Game Theory - History of Economic Thought - Lecture Slides, Slides of Economics

Download Game Theory - History of Economic Thought - Lecture Slides and more Slides Economics in PDF only on Docsity! Game Theory docsity.com What is Game Theory? • Game theory looks at rational behavior when each decision maker’s well-being depends on the decisions of others as well as her own. docsity.com Cournot’s Analysis of Duopoly • In 1838, Antoine Augustin Cournot showed how two firms that make up a duopoly could decide, independently and rationally, how much they should each produce. docsity.com Cournot’s Analysis of Duopoly • First, Cournot developed the crucial marginalist result that a profit-maximizing firm should produce the output at which its marginal revenue (or MR, the additional revenue it would earn from the production and sale of an additional unit of output) is equal to its marginal cost (or MC, the additional cost of an additional unit of output). docsity.com Cournot’s Analysis of Duopoly • Cournot then showed that each firm’s MR depends on the quantity produced by the other firm. • For example, if Firm B floods the market (of, say, milk) by producing a large amount (of milk), the market price (of milk) will be low and, therefore, Firm A’s MR will be low. docsity.com Cournot’s Analysis of Duopoly • The interdependence creates a problem: As Firm A’s decision (about what amount to produce) depends on Firm B’s decision, which depends on Firm A’s decision, which depends on Firm B’s decision, and so on and on, how can we deduce the rational decisions for the two firms? docsity.com Cournot’s Analysis of Duopoly • This puzzle, rooted in interdependence, lies at the heart of game theory and distinguishes it from the theory of rational consumer or firm behavior constructed by marginalists such as Gossen, Jevons, and Thunen. docsity.com Cournot’s Analysis of Duopoly • Gossen and Jevons considered a consumer who has to decide how to spend her income on the various goods that are available for purchase. • The consumer takes the prices of goods as given—as something beyond her control, like the weather—and does not need to think about the decisions of others. docsity.com Cournot’s Analysis of Duopoly • Although Cournot’s 1838 solution was, in the light of John Nash’s invention of Nash Equilibrium more than a century later in 1950, the correct solution to Cournot’s duopoly problem, it did not lead to a general theory of rational behavior by interdependent actors. • There were at least three reasons for this docsity.com Cournot’s Analysis of Duopoly 1. Cournot’s writings were not read widely. 2. He saw himself as solving the narrow problem of duopoly and was unable to see that there was a large class of problems that could be addressed using his technique. 3. Cournot’s justification for his solution had some unattractive underlying assumptions that ended up convincing economists that although Cournot had the right answer, he did not have the right logic to justify his answer. docsity.com VON NEUMANN AND MORGENSTERN: TWO-PLAYER ZERO-SUM GAMES docsity.com Two-Player Zero-Sum Games • This book marks the beginning of modern game theory docsity.com Two-Player Zero-Sum Games • Von Neumann and Morgenstern analyzed rational behavior in two-player zero-sum games, such as the one in Figure 2 below. Figure 2 Betty Left Right Al Top -10, 10 -4, 4 Middle 4, -4 1, -1 Bottom 6, -6 -3, 3 docsity.com Two-Player Zero-Sum Games • Note that the payoffs in each cell add up to zero. • This is why these games are called zero-sum games; one player’s gain is inevitably the other player’s loss. Figure 2 Betty Left Right Al Top -10, 10 -4, 4 Middle 4, -4 1, -1 Bottom 6, -6 -3, 3 docsity.com Two-Player Zero-Sum Games • A referee—think Jeff Probst in Survivor—then reveals all the choices and gives everybody the resulting payoffs. Figure 2 Betty Left Right Al Top -10, 10 -4, 4 Middle 4, -4 1, -1 Bottom 6, -6 -3, 3 docsity.com The Maxmin Solution • Waldegrave, Borel and von Neumann had all discussed a specific rational way to play these games called the maxmin solution. Figure 2 Betty Left Right Min Al Top -10, 10 -4, 4 -10 Middle 4, -4 1, -1 1 Bottom 6, -6 -3, 3 -3 Min -6 -1 docsity.com The Maxmin Solution • This approach argues that each player should look at each of his available options and ask, what is the worst that could happen to me if I choose this option? Figure 2 Betty Left Right Min Al Top -10, 10 -4, 4 -10 Middle 4, -4 1, -1 1 Bottom 6, -6 -3, 3 -3 Min -6 -1 docsity.com The Maxmin Solution • The players’ pessimism is revealed, after the fact, to have been right on the money. Figure 2 Betty Left Right Min Al Top -10, 10 -4, 4 -10 Middle 4, -4 1, -1 1 Bottom 6, -6 -3, 3 -3 Min -6 -1 docsity.com The Maxmin Solution • Von Neumann and Morgenstern showed that every two-player zero-sum game has a unique maxmin (or, saddle-point) solution. – Von Neumann and Morgenstern allowed the use of randomized strategies to get their proof. – Waldegrave had not only introduced the idea of maximin strategies (in 1713) as a rational way to play zero-sum games, he had also shown that allowing the use of randomized strategies could yield a solution when it is impossible to find non-randomized strategies that solve