Question: Consider an auction with one item and three bidders. Each bidder’s private valuation is independently drawn from .

(a) Find an equilibrium of a second-price auction.

(b) Provide the formula of the expected payoff of a bidder with value in [4,10].

(c) Find an equilibrium bidding strategy of the first-price auction using the revenue equivalence theorem.  Edit

Answer: In the “second” price auction, each bidder’s expected payoff is:



EU(v) = v ∗ (v − v/2) = v

2

/2.

→ the same expected utility under 1st price auction.

2. EU(v) = Prob(v)(v − b(v)) = v(v − b(v)).

3. Thus, b(v) = v/2.





1.b.The formula to calculate expected value for betting is fairly simple:

(Amount won per bet * probability of winning) – (Amount lost per bet * probability of losing)

=6

1c. The revenue equivalence theorem states that for certain economic environments, the expected revenue and bidder proÖts for a broad class of auctions will be the same provided that bidders use equilibrium strategies. The purpose of these notes is provide a statement of the result and explain it. The model Seller has a single item to sell. Two potential bidders, both risk-neutral. Each bidder has a value drawn from a uniform distribution on [0; 1]. Standard auctions A ìstandard auctionîis an auction in which: bidders are asked to submit bids; the bidder who submits the highest bid wins the object; bidder i is asked to pay (bi ; bj ). and furthermore there is a Nash equilibrium of the auction game in which (a) the bidders use a strategy b(v) that is increasing in v, so that (b) the bidder with higher value wins; and also (c) a bidder who has value v = 0 ends up paying nothing in equilibrium. The Revenue Equivalence Theorem Theorem 1 If there are two bidders with values drawn from U[0; 1], then any standard auction has an expected revenue 1=3 and gives a bidder with value v and expected proÖt of v 2=2, the same as the second-price auction. The more general version of the theorem, which we wonít prove, asserts that if there are N bidders with values drawn from the same continuous value distribution, then any standard auction will lead to the same revenue and expected bidder proÖt as the second-price auction (the expected revenue may not be 1/3 however). In the N bidder case, the payment rule for bidder i can depend on iís bid bi and all of the competing bids (i.e. pay your bid if your bid is higher than all the rest of the bids would be the Örst price auction payment rule). 1 Theorem 2 If there are N bidders with values drawn from a continuous distribution (e.g. uniform on [a; b]), then any standard auction leads to the same expected revenue, and same expected bidder proÖt, as a second-price auction. Example: Second Price Auction The second price auction is a standard auction. The payment rule has (bi ; bj ) equal to zero if bi < bj , and equal to bj if bi > bj . In equilibrium, each bidder bids his value, so the equilibrium strategy is b(v) = v. In equilibrium the bidder with the higher value will win, and pay the bid (or equivalently value) of the lower-valued bidder. The expected revenue is 1/3. Why 1/3? If we take two draws from a uniform distribution on [0; 1], the higher draw will be on average 2/3 and the lower draw will be on average 1/3. Notice also that if a bidder has value v, he expects to win whenever the other bidder has a value less than v; which happens with probability equal to v. If he does win, he expects to pay v=2. So the expected proÖt of a bidder with value v is U(v) = v (v v=2) = v 2=2. Example: First Price Auction The Örst price auction is also a standard auction. The payment rule has (bi ; bj ) equal to zero if bi < bj , and equal to bi if bi > bj : In equilibrium each bidder submits a bid equal to half his value, so the equilibrium bid strategy is b(v) = v=2. Because in equilibrium the high-valued bidder wins and pays his bid.The expected revenue is again 1/3. Why? On average the higher of the two values is 2/3, and the higher of the two bids is 1/3. A bidder with value v again expects a proÖt v 2=2. Again, the reason is that if a bidder has value v, he expects to win whenever the other bidder has a value below v (because the Örst bidder will bid v=2, and the second bidder will bid less than this whenever its value is below v): So the bidder with value v expects to win with probability v, and if he does win expects to pay v=2, his equilibrium bid. Therefore U(v) = v (v v=2) = v 2=2. Example: Other Standard Auctions There are many other standard auctions. Here are a few examples: All-pay auction: bidders submit bids, the high bid wins and all bidders pay their bids, even losing bidders. The payment rule is (bi ; bj ) = bi . Mixed Örst-price/second-price: High bid wins and winner pays the average of the Örst and second highest bids. So (bi ; bj ) equals zero if bi < bj , and (bi + bj )=2 if bi > bj . Ascending auction: Can be viewed as a standard auction if we think about each bidder giving instructions (ìbid up to some amount bî) to a proxy bidder that executes the strategy. Descending auction: Also can be viewed as a standard auction if we think about each bidder giving an instruction (ìstop the auction if the price falls to bî) to a proxy bidder. There are lots of other auctions that are standard auctions or can be interpreted as such. For instance, an auction where the winner pays half his bid, or his bid plus ten dollars. Or an auction that is ascending until the price reaches $0:50; at which point there is a Önal round of sealed bids.

In First Price Auction The Örst price auction is also a standard auction. The payment rule has (bi ; bj ) equal to zero if bi < bj , and equal to bi if bi > bj : In equilibrium each bidder submits a bid equal to half his value, so the equilibrium bid strategy is b(v) = v=2. Because in equilibrium the high-valued bidder wins and pays his bid.The expected revenue is again 1/3. Why? On average the higher of the two values is 2/3, and the higher of the two bids is 1/3. A bidder with value v again expects a proÖt v 2=2. Again, the reason is that if a bidder has value v, he expects to win whenever the other bidder has a value below v (because the Örst bidder will bid v=2, and the second bidder will bid less than this whenever its value is below v): So the bidder with value v expects to win with probability v, and if he does win expects to pay v=2, his equilibrium bid.

Therefore U(v)= v .(v-v/2) = v2/2  Edit