1. Information, Incentives, and Mechanism Design
Nick Gravin
Course itcs.sufe.edu.cn/~nick
Textbooks available at :
Mechanism Design & Approximation: jasonhartline.com/MDnA/MDnA-ch1to8.pdf
Game Theory, Alive: homes.cs.washington.edu/~karlin/GameTheoryBook.pdf

2. Course structure (Tentative)
Week 1, 2: Mechanism Design and Approximation Overview (Chapter 1, MDA)
Topics: mechanism design, approximation, philosophy thereof, first-price auction, second-price auction, lottery, posted-pricings
Week 2, 3, 4: Equilibrium (Chapter 2, MDA)
Topics: Bayes-Nash equilibirum, dominant strategy equilibrium, single-dimensional agents, BNE characterization, revenue equivalence, uniqueness, revelation principle, incentive compatibility.
Week 4, 5, 6, 7, 8: Optimal Mechanism Design (Chapter 3, MDA)
Topics: single-dimensional mechanism design, surplus-optimal mechanism (VCG), revenue-optimal mechanism (Myerson), amortized analysis, virtual values, revenue curves, revenue linearity.
Week 8, 9, 10, 11: Bayesian Approximation (Chapter 4, MDA)
Topics: reserve pricing, posted pricing, prophet inequalities, correlation gap, monotone hazard rate distributions.
Week 12, 13: Price of Anarchy in games (Chapter 8, GTA)
Topics: selfish routing, existence of equilibrium, affine latencies, network formation games, market sharing games.
Week 14, 15: Stable Matchings and allocations (Chapter 10, GTA)
Topics: applications, Gale-Shapley algorithm, properties of Gale-Shapley algorithm, truthfulness considerations

3. Grading Homework (50%) Final exam (50%) Course report (50%) [bonus]
5 assignments (your score – best 4 out of 5).
Work in pairs (same score for both)
No copying, No late turn in.
Final exam (50%)
Based on problem solving
recommendation: do homework.
Course report (50%) [bonus]
read a research paper/chapter in a book, write a report.

4. Economic & Computational Systems
Individuals take actions
optimize their own (usually selfish) goals
System combines actions to produce an outcome
applies its laws and restrictions.
Systems producing good outcomes:
Spectrum auctions, Residency matching program, Auctions for online advertising, etc.
Bad outcomes:
Financial markets meltdowns, traffic jams, scandals in Olympics games.

5. Example I: Residency Matching
Medical student graduates in USA needs to be assigned to Hospitals for residency.
Preferences:
MGH
> >
Northwestern
Memorial H.
> >
Pennsylvania
hospital
>

6. Example II: Adward auctions
Auction: buyers compete for
better click probability
Click-through rate/
success probability
umbrella
Ads
Slot 1
Slot 2
Slot 3
0.5
0.2
0.1
30 billion $ of revenue for Google!

7. Example III: spectrum auctions
How to reallocate spectrum
Run a double Auction
Buys spectrum:
From inefficient provides
Meet budget constraint
Sells spectrum:
Complex preferences
demand & value uncertainty
Run an auction

8. Example IV: Congestion regulation
People choose:
the fastest route between 𝑠 𝑖 & 𝑡 𝑖
are selfish => suboptimal outcomes
Government:
introduces tolls
to induce socially optimal outcomes
Similar model for selfish routing of packages in a network.

9. Remarks These systems are complex Successful systems have :
individuals’ strategy spaces are large
difficult to optimize over
Successful systems have :
simple rules
individual strategies are simple
This course: trade-off between optimality vs. simplicity
simplicity => robustness, easy computations, practicality.
Means: theory of approximation.
compare the loss of performance of a simple practical mechanism to the complex optimal mechanism.

