Tadelis 13.3 Consider a seller who must sell a single private value good. There
ID: 3321210 • Letter: T
Question
Tadelis 13.3
Consider a seller who must sell a single private value good. There are two potential buyers, each with a valuation that can take on one of three values, i {0, 1, 2}, each value occurring with an equal probability of 1/3 . The players’ values are independently drawn. The seller will offer the good using a second-price sealed-bid auction, but he can set a reserve price of r 0 that modifies the rules of the auction as follows. If both bids are below r then neither bidder obtains the good and it is destroyed. If both bids are at or above r then the regular auction rules prevail. If only one bid is at or above r then that bidder obtains the good and pays r to the seller.
(a) Is it still a weakly dominant strategy for each player to bid his valuation when r > 0?
(b) What is the expected revenue of the seller when r = 0 (no reserve price)? (
c) What is the expected revenue of the seller when r = 1?
(d) What explains the difference between your answers to (b) and (c)?
Explanation / Answer
If r = 0,
----
If r = 1,
pay = 2 (draw)
----
If r = 2,
----
a) A weakly dominany strategy is one which always has a pay-off better or equal to any other strategy and there is atleast 1 more case where this strategy has pay-off equal to some other strategy.
When r > 0, it is also a weakly dominany strategy to bid your actual value because you only pay R (reserve price) if your bid > R and other's is below R. So, bidding your own value would be okay because you are paying a value 'R' , which is less than or equal to own value => Dominant (not sure, weakly or strictly)
Meanwhile, there are strategies (bid highest possible value. here, 2) in this case of other person bidder bidding below R, which will yield the same pay-off (paying R) => Weakly dominant strategy [Asnwer]
b) r = 0 => Exp(revenue) = (1/9)*(sum of all payouts in TABLE 1) = 5/9 [Answer]
c) r = 1 => Exp(revenue) = (1/9)*(sum of all payouts in TABLE 1) = 9/9 = 1 [Answer]
d) As the reserved price = 1 in Part c - when 1 bidder bids 0 but the other bids greater than 0 - the revenue of seller increases from the second-highest bid to the reserved price (as r > second-highest bid)
A | B 0 1 2 0 pay = 0 pay = 0 (B wins) pay = 0 (B wins) 1 pay = 0 (A wins) pay = 1 (draw) pay = 1 (B wins) 2 pay = 0 (A wins) pay = 1 (A wins) pay = 2 (draw)Related Questions
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