There is always an element of luck in everything we do. But in the field of sport sometimes the role of luck might seem a bit too unfair, especially when you are giving it your all. The recent T20 cricket world cup showed an outsized influence of the ‘toss’ – the decision to bat or bowl first, on the outcome of the game. This article talks about a simple yet ingenious suggestion that can neutralise the role of chance of the toss.
“during the recent men’s T20 World Cup in the UAE and Oman, the toss became increasingly problematic and the advantage of winning it and fielding first, when a very heavy dew fell at dusk, became more and more obvious – especially in Dubai where 12 out of 13 games were won by the chasing team…”
“Harris Aziz…Born in Pakistan, he did a Bsc at Lahore University of management sciences, an MSc in maths at Oxford, a PhD at Warwick and is now an associate professor at the University of New South Wales. There he specialises in computational social choice and algorithmic game theory.”
His background coupled with the increasingly data intensive world we live in and the resulting analytics seem to help his clarity of thought on the problem at hand:
“One of the overarching issues that I work on is that many decisions are being made by computers or computer algorithms and one of the overarching themes is how these decisions are made in a fair manner,” he says. “And that was the angle I took when looking at this issue: fairness.”
For Aziz, the toss is not inherently unfair over a long sequence of games as both teams equally have the chance to win it, but it is if you consider it on a match-by match basis. For example, in a one-off game such as the ICC World Test Championship final, the last match of an Ashes series or a T20 World Cup final. The problem is not only advantage to the toss-winning side due to things like the state of the pitch and weather conditions, but also perceived advantage, which can make the losing side feel hard done by and also denigrate the achievement of the winning team.
Aziz’s solution is to move the toss from a two-step formula – toss the coin, winning captain chooses – to a three-step formula: toss, propose, choose. Here, the coin is tossed as normal, then the Unlucky (toss-losing) captain chooses a run handicap to pin to the more favourable option in order to equalise the two choices, before handing back to the Lucky (toss-winning) captain to make the final decision of whether to bat or field. The beauty of the division is that the Unlucky captain will not overestimate or underestimate the runs needed to equal up the decision, as it is the Lucky captain who gets to make the final choice.
“The reason I came up with this method is that it also has really solid mathematical foundations,” says Aziz. “It is inspired by a rule called divide and choose which has been used throughout history to make a very different kind of allocation decision about dividing a divisible resource. Steven Brams and Alan Taylor have written an excellent book on it called Fair Division. Here we don’t have a divisible resource but we do have runs, which are almost a divisible resource, and which we can use to balance things out.”
To non-mathematicians, it is a variation on the age-old solution to squabbling children: let one child cut the cake, and the others choose the slices.
 

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