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Principles of Game Theory

Principles of Game Theory. Lecture 5: Continuous Strategies. Administrative. Quiz 1 results by this afternoon Stop by to pick up the hardcopy after 3 Homework 2 due Thursday Comments from homework 1: zero-sum games. Exam 1 in 2 weeks. Parts 1 and 2 of the textbook (chapters 1-8).

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Principles of Game Theory

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  1. Principles of Game Theory Lecture 5: Continuous Strategies

  2. Administrative • Quiz 1 results by this afternoon • Stop by to pick up the hardcopy after 3 • Homework 2 due Thursday • Comments from homework 1: zero-sum games. • Exam 1 in 2 weeks. • Parts 1 and 2 of the textbook (chapters 1-8)

  3. Games with continuous strategies • Last time: Cournot duopoly model • Drawing the best response functions and looking for places where they all meet will always produce the Nash equilibria of the game • Problem? • Graphing them can be really nasty. Especially with >2 players • A more common approach is “Guess and Verify” if you’re looking for a specific equilibrium.

  4. Hotelling Location Model • Assume there are two stores trying to figure out where to locate in the town (simultaneously) • There’s only one street in the town (Main St) • People live uniformly along Main St • And go to the closest store (same products) • Everyone goes to the store • Firms want to maximize the number of customers. Where should they locate?

  5. Auctions • We’ll (hopefully) spend more time on auctions later in the mini, but we’ve already covered enough to think about finding the equilibria: 2nd price sealed bid auction: • Players? n >= bidders • Each bidder, i, values the object vi • Actions? • Any feasible bid bi in (0,bmax) • Payoff? • If i is the highest bidder:vi – bj where bj is the bid by the next highest bidder • A k-way tie is broken using an appropriate k-sided fair device (coin). • 0 otherwise.

  6. 2nd price auctions There are many equilibria to this game. • Assume • Is bidding your valuation an eq? • What about the following?

  7. Back to Nash • What is (are) the eq of the following? • Any issues?

  8. Risk in Nash • What about the following? • One common complaint about Nash Equilibria is a lack of consideration of risk

  9. Game playing • U: 10 pence • Player 1 moves first by proposing some amount of the 10p to give to Player 2. • Player 2 can either agree and keep it or turn it down. If Player 2 turns it down, both players get 0. • T: 10 pence • Player 1 moves first by proposing some amount, x, of the 10p to give to Player 2. Whatever allocation Player 1 gave is multiplied by 3  3x • Player 2 can allocate the 3x however s/he wants between the two players. • D: 10 pence • Player 1 moves first by proposing some amount of the 10p to give to Player 2. • Player 2 takes what Player 1 gives. Does not move. • BC: 100 pence • Everyone chooses a number in [0,100]. The person who chooses the number closest to 2/3’s of the average of everyone’s numbers gets 100p. If there is a tie, those who tie divide the prize evenly.

  10. Experimental evidence • Often in experiments we observe • “too much” cooperation (prisoners’ dilemma games) • “too much” concession (bargaining games) • Learning. • Is game theory wrong? • Eh… it’s a bad question • Yes and no.

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