Q2.

[eBay's Recommendation] It is hard to imagine that anyone is not familiar with eBay, the most popular auction web site by far. In a typical eBay auction a good is placed for sale, and each bidder places a "proxy bid," which eBay keeps in memory. If you enter a proxy bid that is lower than the current highest bid, then your bid is ignored. If, however, it is higher, then the current bid increases up to one increment (say, $0.01) above the second highest proxy bid. For example, imagine that three people have placed bids on a used laptop of $55, $98, and $112. The current price will be at $98.01, and if the auction ended the player who bid $112 would win at a price of $98.01. If you were to place a bid of $103.45 then the player who bid $112 would still win, but at a price of $103.46, while if your bid was $123.12 then you would win at a price of $112.01. Now consider eBay's historical recommendation that you think hard about the value you impute to the good and that you enter your true value as your bidno more, no less. Assume that the value of the good for each potential bidder is independent of how much other bidders value it.

Argue that bidding more than your valuation is weakly dominated by actually bidding your valuation.

Argue that bidding less than your valuation is weakly dominated by actually bidding your valuation.

Use your analysis to make sense of eBay's recommendation. Would you follow it?

Question 3 [Hawk-Dove] The following game has been widely used in evolutionary biology to understand how fighting and display strategies by animals could coexist in a population. For a typical Hawk-Dove game there are resources to be gained (e.g., food, mates, territories), denoted as v. Each of two players can choose to be aggressive, as 1 Hawk (H), or compromising, as Dove (D). If both players choose H then they split the resources but lose some payoff from injuries, denoted as k. Assume that k > v If both choose D then they split the resources but engage in some display of power that carries a display cost d, with d < v. Finally, if player i chooses H while j chooses D then i gets all the resources while j leaves with no benefits and no costs.

Describe this game in a matrix.

Assume that v = 10, k = 6, and d = 4. What outcomes can be supported as pure-strategy Nash equilibria?