1. Crash Course on Multi-Dimensional Mechanism Design
Matt Weinberg
Princeton University

2. Outline State of affairs for single-item auctions.
Tractable, insightful theory due to [Myerson 81].
Pitfalls of optimal multi-item auctions.
Extremely complex/intractable.
Via examples.
Directions in approximately optimal multi-item auctions.
Brief overviews of recent results.
Very brief overview of related directions in auction design – focus on what’s in EC 2018.

3. Single-Item Auctions Disclaimer:
I’m not planning to motivate the model deeply.
The model is “equivalent” to many others.
I also won’t get into this.
BUT: absolutely feel free to ask questions about these aspects.
Single-Item Auctions

4. Setup: Single Item, Single Bidder
Model:
Seller announces menu: 𝑥 𝐿 , 𝑝 𝐿 𝐿 .
Selecting option L gives item w.p. 𝑥 𝐿 , pay price 𝑝 𝐿 .
Buyer with value 𝑣 drawn from 𝐹 arrives (think 𝐹 = population, 𝑣 = individual).
Buyer selects 𝐿 that maximizes E[utility], denote by 𝐿 𝑣 =(𝑥 𝑣 ,𝑝 𝑣 ).
Utilit y 𝐿 = 𝑣⋅ 𝑥 𝐿 − 𝑝 𝐿 .
Seller goal: maximize expected revenue = 𝐸 𝑣←𝐹 [𝑝(𝑣)].
𝑣←𝐹

5. Background: Single Item, Single Bidder
Seminal Result:
[Myerson 81, Riley/Zeckhauser 83]: optimal menu super simple.
Option one: pay 0 get nothing.
Option two: pay argmax 𝑝>0 𝑝⋅ 1−𝐹 𝑝 ), get item w.p. 1.
𝑣←𝐹

6. Example 1: Single-Item, Single-Bidder
Optimal Auction:
Set price of 1.
Buyer chooses to purchase w.p. 1.
Expected revenue = 1.
(Note: Also optimal to set price 2. Buyer purchases w.p. ½, get revenue 1).
𝑣←𝑈( 1,2 )

7. Example 2: Single-Item, Single-Bidder
Optimal Auction:
Set price of 3.
Buyer chooses to purchase w.p. 1/2.
Expected revenue = 3/2.
𝑣←𝑈( 1,3 )

8. Background: Single Item, Single Bidder
Seminal Result:
[Myerson 81, Riley/Zeckhauser 83]: optimal menu super simple.
Option one: pay 0 get nothing.
Option two: pay argmax 𝑝>0 𝑝⋅ 1−𝐹 𝑝 ), get item w.p. 1.
Corollary: optimal revenue is monotone.
If 𝐹 + stochastically dominates 𝐹, 𝑅𝑒𝑣 𝐹 + ≥𝑅𝑒𝑣(𝐹).
Proof immediate from definition: 1− 𝐹 + 𝑝 ≥1−𝐹 𝑝 for all p.
𝑣←𝐹
𝑣←𝐹 vs 𝐹 +
𝐹 + stochastically dominates 𝐹
∀𝑝,1− 𝐹 + 𝑝 ≥1−𝐹 𝑝

9. Setup: Single Item, Multiple Bidders
Model:
Seller announces mechanism: takes as input a bid 𝑏 𝑖 from each bidder, selects (possibly randomly) a winner, charges bidder 𝑖 price 𝑃 𝑖 ( 𝑏 ).
Want “truthful”: best to bid true value, assuming other bidders do so as well.
Buyers with values 𝑣 𝑖 drawn from 𝐹 𝑖 arrive ( 𝐹 𝑖 = population, 𝑣 𝑖 = individual).
Seller goal: maximize expected revenue = 𝐸 𝑣 ← 𝐹 [ 𝑖 𝑃 𝑖 𝑣 ].
𝑣 1 ← 𝐹 1
𝑣 2 ← 𝐹 2
𝑣 3 ← 𝐹 3

