched the game at home on TV and it was thoroughly enjoyable. My only problem was with one member of the commentary team who to put it diplomatically 'isn't the most knowledgeable'.

At one point he exclaimed: 'teams should always box kick to clear their lines'. Before I use game theory to show that the commentator is here completely wrong let me give some background to rugby. Rugby is a 15 player game where each position on the field has a specific role to play. Whilst the skill sets of each player is in no way as specific as American football (and this is where my comparison of rugby to football starts and ends) it's generally accepted that there are two players on the team who do the kicking: the scrum half and the outside half.

There are various reasons for a team to need to kick the ball, one of the main ones and the one alluded to by the commentator is to relieve pressure when close to your own score line. In this case one of the two mentioned players will (usually) kick whilst some defenders try to block the kick.

Here's a video just on the subject of kicks in rugby :)

Let us simplify the above situation by assuming that a team close to their own score line with the ball (which we'll refer to from now on as the "kicking team") has two strategies available to it: "Box kick" (where the scrum half kicks the ball):

or "Clearance Kick" (where the scrum half passes to the outside half who kicks the ball):

As soon as the scrum half touches the ball the defenders can rush past the offside line to try and block the kick. We assume that the defenders will either anticipate a box kick (rushing the scrum half) or anticipate a clearance kick (rushing the fly half)

I haven't really been able to find any data as to the success of these things so I'll go with my own experience (I'm a scrum half but have played a bit of outside half as well). My success rate at box kicks when they're not anticipated is probably 80% however when teams anticipate a box kick I reckon my success rate probably drops to 20%. To clarify by success rate, I simply mean rate at which the kick successfully relieves pressure.

When an outside half does the kicking I reckon the success rate is probably about 95% when not anticipated and perhaps 70% when it is anticipated.

Assuming this is all a zero sum game this gives the following bi-matrix:

$$\begin{pmatrix}(20,-20)&(80,-80)\\(95,-95)&(70,-70)\end{pmatrix}$$

If the kicking team indeed follows that commentator's advice and always box kicks then the expected utility to the other team as a function of $0\leq q\leq1$ (the frequency with which they anticipate a box kick) is given by:

$$-20q-80(1-q)=60q-80$$

the plot of which is given below:

It's immediate to see that the best response is to choose $q=1$, i.e. to always anticipate a box kick. This gives the kicking team a success rate of clearing their lines of 20%. Can this be improved? 

Well here's a plot of the utility to the kicking team as a function of $p$ and $q$: 

$$u_1(p,q)=-85  p q + 10 p + 25 q + 70$$

So a success rate of 20% is in fact the worst possible utility for the kicking team. As soon as the kicking team deviates from a strategy of always box kicking then their utility will increase (assuming a best response by the defensive team or not).

Using some simple sage code (available here) to compute the Nash equilibria we can equate the equilibrium strategies to be:

$p={5\over17}$ and $q={2\over17}$

At this strategy pairing (where we only box kick 29% of the time) we see that the utility to the kicking team is: 73%. These numbers are to be taken with an obvious amount of caution (remember my original assumptions where based on my personal success rates) but the main message still holds: you should mix your strategies.

If we modify the game slightly taking $r\in[0,100]$ to be the utility gained for an anticipated box kick:

$$\begin{pmatrix}(r,-r)&(80,-80)\\(95,-95)&(70,-70)\end{pmatrix}$$

The following graph plots the equilibrium values for $p$ and $q$ for varying values of $r$:

We see straight away (remember that 80% is the success rate of a box kick against a defense that anticipates a clearance kick) that as long as box-kicking is sometimes the worse option then both teams should mix up their strategies.

These ideas apply to more or less all sports, in American Football you should have a healthy mix of running and passing plays (even if you're a better running team), in tennis you should serve both wide and narrow (even if your wide serve is your stronger serve), in soccer you should take penalty kicks left and right etc...