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. Games of Complete Information

4. Prisoners dilemma Example I If you know what other person does,
what will you do?
A: any action of B
Confess > Silent !
A confess
A silent B
confess
A: -8
B: -8
A: -20
B: 0
silent
A: 0
B: -20
A: -0.5
B: -0.5

5. Dominant Strategy Equilibrium (DSE)
Def [DSE] in a complete information game is a strategy profile such that each player’s strategy is as least as good as all other strategies regardless of the strategies of all other players.
Remark Rarely exists!
Def [NE] Mixed strategy profile 𝐬=( 𝑠 1 ,…, 𝑠 𝑛 ), such that each
𝑠 𝑖 is a best response of player 𝑖 to 𝐬 −𝒊 =( 𝑠 1 ,…, 𝑠 𝑖−1 , 𝑠 𝑖+1 ,…, 𝑠 𝑛 )
Compare to Nash Equilibrium (NE),
which always exists.

6. Chicken game 2 Pure Nash Equilibria: (A swerves, B stays)
(A stays, B swerves)
PNE may not exist
PNE is more likely than DSE.
A stay
A swerve B
stay
A: -10
B: -10
A: -1
B: 1
swerve
A: 1
B: -1
A: 0
B: 0

7. Games of Incomplete Information

8. Incomplete information
Agents have private information (types)
Denoted by types 𝐭=( 𝑡 1 ,…, 𝑡 𝑛 )
Strategy 𝑠 𝑖 ( 𝑡 𝑖 ) of agent 𝑖
𝑠 𝑖 : maps 𝑡 𝑖 ∈ 𝑇 𝑖 →𝐴𝑐𝑡𝑖𝑜 𝑛 𝑖 , or distribution of actions.
Examples: ascending price & 2nd price auctions
𝑡 𝑖 = 𝑣 𝑖 value of agent 𝑖
𝑠 𝑖 ( 𝑣 𝑖 ) – “drop at value 𝑣 𝑖 ”, “bid 𝑏 𝑖 = 𝑣 𝑖 ”
these strategies – truth telling.

9. Dominant strategy Equilibrium (DSE)
Strategy profile 𝐬= 𝑠 1 ,…, 𝑠 𝑛
Def [DSE] is a strategy profile 𝒔 such that, for all 𝑖, 𝑡 𝑖 , and 𝐛 −𝒊
(where 𝐛 −𝒊 generically refers to the actions of all players but i), agent 𝑖’s utility is maximized by following strategy 𝑠 𝑖 ( 𝑡 𝑖 ).
Rem Aside from strategies 𝑠 𝑖 being a map from types to actions DSE for incomplete information ⟺ DSE for full information case.

10. Bayes-Nash Equilibrium (BNE)
Equilibrium: 𝑖 best responds to other agents’ strategies.
what about private types of other agents?
depends on the 𝑖’s believes about types of other agents
common prior assumption (standard in economics).
Def [Prior] agent types 𝐭 are drawn at random from a prior distribution 𝐅 (a joint probability distribution over type profiles) and this prior distribution is common knowledge.
𝐅- may be correlated distribution.
Then 𝑖 does Bayesian update conditional on 𝑡 𝑖 : 𝐅 −𝑖 ​ 𝑡 𝑖
If 𝐅- independent, i.e., 𝐅= 𝐹 1 ×…× 𝐹 𝑛 , then 𝐅 −𝑖 ​ 𝑡 𝑖 = 𝐅 −𝑖 .

11. Bayes-Nash Equilibrium (BNE)
Def [BNE] for a game 𝐺 and common prior 𝐅 is a strategy profile 𝐬=( 𝑠 1 , …, 𝑠 𝑛 ) such that for all agents 𝑖∈ 𝑛 , and all types 𝑡 𝑖
𝑠 𝑖 ( 𝑡 𝑖 ) is a best response, when other agents play 𝐬 −𝑖 ( 𝐭 −𝑖 ) for 𝐭 −𝑖 ∼ 𝐅 −𝑖 ​ 𝑡 𝑖 .
Rem BNE can be regarded as a NE in appropriately defined game
⟹ BNE (in mixed strategies) always exists.

