1. Comp/Math 553: Algorithmic Game Theory Lecture 10
Yang Cai

2. Menu Single-Dim Environment, Myerson’s Lemma Examples
Revenue Maximization (Intro)

3. Reminder

4. Single-dimensional Environment
Definition:
n bidders
Each bidder i has a private value vi (scalar!), her value “per unit of stuff” that she gets.
A feasible set X. Each element of X is an n- dimensional vector (x1, x2, , xn), where xi denotes the “amount of stuff” given to bidder i.
Examples: k-unit auctions, sponsored search

5. Single-dimensional Environment
Examples:
Single-item auction: X is the set of 0-1 vectors that have at most one 1
Selling k units of the same item to unit-demand bidders: X are the 0-1 vectors satisfying Σi xi ≤ k.
Sponsored search auction: X is the set of n- dimensional vectors corresponding to assignments of bidders to slots. If bidder i is assigned to slot j, then the component xi equals the CTR αj of its slot.
What makes a setting single-dimensional is NOT the allocation constraints, but that every bidder’s private information is one scalar.

6. Single-dimensional Sealed Bid Auction
Defined by two rules:
Allocation rule x : Rn  X
Payment rule p : Rn  Rn
Auction Execution:
Collect bids b=(b1 , ..., bn)
[allocation] Implement allocation x(b)
[payments] Charge prices p(b)
Definition: (DSIC) An auction (x, p) is DSIC iff for all i, b-i it is optimal for bidder i to bid his true value:

7. Two important definitions
Definition 1: (Implementable Allocation Rule) An allocation rule x for a single-dimensional environment is implementable if there is a payment rule p s.t. the sealed-bid auction (x, p) is DSIC.
Example: When selling a single indivisible item, the allocation rule that gives the item to the highest bidder is implementable.
Pending Question: Is the Greedy allocation rule implementable in Sponsored Search?
How about giving the item to the second highest bidder in single-item settings? How about the lowest bidder?

8. Two important definitions
Definition 2: (Monotone Allocation Rule) An allocation rule x for a single-dimensional environment is monotone if for every bidder i and bids b−i by the other bidders, the allocation xi(z, b−i) to i is non-decreasing in i’s bid z.
In single-item settings, the allocation rule that gives the item to the highest bidder is monotone, while the allocation rules that give the item to the second highest or the lowest bidder are not monotone.
The Greedy allocation rule for Sponsored Search is monotone.

9. Myerson’s Lemma [Myerson ’81 ] Fix a single-dimensional environment.
An allocation rule x is implementable if and only if it is monotone.
(b) If x is implementable/monotone, there is an essentially unique payment rule such that the sealed-bid mechanism (x, p) is DSIC, given by the formula:
(c) In particular, there is a unique payment function such that the mechanism is DSIC and additionally IR with non-positive transfers (i.e. bi = 0 implies pi(b) = 0, for all b-i).
PROOF ON THE BOARD

10. Myerson’s Lemma Corollaries
Corollary: The greedy allocation rule for sponsored search is implementable. Thus, there is a DSIC auction that maximizes social welfare.
On the other hand, in single-item settings, allocating to the second-highest bidder or the lowest bidder are both not implementable.

11. Application of Myerson’s Lemma

12. Single-item Auction 1 i n …
Bidders v1 vi vn
Auctioneer
Item
Allocation Rule: give the item to the highest bidder.
Payment Rule: Vickrey’s Payment
How about k-unit auction?

13. Sponsored Search 1 i n …
Bidders (advertisers) v1 vi vn 1 j k …
Slots
Auctioneer/Google α1 αj αk
For the purposes of the equation above ak+1 is taken to be 0
Allocation Rule: allocate the slots greedily based on the bidders’ bids.
Payment Rule?

14. SINGLE-ITEM REVENUE-Maximization

15. One Bidder, One Item, Revenue Maximization
Can we meaningfully maximize revenue?
Suppose1 bidder whose value is v.
Highest achievable revenue is v, if we were to offer a take-it-or-leave-it offer of the item at price v.
But v is private...
How can we price the item optimally?
Fundamental issue: fix any price; there is some v for which it fails to extract revenue v.
Bidder’s Private Value is too strong of a benchmark for the auction to compete against.
Solution:
Assume distribution F over bidder’s value is known.
Perform average case analysis

16. One Bidder, One Item, Revenue Maximization
Posting price r gives expected revenue = r (1−F(r))
When F is the uniform dist. on [0,1], optimal choice of r is ½ achieving expected revenue ¼.
The optimal posted price is also called the monopoly price.
Is ¼ the optimal expected revenue of any auction?
The answer is yes, as implied by Myerson’s theorem (next lecture).

17. Two Uniform Bidders, One Item
Two bidders whose values are i.i.d. U[0,1].
Revenue of Vickrey’s Auction is the expectation of the min of the two uniform random variables = 1/3.
What else could we do? Include a reserve price?
Vickrey with reserve ½ has expected revenue 5/12 > 1/3.
Is 5/12 the optimal expected revenue of any auction?
The answer is yes, as implied by Myerson’s theorem (next lecture).

18. Revenue-Optimal Auctions
[Myerson ’ ]
Single-dimensional settings, distribution Fi of every bidder’s private value known to all other bidders, the auctioneer.
Exists Revenue-Optimal auction that is simple, direct, DSIC
Def: An auction with Bayesian Nash equilibrium s1,…,sn is interim IR, if for all bidders i and all vi, the expected utility of bidder i when her value is vi and she plays si(vi) is non-negative in expectation over the other bidders’ values assuming they also use their Bayesian Nash equilibrium strategies s-i.
Direct means 1 round interaction simultaneous and asking to bid the value.
Optimality: The expected revenue of Myerson’s auction when all bidders report truthfully (which is in their best interest to do since the auction is DSIC) is as large as the expected revenue of any other (potentially indirect) interim IR auction, when bidders use Bayesian Nash equilibrium strategies.
Note: Reminder of Bayesian Nash equilibrium, general definition of direct/indirect mechanisms given later in this slide-deck.

19. The Power (?) of Indirect Mechanisms

20. Indirect Mechanisms?
So far considered welfare maximization in single-item, k-unit, sponsored search auctions.
Greedy allocation rule was monotone.
Thus, combined with Myerson’s payment formula can be complemented to a DSIC direct mechanism maximizing welfare in these settings.
So restriction to direct mechanisms did not hurt us.
As we foray into more complex objectives (e.g. revenue, next lecture) or more complex settings (e.g. spectrum auctions, later in this course) could it be that indirect mechanisms are more powerful than direct ones?
Revelation principle: In very general settings, the answer is no!