Lecture Notes 1: Introduction to Game Theory
Games and Decisions
Game theory is relevant any time choices are interdependent – when there are cross-effects between players. In other words, game theory is relevant when, in making choices, you have to take into account how other parties are going to respond to your choices. In defining the class of economic problems where game theory is relevant, we distinguish between decisions and games.
• Decisions are situations where agents make choices without any regard to interaction with others. A good example is the classic consumer choice model from microeconomics – when you go to the store and choose how many apples to buy, you find the optimal bundle for yourself given your budget and the prices. No responses from anyone else to worry about. This is not a game theoretic problem. Another example is monopoly profit maximization – the firm sets the price that maximizes its profit, and that’s it.
• Games are situations in which choices are interdependent. Other agents are impacted by the choices that you make and you are impacted by their choices. This is important. In making choices in the context of a game, you have to think about the relationship between the choices that you make and the choices that other players make.
Brief History of Game Theory
Most of classical microeconomics deals with decisions. Specifically, classical microeconomics has always been good at the following two cases:
• Situations involving one agent – Classical economics is very good at dealing with single-agent optimization problems. Monopoly profit maximization and consumer optimization (e.g. MRS = price ratio) are decisions because they involve only one decision-maker. There is no interactivity.
• Situations involving a very large number of agents – Classical economics also does a good job of analyzing competitive markets. But competitive markets involve decisions, not games. The core assumption in a competitive market is that each agent is too small to impact market conditions. For example, when you go to the store to buy apples, you take the market price as given. You don’t think about the fact that you might push the price of apples up, and how other buyers will react to this, if you buy too many apples.
the first case because there is only one agent relevant for the problem and in the second case because there are so many agents that each one can ignore the interactivity effect of his own choices on the rest of the market. But then what about the intermediate case? What about situations that involve more than one agent, but not so many that each agent can ignore the interactive effect of his own choices? Game theory is born.
Not surprisingly, the first application in economics of what we would today call game theory was to oligopolies – markets involving more than one firm, but not so many that the market is competitive. In these cases, economists recognized early on that the nature of how the firms interact with each other is important in analyzing how oligopoly markets work. In particular, in making choices (e.g. about pricing), firms need a way to model how other firms would react to their choices. Augustin Cournot developed a model of how oligopoly markets operate in 1838, which is an early example of what we would today call a game theoretic model.
These early applications were specific to the problem of oligopolies. It wasn’t until much later that economists began to see how general and rich an area game theory is and how much it can contribute to our understanding of diverse phenomena – and not only to economics. These days, game theory is an increasingly important tool for political science, psychology and even biology. Von Neumann and Morgenstern published their groundbreaking text in 1944, which became the foundation of future work (their early motivation seems to be gambling and parlor games). Many early attempts to generalize game theory and apply it to policy are due to the US Department of Defense, in the context of the Cold War. Here, substantial credit goes to Schelling (1960). John Nash is an eccentric figure who is responsible for much of the early development of the mathematical underpinnings of game theory in the early 1950’s. Today, game theory is a standard part of the toolkit for economists, especially younger economists.
Classification of Games
There are several important distinctions among various classes of games.
• In sequential games, players make their choices in order, observing choices by players who have already moved. Chess is a good example. In simultaneous games, players make choices unaware of the choices of other players. A good example is two teams on a basketball court that have to decide what kind of play they’re going to run.
• In zero-sum games, one player’s gains exactly translate into someone else’s losses. In other words, the net sum of the payoffs is always fixed. Gambling is a good example. In
• In a one-shot game players interact together only one time. In this case, reputation is not important. For example, I might want to undercut a rival’s price and steal his customers if I’m never going to interact with him again. In a repeated game, the same players interact together multiple times. Reputations can be important in this case. I might not want to undercut my rival in this case since he has an opportunity to punish me for it in the future.
• Some games have imperfect information, which can arise from two sources. Strategic uncertainty occurs when players do not know everything about the choices their opponents have made. For example, you might have to bid on an item at an auction without knowing what I am going to bid. External uncertainty occurs when conditions outside the game are unknown but affect the environment. An example might be that the weather impacts the choices of rival firms that operate ski resorts. Another important consideration is asymmetric information, which refers to different information held by different parties. For example, I might be in possession of a new technology that allows me to produce at low cost, but my rival doesn’t know this.
• In a cooperative game, players have the ability to make enforceable agreements with each other before the game begins. For example, business partners might write a contract the forces the buyer to pay up once the seller delivers his product. In a noncooperative game, players are always free to make choices in their own best interest. In other words, noncooperative games require that any agreements between players be self-enforcing. For example, firms are not allowed to make binding contracts to collude on price.
Setup of a Game
Specifying a game involves four key elements.
• Players – Who are the agents involved in the game? Coke and Pepsi might be players in a game involving a price war. In an auction, the players are the bidders.
• Strategies – What choices can the players make? Coke and Pepsi can choose the price that they want to set. Bidders in an auction choose what bids they want to offer.
