# Recitation 7 Notes

## Topics

• What is a game?
• Normal form games
• Equilibria

## Games

Why game theory? Games on networks!

Ex. congestion, international trade, Amazon's new office location, peer effects in school learning, deciding state taxes.
A game is a representation of strategic interaction.

### Example: Prisoner's Dilemma

2 Silent   2 Confess
1 Silent -2, -2 -20, 0
1 Confess 0, -20 -10, -10

### Example: Cournot Competition

How many iPhones should Apple produce?

• Apple produces q1 iPhones at marginal cost \$500.
• Samsung produces q2 Galaxies at marginal cost \$500.
• Price given by inverse demand = 2000 — Q, qq2.
• Apple's profit given by Pq1 — \$500 * q1.
• Samsung's profit given by Pq— \$500 * q2.

## Normal Form Games

Formally, a game consists of 3 elements:

1. The set of players N.
2. The sets of strategies {Si}i∈N.
3. The sets of payoffs {ui: S → ℝ }i∈N.

### Example: Prisoner's Dilemma

• N = {1, 2}
• S1 = {silent, confess}, S2 = {silent, confess}
• u1 : S1 * S2 → ℝ and u2 : S1 * S2  → ℝ are given by the table, where u1 is red and u2 is blue.

2 Silent   2 Confess
1 Silent -2, -2 -20, 0
1 Confess 0, -20 -10, -10

### Example: Cournot Competition

• N = {1, 2}
• S1 = [0, ∞), S2 = [0, ∞)
• We ignore that q must be integers.
• u1 : S → ℝ and u2: S → ℝ given by
u(q1, q2) = (P — \$500)q1 = (\$2000 — q1q2 — \$500)qi

In many cases, the sets of strategies have some structure:

1. Simultaneous games (penalty kicks in soccer).
2. Repeated games (Libor rate manipulation scandal).
3. Sequential games (how should US respond to china's tariffs?).

What happens when there is a game-like situation?
There are many variations...

• Weak prediction: "Dominated strategies are never played."
• Strong prediction: "Mutually optimal strategies are played."

Elimination of strictly dominated strategies

### Example: Prisoner's Dilemma ### Example: Battle of the Sexes No elimination needed.

## Equilibria

Nash equilibrium - A state with no incentive to deviate that can be sustained.

Given the opponents' strategies, what would you do?
"Best response correspondence" Bi : S-iSi

• Bgirl(musical) = {musical}
• Bgirl(soccer) = {soccer}
• Bboy(musical) = {musical}
• Bboy(soccer) = {soccer}

⇒ (M,M) and (S,S) are mutually optimal; "nash equilibria."

When the best response correspondence only has one element, we may instead use the best response function (Bgirl(musical) = musical).

### Example: Cournot Competition

Given Samsung's production q2, Apple wants to maximize its profits u1(q1, q2)=(1500 — q1 — q2)q1. That is, B1(q2) = ½(1500 — q2). Similarly, B2(q1)= ½(1500 — q2).

Nash equilibrium is the fixed point: 