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Game Theory Chapters 1-12: No Strategic Interaction Between Agents Chapter 13 – Part 1: Game Theory Chapters 1-12: No strategic interaction between agents • Consumer’s problem: prices are taken as given • Cost minimization problem: output and input prices are taken as given • Competitive firm’s problem: choose output independently of output decisions of other firms Economic reality: normally payoffs depend on own choices as well as choices of others agents interact strategically when they make choices. Examples: 1. You bargain over the price when buying a second-hand car 2. Market with few sellers. Profits of each seller depends on your price quote and quotes of other sellers 3. Monopolistic market. Entry decision of potential entrant depends on anticipated reaction incumbent monopolist Paul Schure, University of Victoria, Econ 203 Chapter 13 – Part 1: Game Theory Slide 1 Introduction Game Theory • The way to model situations of strategic interaction • Von Neumann and Morgenstern (1944) • Enormous range of economic applications • We only discuss non-cooperative game theory Setup • Defining and describing games The extensive-form representation (the ‘game tree’) The normal-form representation • Equilibria (equilibriums, solutions) for games How do players actually play games? Dominant strategies, minimax strategies, etc Equilibrium concepts: Nash equilibrium, Equilibrium in dominant strategies, Subgame perfect (Nash) equilibrium Paul Schure, University of Victoria, Econ 203 Chapter 13 – Part 1: Game Theory Slide 2 Describing Games Defining and describing Games Game (of strategy): the definition • (Informal definition) A player is engaged in a game of strategy if her payoff depends on her actions as well as on the actions of the other players Player agent Payoff utility Payoffs of all agents together the outcome (of the game) Example: A game of chess Example: Bargaining over buying a car • (Formal definition) A game is comprised of a set of players and an abstract set of ‘rules’ that (a) constraints the actions of the players and (b) defines the outcome that follows from the actions of the players Paul Schure, University of Victoria, Econ 203 Chapter 13 – Part 1: Game Theory Slide 3 Describing Games Describing a game involves a description of • The players • Their possible actions The set of possible actions The sequence of actions The information the players have when they decide • The outcome following the actions, i.e. payoffs for all players This is the extensive-form representation of the game Paul Schure, University of Victoria, Econ 203 Chapter 13 – Part 1: Game Theory Slide 4 Describing Games Another, equivalent way to describe a game • The players • Their possible strategies • The outcome following these strategies This is the normal-form or strategic-form representation Notes • A convenient way to describe the game in its extensive form is to use a “game tree” • If there are 2 players, a convenient way to describe the game in its strategic form is to use a “matrix” • Action = “move” ≠ strategy • Strategy = “plan of action” • Understanding the difference between action and strategy is very important Paul Schure, University of Victoria, Econ 203 Chapter 13 – Part 1: Game Theory Slide 5 Describing Games Example 1: A dynamic game; extensive-form representation Paul Schure, University of Victoria, Econ 203 Chapter 13 – Part 1: Game Theory Slide 6 Describing Games Example 2: A simultaneous-move game; extensive-form rep n. Paul Schure, University of Victoria, Econ 203 Chapter 13 – Part 1: Game Theory Slide 7 Describing Games Example 2: The normal-form representation Toshiba DOS Unix IBM DOS (600,200) (100,100) Unix (100, 100) (200,600) Paul Schure, University of Victoria, Econ 203 Chapter 13 – Part 1: Game Theory Slide 8 Describing Games Class question: derive the normal-form representation of the game of Example 1 • Underscores the difference between actions and strategies IBM has two different strategies and Toshiba four . Paul Schure, University of Victoria, Econ 203 Chapter 13 – Part 1: Game Theory Slide 9 Describing Games There are a thousand ways to classify games 1. Games of perfect and imperfect information • What information do the players have when they move? • Game of perfect information Every player knows all the prior actions at all times Example 1 is a game of perfect information • Game of imperfect information At some stage some player does not know all the prior actions Example 2 is a game of imperfect information • Information sets 2. Simultaneous-move games vs dynamic games • Dynamic game = sequential game (see section 13.9) Paul Schure, University of Victoria, Econ 203 Chapter 13 – Part 1: Game Theory Slide 10 Another example Example 3: The matching pennies game • Simultaneous-move game • Zero-sum game Extensive-form Normal-form representation representation Child 2 Heads Tails Ch.1 Heads (-1,1) (1,-1) Tails (1,-1) (-1,1) Paul Schure, University of Victoria, Econ 203 Chapter 13 – Part 1: Game Theory Slide 11 Class experiment Class experiment: The Prisoner’s dilemma • Experiment Imagine you manage either Ford or GM General Motors High price Low price High price (2,2) (-1,3) Ford Low price (3,-1) (0,0) Write down your name and action/strategy {High price, Low price} on piece of paper Seal your papers. I match players by random draws Paul Schure, University of Victoria, Econ 203 Chapter 13 – Part 1: Game Theory Slide 12 Solving games: the equilibrium Equilibria of games We have now mastered the skill to describe games Question: How do players actually play games? • Difficult question! You can only give an answer when you know the psyche of the players Economists • Assume players consider only their own payoff • Assume specific equilibrium concept (= “mindset”) • Players play an equilibrium Equilibrium of a game (= solution of a game): situation in which no player wishes to change her strategy Paul Schure, University of Victoria, Econ 203 Chapter 13 – Part 1: Game Theory Slide 13 Solving games: the equilibrium So, how do you find the equilibria of a game? 1. Make assumption about the mindset of the players. In formal language: choose an equilibrium concept (solution concept) 2. Compute what is best for each player given this equilibrium concept We shall study three equilibrium concepts in Econ 203 • Dominant strategy equilibrium • Nash equilibrium • Subgame perfect Nash equilibrium Note: In Econ 203 we focus on pure strategies , not on so- called mixed strategies Paul Schure, University of Victoria, Econ 203 Chapter 13 – Part 1: Game Theory Slide 14 Equilibrium in dominant strategies Dominant strategy equilibrium • Players play dominant strategies : strategies that are best independently of the choice of other players • Example: the prisoner’s dilemma experiment • Problem: normally there is no dominant strategy equilibrium • Example: Player 2 Strategy 1 Strategy 2 Pl.1 Strategy 1 (4,4) (4,4) Strategy 2 (0, 1) (6,3) • Partial solution: assume a little more rationality equilibrium after elimination of dominated strategies Elimination of dominated strategies • In example above Strategy 1 is dominated by S2 for Player 2 Paul Schure, University of Victoria, Econ 203 Chapter 13 – Part 1: Game Theory Slide 15 Nash equilibrium The Nash equilibrium (NE) • Set of strategies (one for each player) such that no player wishes to change her strategy given the strategies of the other players The strategy of each player is a so-called best response to the given strategies of the other players • The Nash equilibriu m : Assume there are n players, and that each chooses a strategy si , i =1,...,n . Let the payoff of player i π with strategies ( s1, s2 ,..., sn ) be i (s1, s2 ,..., sn ). * * * The strategy choices ( s1 , s2 ,..., sn ) are called a Nash equilibriu m π * * * ≥ π * * if for all players i we have : i (s1 ,..., si ,..., sn ) i (s1 ,..., si ,..., sn ) for all possible strategy choices si of player i. Paul Schure, University of Victoria, Econ 203 Chapter 13 – Part 1: Game Theory Slide 16 Nash equilibrium Example Nash equilibrium: a coordination game • Who calls the other after the phone line disconnected? Player 2 Call back Wait for other Pl.1 Call back (0,0) (3,6) Wait for (6,3) (0,0) other • Example points to drawback of Nash equilibrium the equilibrium need not be unique • Another drawback of the Nash equilibrium see next slide Paul Schure, University of Victoria, Econ 203 Chapter 13 – Part 1: Game Theory Slide 17 Subgame Perfect Nash Equilibrium (SPNE) The subgame perfect Nash equilibrium Limitation of the Nash equilibrium: Strategies of players can be inconsistent: they can contain non-credible threats • Example Player 1: child Left: go to practice Right: throw fit Player 2: parents Left: persist Right: give in Paul Schure, University of Victoria, Econ 203 Chapter 13 – Part 1: Game Theory Slide 18 Subgame Perfect Nash Equilibrium (SPNE) Subgame Perfect Nash equilibrium • Each player best responds to given strategies of others + Strategies cannot contain non-credible threats • Formally: SPNE = NE which is also NE in all of the subgames Q: How do I find a subgame perfect Nash equilibrium? A: Take game tree and use method called backward induction • Remember, that the SPNE is a set of strategies , not an outcome or a sequence of actions Paul Schure, University of Victoria, Econ 203 Chapter 13 – Part 1: Game Theory Slide 19 Subgame Perfect Nash Equilibrium (SPNE) Example: Backward induction in action Paul Schure, University of Victoria, Econ 203 Chapter 13 – Part 1: Game Theory Slide 20 Notes How to find equilibria of games? • Nash equilibria and equilibria in dominant strategies use the normal form representation whenever possible • Subgame perfect Nash equilibria always use game tree Hints when studying this chapter • Practice helps: Go through many examples and exercises • Empathy helps: pretend you are the player when you think about the player’s optimal action. The equilibrium concept tells you how you think when you are that player • Distinguish actions (moves) from strategies • The most important equilibrium concepts are the Nash equilibrium and subgame perfect Nash equilibrium • Not on exam: “Minimax” strategies, 13.10 (contestable markets), 13.12-13.14 Paul Schure, University of Victoria, Econ 203 Chapter 13 – Part 1: Game Theory Slide 21.
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