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# Now player 1 has no incentive to deviate since her

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Unformatted text preview: eans that her payoﬀ will be v1 − v2 &gt; 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 &lt; 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 &lt; v* ui(bi) B* v* b i B* bi &gt; 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.

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