Zihe Wang

Maximal Revenue with Multiple Goods

Only 1 goodSingle sell VS Bundle sellRandomization is neededLP methodMechanism characterization

Maximal Revenue with Multiple Goods

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!

Only k=1 good

Naive solution ----Sell single separate goodk=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.

Multiple k goods, only 1 buyer -------Single VS Bundle

2 3 4

1/4 1/2 1/4

k=largeSeparate 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!

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.

Multiple k goods, only 1 buyer -------Single VS Bundle

Multiple k goods, only 1 buyer -------Single VS Bundle

From Hart&Nisan(2012)

The optimal mechanism

Buyer utility： b(, )=max{ , , }

Multiple k goods, only 1 buyer -----Randomization is needed

Menu item

q1 q2 s

0.5 0 0.5

0 1 2

1 1 5

The optimal mechanism

The revenue :1/3*0.5+1/3*2+1/3*5=5/2

Multiple k goods, only 1 buyer -----Randomization is needed

Menu item Valuation x where the menu item is chosenq1 q2 s

0.5 0 0.5 (1,0)

0 1 2 (0,2)

1 1 5 (3,3)

The optimal mechanism

Individual rationality and compatiblity constriants

Revenue

Multiple k goods, only 1 buyer -----Randomization is needed

Menu item Valuation x where the menu item is chosenq1 q2 s

(1,0)

(0,2)

(3,3)

IR on (1,0)

IR on (0,2)

IC from (3,3) to (1,0)

IC from (3,3) to (0,2)

(1)*3+(2)*3+(3)+(4):+ + +

= , = =0 , ,

Multiple k goods, only 1 buyer -----Randomization is needed

IR on (1,0) (1)

IR on (0,2) (2)

IC from (3,3) to (1,0)

(3)

IC from (3,3) to (0,2)

(4)

[Hart&Nisan 2012] For every there exists a k-item distribution on such that

and Here correlated has infinite cases.

Multiple k goods, only 1 buyer -----Randomization is needed

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.

Multiple k goods, n buyers, finite cases ----- LP method

Buyer valuation : Allocation rule: qPayment rule (Seller revenue): sBuyer 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 .

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)

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

b is weakly monotone, but s may be not. E.g.

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

Menu item Valuation x where the menu item is chosenq1 q2 s

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

Menu item Valuation x where the menu item is chosenq1 q2 s

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