The Basics of Game Theory: Dominant Strategies and Nash Equilibrium

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This is post 1 of 5 on game theory and it’s application to business and innovation. To view my other posts on game theory, see the list below:

Game Theory Post 2: Location Theory – Hotelling’s Game

Game Theory Post 3: Price Matching (Bertrand Competition)

Game Theory Post 4: JC Penny (Price Discrimination)

Game Theory Post 5: Mixed Strategies

Without a doubt game theory is one of the most powerful and intellectually stimulating subjects I have ever studied. It can be applied to nearly every situation we find ourselves in both personally and professionally. And it can help us know the right decision to make in situations where our choices are affected by the choices of others. But there’s just one problem with game theory: it is taught to be hard.

Say what?

Yes – it’s taught to be hard.

But there’s a secret that the calculus-loving game theory masters don’t bother telling you. That secret is this: there is a basic version of game theory that is brain-dead easy to understand and use.

Seriously?

I know what you’re thinking…you’re probably saying to yourself “Wait, I saw the movie ‘A Beautiful Mind’ about the game theory guy and him writing crazy-looking equations on glass and stuff…that guy was a genius. Most people can’t do what he did.'”

You’re right, most people can’t do what he did and that’s why we’re not even going to try to do what he did. The great part is we don’t need to do what he did in order to understand and use the core of his idea – the essence of which is actually remarkably simple. So simple that anyone can understand and use it.

Let me explain below.

Game Theory is a Sub Set of Economics

To understand game theory it helps to know the core idea behind economics. Most people think economics is about finance, money, public policy and the stock market. While it’s true that the tools within economics (including game theory) are frequently used to study and understand those fields, the core of economics is actually very simple: it is the science behind why people make the decisions they make. In other words, economics is the field of science, social science in fact, behind human decisions.

Economics is just as applicable to why you decided to marry your spouse and eat eggs for breakfast this morning as it is to why you bought stock in Tesla last year. It attempts to analyze and understand all types of human decisions, money related or not.

Here’s a simple example:

Economics of what I should eat for breakfast:
Decision to make Choice A Choice B
What should I eat for breakfast? Cereal Eggs
Value of each choice from 0-10: 3 7

 

So since I value eating eggs for breakfast more than I value eating cereal, so long as those options are available, I’ll choose to eat eggs for breakfast.

Pretty simple isn’t it.

So what does this have to do with game theory?

Game Theory is Simply Analyzing Decisions That Will Affect Other People’s Decisions

Game theory was created as a subset of economics because while economics was good at describing why people made decisions that only affected the individual (microeconomics) or a mass of people (macroeconomics), it was lacking when it came to understanding decisions that involved multiple people where one persons decision would affect the other persons decision. Game theory was created to fill that gap.

At it’s core, game theory is about analyzing decisions that will impact other people’s decisions. Game theorists call these types of decisions “strategies.” The simple premise behind game theory is that you can calculate what is the right decision to make even in multi-person (or multi-player) situations, before needing to make it. If you think about most decisions you make, it’s likely that they have some affect, either large or small, on the decision of others.

For example lets take a story about innovation out of a book I recently read and apply it to game theory. The book is How Google Works by Eric Schmidt. On page 183 the book tells the following story:

“When Google acquired Motorola in 2012, one of the first Motorola meetings Jonathan attended was a three-hour product review, where the company’s managers presented the features and specifications for all of Motorola’s phones. They kept referring to the customer requirements, most of which made little sense to Jonathan since they were so out of tune with what he knew mobile users wanted. Then, over lunch, one of the execs explained to him that when Motorola said “customers,” they weren’t talking about the people who use the phones but about the company’s real customers, the mobile carriers such as Verizon and AT&T, who perhaps weren’t always as focused on the user as they should have been. Motorola wasn’t focusing on its users at all, but on its partners.”

This scenario is quite common in many corporations today. Too often when creating product requirements business leaders confuse the needs of the user with the needs of their customer or partner. Sometimes they are the same person but very often they’re not. In Motorola’s case the customer, meaning the group who pays Motorola for each phone, was the mobile carrier while the user was Verizon’s customer who walked in the store and bought a phone from Verizon. For each product they create Motorola has a choice to make when determining the final product specifications:

  1. Prioritize User Needs First
  2. Prioritize Carrier Needs First

Often the needs of the User will be in harmony with the needs of the Carrier. But there are many cases when they are not in harmony as well. For example suppose Verizon were to require Motorola to pre-install a number of Verizon-specific applications that most users wouldn’t otherwise know about or use unless they came pre-installed on the device. Now also suppose that the user had also asked Motorola to not pre-install Carrier-specific applications because they found them useless and annoying.

