Microeconomics I: Game Theory Introduction: What is Game Theory ...

Microeconomics I: Game Theory

Introduction: What is Game Theory? (see Osborne, 2009, Sect 1.1) Dr. Michael Trost Department of Applied Microeconomics

October 25, 2013 Dr. Michael Trost


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What is game theory?

Game theory is the scientific discipline that studies situations in which decision-makers interact.

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Situations of interaction Situations of interaction are situations in which the well-being of a decision maker depends not only on her own action, but also on the actions of other decision-makers.

Henceforth, such situations are referred to as games.

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Situations of interaction E XAMPLES of situations of interaction: Board and card games (e.g., chess, back gammon, poker, bridge, etc.) Economic games (e.g., firms competing for business, bidders competing in auctions, joint ventures) Political games (e.g., political candidates competing for votes, international trade agreements) Biological games (e.g., animals fighting over preys and territories)

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What is not a game? Two cases: Isolated or insignificant decision maker . (Isolation) Your decisions affect only yourself - Personal issues like whether to go jogging or not, how many movies to watch in a week, what to build in a sand-box. - Price-setting behavior of a monopolistic firm

. (Insignificance) Because there so many decision-makers involved, your decision does not (really) affect others’ decisions. - Buying foreign exchanges or stocks. - Price-taking behavior of a firm (assumption of perfect competition)

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Let’s play a game: The Guessing Game The rules of this game are as follows: . Each of you secretly submits a number from interval [0,100]. The winner is the person whose submitted number is the closest to the two-third of the mean of all submitted numbers. . The winner receives a prize. If there are several winners the prize will be divided equally among them.

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Interaction in the Guessing Game

As its name suggests, the G UESSING G AME induces a situation of interaction: The success of a player’s guess depends essentially on the numbers guessed by the other players.

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Game-theoretic models

Like other sciences, game theory consists of a variety of models. A game-theoretic model is an abstract representation of real-life situations of interaction. Such abstractions allow us to study a wide range of social and biological phenomena and to improve our understanding of the world.

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Rules of a game

A game is a detailed description of a situation of interaction. It describes the rules under which this interaction takes place.

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Rules of a game For example, a game specifies the set of participants (which are referred to as players) the set of actions available by the players the set of outcomes resulting from the available actions the sequence of the players’ moves the information the players have about the past moves of the other players the information the players have about the goals pursued by the other players ...

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Classes of games A class of games is a set of games which have certain rules in common. Following classifications of games are popular in game theory. - Cooperative and noncooperative games - Strategic and extensive games - Games with complete and with incomplete information - Games with perfect and with imperfect information

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Cooperative and noncooperative games Noncooperative games are games in which the players choose independently their actions. The players are not able to enforce a binding agreement on their actions. Cooperative games (also called coalitional games) are games in which the players can form coalitions and engage in a binding agreement on their actions. R EMARK : In this course, we only deal with noncooperative games.

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Strategic games and extensive games

Strategic games (also known as simultaneous move games) describe situations of interactions in which each player moves only ones and the players’ decisions are made simultaneously (i.e, when choosing an action each player is not informed of actions chosen by the other players) E XAMPLES : R OCK -PAPER -S CISSORS, M ATCHING P ENNIES, presidential election

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Strategic games and extensive games

Extensive games (also known as dynamic games) describe situations of interactions in which the players move sequentially. E XAMPLE : T IC -TAC -T OE, M ARIENBAD G AME, poker, eBay auction

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Tic-Tac-Toe

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Marienbad Game

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Complete and incomplete information

A game takes place under complete information if there is common knowledge about the preferences the players have (i.e., every player knows the preferences of every player, every player knows that every player knows the preferences of every player, and so on ad infinitum) . A game takes place under incomplete information if some player is uncertain about the preferences of some other player.

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Perfect and imperfect information

A game takes place under perfect information if each player when she is deciding is informed about the past moves of her opponents. A game takes place under imperfect information if some player is uncertain about the past moves of her opponents.

