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Game Theory

Game Theory

This is part 1 of a 5 part series on Game Theory.

Last week we covered Decision Theory, a set of tools for making decisions using data. Many decisions are not as simple as making a choice, often you are in a situation where your decisions affect others and their decisions affect you. In these circumstances Decision Theory can break down since the uncertainty makes any predicted values hard to estimate.

What can we do? We can play some games!

Game Theory is the study of situations where there is more than one player. Say, for example, you are thinking of lowering the price of your product. Will your competition lower their prices as well? Will your customers choose to buy more? Such a simple decision actually has many players who will make subsequent decisions and in doing so affect you in return. Game theory provides a set of tools and techniques to deal with that complexity and make it easier to build a winning strategy.

The idea that games can be used to model real world scenarios has been around for centuries, but it wasn’t until 1928 when John Von Neumann published a paper on using mathematics to analyze parlor games that Game Theory was born. Today, Game Theory is used to model everything from economics to negotiations because of how well it can help us understand situations where people make many interacting decisions.

We will cover the basics of Game Theory this week with a focus on specific use in answer business strategy questions.

Tomorrow we will get started by seeing how we can take a real life business decision, model it as a game for analysis and build a winning strategy!

“Life is more fun if you play games.” 

Game Theory: Game Strategy

This is part 2 of a 5 part series on Game Theory.

The easiest way to get started with thinking about Game Theory to make decisions is to realize we already started last week! When we covered Decision Theory we were really just talking about a game with a single player – you.

Let’s work with an example. Let us say you are considering raising the price on your product from $100 to $200 to improve your margins and raise revenue. If you model it like a decision (using what we learned last week) it would look as follows:

The outcomes here make it clear we should raise the price and make more revenue!!

However, selling a product is not a single player game as there are at least two players: the seller and the buyer. We need a new way to think about this, which is where Game Theory modeling starts to become useful. Below is the same decision but framed with the decisions for both the buy and the seller at the same time:

This table describes the different decisions faced by the two players, the Seller on the left side and the Buyer across the top. Each cell of the table contains the payoff to each player for the corresponding pair of decisions (the Seller’s payoff is in the lower left corner and the Buyer’s in the upper right). If the Seller raises the price but the Buyer doesn’t buy, the Seller gets nothing but the Buyer saves $200. If the Seller keeps the price and the Buyer buys, the Seller gets $100 and the Buyer spends $100. The payoffs are measured in whatever form of utility best represents the benefit to the players. In the above example, using dollars makes it easy to understand but doesn’t capture the benefit the buyer gets from the product after buying it so it not a very good measure of utility.

In Game Theory, decisions (e.g. “Raise Price”, “Keep Price”) are referred to as strategies. Your job is to choose the strategy that will give you the best payoffs based on what your opponents might do. I’ve updated our example (below) with arrows which represent the path from worse payoffs to better payoffs:

If you have properly modeled the strategies and payoffs, it should be clear which strategies are better than others.

As you can see, Game Modeling is a much more powerful way of thinking about decisions involving more than one person as the incentives become easy to analyze. Tomorrow we will jump into how to do that analysis to choose strategies and win the game!

“We do not stop playing because we grow old, we grow old because we stop playing!”

Game Theory: Zero-sum Games

This is part 3 of a 5 part series on Game Theory.

Winners and Losers

In order to pick a winning strategy, first we need to define the game. That means you need to be confident and correct in the payoffs you assign to each outcome of your game model. Game Theory has a few different kinds of well-known types of games that make understanding payoffs a lot easier.

Zero-Sum Games

In some games, every time one player gains, another loses. These are called zero-sum games because the sum of the outcomes for each player are always zero. A great example is playing poker, since every time one player wins money in a hand, all the other players lose that same amount (in total). Zero-sum games are naturally very competitive as working together has no benefit.

Non-Zero-Sum Games

In these games there is no limit to the payoff any player can earn, so there are strategies where everyone wins together. At the same time, there will be other strategies where one player gains and others lose, so players will need to choose between collaborative strategies and competitive strategies. A great example is hiring employees, since the choice to hire a person and their acceptance of that offer benefits both parties.

After you have your payoffs in place, it’s time to start analyzing! One of the first things to look for is a strategy that, no matter what the other player does, provides a better payoff to you. These are called dominant strategies and make choosing your strategy easy. For example, take the following game model:

No matter what Player 2 does, strategy B always has a higher payout for Player 1. Likewise, no matter what Player 1 does strategy D has a higher payout for Player 2. Hence, B and D are dominant strategies and the players will almost always choose those strategies [1].

Unfortunately, most games lack dominant strategies so you will need to look carefully at the payoff distribution to choose a strategy. But, what if you aren’t sure which strategy your opponent will choose? What if you don’t even know their payoffs? Well, then you have incomplete information and we’ll discuss how to deal with that tomorrow.

