Maximal Revenue with Multiple Goods

1. Maximal Revenue with Multiple Goods Zihe Wang

2. Maximal Revenue with Multiple Goods • Only 1 good • Single sell VS Bundle sell • Randomization is needed • LP method • Mechanism characterization

3. Only k=1 good • Myerson mechanism: The value distribution is uniform on [0,1].The optimal auction is the Vickery auction with reservation price ½. (i)Given the bids v and F, compute virtual value v’(v,F) (ii)Run VCG on the virtual bids v’, determine the allocation and payment Deterministic!

4. Multiple k goods, only 1 buyer -------Single VS Bundle • Naive solution ----Sell single separate good • k=2 Consider the distribution taking values 1 and 2 with equal probability ½. The maximum revenue for single good is 1. The maximum revenue for two goods is 2. If we bundle two goods together and sell. Value is additive. The distribution is The maximum revenue is 3*3/4=2.25! Bundle selling is better than single selling.

5. Multiple k goods, only 1 buyer-------Single VS Bundle • k=large Separate single sell: The value distribution is independent identically uniform on [0,1]. The max revenue for single good is 1/4. The sum of revenue is k/4 in expectation. Bundle sell: The value distribution is a normal distribution on [0,k], concentrated on with probability 99.7%. We set the reserve price as , and get almost revenue in expectation. Bundle selling is better than single selling again!

6. Multiple k goods, only 1 buyer-------Single VS Bundle • Is the bundle selling always better than single selling? NO! Bundling can also be very bad, while single selling is good! • For distribution that takes values 0,1 and 2, each with probability 1/3, the optimal auction can get 13/9 revenue, which is larger than the revenue of 4/3 obtained from either selling the two items separately , or from selling them as a bundle. Optimal auction-----offer to the buyer the choice between any single item at price 2, and the bundle of both items at a “discount” price of 3.

7. Multiple k goods, only 1 buyer-------Single VS Bundle From Hart&Nisan(2012)

8. Multiple k goods, only 1 buyer -----Randomization is needed • The optimal mechanism • Buyer utility： b(,)=max{ , , }

9. Multiple k goods, only 1 buyer -----Randomization is needed • The optimal mechanism • The revenue :1/3*0.5+1/3*2+1/3*5=5/2

10. Multiple k goods, only 1 buyer -----Randomization is needed • The optimal mechanism • Individual rationality and compatiblityconstriants • Revenue

11. Multiple k goods, only 1 buyer -----Randomization is needed • (1)*3+(2)*3+(3)+(4): • +++ =, ==0 , ,

12. Multiple k goods, only 1 buyer -----Randomization is needed • [Hart&Nisan 2012] For every there exists a k-item distribution on such that and Here correlated has infinite cases.

13. Multiple k goods, n buyers, finite cases----- LP method • Let ,,) is bidder i’s valuation vector, denote the bidder i’s valuation for item j. • Let denote the probability that bidder i gets item j when the valuations are x. • Let denote the expected payment of bidder i.

14. Multiple k goods, 1 buyer-----Mechanism characterization • Buyer valuation : • Allocation rule: q • Payment rule (Seller revenue): s • Buyer utility: • The following two definitions are equivalent: 1.The mechanism is Incentive Compatible (truthful). 2.The buyer’s utility b is a convex function of x, and for all x, x’, we have . In particular b is differentiable at x, then .

15. Multiple k goods, 1 buyer -----Mechanism characterization • Any convex function b with for all i defines an incentive compatible mechanism by setting • The expected revenue of the mechanism given by b is • b(x) determine q(x),s(x)

16. Multiple k goods, 1 buyer -----Mechanism characterization • b is weakly monotone, but s may be not. • E.g.

17. Multiple k goods, 1 buyer -----Mechanism characterization

18. Multiple k goods, 1 buyer -----Mechanism characterization