Ch 8&9 exercise solutions
I. 8. 1.a, 1.b
II .8. 1.c
III. 9. 1 IV. 9. 8
I:
(a)
given x+y=1000, we get x=y=1000/2=500. It is a Nash-equilibrium, because none of the drivers have
then a need to deviate.
(b)
since the route C->D is extremely fast, drivers will all choose this dominant strategy. Due to this,
x=y=1000, which achieves a Nash equilibrium.
total cost-of-travel:
former: 500/100+12=17
now: 1000/100+1000/100=20
It increased.
x=y=1000 is the only Nash equilibrium, because in A node, x/100<12 for all x in [1,1000] and in C node
y/100<12 for all y in [1,1000] ⇒ no reason to deviate from route A->C->D->B. Also the calculations go
like that.
II:
(c):
In node A a driver is
indifferent between choosing if x=500,
chooses A->D, if x>500
chooses A->C, if x<500
===> x has to equal 500
In node C driver is similarly indifferent between the routes if y=500 (same idea as in node A)
======> in equilibrium x=500 and y=500, any combination of drivers strategies that satisfies this is a
Nash equilibrium. Notable issue is that the total travel time is fixed to 10 with or without
highway from C to D, because in Nash equilibrium no driver uses that highway.
total cost-of-travel:500/100+5=10
If the government closes the highway, the total time of travel doesn't change.
III:
In a sealed-bid second-price auction, truthful bidding is a dominant strategy so just submit the bid equals
valuation c, no matter how many people participate the auction and what they would do. However, one
should note that this is the case only because the bidders have independent private valuations; if the
situation would be common values situation, then there would be the problem with the Winner’s curse,
because the bidder with the highest valuation would win only because she probably overvalued the
auctioned item.
IV:
(a): The bidder wins the auction if he has the highest valuation and just pays the second highest price.
(b): It does not affect rational bidders’ behaviour, just affect their pay off.
(c):
In this case the valuations are independent, so even if the bidder 3 does behave irrationally, bidders 1
and 2 do not know anything about the valuation of bidder three => bidders 1 and 2 do not want to bid
more than their own valuation. On the other hand, they do not either want to bid less, because then they
would not be able to win. All this means that the bidder 3 is the only one that actually makes negative
profits because of overbidding, while bidders 1 and 2 lose some profit if they win and bidder 3 has got the
second highest bid.
former:
payoff for 1
now
payoff for now
changes
V1 >V2>V3
V1 -V2
V1 >V2>V3
V1 -V2
0
V1 >V3>V2
V1 -(V3+1)/2
(V3+1)/2-V2
V3 >V1 >V2
0
V2 -V1
V1 >V3>V2
V1 -(V3+1)/2
(V3 -1)/2
V3 >V1 >V2
0
V3 -V1
0
0
V1 >V3>V2
other cases
V1 -V3
0