IEOR E4407: Game Theoretic Models for Operations
Recitation 1
Problems
1. Consumers are uniformly distributed along a boardwalk that is 1 mile long. Ice-cream prices
are regulated, and consumers go to the nearest vendor because they dislike walking (assum
IEOR E4407: Game Theoretic Models for Operations
Recitation 1
Sep 13,2013
Solutions
1. Consumers are uniformly distributed along a boardwalk that is 1 mile long. Ice-cream prices
are regulated, and consumers go to the nearest vendor because they dislike w
A Tutorial on the Proof of
the Existence of Nash Equilibria
Albert Xin Jiang
Kevin Leyton-Brown
Department of Computer Science,
University of British Columbia
1 Game-theoretic preliminaries
In this tutorial we detail a proof of Nashs famous theorem on the
The Review of Economic Studies, Ltd.
Joint Projects without Commitment
Author(s): Anat R. Admati and Motty Perry
Source: The Review of Economic Studies, Vol. 58, No. 2 (Apr., 1991), pp. 259-276
Published by: Oxford University Press
Stable URL: http:/www.j
American Economic Association
Knowledge and Equilibrium in Games
Author(s): Adam Brandenburger
Source: The Journal of Economic Perspectives, Vol. 6, No. 4 (Autumn, 1992), pp. 83-101
Published by: American Economic Association
Stable URL: http:/www.jstor.o
Queueing Games
Moshe Haviv
Department of Statistics
The Hebrew University of Jerusalem
Based on To queue or not to queue: Equilibrium behavior in queueing systems
co-authored with Refael Hassin (Kluwer, 2003).
Tel Aviv University
March, 2006
Single server
Lecture 1
Private Value Auctions
Second-Price Auctions
In an SPA, it is a weakly dominant strategy to bid your value.
Consider bidder 1 with value x. Let c be highest competing bid.
x c if c < x
Bid b = x : Payos =
0
if c x
8
< x c if c < b
0
if b < c < x
Lecture 2
Risk Aversion and Asymmetries
Revenue Equivalence Principle
We derived E RI = E RII under the following assumptions:
1. Independence X1 , ., XN are independently distributed
2. Risk neutrality bidders maximize expected prots
Prob [Win] x
E [Paym
Lecture 3
Optimal Mechanisms
Mechanisms
Setup
risk-neutral buyers N = f1, ., N g
values Xi with support Xi = [0, i ]
seller value x0 = 0
s
A selling mechanism is (B , , )
Bi i messages
s
i (b) i probability of winning
s
s
i (b) i expected payment
Mechani
Lecture 4
E cient Mechanisms
E cient Mechanisms
Setup
risk-neutral buyers N = f1, ., N g
values Xi with support Xi = [i , i ]
seller value x0 = 0
s
The highest social welfare is
W (x)
(4 is N
max
Q24
1 dimensional simplex)
Qj xj
j2N
E cient Mechanisms
Se
Lecture 5
Auctions with Interdependent Values
Interdependent Values
Setup
N risk-neutral bidders see signals Xi 2 [0, i ]
Values are
Vi = vi (X1 , ., XN )
with vi (0, ., 0) = 0, E [Vi ] <
Interdependent Values
Setup
N risk-neutral bidders see signals Xi
A Note on Mixed Strategy
November 1, 2012
Consider a two player game with set of strategies S1 = cfw_s1 , . . . , s1 and S2 = cfw_s2 , . . . , s2 .
m
n
1
1
Let the payo be given by matrices A, B Rmn where for any i = 1, . . . , m, j = 1, . . . , n, Aij
(
Recitation - 2 / IEOR4407
1
1. Discussion of Homework 2
2. Problem: Suppose N students go to a restaurant. They agree to split the total cost of the meal.
If a student orders a meal of cost C , and pays X her net payo is C X . Find a Nash
Equilibrium of t
Recitation - 3 / IEOR4407
1
1. Discussion of subgame perfect outcome and equlibrium
2. Joe, Elliot and Nancy are senators who are voting on whether to give themselves a pay raise.
The raise is worth $b, but each person who votes for the raise incurs a cos
Recitation - 4 / IEOR4407
1
1. (Gibbons 2.11)
Observe that there are 2 NEs of the game, namely (T, L) and (M, C). (B, R) is not a NE,
since given player 2 plays R, player 1 is better o playing T. Hence, one should think of a trigger
strategy that would pu
1, (20 Points) Alice and Sylvia collaborate on a class project. Each can contribute any positive
amount of effort; if their contributions are m and g respectively, then Alice earns 2333,; —l— 2m
and Sylvia earns 43; — my. rThe cost to each player for inve