Extensive and Strategic Form Games Econ 171 Reminder: Course requirements • Class website Go to economics department home page. Under Links, find Class pages, then click on Econ 171 • Textbook: Games, Strategies, and Decision Making by Joseph E. Harrington, Jr. • Clicker: Available at campus bookstore i>clicker Register your clicker at www.i>clicker.com/registration Vocabulary for Extensive form games • • • • • Decision Tree Decision Node-Specifies whose turn Branches-Options Terminal Node—End of play Payoffs—For each person at each terminal node. • Strategy—What will you do at each decision node where it is your turn Rock, Paper, Scissors Rock-Paper-Scissors: Full information version, Alice moves first Alice Scissors Rock Paper Bob Rock 0 0 Bob Bob Rock Scissors Paper Paper Rock Scissors Paper 0 1 1 1 0 0 0 0 0 1 0 1 1 0 Scissors 0 0 Clicker Question How many strategies are possible for Bob in the perfect information version of RockPaper-Scissors when Alice moves first A) 3 B) 6 C) 9 D) 12 E) 27 What is a strategy in a game of perfect information? • “A strategy is not a sequence of actions, but rather a catalog of contingency plans, what to do in every situation. ‘’ Harrington, page 34. (Read this section with extra care.) • A strategy is a list stating what you would do at each possible decision node where it is your turn. Strategies for B in perfect information rock, paper, scissors game • A strategy for B in perfect rock, paper, scissors answers 3 questions: – what will I do if I see rock?, – what will I do if I see paper? – What will I do if I see scissors? • There are 3 possible answers to each question. Hence there are 3x3x3=27 possible strategies. • Examples: – Paper if rock, rock if paper, rock if scissors – Or Rock if rock, scissors if paper, paper if scissors And so on… 27 possibilities A tougher game Kidnapping Game Vivica’s strategies • How many strategies are possible for Vivica? A) 2 B) 3 C) 4 D) 8 E) 9 Guy’s Nodes • At how many decision nodes is it Guy’s turn? A) 2 B) 3 C) 4 D) 5 E) 6 Clicker Question • How many strategies are possible for Guy? A) 3 B) 4 C) 6 D) 8 E) 16 Come Again? Why is that • He has 3 nodes. He can do 2 things at each of them. That give us 2x2x2=8 strategies. Kidnap, Don’t Kill if R , Kill if No R Kidnap, Don’t Kill if R, Don’t Kill if No R Kidnap, Kill if R, Kill if No R Kidnap, Kill if R, Don’t Kill if No $ But also, Don’t Kidnap, Don’t Kill if R, Kill if No R, etc. Game theorist’s convention • A strategy lists what you would do at EVERY decision node where it is your turn, even if some things you can choose at an early node prevent you reaching a later node. • This is partly for convenience, but also it turns out to be useful to specify what you would do if you accidentally wound up at any node. Sequences of Action are not strategies • How many sequences of action are possible for Guy? A) 3 B) 4 C) 6 D) 8 E) 16 What are the sequences of actions • Kidnap, Kill • Kidnap, Don’t Kill • Don’t Kidnap Don’t confuse strategies with sequences of actions. Strategies tell us what you would do in any sequence of events. Historical sequences tell you only what did happen. Strategies and imperfect Information Imperfect information • The examples so far had perfect information. • When it is your turn, you know everything that has happened so far. • Sometimes, when it is your turn, you don’t know everything that has happened. • Example: Real rock-paper scissors game has simultaneous moves. Information sets • We deal with this by introducing the concept of information set. • When a player doesn’t know which node of the game she is at, we include all of all the nodes where she might be in a single information set. Rock-Paper-Scissors when players don’t know other’s move Alice Scissors Rock Paper Bob Rock 0 0 Rock Scissors Paper Paper Rock Scissors Paper 0 1 1 1 0 0 0 0 0 1 0 1 1 0 Scissors 0 0 Problem 3 a Player 1 b a Player 2 Player 2 x Player 3 High Low c y Player 3 High Low x y Player2 x y Problem 3 b b a Player 2 Player 2 x y Player 3 High Low c High Low x y Player2 x y Problem 3 c Player 1 b a Player 2 Player 2 x c y x Player2 y x Player 3 Player 3 High Low High High Low Low High Low y What is a strategy in a game of imperfect information? • Recall that In a game of perfect information, a strategy is a list stating what you would do at each possible decision node where it is your turn. • In a game of imperfect information, a strategy is a list stating what you would do at each possible information set where it is your turn. How many strategies does Bob have in simultaneous-move rock-paper-scissors? A) B) C) D) E) 3 6 9 27 4 Games in Strategic Form Details of strategic form game • Set of Players • For each player a strategy set—list of all the strategies that the player could choose. Remember that a strategy tells everything you would do on any occasion when its your turn. • Strategy profile: List of strategies chosen by every player. • Payoff to each player depends on the strategy profile that was chosen. Two player game matrix in strategic form Make a two-by-two table with one row for each strategy that player 1 could choose and one column for every strategy that player 2 could choose. Enter payoffs to players 1 and 2 in appropriate spots. Example: Simultaneous Move Matching Pennies • In this case each player has only two possible strategies. Choose Heads, Choose tails. • Payoff to Player 1 (row chooser) is written first, then payoff to Player 2. Matching Pennies Strategic Form of Game Player 2 Heads Player 1 Heads Tails Tails -1, 1 1,-1 1,-1 -1,1 Rock, Paper, Scissors— Simultaneous Move Rock Rock Paper Scissors 0,0 Paper -1,1 Scissors 1,-1 More complicated game Player 1 C D Player 2 Player 1 G 1 2 E F 3 1 H 2 0 0 0 4 Possible Strategies for Player 1 : What are they? 2 Possible Strategies for Player 2: What are they? Strategic Form Player 2 E Player 1 F C,G 1, 2 3, 1 C,H 0, 0 3, 1 2, 0 2, 0 2, 0 2, 0 D,G D,H The game of Chicken James Dean story. Two macho morons drive their dad’s Buicks down the middle of the road. Swerve or don’t swerve. Alternatively—Two animals both want a resource. Each has two possible strategies. Fight or give up. A fight is very bad for both of them. Chicken a la Extensive Form Player 1 Don’t Swerve Swerve Player 2 Swerve 0 0 Swerve Don’t Swerve 0 1 1 0 Don’t Swerve -10 -10 Strategic form of Chicken Game Swerve Swerve Don’t Swerve 0, 0 Don’t Swerve 0 , 1 1, 0 -10, -10 Player 1 Problem 9 a d c b Player 2 Player 2 x 4 2 Player 1 x 0 6 3 1 1 4 5 2 Player 2 x/x x/y y/x y/y a 4,2 4,2 1,3 1,3 b 2,2 2,2 0,6 0,6 c 3,1 4,2 3,1 4,2 d ? ? ? ? y x 1 3 2 2 x y y y 0 0 So long…at least for now.