Principles of Game Theory

Principles of Game Theory Continuous Strategies: Theory and Applications

Administrative • Quiz 2 next week • Problem Set 2 posted • A bit longer / harder (not impossible)

Topics • Normal form games with >3 players • Mathematical introduction • Introducing notation for what you already know. • Games with continuous strategy spaces

Last time • Normal form games: • Methods to find equilibria

Equilibrium Selection • Multiple equilibria : which one? • Focal points

Multiple players • While aX b matrixes work fine for two players (with relatively few strategies – a strategies for player 1 and b strategies for player 2), we can have more than two players: aX bX … X z

Back to the Basics What are the main components of a game? • Players • Strategies (or you can take Actions as primitive and then work up to Strategies) • Payoffs • Information Environment

Common Notation • Players of an n-player game: • Set of players • Generic players • Often we need to talk about all other players:

Common Notation • Each player i has a strategy space Si with generic element • The set of strategies in the game is just the product space:

Common Notation • Payoffs to each player are dependent on what strategies where selected: • Given the applied nature of the course, we won’t talk about how we get ui– we’ll just assume it’s given. • Given this, a normal form game is simple:

Dominance Now we can be a little more precise in some of our concepts from last time. Strictly dominated:

Nash Equilibrium

Continuous Strategy Spaces • Note that we’ve said nothing about the structure of the strategy spaces. • We will for existence (a bit technical). In general, we don’t need much • Duopoly / Oligopoly competition is a natural game setting. • Consider the classic Cournot model of competition • Firms choose quantity to produce of the same good • Market price is a function of aggregate supply and demand.

Cournot Duopoly • Two firms produce and sell the same product • Each firm chooses how much of the product to produce • Each firm has the following cost function (no fixed costs and constant marginal cost) • C(qi) = cqifor some 0 < c < a • Let qi be firm i’s output • Aggregate output is • Q = q1 + q2 • Assume the market clearing price is given by • P(Q) = a – Q

Cournot Duopoly • Firms want to maximize profits • Profit for i is given by:

Why is it a game? • Who are the players? • Firms 1 and 2 • What are the strategies? • Quantity produced: strategy space for i is Si= [0,∞) • What are the payoffs? • Profits.

Solving it • How would you solve it as a game?

Best Response functions • Definition of a Nash equilibrium: • Each firm i is finding a qi* that solves: • Now we’re back to simple optimization

First Order Conditions Recall So the FOC’s are: And the SOC: But what is ??

Best Response functions • The FOC gives you the best response function

Best Response functions • The FOC gives you the best response function • Mutual Best Response • Nash Equilibrium

Solving for the eq • At the Nash Eq, the following both hold: So Substituting:

Finding the eq quantity • Simple algebra: • From q* it’s straightforward to calculate the aggregate quantity produced, clearing price, and firms’ profits

Next time • More continuous strategies • Bertrand Competition • Experimental evidence • Homework 2 is online. A bit longer; start it now.

Principles of Game Theory