auction and mechanism design

1. An Introduction to Auction
and Mechanism Design
Yitian Chen, IPC Team
chenyitian@jd.com
November 21, 2014
Yitian Auction and Mechanism Design November 21, 2014 1 / 24

2. Outline
Outline
1 Warm-up For Mechanism Design
2 Mechanism Design in Advertising
3 Apply to Supply Chain Management?
4 Question & Answer
Susan Athey
http://www.gsb.stanford.edu/
faculty-research/faculty/susan-athey
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3. Warming-up
Outline
1 Warm-up For Mechanism Design
Negotiation vs Pricing vs Auction
When and why to choose Auction
Equation-based Model vs Agent-based Model
2 Mechanism Design in Advertising
3 Apply to Supply Chain Management?
4 Question & Answer
Yitian Auction and Mechanism Design November 21, 2014 3 / 24

4. Warming-up Negotiation vs Pricing vs Auction
Methods to sell an Item(Items)
Negotiation
Pricing
Auction
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5. Warming-up When and why to choose Auction
When and Why to Auction
When to Auction [1]
A seller has multiple potential buyers
Buyers are symmetric and have independent value
Both seller and buyers are not certain about the true value
Why to Auction [2]
More profits
Less labor cost(Compare to Negotiation)
Auction and Mechanism Design
A seller has a Objective Function(Revenue, Social Value,
Fairness...)
A seller has a selling methods list(Pricing, Negotiation, First Price,
Second Price, VCG)
Mechanism Design: How to choose(create) selling method to
maximize Objective Function
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6. Warming-up Equation-based Model vs Agent-based Model
Model thinking: from the point of Model
Equation-Based Model
Maximize the outcome(Objective function)
Eg.. Generalized Linear Model; Decision Tree...Supervised
Learning
Agent-Based Model
individuals: Agents interact with each other(cooperate, compete)
behaviours: Eg..Every agent is try to maximise to maximize its
profit
outcomes: Eg.. Seller is trying to maximize its revenue(Pareto
optimization)
Yitian Auction and Mechanism Design November 21, 2014 6 / 24

7. Advertising Mechanism
Outline
1 Warm-up For Mechanism Design
2 Mechanism Design in Advertising
Generalized First Price
Generalized Second Price
Locally envy-free Nash Equilibrium
The beautiful VCG Auction
3 Apply to Supply Chain Management?
4 Question & Answer
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8. Advertising Mechanism First price
A laptop example
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9. Advertising Mechanism First price
Generalized first price auction
Bidder Bid Price Order
Dell 6 6 1st
JD 4 4 2nd
Leno 2 2 3rd
MRP 0.5
Yitian Auction and Mechanism Design November 21, 2014 9 / 24

10. Advertising Mechanism First price
Generalized first price auction
Bidder Bid Price Order
Dell 6 6 1st
JD 4 4 2nd
Leno 2 2 3rd
MRP 0.5
Yitian Auction and Mechanism Design November 21, 2014 9 / 24

11. Advertising Mechanism First price
Generalized first price auction
Bidder Bid Price Order
Dell 6 6 1st
JD 4 4 2nd
Leno 2 2 3rd
MRP 0.5
Yitian Auction and Mechanism Design November 21, 2014 9 / 24

12. Advertising Mechanism First price
Generalized first price auction
Bidder Bid Price Order
Dell 6 6 1st
JD 4 4 2nd
Leno 2 2 3rd
MRP 0.5
Yitian Auction and Mechanism Design November 21, 2014 9 / 24

13. Advertising Mechanism First price
The problem of GFP
Yitian Auction and Mechanism Design November 21, 2014 10 / 24

14. Advertising Mechanism Second price
Generalized second price
Bidder Bid Price Order
Dell 6 4 1st
JD 4 2 2nd
Leno 2 0.5 3rd
MRP 0.5
Yitian Auction and Mechanism Design November 21, 2014 11 / 24

15. Advertising Mechanism Second price
Generalized second price
Bidder Bid Price Order
Dell 6 4 1st
JD 4 2 2nd
Leno 2 0.5 3rd
MRP 0.5
Yitian Auction and Mechanism Design November 21, 2014 11 / 24

16. Advertising Mechanism Second price
Generalized second price
Bidder Bid Price Order
Dell 6 4 1st
JD 4 2 2nd
Leno 2 0.5 3rd
MRP 0.5
Yitian Auction and Mechanism Design November 21, 2014 11 / 24

