Basic Game Theory. Economics Auction Theory. Instructor: Songzi Du. Simon Fraser University. September 7, 2016

Transcription

1 Basic Game Theory Economics Auction Theory Instructor: Songzi Du Simon Fraser University September 7, 2016 ECON 383 (SFU) Basic Game Theory September 7, / 7

2 Game Theory Game theory studies the interaction of rational players: make prediction about behavior gives guidance on strategy assess and improve the design (i.e., the rules) of the game Basic ingredients of a game: players information strategies payoffs ECON 383 (SFU) Basic Game Theory September 7, / 7

3 An Example You have a group presentation and an exam tomorrow. Group presentation consists of you and a partner. ECON 383 (SFU) Basic Game Theory September 7, / 7

4 An Example You have a group presentation and an exam tomorrow. Group presentation consists of you and a partner. Expected grade for presentation, both people prepare 100 one person prepares 92 nobody prepares 84 ECON 383 (SFU) Basic Game Theory September 7, / 7

5 An Example You have a group presentation and an exam tomorrow. Group presentation consists of you and a partner. Expected grade for presentation, both people prepare 100 one person prepares 92 nobody prepares 84 Expected grade for exam, study 92 not study 80 ECON 383 (SFU) Basic Game Theory September 7, / 7

6 An Example You have a group presentation and an exam tomorrow. Group presentation consists of you and a partner. Expected grade for presentation, both people prepare 100 one person prepares 92 nobody prepares 84 Expected grade for exam, study 92 not study 80 Suppose you only have time either to prepare for the presentation or to study for the exam (not both) tonight. Likewise for your partner. What should you do? ECON 383 (SFU) Basic Game Theory September 7, / 7

7 Formal setup Suppose there are n players, acting simultaneously. Complete information Payoff function: P i (S 1, S 2,..., S n ) where S i S i is the strategy of player i. ECON 383 (SFU) Basic Game Theory September 7, / 7

8 Formal setup Suppose there are n players, acting simultaneously. Complete information Payoff function: P i (S 1, S 2,..., S n ) where S i S i is the strategy of player i. Strategy S i is player i s best response to (S 1,..., S i 1, S i+1,... S n ) if P i (S 1,..., S i 1, S i, S i+1,..., S n ) P i (S 1,..., S i 1, S i, S i+1,..., S n ) for every strategy S i S i. ECON 383 (SFU) Basic Game Theory September 7, / 7

9 Dominant strategy and Nash equilibrium A strategy profile is a combination of strategies for different players. A strategy S i is the (weakly) dominant strategy for player i if 1 S i is a best response to every strategy profile of other players, 2 for every other strategy S i, there is a strategy profile of other players such that S i is strictly better than S i for player i. ECON 383 (SFU) Basic Game Theory September 7, / 7

10 Dominant strategy and Nash equilibrium A strategy profile is a combination of strategies for different players. A strategy S i is the (weakly) dominant strategy for player i if 1 S i is a best response to every strategy profile of other players, 2 for every other strategy S i, there is a strategy profile of other players such that S i is strictly better than S i for player i. A strategy S i is the strictly dominant strategy for player i if S i is the strict best response to every strategy profile of other players. ECON 383 (SFU) Basic Game Theory September 7, / 7

11 Dominant strategy and Nash equilibrium A strategy profile is a combination of strategies for different players. A strategy S i is the (weakly) dominant strategy for player i if 1 S i is a best response to every strategy profile of other players, 2 for every other strategy S i, there is a strategy profile of other players such that S i is strictly better than S i for player i. A strategy S i is the strictly dominant strategy for player i if S i is the strict best response to every strategy profile of other players. A strategy S i (strictly or weakly) dominates another strategy S i if S i is (strictly or weakly) better than S i for player i given every strategy profile of other players. ECON 383 (SFU) Basic Game Theory September 7, / 7

12 Dominant strategy and Nash equilibrium A strategy profile is a combination of strategies for different players. A strategy S i is the (weakly) dominant strategy for player i if 1 S i is a best response to every strategy profile of other players, 2 for every other strategy S i, there is a strategy profile of other players such that S i is strictly better than S i for player i. A strategy S i is the strictly dominant strategy for player i if S i is the strict best response to every strategy profile of other players. A strategy S i (strictly or weakly) dominates another strategy S i if S i is (strictly or weakly) better than S i for player i given every strategy profile of other players. A strategy profile (S 1, S 2,..., S n ) is a Nash equilibrium if for every player i, S i is a best response to the strategy profile (S 1,..., S i 1, S i+1,... S n ). ECON 383 (SFU) Basic Game Theory September 7, / 7

13 Examples Battle of Sexes Opera Football Opera 3, 2 0, 0 Football 0, 0 2, 3 Stag Hunt Stag Hare Stag 5, 5 0, 3 Hare 3, 0 3, 3 ECON 383 (SFU) Basic Game Theory September 7, / 7

14 A simple first-price auction A single object for auction. Two players, each has a value/willingness-to-pay v i for the object, commonly known. Assume v 1 = 2 and v 2 = 1. Each players submits a bid in {1, 2, 3}. Highest bidder wins, pays his own bid, and gets the object. If there is a tie, bidder 1 wins and pays. ECON 383 (SFU) Basic Game Theory September 7, / 7

15 A simple first-price auction A single object for auction. Two players, each has a value/willingness-to-pay v i for the object, commonly known. Assume v 1 = 2 and v 2 = 1. Each players submits a bid in {1, 2, 3}. Highest bidder wins, pays his own bid, and gets the object. If there is a tie, bidder 1 wins and pays. bid 1 bid 2 bid 3 bid 1 1, 0 0, -1 0, -2 bid 2 0, 0 0, 0 0, -2 bid 3-1, 0-1, 0-1, 0 ECON 383 (SFU) Basic Game Theory September 7, / 7