Presentation Transcript

1. Principles of Game Theory

   Lecture 2: Games with sequential moves Gasper: Principles of Game Theory
2. Administrative Meet with me Sorry if I missed you yesterday. I’ll be around all afternoon today Piazza Homework due Thursday 5pm Email is fine given the small class Any thing else? Gasper: Principles of Game Theory
3. Last time Classification of games Terminology Gasper: Principles of Game Theory
4. Agenda for today Traditional presentation of GT starts with simultaneous move games Initially more confusing but notion of “equilibrium” more immediate Books starts with sequential move games Initially much more intuitive Deeper understanding of equilibrium and strategies more confusing We’ll stick with the book and start sequential move games today and then come back to them Simultaneous move games on Thursday. But first a short case… Gasper: Principles of Game Theory
5. Toys ‘R Us Competitive Landscape Toy & department stores Do not compete with TRU on price (50% markups) General discounters (Wal-Mart, Target) Cannot compete with TRU on selection (9% markups) Toys exhibit high product differentiation Specialized discounters (only TRU) Enjoyed 20% market share nationally Warehouse clubs (Costco, Pace) Introduced in the 1980s By 1989, about 200 items in competition Cause of concern for Toys “R” Us! Gasper: Principles of Game Theory
6. Toys ‘R Us Market research into the future of warehouse clubs Manufacturers estimated 3-5% annual growth rate Toys “R” Us conducted comprehensive but proprietary study Manufacturers were aware of the study, but not its results Toys “R” Us expected warehouse clubs to grow quite fast Gasper: Principles of Game Theory
7. Toys ‘R Us What is the strategic environment Who are the players? (Decision makers) What strategies are available? (Feasible actions) What are the payoffs? (Objectives) What are the rules of the game? What is the time-frame for decisions? What is the nature of the conflict? What is the nature of interaction? What information is available? What would you do? Idea! Issue statement to manufacturers: You may not sell to warehouse clubs without losing Toys “R” Us as a customer Effectively, choose between a $5 billion market (TRU) and a less than $500M market (warehouse clubs) Gasper: Principles of Game Theory
8. Toys ‘R Us Toys ‘R Us strived to change the game: Broker a cooperative agreement among the toy manufacturers Serve as a clearinghouse for complaints Utilize consumer expectations to alter payoffs Annual industry convention & product launch Safety advisories & recalls Availability of compatible products Manufacturers agreed to TRU deal Warehouse club toy sales decreased Pre-agreement growth of 51% per year Overall growth of clubs of over 10% per year (Thanks to Mike Shor for the case info) Gasper: Principles of Game Theory
9. Sequential Moves Sequential move games: The key to sequential move games is anticipating behavior Games with sequential moves are often called extensive form games. Gasper: Principles of Game Theory
10. Sequential moves: Game trees Trees are often used to denote sequential structure Nodes (terminal and non-terminal) will represent choice, or action, nodes will represent chance nodes (“nature’s move”) Terminal node Usually list payoffs as a n-tuple with order being order of play Eg: (5, 7) = Player 1 gets 5; Player 2 gets 7 Branches Branches of choice nodes must be exhaustive (contain all possible actions available to the player at that node) Branches of chance nodes must add to 1 (they will be probabilities) Gasper: Principles of Game Theory
11. Game Tree Example: Gasper: Principles of Game Theory
12. Backward induction “Solving” sequential move games: Backward induction (aka Rollback) Start at the end: at each terminal node conclude what the player would choose Gasper: Principles of Game Theory
13. Backward induction “Solving” sequential move games: Backward induction (aka Rollback) Start at the end: at each terminal node conclude what the player would choose Fold back the tree and repeat. Gasper: Principles of Game Theory
14. Backward induction “Solving” sequential move games: Backward induction (aka Rollback) Start at the end: at each terminal node conclude what the player would choose Fold back the tree and repeat. Gasper: Principles of Game Theory
15. Outcomes vs Strategies The rollback equilibrium outcome: Alice goes and Bob stops But these aren’t strategies Gasper: Principles of Game Theory
16. Outcomes vs Strategies What are the strategies? Complete contingent plan Alice: 2 strategies: Go or Stop Bob: 4 strategies If Alice plays Go, play Go If Alice plays Go, play Stop If Alice plays Stop, play Go If Alice plays Stop, play Stop Gasper: Principles of Game Theory
17. Outcomes vs Strategies Simple, but… The set of strategies that form the equilibrium are not always simply the equilibrium path of play. Strategies are: Specified before the game starts. They’re contingent on the history of play They’re complete – must condition on every possible history. Gasper: Principles of Game Theory
18. More than 2 players How many strategies does each player have? Emily? 2 Nina? 4 Talia? 16. Why? 4 histories 2 actions Gasper: Principles of Game Theory
19. More than 2 actions Each player can have multiple actions: 2x2 tic-tac-toe Gasper: Principles of Game Theory
20. Example Games:Centipede Consider the following 2-person sequential game: A pot of money can be taken by a player when it’s his turn. If he takes it, it’s his to keep. If he passes to player 2, the money grows by 1QR and player 2 now faces the same decision (to keep or pass and let the pot grow). The game lasts at most 10 rounds and if by the end no one takes the pile it disappears. Gasper: Principles of Game Theory
21. Centipede In the book: Gasper: Principles of Game Theory
22. Example Games:“Survivor” There is a pile of 21 stones Each round players sequentially take turns taking 1, 2, or 3 stones. Must take at least 1. Can’t take more than 3. The person who takes the last stone, loses the game Gasper: Principles of Game Theory
23. Homework Do problems in Chapter 3: U2 (look at S2 as a guide) U3 U9 Read chapter 4 Gasper: Principles of Game Theory