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_{2.}
 Apple's profit given by Pq_{1} — $500 * q_{1.}
 Samsung's profit given by Pq_{2 }— $500 * q_{2.}
Normal Form Games
Formally, a game consists of 3 elements:
 The set of players N.
 The sets of strategies {S_{i}}_{i∈}_{N.}
 The sets of payoffs {u_{i}: 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.

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 gamelike 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_{1.}_{}
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: