Game Theory 1. Introduction & The rational choice theory

Transcription

1 Game Theory 1. Introduction & The rational choice theory DR. ÖZGÜR GÜRERK UNIVERSITY OF ERFURT WINTER TERM 2012/13 Game theory studies situations of interdependence Games that we play A group of people is affected by the choices made by every individual within the group Examples: Auctions R&D expenditures Voting at the UN Reducing greenhouse gas emissions Teamwork 2

2 Interesting gquestions in games What do individuals guess about others choices? 3 What action will each person take? What is the outcome of these actions? Is the outcome good for an individual? Is it good for the group? One of you can win a prize Let s play: A guessing g game if there is a tie the prize will be divided equally among the winners 4 Rules of the game: Everyone chooses a number from interval [0.0, 100.0] The winner is the person whose chosen number is the closest to the mean of all chosen numbers multiplied by 2/3

3 Analyzing the guessing g game An experimental study by Rosemarie Nagel (1995) Investigates the depth of reasoning Zero-level belief: No thinking on others actions First-order belief: Thinking only once what numbers others might choose (one-step thinking) Second-order belief: Thinking twice, i.e., thinking on others firstorder beliefs and form own second-order beliefs Most people stop after one to three steps Game theoretical solution requires an infinite number of steps everyone chooses zero 5 Results 6 Original study (Nagel 95) Class of Winter Term 12/13 requency median 30.6 mean 22.2 Relative F Chosen numbers

4 So, what is game theory exactly? Game theory is a formal way to analyze interaction among a group of rational agents who behave strategically 7 Four concepts inherent to game theory: Interaction Group Rational Strategic Example: Teamwork Group: Students jointly preparing a case study 8 Interaction: If I put little effort in the project someone else must work harder to get the work done Strategic: Estimating the likelihood of free-riders in the group Rational: Comparison of the benefits of extra work (better grade) against the costs of extra work (less leisure time)

5 Example: R&D expenditures Group: e.g., pharmaceutical companies 9 Interaction: The first developer of a drug makes the most profits (thanks to the associated patent) Strategic & rational: Choosing how much to spend involves thinking about competitors commitments and possible reactions to own decision i Two cases: One and infinity So, what is not a game? 10 One: Your decisions affect only yourself To go jogging or not, how many movies to watch in a week, A monopoly Infinity: There are so many ypeople p (firms) involved that your decision does not (really) affect others choices Buying stocks Perfect competition

6 Introduction The outline of this course The theory of rational choice Strategic (normal) form games Nash equilibrium in pure and mixed strategies Extensive form games with perfect information Subgame perfect equilibrium Infinitely repeated games Extensive form games with imperfect information Sequential equilibrium 11 What you have to know as a prerequisite Mathematics: Slope of a function linear, quadratic, log and the square-root root functions Basics of optimization 12 First-order characterization of an interior optimum, that the slope of a maximand is zero at optimum Economics: No prerequisites

7 Main textbook for this course: Literature An Introduction to Game Theory by Martin J. Osborne (Oxford U Press) 13 Further (possible) reading Strategies and Games by Prajit K. Dutta (MIT Press) A Course in Game Theory (advanced textbook) by Osborne & Rubinstein (MIT Press) Thinking Strategically t (not a textbook) tb by Dixit & Nalebuff Supporting tutorial There are no extra credits for the tutorial 14 Tutorial starts on October 30 Tuesdays, 12:15 p.m., room 247b/LG1 Exercises will be uploaded a week before The exam will be held on Thursday, February 7, 10:15 a.m., re-exam on February 21, 10:15 a.m.

8 The rational choice theory 15 A rational decision-maker (DM) chooses the best available action according to her preferences (likes and dislikes) No qualitative restrictions for the preferences Rationality lies in consistency of decisions when facing different sets of available actions, not in the nature of likes and dislikes Actions Alternatives that are available in a certain situation 16 Actions are not influenced by the preferences The DM chooses one element of the given set A of available actions The set A could, for example, be the set of bundles of goods that the DM can possibly consume

9 Preferences Assumption 1: You know your preferences When presented with any pair of actions, the DM knows which of the pair she prefers or she knows that she regards both actions as equally desirable 17 Assumption 2: You are consistent with respect to your preferences If the DM prefers the action a to the action b, and the action b to the action c, then she prefers action a to action c Payoff functions A payoff function associates each action with a number 18 Actions with higher numbers are preferred A payoff function u represents a DM s preferences, if for any actions a and b from set A, u(a) > u(b) if and only if the DM prefers a to b

10 Example: A payoff function Three vacation packages to choose from A person prefers the package to Istanbul to other two, which h she regards as equivalent Her preferences between three packages are represented by any payoff function u that assigns the same number to Paris and London and a higher number to Istanbul For example: u(istanbul)=10, u(paris)=u(london)=1, or u(istanbul)=0, ( b ), u(paris)=u(london)=-2( ) ( ) 19 Payoff functions convey ordinal information A DM s preferences convey only ordinal information 20 The preferences may inform that a DM prefers a to b, and b to c, but they do not inform how much the DM prefers a to b, or whether she prefers a to b more than b to c Consequently, a payoff function that represents a DM s preferences also conveys only ordinal information

11 Example: Ordinal information Suppose: u(a)=100, u(b)=10, u(c)=9 21 These numbers do not imply pythat the DM finds a ten times better than b, neither they mean that the DM prefers b to c only slightly The numbers only state t that t the DM prefers a to b and b to c Another payoff function v for which v(a)=3 3, v(b)=2 2, and v(c)=1 1 represents this DM s preferences equally well In general, this DM s preferences are represented equally well by any other payoff function w, for which w(a)>w(b)>w(c) The theory of rational choice 22 Allowing that there are several equally attractive best actions, the rational choice theory states the action chosen by a DM is at least as good, according to her preferences, as every other available action In economics, you can find the theory of rational choice in, e.g., Economic theory of the consumer Theory of the firm

12 What comes next? In the theory of rational choice, a DM controls all variables that affect her 23 In this course, we will study situations in which some of the variables that affect a DM are actions of other DMs Examples: Firms competing for business: each firm controls own price but not others prices but others prices affect its sales Candidates running for a political office: A candidate cares not only about her own political platform but also those of her rivals, which affect her winning chances