17. Advertising Mechanism Second price
GSP Auction rules
Most search engines use Ecpm bid ∗ CTR
instead of Bid
Each advertiser i submits bid Bidi
Positions assigned in order of Bidi ∗ CTRi
Advertiser i’s price Pi is
Bidi+1 ∗ CTRi+1/CTRi
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18. Advertising Mechanism Equilibrium of GSP
Truthful?–Bidding the value
Bidder Value Bid Price Clicks Payoff Order
Dell 6 6 4 200 (6 − 4) ∗ 200 = 400 1st
JD 4 4 2 120 (4 − 2) ∗ 120 = 240 2nd
Leno 2 2 0.5 50 (2 − 0.5) ∗ 50 = 75 3rd
MRP 0.5
Yitian Auction and Mechanism Design November 21, 2014 13 / 24

19. Advertising Mechanism Equilibrium of GSP
Truthful?–Bidding the value
Bidder Value Bid Price Clicks Payoff Order
Dell 6 6 4 200 (6 − 4) ∗ 200 = 400 1st
JD 4 4 2 120 (4 − 2) ∗ 120 = 240 2nd
Leno 2 2 0.5 50 (2 − 0.5) ∗ 50 = 75 3rd
MRP 0.5
Yitian Auction and Mechanism Design November 21, 2014 13 / 24

20. Advertising Mechanism Equilibrium of GSP
Truthful?–Bidding the value
Bidder Value Bid Price Clicks Payoff Order
Dell 6 6 4 200 (6 − 4) ∗ 200 = 400 1st
JD 4 4 2 120 (4 − 2) ∗ 120 = 240 2nd
Leno 2 2 0.5 50 (2 − 0.5) ∗ 50 = 75 3rd
MRP 0.5
Yitian Auction and Mechanism Design November 21, 2014 13 / 24

21. Advertising Mechanism Equilibrium of GSP
Truthful?–If Dell Bidding 3...
Bidder Value Bid Price Clicks Payoff Order
JD 4 4 3 200 (4 − 3) ∗ 200 = 200 1st
Dell 6 3 2 120 (6 − 2) ∗ 120 = 480 > 400 2nd
Leno 2 2 0.5 50 (2 − 0.5) ∗ 50 = 75 3rd
MRP 0.5
Nobody could improve its profit by exchanging its position
We call this kind of equilibrium as Locally Envy-free NE
Yitian Auction and Mechanism Design November 21, 2014 14 / 24

22. Advertising Mechanism Equilibrium of GSP
Truthful?–If Dell Bidding 3...
Bidder Value Bid Price Clicks Payoff Order
JD 4 4 3 200 (4 − 3) ∗ 200 = 200 1st
Dell 6 3 2 120 (6 − 2) ∗ 120 = 480 > 400 2nd
Leno 2 2 0.5 50 (2 − 0.5) ∗ 50 = 75 3rd
MRP 0.5
Nobody could improve its profit by exchanging its position
We call this kind of equilibrium as Locally Envy-free NE
Yitian Auction and Mechanism Design November 21, 2014 14 / 24

23. Advertising Mechanism Equilibrium of GSP
Truthful?–If Dell Bidding 3...
Bidder Value Bid Price Clicks Payoff Order
JD 4 4 3 200 (4 − 3) ∗ 200 = 200 1st
Dell 6 3 2 120 (6 − 2) ∗ 120 = 480 > 400 2nd
Leno 2 2 0.5 50 (2 − 0.5) ∗ 50 = 75 3rd
MRP 0.5
Nobody could improve its profit by exchanging its position
We call this kind of equilibrium as Locally Envy-free NE
Yitian Auction and Mechanism Design November 21, 2014 14 / 24

24. Advertising Mechanism Equilibrium of GSP
Truthful?–If Dell Bidding 3...
Bidder Value Bid Price Clicks Payoff Order
JD 4 4 3 200 (4 − 3) ∗ 200 = 200 1st
Dell 6 3 2 120 (6 − 2) ∗ 120 = 480 > 400 2nd
Leno 2 2 0.5 50 (2 − 0.5) ∗ 50 = 75 3rd
MRP 0.5
Nobody could improve its profit by exchanging its position
We call this kind of equilibrium as Locally Envy-free NE
Yitian Auction and Mechanism Design November 21, 2014 14 / 24