12. Example of BNE Game: 1st price auction for 2 players
𝑡 1 = 𝑣 1 , 𝑡 2 = 𝑣 2 ∼𝑈𝑛𝑖𝑓𝑜𝑟𝑚[0,1] i.i.d.
common prior 𝐅=𝐹×𝐹, where 𝐹 𝑧 = Pr 𝑣∼𝐹 𝑣≤𝑧 =𝑧
Guess BNE: 𝑠 𝑖 𝑧 = 𝑧 2 for 𝑖∈{1,2}.
fix strategy of agent 2 and treat 𝑣 2 , 𝑏 2 as random var.
for any value 𝑣 1 and bid 𝑏 1 calculate utility 𝑢 1 :
E 𝑏 2 𝑢 1 = 𝑣 1 − 𝑏 1 ×Pr 1 wins with bid 𝑏 1
Pr 𝑏 wins = Pr 𝑏 2 b 2 < b 1 = Pr 𝑣 𝑣 < 𝑏 1 =𝐹 2 𝑏 1 =2 𝑏 1
E 𝑏 2 𝑢 1 = 𝑣 1 − 𝑏 1 ⋅2 𝑏 1 -maximized when 𝑏 1 = 𝑣 1 2
Let’s verify.

13. Stages of the Bayesian game
When and what agent 𝑖 knows?
Ex ante (before the game starts)
𝑖 only knows 𝐹 𝑖 ( 𝑣 𝑖 ∼ 𝐹 𝑖 ), and 𝐅 −𝒊 ( 𝐯 −𝒊 ∼ 𝐅 −𝒊 )
Interim (learn private value, before the game)
𝑖 learnt 𝑣 𝑖 , but does not know 𝐯 −𝒊 and 𝐛 −𝒊
Ex post (after the game is played)
𝑖 observes all actions in the game

14. Single-Dimensional Games
Private type 𝑡 𝑖 - value 𝑣 𝑖 ∈ℝ for receiving an abstract service.
Independent types: distribution 𝐅= 𝐹 1 ×…× 𝐹 𝑛
Outcome 𝐱= 𝑥 1 ,…, 𝑥 𝑛 , where 𝑥 𝑖 - indicator if 𝑖 is served.
Payments 𝐩= 𝑝 1 ,…, 𝑝 𝑛 .
Def Linear utility 𝑢 𝑖 = 𝑣 𝑖 ⋅ 𝑥 𝑖 − 𝑝 𝑖 for the outcome ( 𝑥 𝑖 , 𝑝 𝑖 )
Def Game 𝐺:𝐛→(𝐱,𝐩) maps actions to outcomes & payments.
𝑥 𝑖 𝐺 (𝐛)= outcome for 𝑖 when actions are 𝐛.
𝑝 𝑖 𝐺 (𝐛)= payment from 𝑖 when actions are 𝐛
𝑥 𝑖 =1 if served
𝑥 𝑖 =0 otherwise

15. Single-Dimensional Games
Given a game 𝐺 and strategy profile 𝒔
𝑥 𝑖 𝐯 = 𝑥 𝑖 𝐺 (𝑠(𝐯))
𝑝 𝑖 𝐯 = 𝑝 𝑖 𝐺 (𝑠(𝐯))
are allocation rule and payment rule for (implicit) game 𝐺 and 𝐬.
Let’s take agent 𝑖 interim perspective
𝑭 ​ 𝑣 𝑖 , 𝐺, 𝒔 are implicit. We can specify allocation & payment
𝑥 𝑖 𝑣 𝑖 =Pr 𝑥 𝑖 𝑣 𝑖 =1 𝑣 𝑖 ]=𝐄 𝑥 𝑖 𝐯 𝑣 𝑖 ]
𝑝 𝑖 𝑣 𝑖 =𝐄 𝑝 𝑖 𝐯 𝑣 𝑖 ]
𝑢 𝑖 = 𝑣 𝑖 ⋅ 𝑥 𝑖 𝑣 𝑖 − 𝑝 𝑖 ( 𝑣 𝑖 ) [linearity of expectation]