• Payoffs – What is each player’s payoff for all combinations of all strategies? For example, how much profit do Pepsi and Coke make, depending on the prices that each firm sets? Note that your payoffs depend not only on your strategy choices, but also on other players’ strategy choices. This is the essential element of a game.
This last one deserves a brief mention, because it is much misunderstood in game theory and in economics generally. Rationality does not mean that everyone is ruthlessly selfish or out to grab as much money as possible. The notion of a payoff is very general and can include lots of things about people’s preferences, not just money. For example, I could punch you and steal the $20 in your wallet. But just because I’m $20 richer doesn’t mean that this is a rational decision – I’d probably feel guilty about it, and this guilt should be reflected as a lower payoff. As another example, many people give to charity because, even though they lose some money, it makes them feel good. The point is that payoff numbers given in a game can include anything about people’s preferences, not just money. But given the payoffs, people choose strategies to maximize payoff. As a final cautionary note, if it’s a repeated game, then you might be concerned about the long-term payoff and not just the instantaneous payoff.
Given all of this, the starting point in game theory is common knowledge of the rules. Everyone knows the players, the available strategies, the payoffs and that all players are acting rationally. Importantly, common knowledge of the rules does not mean that information must be perfect. There may well be asymmetric or imperfect information, but players need to have a common understanding of the structure of the game.
Given this setup, our goal in this course is to find an equilibrium in the games that we consider – a prediction of play. The nature of this equilibrium will depend on the kind of game under consideration. For example, repeated games and sequential games require a different notion of equilibrium than simultaneous games.
Classic Game Theory Problems
Before we get into technical details, let’s just start with a few fun examples of game theory problems. Most of these have some kind of surprising or not immediately intuitive result. These are usually the most fun kinds of economics problems!
• A basketball team trains a player very hard to shoot three-pointers. After his training is over, his three-pointer scoring per game falls. How can this make any sense? Well, sports matches are interactive. Once the other team figures out that the player is really good at shooting three-pointers, they will respond by blocking him more aggressively. To make matters worse, the newly trained player may even attempt three-pointers less often! This is just like a tennis or soccer player with a perfect shot. He doesn’t want to play it all the time because it’s important to be unpredictable to the other team.
normal reason to offer price discounts is to poach customers from your rivals. But if you know that your rival has an agreement in place to match any sale price that you offer, then you don’t actually get any new customers. Thus, there’s no reason to offer the discount in the first place! Counterintuitively to many, economists see price matching guarantees as a collusive scam by firms to keep each other from undercutting prices.
• A professor announces that a class will be strictly graded on a curve: 20% of the students will get an A, 20% will get a B, etc… The students all get together to make an agreement that they shouldn’t work very hard since it won’t raise their grades overall. Why work hard for the same grade distribution that we get if we’re all lazy? Will the agreement work out? Probably not. While the class as a whole doesn’t gain by working hard, each individual student still has an incentive to work hard to get one of the A’s – especially if other students have said they aren’t going to work hard! Overall, the students as a whole might be better off if they could make such an agreement, but each individual student has an incentive to cheat on it.
• Two roommates let the garbage pile up every day, hoping that the other roommate will clean it up. Every day it gets more disgusting. Surely the roommates would both be better off if one of them would just clean it early on, but both roommates each day might prefer to pay a brinkmanship game, hoping the other roommate will give in first.
• Is it good to be really rigid in bargaining? On one hand, you might be more likely to get concessions if the other party knows that you’ll never give in. On the other hand, you might have a hard time getting people to come to the table at all if they don’t see any benefit for themselves.
• Why is a whole planeload of passengers powerless in the face of a single hijacker? Surely if all the passengers got together, they could overcome the hijacker. Great – you can be the first one in line to give it a try. The same principle applies to subjects of a despotic government. If they all got on board, they could overthrow the regime. So, who wants to be the first one in line at the protest? You might read the fable about “belling the cat” for an amusing account of this problem.
• Airport security screening seems ridiculous – Why do we make 80-year-old nuns go through intensive security checks? Doesn’t it make more sense to “profile”, and more intensively scrutinize people likely to commit terrorist actions? Not necessarily, and for the same reason as the previous example. Do you really want terrorists to know that 80-year-old nuns never get screened at airports?
Outline of the Course
The course is basically split into two halves. The first half will develop the theoretical machinery allowing us to solve for equilibrium in different classes of games. The second half will explore applications of the theoretical techniques developed in the first half of the class.
I. Theory
a. Simultaneous games – Discrete, pure strategies b. Continuous strategy sets
c. Mixed strategies d. Zero-sum games e. Sequential games
f. Games with simultaneous and sequential moves g. Repeated games
h. Evolutionary games (games with a large number of agents, randomly matched, with environment changing – learn by observing what’s successful)
II. Applications
a. Credibility – How to convince people you’re serious b. Brinkmanship – The theory of conflict escalation c. Collective action – Making the world a better place d. Voting – Does the voting rule determine the outcome?
e. Auctions – Buyer and seller strategy in hidden-information environments f. Bargaining – How to negotiate