10. Myerson’s Magic Lemma
Notation: when we refer to an auction (X,P), we mean that:
𝑋 𝑖 ( 𝑏 ) is a random variable that is 1 whenever bidder i gets the item on bids 𝑏 (and 0 otherwise).
𝑃 𝑖 ( 𝑏 ) denotes the price paid by bidder i on bids 𝑏 .
[Myerson 81]: Let (X,P) be any truthful auction. Then expected revenue equals expected virtual welfare. That is:
𝐸 𝑣 ← 𝐹 𝑖 𝑃 𝑖 𝑣 = 𝐸 𝑣 ← 𝐹 [ 𝑖 𝜑 𝑖 𝑣 𝑖 ⋅ 𝑋 𝑖 𝑣 ].
𝜑 𝑖 𝑣 =𝑣− 1− 𝐹 𝑖 (𝑣) 𝑓 𝑖 (𝑣) called virtual value.
LHS: expected revenue. RHS: expected virtual value of winner.
The magic: RHS makes no reference to payments at all!

11. Myerson’s Magic Lemma
[Myerson 81]: Let (X,P) be any truthful auction. Then expected revenue equals expected virtual welfare. That is:
𝐸 𝑣 ← 𝐹 𝑖 𝑃 𝑖 𝑣 = 𝐸 𝑣 ← 𝐹 [ 𝑖 𝜑 𝑖 𝑣 𝑖 ⋅ 𝑋 𝑖 𝑣 ].
𝜑 𝑖 𝑣 =𝑣− 1− 𝐹 𝑖 (𝑣) 𝑓 𝑖 (𝑣) called virtual value.
LHS: expected revenue. RHS: expected virtual value of winner.
If you are interested in doing research on mechanism design, I strongly, strongly recommend getting comfortable with this theorem. Here are two sources:
Chapters 2+3
Lectures 3 +5
One sentence of intuition: If an auction is truthful, then giving item when bid b:
Allows you to make some revenue right now, from the bidder bidding b.
Precludes you from making very high revenue on bids >b, because they can always choose to bid b instead.
Virtual value is a clever “amortized” counting of revenue.

12. Myerson’s Magic Lemma
[Myerson 81]: Let (X,P) be any truthful auction. Then expected revenue equals expected virtual welfare. That is:
𝐸 𝑣 ← 𝐹 𝑖 𝑃 𝑖 𝑣 = 𝐸 𝑣 ← 𝐹 [ 𝑖 𝜑 𝑖 𝑣 𝑖 ⋅ 𝑋 𝑖 𝑣 ].
𝜑 𝑖 𝑣 =𝑣− 1− 𝐹 𝑖 (𝑣) 𝑓 𝑖 (𝑣) called virtual value.
LHS: expected revenue. RHS: expected virtual value of winner.
Immediate Corollary: Let 𝑋 ∗ ( 𝑏 ) always give the item to the bidder with highest virtual value (or throw away the item if all are negative). If there exists a P such that (X*,P) is truthful, then (X*,P) is optimal.
Fact: If 𝜑 𝑖 ⋅ is monotone non-decreasing for all i, then such a P exists.
𝐹 𝑖 is called regular if 𝜑 𝑖 (⋅) is monotone non-decreasing.
Observation: If 𝐹 𝑖 =𝐹 for all i, and F is regular, then the highest virtual bidder = highest bidder.

13. Myerson’s Magic Lemma
[Myerson 81]: Let (X,P) be any truthful auction. Then expected revenue equals expected virtual welfare. That is:
𝐸 𝑣 ← 𝐹 𝑖 𝑃 𝑖 𝑣 = 𝐸 𝑣 ← 𝐹 [ 𝑖 𝜑 𝑖 𝑣 𝑖 ⋅ 𝑋 𝑖 𝑣 ].
𝜑 𝑖 𝑣 =𝑣− 1− 𝐹 𝑖 (𝑣) 𝑓 𝑖 (𝑣) called virtual value.
LHS: expected revenue. RHS: expected virtual value of winner.
Immediate Corollary: Let 𝑋 ∗ ( 𝑏 ) always give the item to the bidder with highest virtual value (or throw away the item if all are negative). If there exists a P such that (X*,P) is truthful, then (X*,P) is optimal.
Theorem [Myerson 81]: Let 𝐹 𝑖 =𝐹 for all i, and F be regular. Then the second-price auction with reserve max 𝑝 𝑝⋅1−𝐹 𝑝 is optimal.
Theorem [Myerson 81]: Let 𝐹 𝑖 be regular for all i. Then there exists a P such that (X*,P) is optimal.