16. Bayes-Nash Equilibrium (BNE)
Proposition When values are drawn from a product distribution 𝐅, single-dimensional game 𝐺 and strategy profile 𝐬 is in BNE only if (⟹) for all 𝑖, 𝑣 𝑖 , and 𝑧
𝑣 𝑖 ⋅ 𝑥 𝑖 𝑣 𝑖 − 𝑝 𝑖 𝑣 𝑖 ≥ 𝑣 𝑖 ⋅ 𝑥 𝑖 𝑧 −𝑝 𝑧 .
The converse (⟸) is true if the strategy profile is onto.
We say a strategy 𝑠 𝑖 (·) is onto if every action 𝑏 𝑖 agent 𝑖 could play in the game is prescribed by 𝑠 𝑖 for some value 𝑣 𝑖 , i.e., ∀ 𝑏 𝑖 ∃ 𝑣 𝑖 𝑠 𝑖 𝑣 𝑖 = 𝑏 𝑖 .
A strategy profile is onto if the strategy of every agent is onto.

17. Characterization of BNE
Theorem When values are drawn from a continuous product distribution 𝐅, single-dimensional 𝐺 and strategy profile 𝐬 are in BNE only if (⟹) for all 𝑖,
[monotonicity] 𝑥 𝑖 ( 𝑣 𝑖 ) is monotone non-decreasing,
[payment identity] 𝑝 𝑖 𝑣 𝑖 = 𝑣 𝑖 ⋅ 𝑥 𝑖 𝑣 𝑖 − 0 𝑣 𝑖 𝑥 𝑖 𝑧 𝑑𝑧+ 𝑝 𝑖 0 ,
where often 𝑝 𝑖 (0)=0.
If the strategy profile is onto then the converse (⟸) also holds.

18. Proof Plan (i) Monotonicity +(ii)Payment identity + onto ⟹ BNE
BNE ⟹ (i) Monotonicity
BNE ⟹ (ii) Payment identity
Let’s fix 𝑖 and drop subscript from notation.
Key idea: deviation of playing 𝑠( 𝑣 ) instead of 𝑠(𝑣)
Proof on the board.

27. Monotonicity + We want to prove that BNE ⟹ 𝑥(𝑣) monotone.
For any 𝑧 2 , 𝑧 1
𝑢 𝑧 1 , 𝑧 2 = 𝑧 1 𝑥 𝑧 2 −𝑝 𝑧 2 ≤ 𝑧 1 𝑥 𝑧 1 −𝑝 𝑧 1 =𝑢 𝑧 1 , 𝑧 1
𝑢 𝑧 2 , 𝑧 1 = 𝑧 2 𝑥 𝑧 1 −𝑝 𝑧 1 ≤ 𝑧 2 𝑥 𝑧 2 −𝑝 𝑧 2 =𝑢 𝑧 2 , 𝑧 2
𝑧 1 𝑥 𝑧 2 + 𝑧 2 𝑥 𝑧 1 ≤ 𝑧 1 𝑥 𝑧 1 + 𝑧 2 𝑥 𝑧 2
⟺ 𝑧 2 − 𝑧 1 𝑥 𝑧 2 −𝑥 𝑧 1 ≥0 +

