When I am getting ready to reason with a man, I spend one-third of my time thinking about myself and what I am going to say and two-thirds about him and what he is going to say.
—Abraham Lincoln
Game theory embodies an essential way of looking at the world, giving us a collection of tools for decision making. It is a field of study by mathematicians, economists, and by managerial strategists as well. However, in this book we will keep the mathematics to a minimum and focus on how game theory can be used to improve our decision making. It will be necessary to learn some jargon along the way, but it must be noted that the point of learning game theory is not about memorizing vocabulary words; it is about training our brain in a new way of thinking.
1.1 How Game Theory Is Different from “Standard” Economics
What was said in the previous paragraph could be said about “standard” economics as well as game theory. Economics is all about making the best choices, and so is game theory. What makes game theory different is that more than one person is making choices, and those choices are interdependent. That is, the best choice of person A depends on the choice of person B, and vice versa.
Let us think about a “standard” example in microeconomics where Joe goes to buy gasoline. The gas station owner might be a monopolist and, by observing Joe’s demand curve for gasoline, decide that the profit-maximizing price is to charge Joe $6 per gallon, whereupon Joe will buy eight gallons of gasoline for a total of $48. However, what if there are two gas stations? The best choice of price for gas station A cannot be discussed without knowing what price gas station B is charging, and vice versa. This situation needs to be analyzed using game theory. Each gas station cannot change their price without taking into account how this action will cause a reaction in their competitor. We will look at these pricing games in more detail in Chapter 10.
When it comes to games we generally classify them into three types: games of skill, games of chance, and games of strategy. A game of skill is one where the outcome depends primarily on a particular talent or training; for example, running a 100 meter race. A game of chance is one where the outcome is primarily determined by a random outcome (what game theorists often call an “act of nature”), such as a dice game or flipping a coin. A game of strategy is one where the outcome is determined by the choices players make. Game theory is primarily concerned with games of strategy: It is about how to make the best choice when your best choice both affects, and is affected by the choices of others. However, it is obvious that many games have aspects of two or even all three types of games. Most sports such as basketball rely primarily on skill, but strategy and even chance play significant roles in the outcome of sports contests.
So while “standard” economics is about a person or firm’s optimal decision making, game theory is about interrelated decision making, looking ahead and anticipating how your action will affect others’ actions, and in turn how those actions will affect you. The main lesson that we will see time and time again is that in a game of strategy what the other players believe and choose to do are just as important as your own choices and beliefs.
1.2 A Brief History
Where does one begin to trace the ideas of game theory? Certainly one could go back to General Sun Tzu (ca. 1500 BC) or earlier. However, here we will focus on more traditional economic lines of thought. One could argue that Adam Smith laid many of the foundations of game theory in his 1776 Wealth of Nations. Many of his arguments were driven by the idea that humans make rational, self-interested decisions that are necessarily interdependent. Although the following quote from Book 1, Chapter 2 is perhaps overused, it is because it is instructive on so many levels:
In civilized society he stands at all times in need of the cooperation and assistance of great multitudes, while his whole life is scarce sufficient to gain the friendship of a few persons. In almost every other race of animals each individual, when it is grown up to maturity, is entirely independent, and in its natural state has occasion for the assistance of no other living creature. But man has almost constant occasion for the help of his brethren, and it is in vain for him to expect it from their benevolence only. He will be more likely to prevail if he can interest their self-love in his favour, and show them that it is for their own advantage to do for him what he requires of them. Whoever offers to another a bargain of any kind, proposes to do this. Give me that which I want, and you shall have this which you want, is the meaning of every such offer; and it is in this manner that we obtain from one another the far greater part of those good offices which we stand in need of. It is not from the benevolence of the butcher, the brewer, or the baker, that we expect our dinner, but from their regard to their own interest. We address ourselves, not to their humanity but to their self-love, and never talk to them of our own necessities but of their advantages.
And thus, the rationale for the analysis of strategic interaction was laid down. Famous examples of strategic interdependent analysis were conducted by Antoine Augustine Cournot (1838) and Joseph Louis François Bertrand (1883), who made early arguments about ideal strategy when setting prices against a competitor. However, most people date the birth of game theory to 1928, when the brilliant mathematician and computer scientist John von Neumann wrote the article “On the Theory of Games of Strategy.” Later, von Neumann and Oskar Morgenstern published the book Theory of Games and Economic Behavior. This work was groundbreaking in many ways, though limited, because it largely analyzed “zero-sum” games. A zero-sum game is one where anything lost by one player is exactly what is gained by another. For example, if we flip a quarter and you guess heads or tails to win it from me, then either you guess wrong (I lose 0, you win 0, 0 + 0 = 0 (zero sum)) or you guess right (I win –25 cents, you win 25 cents, –25 + 25 = 0 (again, zero sum)).
Most games are not zero sum. As an example, take two gas stations located next to each other, which are currently making $10,000 per day, both charging high prices. Another option could be that they both charge low prices, both making $1,000. The payoffs would almost never add up to zero; instead they add to $20,000 in one case and $2,000 in the other (and of course, there are many other possibilities we omitted).