14. Myerson’s Magic Lemma
[Myerson 81]: Let (X,P) be any truthful auction. Then expected revenue equals expected virtual welfare. That is:
𝐸 𝑣 ← 𝐹 𝑖 𝑃 𝑖 𝑣 = 𝐸 𝑣 ← 𝐹 [ 𝑖 𝜑 𝑖 𝑣 𝑖 ⋅ 𝑋 𝑖 𝑣 ].
𝜑 𝑖 𝑣 =𝑣− 1− 𝐹 𝑖 (𝑣) 𝑓 𝑖 (𝑣) called virtual value.
LHS: expected revenue. RHS: expected virtual value of winner.
What if 𝑭 𝒊 irregular? (Answer below is brief, see sources for further detail).
Requires “ironing.” Basically this “forces” 𝜑 𝑖 ⋅ to be monotone.
[Myerson 81]: Let (X,P) be any truthful auction. Then expected revenue is upper bounded by expected ironed virtual welfare. That is:
𝐸 𝑣 ← 𝐹 𝑖 𝑃 𝑖 𝑣 ≤ 𝐸 𝑣 ← 𝐹 [ 𝑖 𝜑 𝑖 𝑣 𝑖 ⋅ 𝑋 𝑖 𝑣 ].
𝜑 𝑖 (𝑣) called ironed virtual value. No closed form, but monotone.
LHS: expected revenue. RHS: expected ironed virtual value of winner.

15. Myerson’s Magic Lemma
What if 𝑭 𝒊 irregular? (Answer below is brief, see sources for further detail).
Requires “ironing.” Basically this “forces” 𝜑 𝑖 ⋅ to be monotone.
[Myerson 81]: Let (X,P) be any truthful auction. Then expected revenue is upper bounded by expected ironed virtual welfare. That is:
𝐸 𝑣 ← 𝐹 𝑖 𝑃 𝑖 𝑣 = 𝐸 𝑣 ← 𝐹 [ 𝑖 𝜑 𝑖 𝑣 𝑖 ⋅ 𝑋 𝑖 𝑣 ].
𝜑 𝑖 (𝑣) called ironed virtual value. No closed form, but monotone.
LHS: expected revenue. RHS: expected ironed virtual value of winner.
Theorem [Myerson 81]: For all cases, optimal mechanism has following format:
Exist monotone virtual transformations 𝜙 𝑖 : 𝑅 + →𝑅.
Awards item to argma x 𝑖| 𝜙 𝑖 𝑏 𝑖 >0 𝜙 𝑖 𝑏 𝑖 .
Prices exist to guarantee truthfulness.

16. Background: Single Item, Multiple Bidders
Seminal Result:
[Myerson 81]: Magic Lemma! Revenue = virtual welfare.
Special case: 2nd-price auction with reserve = max 𝑝 𝑝⋅1−𝐹 𝑝 is optimal.
General case: maximize ironed virtual welfare.
𝑣 1 ← 𝐹 1
𝑣 2 ← 𝐹 2
𝑣 3 ← 𝐹 3

17. Background: Single Item, Multiple Bidders
Seminal Result:
[Bulow/Klemperer 96]: “increased competition better than market research.”
If 𝐹 𝑖 =𝐹, regular, OPT 𝐹,𝑛 ≤2𝑛𝑑.price(𝐹,𝑛+1).
Better to recruit extra bidder than learn ma x 𝑝 𝑝⋅1−𝐹 𝑝 .
𝑣 1 ←𝐹
𝑣 2 ←𝐹
𝑣 3 ←𝐹
𝑣 4 ←𝐹

18. Summary: Single Item Optimal Mechanism for One Item: Deterministic
Monotone: better distributions more revenue.
One bidder: sets price.
Many iid regular bidders: 2nd price auction with reserve.
One extra bidder > knowing reserve.
Many non-iid irregular bidders: maximize ironed virtual welfare.
𝑣 1 ←𝐹
𝑣 2 ←𝐹
𝑣 3 ←𝐹