25. Advertising Mechanism Equilibrium of GSP
Locally Envy-free NE
Definition: An equilibrium is locally
envy-free if no player can improve his
payoff by exchanging positions
Motivation: squeezing – if an equilibrium
is not locally Envy-free NE, there might be
an incentive to squeeze.
Inequality of LEF-NE
(Vk − Pk ) ∗ Xk ≥ (Vk − Pk−1 ∗ Xk−1)
No incentive to move below
(Vk − Pk ) ∗ Xk ≥ (Vk − Pk+1 ∗ Xk+1)
No incentive to move above
Hal Varian
http://people.ischool.
berkeley.edu/~hal/
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26. Advertising Mechanism The beautiful VCG Auction
Think Second price with social value
Bidder Value Clicks social value
Dell 6 200 ?
JD 4 120 ?
Assume there is only one ad slot for sales.
Dell participate in the auction, Dell win the slots and the total
value = 6 ∗ 200 = 1200, others’s social value is 0
If Dell does not participate in the auction, JD win the slots and the
total social value = 4 ∗ 200 = 800
the hurt to others’s 800 − 0 = 800, the price is the average hurt
800/200 = 4
Yitian Auction and Mechanism Design November 21, 2014 16 / 24

27. Advertising Mechanism The beautiful VCG Auction
Think Second price with social value
Bidder Value Clicks social value
Dell 6 200 ?
JD 4 120 ?
Assume there is only one ad slot for sales.
Dell participate in the auction, Dell win the slots and the total
value = 6 ∗ 200 = 1200, others’s social value is 0
If Dell does not participate in the auction, JD win the slots and the
total social value = 4 ∗ 200 = 800
the hurt to others’s 800 − 0 = 800, the price is the average hurt
800/200 = 4
Yitian Auction and Mechanism Design November 21, 2014 16 / 24

28. Advertising Mechanism The beautiful VCG Auction
Think Second price with social value
Bidder Value Clicks social value
Dell 6 200 ?
JD 4 120 ?
Assume there is only one ad slot for sales.
Dell participate in the auction, Dell win the slots and the total
value = 6 ∗ 200 = 1200, others’s social value is 0
If Dell does not participate in the auction, JD win the slots and the
total social value = 4 ∗ 200 = 800
the hurt to others’s 800 − 0 = 800, the price is the average hurt
800/200 = 4
Yitian Auction and Mechanism Design November 21, 2014 16 / 24

29. Advertising Mechanism The beautiful VCG Auction
Think Second price with social value
Bidder Value Clicks social value
Dell 6 200 ?
JD 4 120 ?
Assume there is only one ad slot for sales.
Dell participate in the auction, Dell win the slots and the total
value = 6 ∗ 200 = 1200, others’s social value is 0
If Dell does not participate in the auction, JD win the slots and the
total social value = 4 ∗ 200 = 800
the hurt to others’s 800 − 0 = 800, the price is the average hurt
800/200 = 4
Yitian Auction and Mechanism Design November 21, 2014 16 / 24

30. Advertising Mechanism The beautiful VCG Auction
Think Second price with social value
Bidder Value Clicks social value
Dell 6 200 ?
JD 4 120 ?
Assume there is only one ad slot for sales.
Dell participate in the auction, Dell win the slots and the total
value = 6 ∗ 200 = 1200, others’s social value is 0
If Dell does not participate in the auction, JD win the slots and the
total social value = 4 ∗ 200 = 800
the hurt to others’s 800 − 0 = 800, the price is the average hurt
800/200 = 4
Yitian Auction and Mechanism Design November 21, 2014 16 / 24

31. Advertising Mechanism The beautiful VCG Auction
Think Second price with social value
Bidder Value Clicks social value
Dell 6 200 ?
JD 4 120 ?
Assume there is only one ad slot for sales.
Dell participate in the auction, Dell win the slots and the total
value = 6 ∗ 200 = 1200, others’s social value is 0
If Dell does not participate in the auction, JD win the slots and the
total social value = 4 ∗ 200 = 800
the hurt to others’s 800 − 0 = 800, the price is the average hurt
800/200 = 4
Yitian Auction and Mechanism Design November 21, 2014 16 / 24

32. Advertising Mechanism The beautiful VCG Auction
The VCG Auction
Definition:Wv = total value received by
agents, where v is the vector of values
Payment rule: VCG payment of is W−i
(0, X−i) - W−i (x) = value to others when i
bids 0(does not bid), minus that when i
bids truthfully(externality to others).
Important: VCG is socially optimal-
maximize the social welfare.
Vickrey
http://en.wikipedia.org/
wiki/William_Vickrey
Yitian Auction and Mechanism Design November 21, 2014 17 / 24