To help us predict what rational people in all types of games will do—including non zero-sum games—John Nash formulated and presented his now famous “Nash Equilibrium” concept in several works between 1949 and 1951. This concept will be discussed further in Chapter 2. Since that time, many researchers have refined Nash’s original idea and applied it to myriad real-world problems. Experimental economists have tested it to see if people really behave as his equilibrium concept would predict. The short answer is that sometimes people do not, and throughout the book we will discuss examples of this, and try to point out why this happens.
1.3 A Few General Principles: Payoffs, One-Shot, Common Knowledge, and Rationality
It’s All About the Payoffs
When we analyze strategic situations (games), it is very important that we understand the motivations of the players involved. Most of the time we will represent how good an outcome is to a player with a number. This number could represent happiness, “utility,” money, or anything of value to a player, where a higher number is better. Suppose for a particular outcome of a game that my “payoff” is 50 and your payoff would be 10. A very important assumption that we will stick to throughout this book is that I do not care about your payoff of 10 and you do not care about my payoff of 50. All I care about is what I get and all you care about is what you get. Of course, there are many real-world situations where this might not be the case—for example, people tend to care about how much someone else gets relative to themselves. In these cases, we will simplify the situation so that the numerical payoff we see in our analysis includes what you get, as well as how you feel about what the other player gets, and vice versa. Calculating payoffs in this way will simplify our analysis.
One Shot
We will also normally begin by assuming that the games we discuss happen only once. We do this because analyzing repeated interactions becomes very complicated quickly, and it is not possible to solve repeated games without adding a vast array of assumptions and mathematics. We will discuss some important implications and intuitions behind repeating games over time in Chapter 6, but will normally simplify our discussion to the simpler “one-shot” world for most of the book.
Common Knowledge
Another assumption that we make is that all of the players know everyone’s payoffs, and everyone knows that everyone knows this, and everyone knows that everyone knows, ad infinitum. This kind of assumption is important, because any gap in who knows what about who knows what can create interesting wrinkles.
Example: “Friends” on Television
In the popular TV sitcom Friends, we see the best example of where different “levels” of knowledge are crucial. In the episode “The one where everyone finds out,” two characters, Chandler and Monica, are “secretly” dating, but Joey, Phoebe, and Rachel all know about it. Joey, Chandler, and Monica are in a state of perfect information—meaning that everyone in this group knows what each other know, and they know that each other know, and…. However, even though Rachel and Phoebe know, Chandler and Monica are unaware that they know. So, it is Rachel and Phoebe who have the real secret now.
Because Rachel and Phoebe secretly know about the relationship, they decide to have some fun. Phoebe pretends to seduce Chandler to see what happens, because he is pretending to be single. Chandler is flattered at first but then figures out that Phoebe knows. However, Phoebe is now in the dark about this level of knowledge, and so Chandler shocks Phoebe by pretending to seduce her in return! Each time one of them realizes that the other has figured out the last secret, the power shifts!
Rationality
Finally, we will make the assumption that players make choices in a way that they think will benefit them the most. Players will try to use logic to anticipate other players’ actions and make the best choice in response. This is really the key to most of game theory: thinking ahead to the choices or reactions that other players are likely to make, and then using that information to strategically improve your actions.
Now, even though we analyze games assuming that people are rational, economists are paying more and more attention to situations where humans are often irrational. For example, if presented with the opportunity to purchase an extended warranty for a car for $800 that only applies to the fourth and fifth years of car ownership, almost all will decline to purchase this. However, when added as an option that only adds $15 per month to a $300 monthly car payment (for 60 months), many people jump at the chance. We will point out situations where irrationality is an important factor from time to time. However, it is necessary to first understand rational and predictable behavior before one can appreciate the importance of irrationality.
Further Reading
For links to summaries and YouTube clips of the “Friends” episode discussed in the chapter, go to gametheory.burkeyacademy.com. You will also find other helpful information, including additional links, commentary and examples.
Dixit, A. K., & Nalebuff, B. J. (1993). Thinking strategically. W. W. Norton.
Academic Papers and Books Using Lots of Math: Below I give references for the work of the early researchers I mentioned in the text. For most readers, they will be of very little interest—but if you are interested, go for it!
Bertrand, J. (1883). Book review of theorie mathematique de la richesse sociale and of recherches sur les principles mathematiques de la theorie des richesses. Journal de Savants, 67, 499–508.
Cournot, A. A. (1838). Recherches sur les principes mathématiques de la théorie des richesses. L. Hachette.
von Neumann, J. (1928). Zur Theorie der Gesellschaftsspiele. Mathematische Annalen, 100(1), 295–320. English translation: On the theory of games of strategy. In A. W. Tucker & R. D. Luce (Ed.), Contributions to the theory of games (1959, Vol. 4, pp. 13). Princeton, NJ: Princeton University Press.
von Neumann, J., & Morgenstern, O. (1944/2007). Theory of games and economic behaviour. Princeton, NJ: Princeton University Press.