Introduction to Game Theory

Gabriel Carroll, Econ 180, Fall 2014

September 22, 2014

1 Logistics

Go through syllabus overview, emphasizing:

mathematical background (no econ needed);

differences from other game theory classes;

grading policy, including midterm date (Mon Oct 27) and final exam (Fri Dec

12, 8:30 11:30 AM)

Textbooks

Tadelis is the most advanced undergrad book, its good but not deep enough

Fudenberg-Tirole is the standard grad book, not ideal for a first course

So the main source is the lectures; Ill post outlines and Xin will write more

detailed notes as class goes along

Will play a few games in class, with Xin administering

Ill try to learn who everyone is, but it will take a while; you are encouraged to come

to office hours

2 What are games?

Classic kind of problem in economic theory: decision problem one player makes

a decision

1

Example: a monopolist setting price p [0,) to maximize profit (pc)D(p),

where cost c > 0 is constant and D(p) is some continuous, decreasing, positive-

valued function

Game theory studies situations where two or more decision-makers interact, and

each ones preference may depend on what the others are doing

Example: two firms in Bertrand competition. Each sets a price p1, p2, and

all customers buy from the lowest-price firm, or they split evenly if prices are

equal. So firm 1s profits are (p1c)D(p1) if p1 < p2, (p1c)D(p1)/2 if p1 = p2,

and 0 if p1 > p2, and likewise for firm 2. Particular choices of c and D(p) (say,

c = 1 and D(p) = 1/(1 + p)) make this into a specific game

Firm 1 wants to just undercut whatever it thinks 2 will charge (unless 2 will

charge c)

Many more examples of situations modeled as games....

Industrial organization: firms deciding what products to develop, how much

to advertise

Labor economics: workers deciding how hard to compete for a promotion;

workers getting education to signal their ability to employers

Development economics: farmers deciding whether to experiment with a new

crop, or wait for their neighbors to experiment; deciding whether to share food

with neighbor in a famine year, hoping the neighbor can pay back later

Macroeconomics: firms setting prices based on expectations of future inflation

Finance: traders deciding how much of various assets to buy and sell, depending

on their private knowledge

Political science: legislators bargaining over how to divide a budget; countries

deciding how much to spend fighting a war; candidates choosing their platforms

to try to get elected

Urban planning: commuters deciding which road (or train, boat...) to take

Computer science: routers deciding where to send network traffic; hackers

looking for security holes

Evolutionary biology: animals level of aggression / territoriality

2

3 Formal definition and examples

Ingredients in a (normal-form / strategic-form) game G = (N,A, u):

N = {1, . . . , n} set of players

A = A1 An where Ai is is actions (or strategies)

u = (u1, . . . , un), ui : A R set of payoff functions the thing each player

wants to maximize

Notation a for action profile, ai for actions of opponents, Ai for set of action

profiles of opponents, (ai, ai)

Examples describe all the ingredients

Bertrand competition, as before

Coordination game:

A B

A 1, 1 0, 0

B 0, 0 1, 1

(explain matrix notation for finite two-player games)

Anti-coordination or matching pennies game (applications could include gen-

erals choosing where to attack and defend, or two firms, with one stronger than

the other, deciding what products to market):

H T

H 1,1 1, 1

T 1, 1 1,1

Prisoners dilemmaC D

C 2, 2 0, 3

D 3, 0 1, 1

Applications include any kind of simultaneous favor exchange, e.g. offering to

do an errand for a friend thats convenient for you; or not littering in the park.

Usually associated with the silly prisoner story

3

4 Dominance

We want to predict how people will behave in games

One prediction: people wont do something if theres something else obviously bet-

ter. Specifically, say action ai strictly dominates a

iif ui(ai, ai) > ui(a

i, ai) for

every ai Ai.

We usually are comfortable predicting that people wont play something strictly

dominated

Example: in prisoners dilemma, C strictly dominates D. So we predict (D,D) as

the unique outcome. Note that both players nonetheless prefer (C,C).

If players can foresee others rationality, we can sometimes predict more. Example:

the follow your crush to lunch game. 1 = you, 2 = your crush, A and B are places

to eat lunch. You like A better, but mostly just want to be near your crush; crush

likes B better and doesnt care about you.

A B

A 4, 0 1, 1

B 0, 0 3, 1

Iteratively deleting dominated strategies gets us to (B,B).

Will study this process in more detail next week....

5 Nash equilibrium

In many games, nothing is dominated. But would still like to make some kind of

prediction.

Some predictions better than others. For example, in coordination game, if we

confidently predicted (A,B), we shouldnt be so confident, since either player would

like to change.

N ash equilibrium: a strategy profile a A such that ui(a) ui(a

i, ai) for all i and

all ai Ai.

4

This is the most fundamental concept in game theory

A consistent prediction for what players will do

Solve some of the earlier examples....

Bertrand competition: only Nash equilibrium is (c, c). First, check this is an

NE. Then, show there are no others. (Sketch: WLOG can focus on profiles

with p1 p2; if p1 < c then 1 would rather charge more, if p1 > c then 2 would

rather undercut 1, and if p1 = c and p2 > c then 1 would rather charge more.

So only (c, c) can be NE)

Prisoners dilemma: only (D,D)

Coordination game: (A,A) and (B,B)

Matching pennies: none!

As we can see, games can have multiple NE, or none

In the matching pennies game, there is nonetheless an intuitive prediction: wed

expect to see each player choosing H and T each half the time.

There is an extension of NE that gives us this prediction, but first we need a little

detour....

5