In a world of imagination, let’s apply consumer and producer sides of economics in cricket. In economics there is a concept known as production function. It shows the relation between the input and output. In the case of cricket, inputs include players of both teams, and the number of balls. Output refers to the runs they score.

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, the 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 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 case of players, each team has 11 players, the number of players are 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 case of number of balls, each team has the same number of balls, but the number of runs scored depends on the players’ productivity. Given the productivity of other team in terms of bowling, 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 depends 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 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 that they have scored or by winning the overall match?

For winning, we need to consider the runs of second team as well. Therefore, team 1 will get disutility form 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 run scored by the same team. Also, there are other factors, let’s discuss them one by one.

Let us 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 first batting team depends on following factors–

· Runs scored till ball i.

· Remaining balls.

· 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 second situation, which shows higher the scores, higher will be the utility, given the number of balls.

Utility also depends on numbers of the remaining balls. 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 number of remaining balls decreases, utility falls. There is a direct relation between utility and number of remaining balls.

Utility also depends on 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 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 lost at 70th ball, 2nd lost at 150th ball, 3rd lost at 180th ball, etc. As soon as your team lost 1st wicket your utility decreases by 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 number of wicket lost and it goes from 1 to 10, you can lose 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, does the runs of team y affects your utility? The obvious answer is yes. If team y scores more runs, 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 utility of team x and runs scored by team y. Remember that match is not finished yet, that means, team x utility is now depends on these factors –

· Scores by team x.

· score by team y till ball i.

· number of remaining balls for team y

· number of wicket lost by team y.

Let’s take this one by one-

Utility of team x depends on

· scores by team x: higher the score higher the utility.

· score by team y till balli: higher the score of team y, lower the utility of team x.

· number of remaining balls for team y: more the balls left for team y, lower the utility for team x.

· Number of wickets lost by team y: again, similar to earlier argument, if 1st player get out, team x will get more utility as compared to getting out of 8th player. 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 balli – number of remaining balls for team y + (1/logxi).

Therefore, we can write 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 balli – number of remaining balls for team y + (1/logxi)

Case – 3: __When 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

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 balli + 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 runs or if they (team x) have more number of remaining balls. Team y will get more utility if team y is able to take wickets of team x. Again, 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 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.

· Number of remaining balls: more the remaining balls, more the utility.

· Number of wicket 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 balli + 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 balli + 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 balli + number of remaining balls for team y – (1/logxi) – scores by team x

Case – 3: __When matchis over:__

UY(X,Y)= Runs by team x – Runs by team y, If runs by team x > runs by team y.

OR

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 lover.__