[1] But not always. You may have noticed that this is actually a famous game called the Prisoner’s Dilemma. It is a well- studied game type because there are clear dominant strategies (B and D) but the total outcome for both players would be best if they both choose to cooperate and choose the non-dominant strategy (A and C). It’s really fascinating and I suggest reading more if you are interested.

“A delayed game is eventually good, a bad game is bad forever.” 

Game Theory: Incomplete Information

This is part 4 of a 5 part series on Game Theory.

So far, we’ve talked about games where you know everything: all of the player strategies and all of their payoffs. But we are modeling the real world. How often do you know all the options your competition has? How well do you know the costs and benefits of each of their choices? It’s unlikely you know them any better than they know yours, which is not very well.

Most of the time when you are using Game Theory, you might have either incomplete or imperfect knowledge to work from. These sound the same but they are subtly different!

  • Incomplete Information means there are things you simply don’t know, such as the opponent’s strategies or payoffs.
  • Imperfect Information means you won’t know when or if an opponent makes a move.

When you sit down and model your game, unlike the clean and perfect models I’ve presented so far it’s likely yours will look something like this:

The best way to fill in the gap is to reason through the strategy or payoff based on what you know about the strategy or the player. This can be easier than it sounds as, like in Decision Trees, there are likely a small number of possible outcomes. For example, if you lower the price of your product your competition can only respond in one of three ways: lowering their price, keeping their price or raising their price. Likewise, it might be hard to know the specific payoffs for prices changes to your competition, but you can likely understand the relative payoffs (raising prices produces more revenue while lowering prices reduces it).

If you cannot fill in the gaps then you might want to think about the game differently. So far, the matrix model for games we have been using is called “normal” form. There is another form for game models called “extensive” form, which also captures the order of decisions by players and can help simplifying games with missing information. An extensive form representation of the above example would look as follows if Player 1 goes first:

In extensive form modeling, we start with the first player to make a move (in this case Player 1) and create a graph where the edges are their possible strategies (in this case A or ?, i.e., unknown). Each of those lead to a point where the next player (Player 2) chooses between their strategies which in turn lead to payoffs. This captures the turn-based nature of games where one player makes a move and the other player reacts to it.

As you can see, this greatly simplifies our incomplete information  problem because we do not need to worry about all of the unknowns together! We only care about unknowns if the path of decisions that lead to them are likely to occur. For example, if we are Player 2 and Player 1 chooses strategy A, we are going to choose strategy D regardless of what the unknown payoff would be for Player 1 if we had chosen strategy C – because we didn’t choose it!

There are more advanced and formal methods to calculate the unknown payoffs probabilities that we don’t have time to go into today, but in my experience for practical decision making you can get most of the way through simple reasoning and pruning of likely outcomes.

Tomorrow we’ll take everything we’ve covered one step further by covering games with more than two players!

“It may be that all games are silly. But then, so are humans.” 

Game Theory: Multiplayer Games

This is part 5 of a 5 part series on Game Theory.

So far this week I have, conveniently, only covered games with two players. That is in part because it’s easier to understand Game Theory concepts with simpler games, and in part because I’m lazy. Still, I shall rise to the occasion today so we can cover multiplayer games!

Of course, in the real world, most game models will have many players. Even the simple examples I have used this week, changing the price for a product, include at least three players: the seller, the buyer and the competition. How do you choose a strategy when there are so many players?

The good news is that all of the strategies we discussed work for games of any size. There is nothing in any of the techniques that changes with more than two players, other than the complexity of modeling and analysis. It can be harder to understand the payoffs with three players, since instead of 4 possible outcomes now you will have 8 (since each player’s decisions affect the others). Still, with practice this can become just as easy.

Identifying dominant strategies becomes more important with more players since it is an easy shortcut in the analysis process. If there are no dominant strategies, you can still find the Nash Equilibrium, which is the point where all players are choosing the best possible strategies and will not change them even if every other player has chosen their own best strategy. This is a form of steady-state in games which dictates how many parties might fall into a predictable behavior. I don’t have time to go into how to determine the Nash Equilibrium for your game, but I suggest reading up on it if you are interested.

In Review: Game Theory is a powerful set of tools for understanding and deciding between different options where the decisions of many people interact. These tools make it a lot easier to cut through the complexity and make a data driven decision, and apply to a surprisingly wide variety of problems ranging from product changes to hiring decisions. I hope you have found our brief introduction useful!

Next Week: We’ve spent a lot of time on the process of making decisions, let’s get back to using your data. Customer Personas are a great way to better understand your customers and answer the age old question: Who are they? We’ll cover how to find personas in your data and the wealth of benefits they yield for you when you do.

“Games lubricate the body and mind.” 

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