a game. docsity.com The Maxmin Solution • Although von Neumann and Morgenstern showed economists how players could make rational choices in two-player zero-sum games, their achievement did not lead to any significant use of game theory in economics. docsity.com JOHN FORBES NASH docsity.com Nash Equilibrium • It was John Nash (in 1950) who provided the crucial breakthrough idea, called Nash Equilibrium, that enabled the analysis of rational behavior in games that are not necessarily zero-sum and may have any number of players. docsity.com A Beautiful Mind • John Forbes Nash shared the 1994 Nobel Memorial Prize in Economics with Reinhard Selten and John Harasanyi • The story of his extraordinary academic discoveries, his descent into madness, and his miraculous recovery is told in A Beautiful Mind by Sylvia Nasar. docsity.com Nash Equilibrium • Consider the following game. • What is its Nash Equilibrium? Figure 3 Betty Left Center Right Al Top 3, 1 2, 3 10, 2 High 4, 5 3, 0 6, 4 Low 2, 2 5, 4 12, 3 Bottom 5, 6 4, 5 9, 7 docsity.com Nash Equilibrium • Note first that this is not a zero-sum game. • For example, if Al plays Top and Betty plays Center, Al gets $2 and Betty gets $3. • These gains are mutual gains and are not obtained at each other’s expense. Figure 3 Betty Left Center Right Al Top 3, 1 2, 3 10, 2 High 4, 5 3, 0 6, 4 Low 2, 2 5, 4 12, 3 Bottom 5, 6 4, 5 9, 7 docsity.com Nash Equilibrium • Check that the only Nash equilibrium is “Al plays Low and Betty plays Center”. Figure 3 Betty Left Center Right Al Top 3, 1 2, 3 10, 2 High 4, 5 3, 0 6, 4 Low 2, 2 5, 4 12, 3 Bottom 5, 6 4, 5 9, 7 docsity.com Nash Equilibrium • This is not a Nash equilibrium because, although Left is indeed Betty’s best move when Al plays High, Al’s best move when Betty plays Left is Bottom, not High. Figure 3 Betty Left Center Right Al Top 3, 1 2, 3 10, 2 High 4, 5 3, 0 6, 4 Low 2, 2 5, 4 12, 3 Bottom 5, 6 4, 5 9, 7 docsity.com Nash Equilibrium • Nash was able to show that virtually any game-like situation that one could think of is guaranteed to have at least one Nash equilibrium. – Like von Neumann and Morgenstern, Nash assumed that players could use randomized strategies. docsity.com Nash Equilibrium • In other words, for virtually any problem in the social sciences, the Nash equilibrium concept makes it possible to say something definite about the likely outcome; one does not have to throw up one’s hands in despair. • For this reason, it is not an exaggeration to say that Nash equilibrium is the fundamental unifying concept for all social sciences. docsity.com Reinhard Selten and Equilibrium Selection • Reinhard Selten showed a way out of these difficult situations by demonstrating that some Nash equilibria have unattractive properties. • Consequently, where there are multiple Nash equilibria, we may be able to reduce the number of equilibria by throwing out the ones with the unattractive properties. docsity.com Reinhard Selten and Equilibrium Selection • Consider the following game expressed in tree form (or, extensive form) l r L R Al’s move Betty’s move (8, 10) (5, 1) (11, 5) Figure 4 docsity.com Two Nash Equilibria • It can be checked that this game has two Nash equilibria: 1. (NE1) Al chooses r and Betty chooses R, and 2. (NE2) Al chooses l and Betty chooses L. l r L R Al’s move Betty’s move (8, 10) (5, 1) (11, 5) Figure 4 The problem with having two Nash equilibria is that one cannot say anything definite about the likely outcome of this game. docsity.com Subgame Perfect Equilibria • In this way, Selten was able to show how the problem of multiple equilibria can be made managable, at least in some cases, by eliminating those Nash equilibria that are not subgame perfect. l r L R Al’s move Betty’s move (8, 10) (5, 1) (11, 5) Nash Equilibrium, but not Subgame Perfect Subgame Perfect Nash Equilibrium Figure 4 docsity.com Two Nash Equilibria • It can be checked that there are two Nash equilibria: – (NE1) Al plays T and Betty plays L, and – (NE2) Al plays B and Betty plays R. Figure 5 Betty L R Al T 6, 10 8, 10 B 5, 1 11, 5 docsity.com Two Nash Equilibria • However, Selten argued that NE1 is unlikely to occur. • To see why, note that if Al plays T, L and R are equally good for Betty. • Nevertheless, we cannot realistically expect Betty to play L. • She will instead play R because, if Al plays T, R is no worse for Betty than L, and if by chance Al plays B then R is definitely better for Betty. Figure 5 Betty L R Al T 6, 10 8, 10 B 5, 1 11, 5 docsity.com JOHN HARSANYI AND INCOMPLETE INFORMATION docsity.com John Harsanyi and Incomplete Information • Both Nash and Selten received the Nobel Memorial Prize in Economics in 1994. • The third game theorist who got the Nobel that year was John Harsanyi. • Harsanyi showed how to solve games under incomplete information. docsity.com Complete and Incomplete Information • In the games discussed so far—Figures 2-5— there is nothing uncertain about the preferences of Al and Betty. • The payoffs to Al and Betty in the various outcomes reflect their preferences over those outcomes. • Therefore, Al knows what sort of person Betty is—what Betty likes or dislikes about the various possible outcomes—and Betty knows what sort of person Al is. docsity.com