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

Matching Pennies On blackboard, draw the extensive form game tree for matching pennies with perfect information where A moves first. Then do the same for simultaneous move version.

Clicker question How many strategies are available to the player that moves second in the perfect information version of matching pennies? A)2 B)4 C)6 D)8

Rock, Paper, Scissors

Rock-Paper-Scissors First let’s draw the game tree on the blackboard for the game played with perfect information, where Player A moves first. – Let Payoff be 1 if you win, -1 if you lose, 0 if you tie. How would we draw the game tree for this game if the players move simultaneously?

Clicker Question How many (pure) strategies are possible for Player B in the perfect information version of Rock-Paper-Scissors A)3 B)6 C)9 D)12 E)27

Strategies for B in perfect information rock, paper, scissors game A strategy for B in perfect information 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

What is a strategy? “A strategy is not a sequence of actions, but rather a catalog of contingency plans, what to do in every situation. ‘’ Harrington, page 34. For games with perfect information, this must specify a player’s action at each node at which it would be that player’s turn.

Information sets in imperfect information games Information set—A collection of nodes such that you the player whose turn it is does not know which of these nodes he is currently at. See Blackboard Examples: – Simultaneous move Rock, paper, scissors – Simultaneous move Matching pennies

Strategy in game with incomplete information A strategy must specify a player’s action at each information set at which it would be that player’s turn.

Clicker question How many (pure) strategies are available to each player in the simultaneous move matching pennies game? A)2 B)4 C)6 D)8

Clicker question How many strategies are available to each player in the simultaneous move rock-paper-scissors game? A)2 B)3 C)6 D)9 E)27

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.

Simultaneous move Matching Pennies Strategic Form of Game -1, 1 1,-1 -1,1 Heads Tails Heads Tails Player 2 Player 1

Rock, Paper, Scissors— Simultaneous Move 0,0-1, 11,-1 0,0-1,1 RockPaperScissors Rock Paper Scissors

More complicated game Player 1 Player D C F E GH 4 Possible Strategies for Player 1 : What are they? 2 Possible Strategies for Player 2: What are they?

Strategic Form 1, 2 3, 1 0, 0 3, 1 2, 0 Player 2 EF Player 1 C,G C,H D,G D,H

Prisoners’ Dilemma Game 10, 10 0, 11 11, 0 1, 1 CooperateDefect Cooperate Defect PLAyER 1 PLAyER 1 Player 2

Two rounds of Prisoners’ Dilemma Players 1 and 2 play two rounds of prisoners’ dilemma. Draw the game tree with information sets if they move simultaneously in each round, but can see results of round 1 before starting round 2. Now draw the game tree with information sets if they play two rounds but only see other’s plays at the end of the game.

Clicker Question Players A and B play two rounds of simultaneous move prisoners’ dilemma. They don’t get to see how the other player played until both rounds are over. How many strategies are possible for each player? A)2 B)4 C)8 D)16 E)32

Clicker Question Players A and B play two rounds of simultaneous move prisoners’ dilemma. Each gets to see the other’s move in round 1 before choosing an action for round 2. How many strategies are possible for each player? A)2 B)3 C)4 D)16 E)32

The game of Chicken James Dean story. 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. How do we make an interesting game of this?

0, 0 0, 1 1, 0-10, -10 Swerve Don’t Swerve

So long…at least for now.

Common Knowledge of a fact Three ladies in a railway car. All have dirty faces. They can see each other’s faces, but not their own. Each would blush visibly if she knew her own face was dirty. All are brilliant logicians and they all know this. The conductor comes into the car and announces for all to hear. “Someone in this car has a dirty face.”

Common Knowledge Why should this news matter? All three can see two dirty faces. In fact, all three know that the others can see at least one dirty face. Lady 1 says, Suppose that my face is clean. Then Lady 2 will see exactly one dirty face—that of Lady 3. Lady 2 will reason, “if my own face is clean, then Lady 3 will see 2 clean faces.” If Lady 3 saw 2 clean faces, she would know her face was dirty and would blush. If Lady 3 doesn’t blush, lady 2 would conclude that her own face is dirty and would blush. Therefore if Ladies 2 and 3 don’t blush, Lady 1 must conclude that her own face if dirty.

Moral of the story In this example, all of the ladies know that there is a dirty face in the car, but until the conductor comes, none of them know that the others all know that there is a dirty face. This additional bit of common knowledge leads them to a conclusion otherwise unavailable.