10. Let us isolate this issue:
Example IV: a model
Network of routers 𝑁=({𝑟𝑜𝑢𝑡𝑒𝑟𝑠}, {𝐿𝑖𝑛𝑘𝑠})
𝑛 strategic users { 𝑠 1 , 𝑡 1 … 𝑠 𝑛 , 𝑡 𝑛 }
Routers/links: capacity constraints
We need a mechanism for congestion control!
Issues:
dynamic demands
complex network structure
strategic user behaviour
etc.
Let us isolate this issue:
Competition for 1 link

11. Single-item Allocation Problem
Definition [single-item problem] is given by
1 indivisible item available
𝑛 strategic agents competing for the item
each agent 𝑖 has a value 𝑣 𝑖 ≥0 for receiving the item.
Objective: maximize social surplus (social welfare), i.e.,
max 𝑣 𝑗 , 𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡 𝑗 – gets the item.
Optimal solution: max 𝑖 𝑣 𝑖 - allocate to the highest value
Notation: 𝑣 (1)

12. Mechanism I 𝑀 1 : You (say, value 100 RMB):
ask agents to report their values
⇒ agent 𝑖 reports 𝑏 𝑖 ,
select the agent 𝑖 ∗ with the highest report
⇒ 𝑖 ∗ = 𝑎𝑟𝑔 max 𝑖 𝑏 𝑖
allocate the item to 𝑖 ∗
You (say, value 100 RMB):
How do you bid?
You bid highest number you can think of!
How to address:
Bids are meaningless?
Pay for overbidding?

13. No payments: Lottery Definition [Lottery]
select a uniformly random agent, and
allocate the item to this agent.
Theorem Lottery is an 𝑛 approximation.
Proof. The expected social welfare (SW) = 1 𝑛 𝑖∈[𝑛] 𝑣 𝑖
The optimum = 𝑣 (1) = max 𝑖∈[𝑛] 𝑣 𝑖 ≤ 𝑖∈ 𝑛 𝑣 𝑖 ≤𝑛⋅(𝑆𝑊).
Bound is tight, when 𝑣 1 =1, 𝑣 2 = 𝑣 3 =…= 𝑣 𝑛 =𝜀.

14. Single-item Auction
Def [Single-item auction] Solution to the allocation problem
that solicits bids,
picks a winner,
determines payments.
Def [First-price auction]
ask agents to report their values (⇒ agent 𝑖 reports 𝑏 𝑖 ),
select 𝑖 ∗ with highest report ( ⇒ 𝑖 ∗ = 𝑎𝑟𝑔 max 𝑖 𝑏 𝑖 ),
allocate the item to agent 𝑖 ∗
charge this winning agent her bid 𝑏 𝑖 ∗ .
Used in government procurements
Not easy to calculate your best strategy
A natural example:
Social welfare 𝑣 𝑖 ∗ = 𝑣 𝑖 ∗ − 𝑏 𝑖 ∗ + 𝑏 𝑖 ∗
agent’s utility seller’s ,

15. How to play? Def [utility] agent 𝑖 derives utility 𝑢 𝑖
𝑢 𝑖 = 𝑣 𝑖 − 𝑏 𝑖 , if she wins in the auction
𝑢 𝑖 =0, if she does not get the item
Assumption: agents want to maximize their utility
if responding to others ⇒ play (pure) Nash Equilibrium.
uncertainty about bids ⇒ maximize utility in expectation.
Experiment:
20 people got 𝑣 𝑖 ∼𝑈[0,4] usd
randomly split into 10 pairs and played first-price auction
winner of each auction would get 𝑣 𝑖 − 𝑏 𝑖 usd.
Over-biding compared to equilibrium ⇒ (Assumption?)

16. Ascending-price Auction
Def [ascending-price auction] a.k.a. English auction.
(gradually) raise an offered price up from zero,
let agents to drop out if don’t want to win at the offered price,
stop at the price 𝑝 ∗ : the second-to-last agent drops out,
item → the last remaining agent, charge her 𝑝 ∗ .
Agent’s strategy (value 𝑣):
drop at price 𝑝<𝑣 ⇒ certainly don’t get the item ⇒ bad idea.
continue above 𝑝>𝑣 ⇒ if win, pay more than 𝑣 ⇒ bad idea.
⇒ dominant strategy: drop at 𝑝=𝑣.