33. Apply to Supply Chain Management?
Outline
1 Warm-up For Mechanism Design
2 Mechanism Design in Advertising
3 Apply to Supply Chain Management?
Procurement Problem
Dynamic graph
4 Question & Answer
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34. Apply to Supply Chain Management? Procurement Problem
ICP team is going to purchase 100 cups
Supplier Supply Num Unit price Sold partly
Ada 80 2 Yes
Bob 100 3 No
Carol 60 5 Yes
The outcome is determine by the bid:
Supplier Ada and Carol win the auction
Ada gets 80 cups supply and Carol gets 20
cups supply (80 ∗ 2 + 5 ∗ 20) ≤ (3 ∗ 100)
Price of Ada: If Ada does not participate,
Bob wins the auction, the total social value is
100 ∗ 3 = 300
if Ada participate, others’ social value: Carol
20*5 = 100, the social value hurt to others is
(300-200) price = (300-100)/80=2.5
Price of Carol : (3*100-80*2)/20=7
Yitian Auction and Mechanism Design November 21, 2014 19 / 24

35. Apply to Supply Chain Management? Procurement Problem
ICP team is going to purchase 100 cups
Supplier Supply Num Unit price Sold partly
Ada 80 2 Yes
Bob 100 3 No
Carol 60 5 Yes
The outcome is determine by the bid:
Supplier Ada and Carol win the auction
Ada gets 80 cups supply and Carol gets 20
cups supply (80 ∗ 2 + 5 ∗ 20) ≤ (3 ∗ 100)
Price of Ada: If Ada does not participate,
Bob wins the auction, the total social value is
100 ∗ 3 = 300
if Ada participate, others’ social value: Carol
20*5 = 100, the social value hurt to others is
(300-200) price = (300-100)/80=2.5
Price of Carol : (3*100-80*2)/20=7
Yitian Auction and Mechanism Design November 21, 2014 19 / 24

36. Apply to Supply Chain Management? Procurement Problem
ICP team is going to purchase 100 cups
Supplier Supply Num Unit price Sold partly
Ada 80 2 Yes
Bob 100 3 No
Carol 60 5 Yes
The outcome is determine by the bid:
Supplier Ada and Carol win the auction
Ada gets 80 cups supply and Carol gets 20
cups supply (80 ∗ 2 + 5 ∗ 20) ≤ (3 ∗ 100)
Price of Ada: If Ada does not participate,
Bob wins the auction, the total social value is
100 ∗ 3 = 300
if Ada participate, others’ social value: Carol
20*5 = 100, the social value hurt to others is
(300-200) price = (300-100)/80=2.5
Price of Carol : (3*100-80*2)/20=7
Yitian Auction and Mechanism Design November 21, 2014 19 / 24

37. Apply to Supply Chain Management? Procurement Problem
ICP team is going to purchase 100 cups
Supplier Supply Num Unit price Sold partly
Ada 80 2 Yes
Bob 100 3 No
Carol 60 5 Yes
The outcome is determine by the bid:
Supplier Ada and Carol win the auction
Ada gets 80 cups supply and Carol gets 20
cups supply (80 ∗ 2 + 5 ∗ 20) ≤ (3 ∗ 100)
Price of Ada: If Ada does not participate,
Bob wins the auction, the total social value is
100 ∗ 3 = 300
if Ada participate, others’ social value: Carol
20*5 = 100, the social value hurt to others is
(300-200) price = (300-100)/80=2.5
Price of Carol : (3*100-80*2)/20=7
Yitian Auction and Mechanism Design November 21, 2014 19 / 24

38. Apply to Supply Chain Management? Procurement Problem
ICP team is going to purchase 100 cups
Supplier Supply Num Unit price Sold partly
Ada 80 2 Yes
Bob 100 3 No
Carol 60 5 Yes
The outcome is determine by the bid:
Supplier Ada and Carol win the auction
Ada gets 80 cups supply and Carol gets 20
cups supply (80 ∗ 2 + 5 ∗ 20) ≤ (3 ∗ 100)
Price of Ada: If Ada does not participate,
Bob wins the auction, the total social value is
100 ∗ 3 = 300
if Ada participate, others’ social value: Carol
20*5 = 100, the social value hurt to others is
(300-200) price = (300-100)/80=2.5
Price of Carol : (3*100-80*2)/20=7
Yitian Auction and Mechanism Design November 21, 2014 19 / 24

