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Economics of Cricket

Economics of cricket

Posted on August 28, 2019 by Sahil Arora  | Academics | Blog

In an imaginary world, let’s apply consumer and producer sides of economics to cricket. These basic concepts are taught in any BA Economics course. In economics, there is a concept known as production function. It shows the relation between input and output. In the case of cricket, inputs include players of both teams and the number of ball and output refers to the runs they score.

Economics (Hons) courses teach concepts that are practically applicable to multiple disciplines, in this case, we are applying it to sports specifically cricket. To develop an analogy around consumers’ theory in economics, let us assume that there are two sets of consumers – the two cricket teams. These two teams are also producers. Thus, in our model, consumers are also producers.

From the consumer’s perspective, let’s assume that there are only two sets of consumers, who serve as producers as well. They consume by producing their own product, but their output depends on each other’s productivity.

Every team wants to maximize their output and their utility. Teams can produce more output i.e. score more runs either by increasing the number of inputs or by using inputs more efficiently. This means that either the inputs to the production function can be changed (increased), or the production function itself could change to allow for greater output from the same input. An example would be if we compare MS Dhoni with Sarfaraz Ahmed, we all can say that there is a higher possibility for team India to produce more output (score more runs) because India has a more efficient player as compared to Pakistan.

Here we are using three inputs – players of team X, players of team Y and number of balls. In the case of players, each team has 11 players, the number of players is fixed but their productivity and efficiency matters. Here we can take productivity and efficiency as a player’s quality in terms of bowling, batting, fielding, stamina, etc. In the case of the number of balls, each team has the same, but the number of runs scored depends on the players’ productivity. Given the productivity of the other team in terms of bowling, the efficiency of the player of the opposite team will be measured by runs. This means that runs scored by one team on a particular ball depend on the productivity of both – the team which is batting, and the one that is bowling. The objective of the team (which is bowling) is not to make the other team’s output to increase.

Hence production functions for both teams are given by- Fx(X, Y, number of balls) and Fy(X, Y, number of balls). And the output is measured by runs scored by both teams.
Therefore, we can write the production function for both teams as-
Scores by team x= Fx(X, Y, number of balls)
Scores by team y= Fy(X, Y, number of balls)

Each team wants to maximize its utility. But the question is, are they only getting utility from the runs scored or by winning the overall match?

For winning, we need to consider the runs of the second team as well. Therefore, team 1 will get disutility from the runs scored by the second team. More the runs scored by one team, lower will be the utility for another. It means that there is an inverse relation between utility by team 1 and runs scored by team 2, and there is a direct relation between utility and runs scored by the same team. Also, there are other factors, let’s discuss them one by one.

Let's assume that team 1 is batting first.
We can write utility function for team X if the match is over as-
Ux(X,Y)= Fx(X,Y, number of balls) - Fy(X,Y, number of balls)
OR
Ux(X,Y) = scores by team x - scores by team y.

We can only use this utility function when both the teams have played the match and the match is finished, as we need scores of both teams. Now take a case where you support team x or you are one of the players of team x. If you bat first, you will get a utility if you or your team will score runs. That means, during the match also, you will be getting utility, therefore, we need to use different utility function.

In this case utility function for the team that bats first depends on the following factors–

  • Runs scored till ball i.
  • Remaining balls.
  • The number of wickets lost till ball i.

Utility depends on how much runs team has scored till ball i. Let’s take two situations-

  1. Team has scored 100 runs in 70 balls.
  2. Team has scored 101 runs in 70 balls.

We as team members prefer the second situation, which shows higher the scores, higher will be the utility, given the number of balls.

The utility also depends on the number of balls remaining. Let’s take two situations again –

  1. Team has scored 240 runs in 179 balls.
  2. Team has scored 240 runs in 200 balls.

Given two situations, you always prefer situation 1. The reason being we are hopeful of scoring more (at least one) runs in 11 (200-179) balls. That means situation 1 gives you more utility as compared to situation 2. Therefore, as the number of remaining balls decreases, utility falls. There is a direct relation between utility and number of remaining balls.

The utility also depends on the number of wickets lost till ball i. There is an inverse relation between utility and number of wickets lost till ball i. If there is a loss of 1 wicket at ball i (say i is 50), then the utility of batting team will fall due to this loss, hence there is an inverse relation. But we need to take a decreasing function because of the following reason.

