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Unformatted text preview: eans that her payoﬀ will be v1 − v2 > 0, and all other payoﬀs
will be 0. Now, player 1 has no incentive to deviate, since her utility
can only decrease.
Likewise, for all other players vi �= v1 , it is the case that in order for
vi to change her payoﬀ from 0 she needs to bid more than v1 , in which case her payoﬀ will be vi − v1 < 0. Thus no incentive to deviate from for any player. 5 Game Theory: Lecture 3 Examples Second Price Auction (continued)
Are There Other Nash Equilibria? In fact, there are also unreasonable
Nash equilibria in second price auctions.
We show that the strategy (v1 , 0, 0, ..., 0) is also a Nash Equilibrium.
As before, player 1 will receive the object, and will have a payoﬀ of
v1 − 0 = v1 . Using the same argument as before we conclude that
none of the players have an incentive to deviate, and the strategy is
thus a Nash Equilibrium.
It can be veriﬁed the strategy (v2 , v1 , 0, 0, ..., 0) is also a Nash
Equilibrium.
Why? 6 Game Theory: Lecture 3 Examples Second Price Auction (continued)
Nevertheless, the truthful equilibrium, where , bi = vi , is the Weakly
Dominant Nash Equilibrium
In particular, truthful bidding, bi = vi , weakly dominates all other
strategies.
Consider the following picture proof where B ∗ represents the
maximum of all bids excluding player i ’s bid, i.e.
B ∗ = max bj ,
j � =i and v∗ is player i’s valuation and the vertical axis is utility. ui(bi)
v*
b i = v* ui(bi)
B* bi v*
bi < v* ui(bi)
B* v* b
i B*
bi > v*
7 Game Theory: Lecture 3 Examples Second Price Auction (continued)
The ﬁrst graph shows the payoﬀ for bidding one’s valuation. In the
second graph, which represents the case when a player bids lower
than their valuation, notice that whenever bi ≤ B ∗ ≤ v ∗ , player i
receives utility 0 because she loses the auction to whoever bid B ∗ .
If she would have bid her valuation, she would have positive utility in
this region (as depicted in the ﬁrst graph).
Similar analysis is made for the case when a player bids more than
their valuation.
An immediate implication of this analysis is that other equilibria
involve the play of weakly dominated strategies. 8 Game Theory: Lecture 3 Mixed Strategies Nonexistence of Pure Strategy Nash Equilibria
Example: Matching Pennies.
Player 1 \ Player 2 heads
tails
heads
(−1, 1) (1, −1)
tails
(1, −1) (−1, 1)
No pure Nash equilibrium. How would you play this game? 9 Game Theory: Lecture 3 Mixed Strategies Nonexistence of Pure Strategy Nash Equi...
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This document was uploaded on 03/19/2014 for the course EECS 6.254 at MIT.
 Spring '10
 AsuOzdaglar

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