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Game-theoretic solution concepts A game describes the set of actions a player can do, but does not specify the actions that the player do take. A solution of a game determines the set of actions that may be realized by the players and a solution concept for a class games determines for each game of this class the set of actions that may be realized by the players. Game theory aims to provide reasonable solutions for classes of games and to examine their properties.

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The Nash equilibrium concept The most prominent solution concept of game theory is the solution concept introduced by John F. Nash (1950), which is known nowadays as the Nash equilibrium concept.

John F. Nash (born on June 13, 1928) received, together with John C. Harsanyi and Reinhard Selten, the Sveriges Riksbank Prize in Economic Sciences in Memory of Alfred Nobel in 1994 for their “pioneering analysis of equilibria in the theory of noncooperative games”.

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The Nash equilibrium concept

The actions chosen by the players constitute a Nash equilibrium if they satisfy following property of stability: None of the players has an incentive to deviate from her action provided that the other players have already realized these actions. Q UESTION : What’s the Nash equilibrium of the G UESSING G AME? Is it reasonable?

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Positive and normative game theory Game theory is used to address positive issues of situations of interaction. - Why do interacting decision-maker behave as they do? (explanation) - How will decision-makers behave in situations of interaction? (prediction)

normative issues of situations of interaction. - How should interacting decision-maker behave? (recommendation)

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Positive applications of game theory

For example, game theory allows us to understand - why public goods (e.g., defense or flood protection) are often provided by state. - why arrangements of production cartels are often broken by their members. - how the size of sunk costs will affect the price setting of a monopolistic firm.

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Normative applications of game theory

For example, game theory helps us to figure out - the winning (or at least non-losing) strategies for simple recreational games (e.g., T IC -TAC -T OE, M ARIENBAD GAME ). - the profit-maximizing bids in auctions.

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Outline of this course I The theory of rational choice (a) Preferences and utility function (b) Expected utility function

II Simultaneous move games with complete information (a) Nash equilibrium in pure strategies (b) Nash equilibrium in mixed strategies

III Simultaneous move games with incomplete information (a) Bayesian games and Bayes-Nash equilibrium (b) Auctions

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Outline of this course

IV Dynamic games with complete information (a) Extensive form games and subgame perfect equilibrium (b) Bargaining games

V Dynamic games with imperfect information (a) Repeated games and the Folk Theorems (b) Signalling games and sequential equilibrium

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Lectures

D ATE : The two-hour lectures game theory start on October 25 and take place on Fridays, 8:30 a.m., room LG 2/213. A NNOUNCEMENTS : An additional two-hour lecture (instead of a tutorial) is given on October 25, Friday, 10:15 a.m., room LG 2/213. In the last week of lectures a mock exam will be discussed.

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Tutorials to the lectures In addition to the lecture tutorials are offered. D ATE : The two-hour tutorials start on November 1 and take place on Fridays, 10:15 a.m., room LG 2/213. E XERCISES : Exercise sheets for tutorials will be uploaded a week before. We advise you to go carefully through these exercises before we will discuss them in tutorials. The formation of learning groups is highly recommended. R EMARK : There are no extra credits for the tutorials.

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Literature

Main textbook for this course: Osborne, M. (2009), An Introduction to Game Theory, 2nd edition, Oxford University Press, Oxford.)

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Literature Further possible readings: Heifetz, A. (2012), Game Theory: Interactive Strategies in Economics and Management, Cambridge University Press, Cambridge. (Introductory textbook) McCarty N. and Meirowitz, A. (2007), Political Game Theory: An Introduction, Cambridge University Press, Cambridge. (Advanced textbook) Osborne, M. and Rubinstein, A. (1994), A Course in Game Theory, MIT Press, Cambridge. (Advanced textbook)

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Acknowledgments

¨ ur ¨ Gurerk ¨ ¨ I am indebted to Ozg and Manfred Konigstein for their permission to paste parts of their teaching materials into these lecture notes.

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