39. Apply to Supply Chain Management? Procurement Problem
ICP team is going to purchase 100 cups
Supplier Supply Num Unit price Sold partly
Ada 80 2 Yes
Bob 100 3 No
Carol 60 5 Yes
The outcome is determine by the bid:
Supplier Ada and Carol win the auction
Ada gets 80 cups supply and Carol gets 20
cups supply (80 ∗ 2 + 5 ∗ 20) ≤ (3 ∗ 100)
Price of Ada: If Ada does not participate,
Bob wins the auction, the total social value is
100 ∗ 3 = 300
if Ada participate, others’ social value: Carol
20*5 = 100, the social value hurt to others is
(300-200) price = (300-100)/80=2.5
Price of Carol : (3*100-80*2)/20=7
Yitian Auction and Mechanism Design November 21, 2014 19 / 24

40. Apply to Supply Chain Management? Dynamic graph
Buying a path in a network
Many problem can be modelled as a directed
graph G = (V, G).
Each link E e has a cost Ce ≥ 0
Each edge owner can submit a bid.
We want to buy a path between two
vertices(s, t).
Each edge owner havs incentive to change
the cost.
Buy a Path from a to e
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41. Apply to Supply Chain Management? Dynamic graph
Solution with VCG Auction
Find the shortest path
(a − b − d − e)(Dijkstra Algorithm)
If a − b does not participate, choose
(a − d − e)
The social value when (a − b) does
not participate 7+4=11
The others’ social value when a − b
participate (b − d) and (d − e) 2+4=6
the price of edge (a − b) 11-6=5
The solution maximise the social
value(buyers and sellers’ total profit)
Yitian Auction and Mechanism Design November 21, 2014 21 / 24

42. Apply to Supply Chain Management? Dynamic graph
Solution with VCG Auction
Find the shortest path
(a − b − d − e)(Dijkstra Algorithm)
If a − b does not participate, choose
(a − d − e)
The social value when (a − b) does
not participate 7+4=11
The others’ social value when a − b
participate (b − d) and (d − e) 2+4=6
the price of edge (a − b) 11-6=5
The solution maximise the social
value(buyers and sellers’ total profit)
Yitian Auction and Mechanism Design November 21, 2014 21 / 24

43. Apply to Supply Chain Management? Dynamic graph
Solution with VCG Auction
Find the shortest path
(a − b − d − e)(Dijkstra Algorithm)
If a − b does not participate, choose
(a − d − e)
The social value when (a − b) does
not participate 7+4=11
The others’ social value when a − b
participate (b − d) and (d − e) 2+4=6
the price of edge (a − b) 11-6=5
The solution maximise the social
value(buyers and sellers’ total profit)
Yitian Auction and Mechanism Design November 21, 2014 21 / 24

44. Apply to Supply Chain Management? Dynamic graph
Solution with VCG Auction
Find the shortest path
(a − b − d − e)(Dijkstra Algorithm)
If a − b does not participate, choose
(a − d − e)
The social value when (a − b) does
not participate 7+4=11
The others’ social value when a − b
participate (b − d) and (d − e) 2+4=6
the price of edge (a − b) 11-6=5
The solution maximise the social
value(buyers and sellers’ total profit)
Yitian Auction and Mechanism Design November 21, 2014 21 / 24

45. Apply to Supply Chain Management? Dynamic graph
Solution with VCG Auction
Find the shortest path
(a − b − d − e)(Dijkstra Algorithm)
If a − b does not participate, choose
(a − d − e)
The social value when (a − b) does
not participate 7+4=11
The others’ social value when a − b
participate (b − d) and (d − e) 2+4=6
the price of edge (a − b) 11-6=5
The solution maximise the social
value(buyers and sellers’ total profit)
Yitian Auction and Mechanism Design November 21, 2014 21 / 24

46. Question & Answer
Outline
1 Warm-up For Mechanism Design
2 Mechanism Design in Advertising
3 Apply to Supply Chain Management?
4 Question & Answer
Yitian Auction and Mechanism Design November 21, 2014 22 / 24

47. Question & Answer
Question & Answer
Yitian Auction and Mechanism Design November 21, 2014 23 / 24

48. Question & Answer
References
[Auction Theory, 2002] Vijay Krishna
[Auction Versus Negotitations] Jeremy Bulow and Paul Klemperer
The American Economic Review, Vol. 86, No. 1. (Mar., 1996), pp.
180-194.
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