It just so happens that this scenario is perfect for doing game theory analysis. With this as the background, we can start to apply game theory.

To analyze this situation let’s put ourselves in the shoes of Motorola and try to predict using game theory what a competing mobile phone manufacturer will do. Note that we’re not going to analyze what Verizon or the end user is going to do – we already know what they want because they told us. The important piece of the puzzle that we don’t understand is what a competitor will decide to do in the market given they are subject to basically the same conditions as Motorola.  This is a classic example of a competitive situation with an uncertain outcome that game theory can help us clarify.

To do the analysis, all we need to do is follow these simple steps:

Step 1: Define the Players

In every game or multi-person interaction, you will have multiple players. The first step to constructing a game theory analysis is to write down the names of the players involved. For simplicity, it’s best to keep the number of players down to two. Adding more players than two becomes extremely complicated so if your game has more than two players, try to group the players into two broad groups with similar goals.

In continuing the example above, all we have is Motorola and for the sake of argument let’s call the competitor Samsung.

Here’s our list of players:

  1. Motorola
  2. Samsung

Simple right? Now on to step two…

Step 2: List the Most Relevant Choices Available to Each Player

This part is pretty simple as well. All we do is take each player and list the choices available to each player. To do this I’ll simply copy the choices mentioned earlier.

  1. Motorola
    • Prioritize User Needs First
    • Prioritize Carrier Needs First
  2. Samsung
    • Prioritize User Needs First
    • Prioritize Carrier Needs First

Note that I created the choices to be symmetrical meaning that the same choices are available for each player. This doesn’t always need to be the case (and in many cases won’t be) but for this simple example keeping the choices the same between the two players works out quite well.

Okay, now that we have the players and the choices available to each player listed, there are just a couple more steps before we can start the analysis.

Step 3: Create The Scenarios Matrix

Most people who explain game theory (college professors, etc.) skip this step and jump straight to figuring out the payoff matrix. I’ve found that to be a mistake because often the most challenging part of game theory is simply creating an accurate payoff matrix. By creating a scenarios matrix first, we make it easy to create a payoff matrix.

So what is a scenarios matrix?

A scenarios matrix shows the players and list of choices available to the players in a table or matrix format. The cells inside the matrix represent the specific scenarios that can play out. To setup a scenarios matrix simply take the player names and choices available to each player and list them in a table like below:

game-theory-scenarios-matrix-setup3

Now with the scenarios matrix setup, it’s helpful to think through and write out each scenario within each blank cell. For example, if “Motorola” chose “User Needs First” and “Samsung” chose “User Needs First” as well (in other words they both decide to not include carrier applications), then the scenario to write in the first blank cell to the right of “Motorola” and “User Needs First” and underneath “Samsung” and “User Needs First” would be “Motorola and Samsung both choose to omit carrier applications.”

If you do this for all blank cells, you get the following completed scenarios matrix:

game-theory-scenarios-matrix-completed3

The great part of having a scenarios matrix is it illuminates all the possible choice combinations available and lets you think through each of them one-by-one. In many cases, this step turns out being the most valuable step in the process. But there are a few more steps before we reach the conclusion of the analysis.

Step 4: List How Much Each Player Values Each Choice

This is the step that is usually the most challenging to figure out with a reasonable level of accuracy. But there are a few ways to do it that are relatively easy and straightforward. In our example, since we are playing the part of Motorola we should be able to do market research and make reasonable relative estimates of the payoffs we might enjoy under each scenario.

In this case, lets imagine there are 10 consumers who are looking for a smartphone and we decide to survey them to determine their likelihood of buying a Motorola phone under each scenario. To do this, all we need to do is create questions that relate to each scenario. For example the survey could say:

  1. Question 1: What are your preferences regarding Verizon specific applications? (this question is not a specific scenario but is used to validate/invalidate the basic premise of the analysis, that users think carrier-specific apps are useless and annoying)
    1. I like them – 0
    2. I don’t like them – 8
    3. I am indifferent to them – 2
  2. Question 2: Suppose you were to were to walk into a Verizon store and see two essentially identical phones, Phone M and Phone S, from different manufacturers that both meet your needs and neither phone has Verizon specific applications included, which phone would you choose? (in practice this question is unnecessary because we know it should follow a random distribution and split 50/50 but I include it here for sake of being comprehensive with this example)
    • Phone M – 5
    • Phone S – 5
  3. Question 3: Suppose Phone M and Phone S are still essentially identical except for one difference, one phone includes Verizon-specific applications while the other does not, which phone would you buy?
    • The one without Verizon apps – 8
    • The one with Verizon apps – 2 (note: I recognize that these two could go either way but let’s put them with Phone S and assume that the Verizon salesman swayed them)
  4. Question 4: Suppose Phone M and Phone S were essentially identical and both included Verizon-specific applications, which phone would you buy?
    • Phone M – 3
    • Phone S – 3
    • I would not buy either – 4

