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.

Meaningless!

This story is inspired by the mindless craze for Salsa and Ballroom dancing here in UW among the desi boys. The conversations are real, because they have happened, sometimes many times over, mainly after consuming couple of pitchers of Spotted Cow.  Infact these conversations have been repeated so many times that we have nothing much to add if it comes up. The story is imagination. Rajesh’s dialogue was inspired by this blog post by Arvind Iyer.

We were driving back to Stony Brook after a wonderful Diwali at my brother’s house, everyone except the driver, still reminiscing about the delightful Diwali spread that Manni managed to cook. Slowly, the conversation drifted to the cool dude that my brother keeps talking about, the dude who took Salsa classes and got himself the American babe. I was tired of their story, but for the other “makkal”, it was awe-inspiring. Having retold the story while chewing Paan after Diwali lunch, the “makkal” hope levels had like multiplied thousand times. If he could do it, we do stand a chance. iPhones were out and university listings for Salsa classes and Ballroom dancing, and plans were being hatched to write the most romantic of romantic stories.

Rajesh, navigating the interstates, instructing me on exits and speed-limits, turned back, and in the most philosophical of tones, stated the obvious. Everyone knew it, the iPhones and the planning was just the idea of some fun. But still, he rattled on, “So this is our plan? Our lives are so pathetic, that we plan laboriously about how we bump into a girl, but we are so incompetent at doing it that we don’t even plan for the eventuality that one of our coincidence trip actually works out”. The laughing stopped for a moment. Truth does that sometimes. And then, sitting right behind me, GK, let rip one of his stories.

Now, we have nothing against GK’s stories. Some are fabulous, some are rotten. He thinks some stories have a deeper meaning, but most of us at the end of it go “Ok, so what next”, and some he thinks are too funny, and he gets ready to let out a huge roar of laughter, and most of us don’t even understand it. We let him go on with is stories, as sometimes, it is the fodder with which we get to attack him back.

So GK continued, this reminds me of my cousin, Don’t you remember him, the guy who won $1000 at Roulette in Vegas. Anyway, this was long before he came to the US. He was just like most of us here, born and raised up in Madras, lived there all his life, took the long College bus ride to college, sat through boring lecture and tolerated the prison rules of college, yada, yada, yada. But, then, instead of getting a job at TIDEL park or even Electronic City, he managed to get one in Bandra-Kurla Complex in Bombay. That put all of his family members in a quandry. Bangalore is OK, but Bombay, he does not even know Hindi. Somehow, the pathetic job scene, you know, the dot-com burst and all, forced my dear cousin to go to Bombay.

Fortunately, there was one other Iyer paiyyan who had landed the same job, and those two together decided that they will take an apartment in Vashi. Now, Vashi is in New Bombay, and to get to BKC, arey, the Bandra Kurla Complex, one has to take the Harbour line train from Vashi to Kurla and then a bus to the workplace. You all know, how train services are in Bombay. So anyway, my cousin decides to go to Bombay, and peri-amma, in a fit of anxeity, hooks onto the Maami network, and finds the phone number of someone who knows someone who is related in someway to her. She gives my cousin the contact’s phone number and tells him that some relative who knows someone has told Sowmya in Mumbai about my cousin’s arrival, and that he should call her and not hesitate to take any help. My cousin diligently copies the number into his contact book and starts preparing for the journey.

Much like most of us, he also had the plans for a coincidental bumping into with his dream lady. His plan was to take a book to read wherever he went, and look for a girl reading the same book or the same author. Not that he liked reading. He would skim through some chapters and through some history of the author and just take the book along with him. And, he frequented the roadside book-kadai’s frequently, gathering data about popular books, so that he could be seen with one of those. You know, increase the probability.

So, anyway, he reaches Bombay, settles into his apartment, and on the first Monday to work, carefully selects a Sidney Sheldon novel, carelessly selects a shirt and pant that hardly match, and leaves to Vashi station. He acted immersed in the book, acted that he had been in Bombay all his life, acted that getting into a train with 1000 other people were second nature to him. But, he slowly, unwittingly walked and stood beside the location where the ladies coach of the 12-car train would arrive. He stole nervous glances from his, supposedly voracious reading, kept his eye hooked to a single word on the page, ears tuned to the lady announcing the trains, hand slipped into his pocket, holding the wallet tight. Amidst all this activity of looking cool, hiding fear, a girl tapped on his back. He looked back, saw her face, looked down at her hands, and saw the same Sidney Sheldon in her hand, let out a wry smile, managed to utter Same book and continued to stare at his shoes. The girl simply said, you are standing in the wrong place. This is where the ladies compartment will come. My cousin was totally embarrassed now, and continuing to stare at his shoes, slowly walked away to where the concentration of men seemed higher. Soon the train came, and he was off to his first day at work.

Slowly, his life fell into a pattern, and he would reach Vashi everyday, hoping to get a glimpse of his Book-mate. All the time, the Romantic story generator inside his head was spinning yarns and yarns of story. In a crowded city, everyone is not Madhavan to calculate probabilities.

By now, it had been a month since my cousin had shifted to Mumbai. His dream story had gotten complicated, and as he said to his Iyer paiyyan roomie, it was just a matter of another accidental meeting at the Vashi station. Amidst the excitement of Bombay, he had forgotten to call Sowmya, and as the news spread in the Maami network, all the someones were angry that the boy did not have the courtesy to even call her. To prove her point to my periamma, the someone Maami who was the source of the contact, instructed Sowmya to go meet my cousin. Sowmya reluctantly agreed.

One humid evening, as my cousin and the nice Iyer paiyyan were lamenting the lack of proper Tamil food in Vashi and debating over the yucky Madras Mess where they atleast serve some Sambhar or the cheap North Indian eatery down the road where they had to eat greasy paratha and paneer, the bell rang. My cousin, opened the door,  and was pleasantly startled, to see the Sidney Sheldon girl and some dude waiting at their door. Tongue-tied, he barely managed to ask about them, when Sowmya replied that she was so-and-so’s daughter who knew someone who knew someone who knew your mother. He invited her in. Somewhere, deep inside his head, a smaller version of himself, did a nice jig, and my cousin thanked God for life’s wonderful coincidences, and assumed that such a meeting was a sign, like how Madhuri had thought about it in DTPH. The excitement was short-lived, as Sowmya, introduced the dude waiting with her as her husband. What followed was standard small talk, false promises of keeping in touch, a sugary milky tea, that the Iyer paiyyan made, and the silent noise of shattering of million dreams.

Thus, GK finished the story, that took up 15 exits. As usual, we went “What? Nice story, but what was the point?”

Rajesh quipped, “Exactly, what I was saying”.

Thankfully, we arrived in a few minutes, returned the rental and were off to our bedrooms, to sleep and dream.