This is an interesting model that I learnt while taking an online course ‘Model Thinking’ by Scott E Page on Coursera.org. Many of the statistical/mathematical models are very academic in nature with very limited use in the day-to-day life. However I can see potential of this model to understand/analyze many of real life situations that I will explain further.
In short, Markov process is a mathematical model that attempts to analyze transition between two states that may initially seem random. What does that mean? For example lets say in an organization, employees can be in 2 states. Motivated/Hardworking OR Demotivated/Lazy. Lets say after the quarterly performance appraisals, 20% of the Hardworking employees become Lazy and 25% of Lazy employees become Hardworking. The question, this model attempts to answer is after a foreseeable future say after 5 years, is it possible to estimate total number of Hardworking Employees and Lazy employees and if there is a point of equilibrium for such a situation.
Few conditions should exist for Markov process to work –
- Finite set of states (e.g. Hardworking and Lazy)
- Fixed transition probabilities that remain fairly stable throughout the observation period (e.g. 20% and 25%)
- Possibility of freely getting from one state to another throughout the series of transition (e.g. Hardworking becoming Lazy and vice versa)
Now, let us try to solve the above problem. First we should capture the data in Markov Transition Matrix.
|H(t) – Hardworking at time (t) i.e. starting point||L(t) – Lazy at time (t) i.e. starting point|
|H(t+1) – Hardworking after (t+1) i.e. after the first quarterly performance appraisal||0.8 (i.e. since 20% of hardworking employees become Lazy, only 80% are left Hardworking)||0.25 (i.e. 25% of Lazy employees become hardworking)|
|L(t+1) – Lazy after (t+1) i.e. after the first quarterly performance appraisal||0.2 (i.e. 20% of Hardworking employees become Lazy)||0.75 (i.e. since 25% of Lazy employees become Hardworking, only 75% are left Lazy)|
The above Matrix is multiplied by the sample size.. In this example, let us assume we have 100 employees to start with and all are Hardworking. To get the number of Hardworking and Lazy employees after the first quarterly performance appraisal (i.e. t+1), we have to Multiply .8 x 100 + .25 x 0 to get number of Hardworking employees and multiply .2 x 100 + .75 x 0 to get number of Lazy employees. This process is repeated multiple times to find equilibrium.
To calculate the state of equilibrium, we use the following equation –
In simple language the equation is – the percentage (p) of employees ending up as Hardworking is equal to 80% of (p) plus 25% of rest of the employees (1-p). i.e. Percentage (p) of Employees staying Hardworking:
p = .8p + .25(1-p)
p = .8p + .25 – .25p
p = .25 + .55p
p – .55p = .25
.45p = .25
p = .25/.45
p = .55555
i.e. ~55% of employees will stay Hardworking in the equilibrium state and (1-p) i.e. ~45% will become Lazy in the equilibrium state. This means beyond this state every (t+1) iteration will result in equal number of employees moving between two states. Let us test this.
Number of Hardworking Employees turning into Lazy employees at (t+1) after the equilibrium = 20% of 55 = 11
Number of Lazy Employees turning into Hardworking employees at (t+1) after the equilibrium = 25% of 45 = 11 (rounded)
Some other scenarios where this model can be applied are –
- Customers changing phones ( iPhone and Windows)
- Health of people in a given region (Healthy and Sick)
- Political Preferences (Right and Left)
- Migration of people from villages to cities and vice versa
- State Countries to be in the states of Free, Partially Free and Not Free/Dictatorial Rule. It is very intriguing to see the model predication and the actual data is so close (per the table below). Also disappointing to see that there will not be a free world in the near future.