### 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
*q*_{1}iPhones at marginal cost $500. - Samsung produces
*q*_{2}Galaxies at marginal cost $500. - Price given by inverse demand
*P*= 2000 —*Q*,*Q*=*q*_{1}+*q*_{ } - Apple’s profit given by
*Pq*_{1}*—*$500 **q*_{ } - Samsung’s profit given by
*Pq*_{2}— $500 **q*_{ }

### Normal Form Games

Formally, a game consists of 3 elements:

- The set of players
*N.* - The sets of strategies {
*Si*}i∈*N.* - The sets of payoffs {
*ui*:*S*→ ℝ }i∈*N.*

#### Example: Prisoner’s Dilemma

*N*= {1, 2}*S*_{1}= {silent, confess},*S*_{2}= {silent, confess}*u*_{1}:*S*_{1}**S*_{2}→ ℝ and*u*_{2}:*S*_{1}**S*_{2}→ ℝ are given by the table, where*u*_{1}is red and*u*_{2}is blue.

2 Silent | 2 Confess | |
---|---|---|

1 Silent | -2, -2 | -20, 0 |

1 Confess | 0, -20 | -10, -10 |

#### Example: Cournot Competition

*N*= {1, 2}*S*_{1}= [0, ∞),*S*_{2}_{=}[0, ∞)- We ignore that
*q*must be integers.

- We ignore that
*u*_{1}:*S*→ ℝ and*u*_{2}:*S*→ ℝ given by

*u*_{i}(*q*_{1},*q*_{2}) = (*P —*$500)*q*_{1}= ($2000 —*q*_{1}—*q*_{2}— $500)*q*_{i}

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

- Simultaneous games (penalty kicks in soccer).
- Repeated games (Libor rate manipulation scandal).
- 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” B_{i} : *S*_{-i} → *S*_{i}

- B
_{girl}(musical) = {musical} - B
_{girl}(soccer) = {soccer} - B
_{boy}(musical) = {musical} - B
_{boy}(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 (B_{girl}(musical) = musical).

#### Example: Cournot Competition

Given Samsung’s production *q*_{2}, Apple wants to maximize its profits

*u*_{1}(*q*_{1}, *q*_{2})=(1500 — *q*_{1} — *q*_{2})*q*_{}

_{\n<section data-uuid=}

That is, B_{1}(*q*_{2}) = ½(1500 — *q*_{2}). Similarly, B_{2}(*q*_{1})= ½(1500 — *q*_{2}).

Nash equilibrium is the fixed point: