CO 456: Pivoting and Lemke-Howson Examples
Sept 28
My laptop is in for repairs, so I didnt have time to prepare a slideshow for this lecture. Here are some
notes that follow the lecture topic.
1
Quick Review
Recall the polyhedra P and Q for a 2-player gam

CO 456: Assignment 1
Due Wed, September 26 in class.
Include your name and ID number.
Write the number of pages on the front page and number each page.
Problem 1 - 5 marks
If k = n k , then the only PNE is when all citizens vote. To see that this is a PNE

CO 456: Assignment 1
Due Wed, September 26 in class.
Include your name and ID number.
Write the number of pages on the front page and number each page.
Problem 1 - 5 marks
Two candidates A and B compete in an election. There are n citizens and k support c

CO 456: Project Solutions
Problem 1.
Consider this example with n players and n stores. We have v1 = n +1 and vj = 1 for all 2 j n.
Now, OP T = 2n because we could assign player i to store i and have each store visited. I claim
that the only equilibrium h

CO 456: Project
The entire project will be marked out of 50 marks and accounts for 5% of the overall grade. It
may be completed individually or in a group of two. Projects with two students will be marked
the same as a project with one student, and both s

CO 456
Coalition Games
Zachary Friggstad
October 31, 2012
Last Time
We introduced the core: the set of all actions aN for the grand
coalition such that no coalition S has an action that all of its
members prefer to aN .
We spent quite a while on a larger

CO 456
Coalition Games and the Core
Zachary Friggstad
October 29, 2012
Coalition Games
Reminder:
In a coalition game, every nonempty subset of players (a coalition)
has a set of actions and the players have preferences/payouts over
all actions in which th

CO 456
Extensive Games and Coalition Games
Zachary Friggstad
October 26, 2012
Last Lecture
The one deviation principle: to check that given strategies form an
SPE in a nite horizon game, it suces to check that no player can
improve their payo by changing

CO 456
More on Extensive Games
Zachary Friggstad
October 24, 2012
Subgames
There is a very satisfying resolution to this problem of articial
equilibria. To do this, we rst introduce the notion of a subgame.
Let be an extensive game with player function P

CO 456
More on Extensive Games
Zachary Friggstad
October 22, 2012
Extensive Games Summary
We were discussing extensive games: games where the players take
turns playing. A player may use the knowledge of previous actions
to determine their play.
One can v

CO 456
Extensive Games
Zachary Friggstad
October 19, 2012
Extensive Games
So far, the model we have studied has been where all players make
their moves simultaneously.
In many cases, they take turns: the actions of a player can be
decided with previous pl

CO 456
Congestion Games
Zachary Friggstad
October 17, 2012
Best Response Dynamics
We turn our attention to proving the existence of an equilibrium.
This is a bit dierent than in previous problems we have discussed.
The objective function can get worse aft

CO 456
Congestion Games
Zachary Friggstad
October 15, 2012
Congestion Games
Recall that in a congestion game we have:
A directed graph G = (V , E ).
Non-decreasing edge capacity functions ce : R0 R0 for
each edge e E .
A set of k players, each given by a

CO 456
Price of Anarchy in Load Balancing and
Introduction to Congestion Games
Zachary Friggstad
October 12, 2012
Load Balancing
Recall that we have m machines with speeds sj and n jobs with
weights wi . The jobs are the players and their action is to cho

CO 456
Introduction to Pure Nash Equilibrium
Zachary Friggstad
September 12, 2012
Pure Nash Equilibrium
Our rst topic in the course will be analyzing of how experienced,
single-minded players behave in a strategic game.
Informally, our assumptions mean th

CO 456
More Pure Nash Equilibrium
Zachary Friggstad
September 14, 2012
Best Responses
Suppose ai is an action prole for all players except player i .
Denition
ai Ai is a best response to ai if ui (ai , ai ) ui (ai , ai ) for all
other actions ai Ai .
An o

CO 456: Assignment 5
Due Wed, Nov 21 in class.
Include your name and ID number.
Write the number of pages on the front page and number each page.
Problem 1 - 8 marks
The following two problems are worth 4 marks each.
1. Run the Top Trading Cycles algorith

CO 456: Assignment 4 Solutions
Problem 1
1. Suppose player i changes their path Pi to Pi to strictly lower their penalty. Every edge e Pi Pi
contributes the same to both ui (f ) and ui (Pi , fi ). Every e Pi Pi contributes ri ce (fe ) to ui (f ),
but none

CO 456: Assignment 4
Due Wed, Nov 7 in class.
Include your name and ID number.
Write the number of pages on the front page and number each page.
Problem 1 - 6 marks
Answer the following questions for congestion games.
1. Show that the best response dynami

CO 456: Assignment 3
Due Wed, October 24 in class.
Include your name and ID number.
Write the number of pages on the front page and number each page.
Throughout the entire assignment, the price of anarchy/stability will be with respect to pure strategies

CO 456: Assignment 3
Due Wed, October 24 in class.
Include your name and ID number.
Write the number of pages on the front page and number each page.
Throughout the entire assignment, the price of anarchy/stability will be with respect to pure strategies

CO 456: Assignment 2
Due Wed, October 10 in class.
Include your name and ID number.
Write the number of pages on the front page and number each page.
Problem 1 - 5 Marks
Consider the following 2-player strategic game.
The best responses have been highlig

CO 456: Assignment 2
Due Wed, October 10 in class.
Include your name and ID number.
Write the number of pages on the front page and number each page.
Problem 1 - 5 Marks
Consider the following 2-player strategic game.
p1 \ p2
X
Y
A
(1, 1) (0, 0)
B
(1, 1)

CO 456
The Lemke-Howson Algorithm: Part I
Zachary Friggstad
September 19, 2012
One Last Zero-Sum Comment
We proved that any pair of optimal primal and dual solutions to
the LP from last lecture form a mixed Nash equilibrium.
The converse is trivial to ver

CO 456
Zero-Sum Games
Zachary Friggstad
September 19, 2012
Another Corollary
The following promised result is a trivial corollary of the previous
lectures result. We may let an action ai Ai correspond to the
mixed strategy i with i (ai ) = 1 and i (y ) =

CO 456
More on Mixed Equilibrium
Zachary Friggstad
September 19, 2012
Bach or Stravinsky
p1 \ p2
B
S
B
(2, 1) (0, 0)
S
(0, 0) (1, 2)
We have found 3 mixed Nash equilibrium.
The pure strategy (B , B ).
The pure strategy (S , S ).
The mixed strategy 1 (B )

CO 456
Introduction to Mixed Strategies
Zachary Friggstad
September 17, 2012
Mixed Strategies
As we have seen, not all strategic games have a PNE.
However, there are other interesting types strategies that we can
consider. Suppose each player randomly pic

CO 456
Load Balancing
Zachary Friggstad
October 10, 2012
Load Balancing
Recall the load balancing game on uniformly related machines.
We have a set of machines cfw_1, . . . , m and a set of jobs cfw_1, . . . , n.
Each job i is a player with actions Ai = c

CO 456
The Price of Anarchy and Stability
Zachary Friggstad
October 3, 2012
Ineciency of Equilibria Example
A set of n players are competing for bandwidth across a single
channel. The action available to each player i is to request some
fraction xi [0, 1]

CO 456
Convex Coalition Games and The Housing
Exchange Game
Zachary Friggstad
November 2, 2012
Last Time
Coalition games with transferrable utility: a coalition S creates
value v (S ) and their actions are to divide it between their
members.
Lemma
An acti