17. Ascending-price Auction
View at the system:
every agent follows dominant strategy
second-highest bidder 𝑣 (2) will drop
highest bidder gets the item, pays 𝑣 (2) ⇒ utility 𝑣 (1) − 𝑣 (2)
SW: 𝑣 (1) = 𝑣 (1) − 𝑣 (2) + 𝑣 (2) ⇒ optimal social surplus.
Theorem The ascending-price auction maximizes the social surplus in dominant strategy equilibrium.
Question: How to extend this to routing example?
Issues:
too slow to implement in practice
ascending prices don’t find socially optimal routing scheme.

18. Second-Price Auction Def [Second-price auction] a.k.a. Vickrey auction
accept sealed bids,
item → the agent 𝑖 ∗ with the highest bid,
charge 𝑖 ∗ the second-highest bid.
Strategically equivalent to the ascending price auction
Theorem Truthful bidding is dominant strategy.

19. Truthfulness Theorem Truthful bidding is dominant strategy.
Proof Fix all values 𝑣 1 ,…, 𝑣 𝑛 . Let 𝑣 𝑖 = max 𝑗≠𝑖 𝑣 𝑗 . For agent i:
Case 1. bid 𝑏 𝑖 > 𝑣 𝑖 ⇒ wins, pays 𝑣 𝑖 ⇒ 𝑢 𝑖 = 𝑣 𝑖 − 𝑣 𝑖 .
Case 2. bid 𝑏 𝑖 < 𝑣 𝑖 ⇒ loses ⇒ 𝑢 𝑖 =0.
𝑖 max utility: if 𝑣 𝑖 > 𝑣 𝑖 wants case 1, if 𝑣 𝑖 < 𝑣 𝑖 wants case 2.
Bidding 𝑏 𝑖 = 𝑣 𝑖 achieves that ⇒ dominant strategy.
Corollary Second-price auction maximizes social surlplus.
Observations:
𝑣 𝑖 - infimum of bids 𝑖 can make and still win;
𝑣 𝑖 - agent 𝑖’s critical value = the price charged to 𝑖;
𝑣 𝑖 - a function only of the other agents’ reports.

20. General routing problem
Def [second-price routing mech.] a.k.a. Vickrey-Clarke-Groves
solicit sealed bids,
find the set of messages that can be routed simultaneously with the largest total value,
charge the agents of each routed message their critical values.
Theorem The second-price routing mechanism has truthful bidding as a dominant strategy.
Corollary The mechanism maximizes the social surplus.
Proof similar to 2nd-price auction

21. Can let only 1 message through
Example C
Optimal value: 5$
(1) 𝐴→𝐵 & (4) 𝐶→𝐷
Can let only 1 message through A D
What are the payments for
senders of messages (1) and (4)?
𝑣 1 (𝐴→𝐵)=1$
𝑣 2 (𝐴→𝐷)=2$
𝑣 3 𝐵→𝐶 =3$
𝑣 4 (𝐶→𝐷)=4$
Critical value for (1)? B

22. Example C 𝑣(𝐴→𝐵,𝐶→𝐷)= 0$+4$ >𝑣 𝐵→𝐶 =3$ A D
What are the payments for
senders of messages (1) and (4)?
𝑣 1 (𝐴→𝐵)=0$
𝑣 2 (𝐴→𝐷)=2$
𝑣 3 𝐵→𝐶 =3$
𝑣 4 (𝐶→𝐷)=4$
Critical value for (1)? B
What if (1) bids 0, does she win?
Yes.
⇒ 𝑝 𝐴→𝐵 =0$

23. Example C A D What are the payments for
senders of messages (1) and (4)?
𝑣 1 (𝐴→𝐵)=1$
𝑣 2 (𝐴→𝐷)=2$
𝑣 3 𝐵→𝐶 =3$
𝑣 4 (𝐶→𝐷)=4$
Critical value for (4)? B

