# Computational Game Theory

### Homework number 1

1.

Auction and Nash Equilibrium: a.

Consider a first price auction where each participant pays \$1 to participate in the auction.

Are there deterministic Nash Equilibriums?

(Assume that a player can choose not to participate.) b.

Consider a second price auction where the participant that bids the second price receives

\$1 (if there are more then one, they share it).

What are the deterministic Nash Equilibriums?

(Assume that all the valuations and the bids are positive integer numbers. In case of a tie, the player with the higher

ID wins.)

2.

Job Scheduling: Show how to compute efficiently a Nash equilibrium for n jobs and m (different speed) machines.

(Hint: consider the jobs in a sorted order.)

3.

PoA and Job Scheduling: In unrelated machines a job has a different load on each machine, i.e., w i

(J) is the load og J on machine M i

.

Show that for unrelated machines the price of anarchy can be unbounded.

Bonus: For identical machines and unit size jobs, for correlated equilibrium, give an example where the price of anarchy is high (There is an example with √m, when m=n).

Can you give also an upper bound?!

4.

Selfish Routing: a.

Show that if the link delay function is l e

(x) = a x , for a constant a> 0, then OPT=NASH. b.

Consider a quadratic delay function is l(x) = ax 2

+ bx +1 , for some constants a and b . Derive an example that NASH differs from OPT (as much as you can).

### The homework is due April 25 (in class)

hillel_f@yahoo.com

amirep@tau.ac.il

olonetski@tau.ac.il

י נורטקלא ראוד yuvalne@txtechnion.ac.il

fattalsh@post.tau.ac.il

michalros@gmail.com

rosenric@rau.ac.il

oren@poleg.org

mayaben@post.tau.ac.il

lotem@eng.tau.ac.il

hila_p@yahoo.com

yaakov.hoch@weizmann.ac.il

rubianat@post.tau.ac.il

bronstein@post.tau.ac.il

stloyare@tau.ac.il

lgavish@tau.ac.il

bruckner@post.tau.ac.il

zachdres@post.tau.ac.il

pyalsch@gmail.com

orenmohr@yahoo.com

eyaldav1@post.tau.ac.il

liorg@post.tau.ac.il

pisotsky@gmail.com

elih@post.tau.ac.il

liza.potikha@gmail.com

anat@eng.tau.ac.il

galca8@yahoo.com

guy_tennenbaum@yahoo.com

yedaelwa@post.tau.ac.il

רצנ

לתפ

בידא

ןזור

ןזור

### תואצרה ימוכיס

החפשמ םש םש

יטרפ

לבוי

רחש

ידע

לכימ

יקיר

האצרה

.

1

.

2

גלופ

ןלפק

לבות

ךוה

ןיבור

ןייטשנורב

וקנרלוטס

שיבג

ןרוא

הימ

םטול

ירמע

הליה

בקעי

ןתנ

באז

ירדנא

רואיל

.

.

.

.

3

4

5

6

רנקורב

ןורהא יחצ

רדיינש

רומ

דוד

שיבג

יקדיסיפ

ןח ןבא

החיטופ

ןירפלה

ןיידבור

םואבננט

ןמדלו

רשיילפ

ןייטשפא

יקצנולוא

ןורש

חצ

לייא

ןרוא

לייא

רואיל

ייגרס

היליא

הזיל

תנע

םוילג

איג

לאעדי

ללה

רימא

הנלאבס

.

.

.

.

.

7

8

9

10

11

.

12 asapha@smile.net.il

ןונרא ףסא liors@post.tau.ac.il

אריפש רואיל