Principles of Game Theory

Principles of Game Theory Lecture 6: Linking Simultaneous & Sequential Games

Administrative • Homework 2 due 5pm • Typo in the last problem – check piazza • Quiz 2 on Sunday.

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.

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.

Stage Games There are games that are both sequential and simultaneous • Classic examples include things like repeated games • Anything where there are multiple stages: investment and pricing

Stage Game Investment and pricing: • 2nd stage aka “subgame”

Order of moves • We’ve already seen examples of 1st and 2nd mover advantages. • Consider the simple Prisoner’s Dilemma:

Simultaneous  Sequential • What happens if the husband moves first?

Simultaneous  Sequential So why the big difference? • Hint: it’s not just because the husband gets to move first. Note the same outcome occurs if the wife moves first. • Very important:it’s that in moving first the husband knows that the wife can (and will) condition her actions on what he does

Another example What is (are) the equilibrium(ia) here:

Fed/Congress example What happens if the fed moves first?

Fed/Congress example When Congress moves first?

Fed/Congress • So we’ve seen examples of 1st mover and second mover, but the Fed/Congress game is a little different. • In this game both players want Congress to move first: payoffs of (3,4) vs (2,2). • We’ll see later that having Congress verbally say what it’s going to do isn’t enough. It must be a credible commitment. • So going from Simultaneous to Sequential can change the equilibrium outcome. • Figuring out what changes when you change the “rules” of the game can be very important

Sequential Games with Imperfect information • Imperfect vs incomplete information • Incomplete information: players are unaware of the payoffs of their the other players (or the players “types”) • We’ll move to this after the exam. All games so far are games of complete information. • Imperfect information: players are simply unaware of the actions chosen by other players. • They know who the other players are what their possible strategies/actions are, and the preferences/payoffs of these other players. • Information about the other players in imperfect information is complete.

Imperfect Information What happens when player 2 knows that player 1 moved first but doesn’t know what he did? • Before we said it would be “as if” it were simultaneous. Now we’ll see why. • What makes a sequential game different? • Players can condition their behavior on the history of play – or their information up to that node.

Imperfect information Tennis example from the book (fig 6.9) • N. doesn’t know what E. did:

Imperfect Information Imperfect information in Decision Analysis:

Congress/Fed • Back to the Congress Fed example: 3, 4

Congress/Fed • When the 2nd player can’t condition behavior on any history, we’re back to the original normal form game:

Representing Sequential Games as Simultaneous Most of you are probably thinking that extensive form games are easier to solve than normal form games. Well… this is where they get tricky (and often presented after normal form games) • When players can condition behavior, all players know this and it opens up the possibility for more equilibria

Subgame perfection • What are the strategies for each player below: 3, 4

Subgame perfection • Recall how we constructed normal form games: • A matrix of strategies • We now have 2 equilibria!?

Subgame perfection • There are 2 equilibria to this game • But only 1 of them is what we call a subgame perfect eq 3, 4

Subgames • How many subgames?

Subgame Perfection • An equilibrium is called a subgame-perfect equilibrium if it is an equilibrium in every subgame. • 2 equilibria:

Subgame Perfection • 2 Nash equilibria: • But only 1 subgame perfect eq

Principles of Game Theory