24. Example C 𝑣(𝐴→𝐵,𝐶→𝐷)=1$+1$ <𝑣 𝐵→𝐶 =3$ A D What are the payments for
senders of messages (1) and (4)?
𝑣 1 (𝐴→𝐵)=1$
𝑣 2 (𝐴→𝐷)=2$
𝑣 3 𝐵→𝐶 =3$
𝑣 4 (𝐶→𝐷)=1$
Critical value for (4)? B
What if (4) bids 1, does she win?
No.
⇒ 𝑝 𝐶→𝐷 >1$

25. Example C 𝑣(𝐴→𝐵,𝐶→𝐷)=1$+2$ =𝑣 𝐵→𝐶 =3$ A D What are the payments for
senders of messages (1) and (4)?
𝑣 1 (𝐴→𝐵)=1$
𝑣 2 (𝐴→𝐷)=2$
𝑣 3 𝐵→𝐶 =3$
𝑣 4 (𝐶→𝐷)=2$
Critical value for (4)? B
What if (4) bids 2, does she win?
Tie.
⇒ 𝑝 𝐶→𝐷 =2$

26. VCG: collusion C A D What are the payments for
senders of messages (1) and (4)?
𝑣 1 (𝐴→𝐵)=1$
𝑣 2 (𝐴→𝐷)=2$
𝑣 3 𝐵→𝐶 =3$
𝑣 4 (𝐶→𝐷)=4$ B

27. VCG: collusion C 𝑣(𝐴→𝐵,𝐶→𝐷)=5$+5$ >𝑣 𝐵→𝐶 =3$ A D
What are the payments for
senders of messages (1) and (4)?
𝑣 1 (𝐴→𝐵)=5$
𝑣 2 (𝐴→𝐷)=2$
𝑣 3 𝐵→𝐶 =3$
𝑣 4 (𝐶→𝐷)=5$ B
What if (1) & (4) both bid 5?
Win,
𝑝 𝐴→𝐵 = 𝑝 𝐶→𝐷 =0$

28. Posted Pricing (single item)
Question: What if even 2nd-price auction is too complicated?
Def [uniform-pricing] for a given price 𝑣 , mechanism serves the first agent willing to pay 𝑣 (break ties in arrival order).
E.g., 𝐹=𝑈 0,1 , 𝐹 𝑧 =𝑧, 𝐹=𝑈 0,2 , 𝐹 𝑧 = 𝑧 2
Assumption: Distributional knowledge of demand.
𝑣 𝑖 ∼𝐹 i.i.d. for all 𝑖∈[𝑛], where 𝐹 𝑧 = Pr 𝑣∼𝐹 [𝑣≤𝑧] is a known cdf (cumulative density function).

29. Posted pricing (single item)
2nd price auction: ax ante probability of 𝑖 winning is 1 𝑛
Theorem
For values drawn i.i.d. from any distribution F, the uniform pricing of 𝑣 = 𝐹 −1 (1− 1 𝑛 ) is an 𝑒 𝑒−1 ≈1.58 approximation to the optimal social surplus.
Proof On the board
Before the values are drawn
Mimic by posted price: 𝑣 = 𝐹 −1 (1− 1 𝑛 )

30. Discussion Vickrey-Clarke-Groves (VCG) routing. Algorithmic theory:
Can be tricky to compute the optimal solution.
Actually, for Border Gateway Protocol (BGP) is NP-hard.
Algorithmic theory:
if computing the optimal solution is intractable
try instead to compute an approximately optimal solution.
Question: Can we use an approximation algorithm and still
retain the dominant-strategy incentive property?
Question: Is there a general theory for designing approximation
mechanisms from approximation algorithms?

31. Theory: what we want
Informative: pinpoints salient features of the environment and characteristics of the good mechanisms.
Prescriptive: gives concrete suggestions for how a good mechanism should be designed.
Predictive: The mechanisms that the theory predicts should be the same as the ones observed in practice.
Tractable: The theory should not assume super-natural ability for the agents or designer to optimize.

32. Philosophy of approximation
Picasso’s “Bull study”