19. Summary: Single Item
Main Takeaway: Rich theory of optimal mechanisms + extensions.
𝑣 1 ←𝐹
𝑣 2 ←𝐹
𝑣 3 ←𝐹

20. Multi-Item Auctions

21. Background: Multiple Items
Model:
Seller announces menu: ( 𝑥 𝐿 , 𝑝 𝐿 ) 𝐿 .
Selecting option 𝐿 gives item 𝑗 w.p. 𝑥 𝐿,𝑗 , pay price 𝑝 𝐿 .
Buyer with value 𝑣 drawn from D arrives (think D = population, 𝑣 = individual).
Buyer selects 𝐿 that maximizes E[utility], denote by 𝐿( 𝑣 ).
This talk: Utilit y 𝐿 = 𝑣 ⋅ 𝑥 𝐿 − 𝑝 𝐿 . Buyer valuation additive (𝑣 𝑆 = 𝑖∈𝑆 𝑣 𝑖 ).
Seller goal: maximize expected revenue = 𝐸 𝑣 ←D [ 𝑝 𝐿 𝑣 ].
<𝑣 1 , 𝑣 2 > ←D

22. Two Items – Dependency is Necessary!
Optimal Auction?
𝑣 1 ←𝑈( 1,2 )
𝑣 2 ←𝑈( 1,2 )
Value for apple doesn’t change whether or not get orange.
Because valuation is additive.
Even if knew value for the orange, no info about apple.
Because values are independent.
No dependency whatsoever between items.
So sell each separately at optimal price 1?
Make revenue of 1 per item. 2 total.
Not Optimal!

23. Two Items – Dependency is Necessary!
Optimal Auction?
𝑣 1 ←𝑈( 1,2 )
𝑣 2 ←𝑈( 1,2 )
Better idea: bundle the fruits together, set price of 3.
i.e. offer consumer following options:
- Get apple and orange, pay 3.
- Get nothing, pay nothing.
With probability 3/4, fruit basket sells.
Make expected revenue 9/4 > 2.

24. Two Items – Randomization is Necessary!
<𝑣 1 , 𝑣 2 > ← correlated
Optimal Auction:
Requires randomization [Daskalakis/Deckelbaum/Tzamos 13].
Offer consumer following options (unique optimal mechanism):
Get apple and orange, pay 4.
Get orange with probability 1, apple with probability 1/2, pay 2.5.
Get nothing pay nothing.
[Briest/Chawla/Kleinberg/W. 10, Hart/Nisan 13]: If values are correlated, gap can be infinite. Rev = ∞. All deterministic mechanisms get < 1.
𝑣 1 ←𝑈( 1,2 )
𝑣 2 ←𝑈( 1,3 )

25. Two Items – Market Research >> Competition!
Question:
Which is better, OPT(D,1) or 2nd.price(D, 2)?
Answer:
Correlated D, two items, OPT(D,1) > 2nd.price(D,n) for all n. [Hart/Nisan 13].
Both marginals regular.
Even if D = × 𝑗=1 𝑚 𝐹, could have OPT(D,1) > 2nd.price(D, ln(m) ) [Hart/Nisan 12].
𝐹 regular.
<𝑣 11 , 𝑣 12 > ←D
<𝑣 21 , 𝑣 22 > ←D

26. Two Items – Non-Monotonicity!
Question:
Which is better, selling fruit to 𝐹×𝐹, or selling fruit to 𝐹 + × 𝐹 + ?
Answer:
Sometimes, 𝑂𝑃𝑇 𝐹×𝐹 >𝑂𝑃𝑇( 𝐹 + × 𝐹 + ) [Hart/Reny 13].
Two-item auctions are not monotone: If everyone in 𝐹 + × 𝐹 + likes both fruits more than counterpart in 𝐹×𝐹, might make less revenue selling them.
𝑣 1 ←𝐹 vs. 𝐹 +
𝑣 2 ←𝐹 vs. 𝐹 +
𝐹 + stochastically dominates 𝐹
∀𝑝,1− 𝐹 + 𝑝 ≥1−𝐹 𝑝

27. Summary: Multiple Items
Optimal Mechanism for multiple items:
Could be randomized, even when values iid.
Could be infinitely better than best deterministic if values correlated.
Market research could be better than extra bidders, even when values iid.
Could be better than any finite extra bidders if values correlated.
Non-monotone: better distributions might yield less revenue.
All sorts of other complexities (not all covered here).
<𝑣 1 , 𝑣 2 > ←D