Take a case where 1st wicket is lost at the 70th ball, 2nd is lost at 150th ball, 3rd is lost at 180th ball, etc. As soon as your team lost the 1st wicket, your utility decreases by a larger amount as compared to 8th wicket lost by the team. The reason being, normally players are arranged in descending order of their productivity of batting. If you lose the most productive player your utility will fall more than as compared to losing a less productive player.

Therefore, utility function can be written as –

Ux(X,Y)= runs scored by team x till ball i + number of remaining balls – (1/logxj) where “i” goes from o to 300, given it is a 50 overs match and “j” is the number of wickets lost and it goes from 1 to 10, you can lose a maximum of 10 wickets.

Now think of a situation where batting team’s (team x) batting is over and balling team (team y) starts batting. In this situation, we need to think what could be the utility function for team x? Now think like you belong to team x, do the runs of team y affect your utility? The obvious answer is yes. If team y scores more runs, the utility for team x will fall. In the end, the team with the higher runs will win. Therefore, we can say that there is an inverse relation between the utility of team x and runs scored by team y. Remember that match is not finished yet, that means, team x's utility is now dependent on these factors –

  • Scores by team x.
  • score by team y till ball i.
  • number of remaining balls for team y
  • the number of wickets lost by team y.

Let’s take this one by one-

The utility of team x depends on

  • scores by team x: higher the score higher the utility.
  • score by team y till ball i: higher the score of team y, lower the utility of team x.
  • The number of remaining balls for team y: more the balls left for team y, lower the utility for team x.
  • The number of wickets lost by team y: again, similar to an earlier argument, if 1st player gets out, team x will get more utility as compared to if the 8th player gets out. Therefore, we have used a decreasing function (1/logxj).

Therefore, when team y is batting, we can get the utility for team x from the following utility function – Ux(X, Y)= team x score - score by team y till ball i - number of remaining balls for team y + (1/logxi).

Therefore, we can write the utility function for team x (batting first) in three different cases.

Case- 1: When team x starts batting:
Ux(X,Y)= runs scored by team x till ball i + number of remaining balls – (1/logxj)

Case- 2: When team y starts batting:
Ux(X, Y)=team x score - score by team y till ball i - number of remaining balls for team y + (1/logxi)

Case – 3: When the match is over:
Ux(X,Y) = scores by team x - scores by team y.

Now our task is to see the utility for team y.
Utility after match ends will be given by –
UY(X, Y)= Runs by team x – Runs by team y, If runs by team x > runs by team y.
OR
The number of wickets left, if runs by team x < runs by team y.
But what is the utility for team y when team x is batting? In this situation, the utility for team y would be given by –
Uy(X,Y)= - [score by team x till ball i + number of remaining balls - (1/logxi)], Where j refers to number of wicket lost.

That is exactly negative of team x utility. The reason being, team y is getting lower utility if team x scores more run or if they (team x) have more number of remaining balls. Team y will get more utility if team y can take wickets of team x. Again, the number of wickets lost is a decreasing function with the same logic which is given above.

Now let’s take a case where team x batting is over and team y starts batting. In this case, the utility for team y would be derived from –

  • Scores by team x: higher the scores of team x, lower will be the utility for team y.
  • Scores by team y: higher the runs, higher will be the utility.
  • The number of remaining balls: more the remaining balls, more the utility.
  • The number of wickets lost: less the number of wicket lost, more will be the utility.

Therefore, we can write team y utility function as – Uy(X, Y)= score by team y till ball i + number of remaining balls for team y - (1/logxi) - scores by team x. Hence, for all three cases, team y also have different following utility functions-

Case -1: When team x starts batting:
Uy(X,Y)= - [score by team x till ball i + number of remaining balls for team X - (1/logxi)], Where j refers to number of wicket lost.

Case- 2: When team y starts batting:
Uy(X, Y)= score by team y till ball i + number of remaining balls for team y - (1/logxi) - scores by team x

Case – 3: When the match is over:
UY(X, Y)= Runs by team x – Runs by team y, If runs by team x > runs by team y. OR The number of wickets left, if runs by team x < runs by team y.

Use the concepts of economics, maximize your utility by being a cricket player.

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