Now all we need to do is list these values in a payoff matrix such as the one below. To make it easy to keep track of the payoffs for each I’ve color coded the payoff values to correspond with either Motorola or Samsung. All we need to do is enter the values for each cell following the logic in the diagram below:

game-theory-payoff-matrix-setup

Once we do this for all the open payoff cells, we get the following:

filled-out-game-theory-payoff-matrix-example1Now that we have created a complete payoff matrix we can start to solve the game!

Step 5: Look For Dominant Strategies

This is where game theory gets really interesting. The first step in this analysis is to determine if any of the choices (or strategies) for either player are dominant over the other choices.

Dominant Strategies

The definition of a dominant strategy is a choice that is preferable for one player no matter what their opponent chooses to do. To determine if there is a dominant strategy for Motorola, we first start by comparing the possible outcomes for Motorola if they decide to put user needs first versus the possible outcomes if they decide to put carrier needs first. In this example we can see that if Motorola chooses to put user needs first, the possible payoffs they will get is either 5 or 8. If they choose to put carrier needs first, the possible payoffs they will get is either 2 or 3. So no matter what Samsung decides to do, if Motorola puts user needs first they will be better off than if they choose to put carrier needs first.

game-theory-basics-dominated-strategy-example1

This is an important finding because now we can eliminate the option of putting carrier needs first for Motorola because we know that’s not a rational choice.

Now let’s analyze Samsung’s options and determine if they have a dominant choice as well. In Samsung’s case, their payoffs are symmetrical with Motorola’s because earlier we assumed everything about the two phones was identical except for the possible addition of carrier-specific applications. Below is the comparison of Samsung’s options with irrational (or otherwise called “dominated”) choices marked as red.

game-theory-payoff-matrix-dominated-strategy-example3

Note that if Samsung chose Carrier Needs First the best possible outcome is a 3 whereas if they chose User Needs First, the worst possible outcome is a 5. Hence choosing User Needs First is the dominant strategy for Samsung as well.

Nash Equilibrium

So if both Motorola’s and Samsung’s dominant strategy is to put User Needs First then that’s what game theorists call the Nash Equilibrium of the game. Nash equilibrium is simply the set of choices players make wherein players can do no better by choosing an alternative strategy. This may sound counter intuitive at first because we know from the payoff matrix that Motorola could do better than 5 if they chose to put user needs first and Samsung chose to put carrier needs first. But that’s not Nash equilibrium because we know Samsung won’t choose carrier needs first given they can get a better payoff by choosing user needs first. Hence equilibrium only occurs as the optimum strategy given known best responses from the other players.

game-theory-nash-equilibrium1There are some cases where dominant strategies are not present and when multiple Nash equilibria  can arise. The analysis of those types of games is more complicated but they can still be solved and I’ll explain how to analyze those in a different post.

Now that we know the solution to this particular game, the question arises, does Motorola and Samsung choose to include carrier-specific applications on some phones in the real world? The short answer is yes but the real answer is both companies are heading more and more toward the Nash equilibrium we predicted. For example, prior to Google’s purchase of Motorola in 2011, Motorola would often cave to carrier demands by offering phones that were exclusive to specific carriers and full of carrier-specific applications. However, during the time Google owned Motorola they released several new phones under Google’s leadership – most of which worked with multiple carriers and had award-winning specifications and prices. In essence, Google helped move Motorola toward equilibrium while Samsung was already well down that path.

The last thing I’ll mention is that when doing this type of analysis you’ll need to know that, like most quantitative-based analyses, the prediction of the analysis is only as good as the input data. If you have precise and accurate data to input into the game theory framework, you’ll get precise and accurate predictions out of it. Most of the time though it will be challenging to gather perfect information so do the best you can and make assumptions where necessary (like we did in this example).

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One Response to "The Basics of Game Theory: Dominant Strategies and Nash Equilibrium"
  1. kephas kelly says:

    nice stuff

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