28. What has EC been up to?

29. Approximately optimal mechanisms
Setting: 1 additive buyer. m items. Independent items.
One simple way to sell items: sell them separately (SRev).
Another simple way to sell items: bundle them together (BRev).
[Hart/Nisan 12]: Exist instances with 𝑆𝑅𝑒𝑣≤𝑂𝑃𝑇/ log 𝑚 .
[Hart/Nisan 12]: Exist instances with 𝐵𝑅𝑒𝑣≤𝑂𝑃𝑇/𝑚.
[Li/Yao 13]: all instances, 𝑆𝑅𝑒𝑣≥𝑐⋅𝑂𝑃𝑇/ log 𝑚 . (some constant c).
[Babaioff/Immorlica/Lucier/W. 14]: all instances, max 𝑆𝑅𝑒𝑣,𝐵𝑅𝑒𝑣 ≥𝑂𝑃𝑇/6.

30. Approximately optimal mechanisms
Lots of extensions:
[Rubinstein/W. 15]: “subadditive bidder with independent items.”
[Eden/Feldman/Friedler/Talgam-Cohen/W. 17]: “limited complementarity.”
[Liu/Psomas 18]: “dynamic auctions.”
[Yao 15]: multiple additive bidders.
[Chawla/Miller 16]: multiple bidders “gross substitutes with independent items.”
[Cai/Devanur/W. 16]: reinterpret via LP duality.
[Cai/Zhao 17]: multiple bidders “subadditive with independent items.”
If you are interested in doing research in this area, I strongly recommend:
Get familiar with techniques in [Hart/Nisan 12, Li/Yao 13, BILW 14].
Especially “core-tail decomposition.”
Get comfortable with LP duality, will help unify different ideas [CDW 16].
Ask creative questions! (see next slide).

31. Twists on approximately optimal mechanisms
“Competition Complexity:” Additional bidders to beat OPT with something simple?
Start here: [Bulow/Klemperer 96, Eden/Feldman/Friedler/Talgam-Cohen/W. 17].
[Feldman/Friedler/Rubinstein 18]: improved guarantees – session 7A!
Structure of optimal mechanisms in special cases:
Start here: [Daskalakis/Deckelbaum/Tzamos 17].
[Babaioff/Nisan/Rubinstein 18]: deterministic mechanisms, 2 items – session 7A!
Sample Complexity: How much market data before good revenue?
Start here: [Cole/Roughgarden 14].
[Balcan/Sandholm/Vitercik 18, Babaioff/Gonczarowski/Mansour/Moran 18] – session 4A!
Prior Independence: What about no market data?
Start here: [Hartline/Roughgarden 09].
[Tang/Zeng 18, Besbes/Allouah 18] – session 8A!

32. …and everything else
Dynamic/repeated Auctions: What if interact with same buyer many times?
Start here: [Papadimitriou/Pierrakos/Psomas/Rubinstein 16]
[Mirrokni/Paes Leme/Tang/Zuo 18, Agrawal/Daskalakis/Mirrokni/Sivan 18] - session 4A!
[Feng/Podimata/Syrgkanis 18, Braverman/Mao/Schneider/W. 18] – session 8A!
Correlated/Interdependent values: What if values aren’t independent?
Start here: [Ronen 01, Milgrom/Weber 82].
[Dobzsinski/Uziely 18] – session 7A!
[Eden/Feldman/Fiat/Goldner 18] – session 6A!
Two-sided markets: Many buyers/sellers, all strategic.
Start here: [Myerson/Satterthwaite 83].
[Babaioff/Cai/Gonczarowski/Zhao 18] – session 6A!
Mechanism design + Behavioral Game Theory: What if buyers aren’t fully rational?
Start here: [Li 17].
[Akbarpour/Li 18] – session 6A!

33. Thanks for listening! Takeaways
If you want to do research in this area:
Absolutely get comfortable with [Myerson 81].
Single-item auctions very well understood.
Likely want to get comfortable with [Hart/Nisan 12, Li/Yao 13, BILW 14].
Optimal multi-item auctions weird. Approximately optimal auctions fun!
Check out the directions referenced previously and see what excites you.
“Approximation” can mean many different things – all interesting.
Why do research in this area?
Tons of open directions, extremely well-represented at EC (see previous slides).
Transferable: tools from one direction often useful in others (note major author overlap on previous slides).
Fun! Similar to Approximation Algorithms, but with lots of fun twists.
Thanks for listening!