Zihe Wang

Only 1 good Single sell VS Bundle sell Randomization is needed LP method Mechanism characterization

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!

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 /41/21/4

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.

From Hart&Nisan(2012)

Menu item q1q2s

Menu itemValuation x where the menu item is chosen q1q2s 0.50 (1,0) 012(0,2) 115(3,3)

Menu itemValuation x where the menu item is chosen q1q2s (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)

IR on (1,0) IR on (0,2) IC from (3,3) to (1,0) IC from (3,3) to (0,2)

Menu itemValuation x where the menu item is chosen q1q2s

Menu itemValuation x where the menu item is chosen q1q2s