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).

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