14.15J | Spring 2018 | Undergraduate

Networks

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 P = 2000 — Q, Qq1q
  • Apple’s profit given by Pq1 $500 * q
  • Samsung’s profit given by Pq2 — $500 * q

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       
    ui (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

 

2 by 2 table with three options crossed out.

 

Example: Battle of the Sexes

2 by 2 table with two options circled.

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)q

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That is, B1(q2) = ½(1500 — q2). Similarly, B2(q1)= ½(1500 — q2).

Nash equilibrium is the fixed point:

Mathematical equation.

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