# Simulation Based Estimation — How You Estimate as if You were Playing Monopoly

# Introduction

The world can sometimes be like a board game.

Imagine if the economy is exactly like a board game, say Monoploy or Civilization. People do decisions according to a rule, and as an economist, your goal is to get inference about how certain design might influence the result. You as a statistician, or econometrician, want to estimate details like “how many real estate will one be willing to purchase if the jail is right next to it” or something like that. What is you best approach?

Well, traditional econometric approach might come up with some simple linear functional form like (Just for the sake of demonstration)

`num_of_real_estate = a + b1*distance_to_jail + b2*num_of_player + e`

But there is another type of estimation which, might not be the best approach in practice due to complexity, but it sure is something intuitive and fun to know.

Come on, it’s a board game! Why don’t we just let the computer play the game for several times and simulate the results, then adjust the “parameters” that was coded to match the result of real world?

This is commonly know as the **Simulation Based Model. **In this article, I want to introduce this concept as simple as possible.

# Oligopoly in Airline

Let’s start with a scenario of the “airline entrance”.

For each route(for example from Seattle to LA) there might be airlines that enter and some that don’t. The decision from the airline company might include how large the population of the endpoint is, or the distance of the route. There are, however, another factor that stopped a firm from entering the market — the number of airlines already in the route.

Number of airlines in a certain route (we call it a market) can affect the profit of a company. After all, the demand doesn’t change much, and an extra company divides the customers and hence decreases the profit.

If an ariline believes that entering is profitable, it will eventually enter. However if entering causes the average profit for each airline to decrease below 0, the best decision for the airline is to not enter.

Imaging yourself facing a decision problem when playing this board game.

Which airline will enter depends on some **luck**. Some unexpected factor of an airline might grant it with some extra profit, and hence survives in this battle of entering.

To estimate this, let us construct a model.

# Model for Estimation

What we have as data is a list of markets (each routes) and the airlines that provides service in the market.

We want to simulate, for each market and each airlines within it, whether it will enter or not. For each market, want to take into account

- The market’s effect
- The airline’s own effect
- The number of existing airlines
- Some other shock

All of these jointly decides the profit of each airline in each market. Only if one earns a positive profit will one enter.

Let us conclude the above ideas into the following graph

So basically we have to estimate, for every market, for every airline, its profit, and compare it with the real world data.

Is everything perfectly determined? No! We have to take into account the “luck” (or unluckiness) observed by the airline, but not for economist, which is colored in orange in the graph.

The common method of dealing with these “luckiness” is by doing an ** Monte-Carlo** simulation for each market.

# Monte-Carlo Simulation

The idea is pretty simple.

Since we don’t know what “luck” they encounter exactly, we **try random lucks** into each of our model several times, and see the **average behavior**. If the parameters in our model is correct, we will have the highest probablility, or in the terminology of a statistician, the *likelihood will be maximized*.

The following is a visualization of the process for one market.

We have to do this for every market, in which 100 times of the Monte-carlo simulation is done.

I haven’t describe the details of simulating an entrance of an airline yet, so let us get into it a bit more. After all, we are playing a game.

# Simulation Detail — Who will enter?

This is actually very intuitive.

## The story

First, every airline assumes that no body enters. Under this situation, airlines decide to enter the market or not. Assume that airline 1, 3, 4 will enter.

However, the profit for each airline depends on the number of incumbents in the market. In this case airline 4 realizes that he has no chance, hence leaving the competition. There are only 1 and 3, which is 2 airlines left.

Given that there are only 2 airlines, airline 1 and airline 3 (I apologize for not naming my airlines as ABC) reconsider their profit, and airline 3 realizes that he won’t earn any profit, and hence leaving the market.

The survived winner is airline 1. Congrats!

## The algorithm

We can simulate the above procedure by iteratively adding the number of airlines that each airline expects to have entered, until adding one more airline will make the surviving airlines quit the market, such that the total airlines that enter is less than the estimated entrance number.

This procedure is clearly demonstrated below

I hope now you see how fun (and tedious when coding )this estimation method can be.

**Method of simulated moments (MSM)**

After we have the result of simulation given a set of parameter that we proposes, we can check that under this specification of parameter, how big is the probability that the data we collected will occur. We want to maximize this probability. This is idea is called the **Maximum simulated likelihood (MSL)**.

Similarly, if we are not sure how we can construct our likelihood, we might just want to match things like the mean, variance, or anything that we called “moments”, and in that case the method is called the **Method of simulated moments (MSM).**

Here I am not sure how I can construct the probability, therefore I will implement the MSM method.

We have the following moments, or equivalently in this case, the averages

- The expected different between the total number of airlines in each market should be 0
- The expected different between the numbers of entrance for airline 1,2,3,4 in each market should also be 0.

We can then perform the standard **Generalized Method of Moments (GMM).**

Since this is not an article about GMM, I will not spend too much in describing the details. I might publish another one later on.

# Conclusion

I tried to explain some fun part of econometrics in the literature. This article is basically adapted from Berry (1992), and it is pretty hard to read (there are some microfoundations and theoretical results of the estimation model).

Nevertheless, I found the idea interesting and intuitive, while simultaneously being kind of important in the literature of econometrics, since it shed light on how one can estimate with theoretical ideas backing the phenomenons.

Hope this article isn’t too technical and theoretical, and can be enjoyed by non-econometrician.

Technical details and full code can be found in https://rpubs.com/tedb05202014/912839

# Reference

Berry, Steven T. “Estimation of a Model of Entry in the Airline Industry.” *Econometrica* 60, no. 4 (1992): 889–917. https://doi.org/10.2307/2951571.