4th and 2: Game theory in Cricket

I was a casual football fan. The Sunday Night football game between Colts and Patriots was the tipping point. Now I am a Football fan 🙂

Awesome games can do that to you. But the game also started furious debate among many football followers in the US on Patriots coach Bill Billicheck’s call to go for it at 4th and 2.

Pundits have called it stupid and the worst decision ever.

Statisticians have called it the Correct decision and a brave one.

The jury is still out.

But that leads me to this situation in an ODI that I am thinking about. As with any hypothetical situation, I will try and outline my assumptions.

I will consider a particular outcome of the game as an realization of a random process. So, I want any of you reading to think of this as an maximization of the expectation, i.e., finding the choice that will maximize the percentage of favourable outcome (winning) if the same situation repeated itself infinite times.

I will assign some probabilities to events, which again is hypothetical, but confirms to some intuition and some observed results in some games.

Assume that Sachin, batting on, having scored a magical century is partnered by Ishant Sharma (or some other equally inept No.11). McGrath is slotted to bowl the 50th over. India needs 10 runs and one ball is remaining in the 49th over. This is the last wicket.

With the equation being 10 of 7, Sachin is facing. Let us assume that the probability that Sachin facing McGrath and getting 9 of 6 balls is much higher than Probability of Ishant facing McGrath and getting 6 of 6. (This assumption is valid somewhat. Sachin in form, even if he gets a dot-ball, has the class to hit a boundary or two. Ishant, has higher probabilities of playing dot-balls and getting out to McGrath trying to rotate strike. The probability of an ill-advised run, like the one Alan Donald and Lance Klusener took in the ’99 WC also increases when Ishant is facing McGrath in 6/6 situation).

Sachin is facing the last ball of the 49th. What does Ponting do?

  1. Bring in the field to prevent a single. Risk Sachin hitting a boundary but in the process give McGrath the chance to get Ishant out and win the game, with Ishant facing McGrath in a 6/6 situation
  2. Keep a safe field, with fielders posted on the boundary. Allow Sachin to take the single, and give him the chance to dictate the game with Sachin facing McGrath in 9/6 situation.

In my mind, this is an interesting question. What does Sachin do in that situation. Score as much as possible from the last ball of the 49th, or keep strike for the 50th over?

There are lot more such statistical situations in Cricket, which coaches can use to their advantage. The best example is when to take the Batting power-play. Right now, Australia wants to take it in the 43rd or the 42nd over, sometime when they would anyway start hitting. India tends to take in at the 35th when the ball is changed (to take advantage of a harder ball). But even when the option of taking the power play anytime is available, most teams are playing with a fixed strategy. This is bad, for it does not allow statisticians to gather information regarding power-play strategy.

The current strategy is playing it safe. The pay-off of keeping the power-play to the final overs is really high when you have set-batsmen going into the 40th over. It fails badly when the set-batsmen (mostly because he is really tired) gets out late in the 30th over, or for some reason the two new players are there in the crease when the power-play is taken. Captains have to have a flexible strategy and identify batsmen with whom they would ideally want to play a power-play and if they are in, and looking good, take the power-play and maximize the returns. Or if they are chasing, take it sooner than later so that hitters in the middle order like Pietersen, Yuvraj etc can make the chase easier for the batsmen to follow rather than allowing them to play and get out with the run-rates getting steeper and leaving your power-play with the lower order.

Fielding captains too have choices. In the given fixed strategy of the batting side, they are better off getting their best bowlers an over or two before they know the time when the opposition is going to take the power-play.  An wicket or two before the power-play is priceless. Instead, now it is predictable that the opposition takes the power-play and then the fielding captain brings in his best bowler. The fielding team is also trying to minimize damage.

In game theory parlance, both teams are playing their security strategy, i.e., they are playing that strategy for which they have the “best” worst case result. Instead, they will be better off playing an  aggressive strategy, in which they maximize their “best” best case result.

A lot more can be done with the power-plays, but the captains are taking the safest choices. The game still remains static, except that we have new words and new signs from the umpires.  We need a truly revolutionary caption who has the guts to throw caution to the wind. He may lose a game or two, but in the long run he is going to win much more than he loses. Then the game becomes truly dramatic.

The one thing that can make ODI decision making even more dramatic is if the fielding side is given equal decision making capabilities. The bowling power-play is useless. No Bowler would like a fielding restriction. More interesting would be to get away with bowling power-play and allow two bowlers to bowl 12 overs each. That would allow teams to pack in batsmen in the team and the fifth bowler 10 overs get reduced to 6 and the fielding captain has the option of unleashing his best bowlers just before power-plays to prize out the in-form batsmen.

Definitely, more dynamic than the changes now that still has not made the ODI more dynamic.

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  1. #1 by andyuncut on November 19, 2009 - 7:30 am

    I am all for removing the cap of 10 overs for any bowler. Let at least 2 bowlers bowl 12 or even 15 overs! It will also somewhat offset the advantage batsmen enjoy in various ways in one-dayers.

    Regarding what to do in 10/7 scenario, my take is – If Sachin is in form, take a single and back yourself to hit 9 in 6. If you are also patchy and the team has been losing matches, try to hit a 4 and make it 6 out of 6.

    • #2 by Kaushik on November 19, 2009 - 11:48 am

      You still are talking about the safest options. Especially, when the team has been patchy and losing, it makes more sense for Sachin to take the single and keep strike, because Sachin on strike has more probability of winning, although a 4 of 48.6 will calm some nerves in the dressing room.

      My point is no captain in the world, including Ponting will risk brining the field in to deny a single to Sachin at the expense of opening up the boundary. Everybody is programmed to minimize the worst case scenario. That is why, we hardly see Indian captains declaring aggressively in test matches (even Sangakarra delayed it yesterday…250 was lead enough). The backlash for losing after an declaration which leaves you some chance of winning but also the other team a small window of comeback is much more than drawing the game after reducing your chances of a victory but shutting out the opposing team. Most test matches in India could have a result if the captain thinks a 60-30-10 Win-draw-lose % at the middle of the 4th day is much better than a 45-55-0 W-D-L %. Most captains want to take the second option. That is why Australia won so much. They had a good team, no doubt, but their team always played the percentages to winning the game, no matter what time. So you see them scoring quickly, setting wicket taking fields even when the other team was 300-3 and declaring if the payoff was such that losing was less -% was less than 30% or so.

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