1. Netw
work struccture a) In the figurre below th
he blue nod
des are men
n and the reed nodes arre women. Does t
this netwo
rk have a h
high degreee of homoph
hily based on gender?? b) A
Assume thaat everyone in the exaam room kn
nows each other, and
d that they aare f
friends wit
th everyonee sitting neext to them and enemiies with everyone elsee. Is t
this netwo
rk (structu
urally) balanced? Is it w
weakly ballanced? c) Which nod
W
des in the fo
ollowing neetwork hav
ve the strong
g triadic cllosure prop
perty?
d) Are the Pag
A
geRank vallues in the ffigure below equilibriium values?
2. Game theory a) Does the game below have any strictly dominant strategies? If so, which? b) Find all (pure strategy) Nash equilibria for the game in a). c) Show an example of a game which has exactly one (pure strategy) Nash equilibrium but no dominant strategies. 3. Sponsored search markets A search engine uses VCG to sell three ad slots. Slot a has a clickthrough rate of 2, slot b a clickthrough rate of 5 and slot c a clickthrough rate of 1. There are three advertisers interested in the slots: the revenue per click of advertiser x, y and z is 1, 2 and 3, respectively. a) It the advertisers are rational, what are their bids in this case? b) Which advertiser buys which slot, and how much do they pay? 4. Voting Find an ordering of the alternatives that shows that the following collection of preferences are single‐peaked: Indian ≻Ann Italian ≻Ann Japanese ≻AnnFransk ≻Ann Chinese French ≻Bob Indian ≻Bob Italian ≻Bob Japanese ≻Bob Chinese Italian Cath Indian Cath French Cath Japanese Cath Chinese