28. Payment identity BNE ⟹ 𝑝 𝑣 =𝑣⋅𝑥 𝑣 − 0 𝑣 𝑥 𝑧 𝑑𝑧+𝑝 0 .
Let 𝑧 2 > 𝑧 1 , then
𝑢 𝑧 1 , 𝑧 2 = 𝑧 1 𝑥 𝑧 2 −𝑝 𝑧 2 ≤ 𝑧 1 𝑥 𝑧 1 −𝑝 𝑧 1 =𝑢 𝑧 1 , 𝑧 1
⟹𝑝 𝑧 2 −𝑝 𝑧 1 ≥ 𝑧 1 𝑥 𝑧 2 −𝑥 𝑧 1
𝑢 𝑧 2 , 𝑧 1 = 𝑧 2 𝑥 𝑧 1 −𝑝 𝑧 1 ≤ 𝑧 2 𝑥 𝑧 2 −𝑝 𝑧 2 =𝑢 𝑧 2 , 𝑧 2
⟹𝑝 𝑧 2 −𝑝 𝑧 1 ≤ 𝑧 2 (𝑥 𝑧 2 −𝑥( 𝑧 1 ))
𝑧 2 𝑥 𝑧 2 −𝑥 𝑧 1 ≥𝑝 𝑧 2 −𝑝 𝑧 1 ≥ 𝑧 1 𝑥 𝑧 2 −𝑥 𝑧 1

34. Characterization: conclusions
We did not assume:
game is a single-round sealed-bid auction
can be any wacky multi-round game
only a winner makes payments
can be a game with everyone paying
We show that all BNE have a nice form.

35. Characterization: DSE
Theorem 𝐺 and s are in DSE (Dominant strategy equilibrium) only if for all 𝑖 and 𝐯,
[monotonicity] 𝑥 𝑖 ( 𝑣 𝑖 , 𝐯 −𝑖 ) is monotone non-decreasing,
[payment identity]
𝑝 𝑖 𝑣 𝑖 , 𝐯 −𝑖 = 𝑣 𝑖 ⋅ 𝑥 𝑖 𝑣 𝑖 , 𝐯 −𝑖 − 0 𝑣 𝑖 𝑥 𝑖 𝑧, 𝐯 −𝑖 𝑑𝑧+ 𝑝 𝑖 0, 𝐯 −𝑖 ,
where 𝑧, 𝐯 −𝑖 is the valuation profile with i-th coordinate 𝑣 𝑖 ←𝑧.
If the strategy profile is onto then the converse (⟸) also holds.

36. Dominant strategy equilibrium (DSE)
Proof Apply characterization for Bayes-Nash equilibria.
DSE is stronger equilibrium concept than BNE.
𝐷𝑆𝐸⊂𝐵𝑁𝐸, and in 1st price auction there is a 𝐵𝑁𝐸∉𝐷𝑆𝐸.
Proof follows from characterization of BNE
fix 𝐯 −𝑖 to be point mass distributions.
apply BNE characterization.
Rem When game is deterministic, i.e., 𝑥 𝑖 𝐯 ∈{0,1}, then by monotonicity condition 𝑥 𝑖 𝑧, 𝐯 −𝑖 must be a step function.

37. Deterministic games
Corollary A deterministic game 𝐺 and deterministic strategies 𝐬 are in DSE only if for all 𝑖 and 𝐯,
(i) [step-function] 𝑥 𝑖 ( 𝑣 𝑖 , 𝑣 −𝑖 ) steps from 0 to 1 at some 𝑣 𝑖 ( 𝑣 −𝑖 ),
(ii) [critical value]
If the strategy profile is onto then the converse also holds.
Rem There is uncertainty about tie breaking.
E.g., in 2nd price auction who gets the item when 𝑣 1 = 𝑣 2 ?
Natural rules: lexicographical or randomized tie-breaking.

38. Revenue Equivalence
Corollary For any two mechanisms where 0-valued agents pay nothing, if the mechanisms have the same BNE outcome then they have same expected revenue. Explanation: 𝑥(⋅) in the BNE of both mechanisms is the same, then the [payment identity] tells that expected payments must be equal.

39. Revenue Equivalence: application
Let’s compare revenue of the 1st and 2nd price auctions.
Assume that values are distributed i.i.d.
Equilibrium outcome in 2nd price auction:
agent with the highest value wins
Equilibrium outcome in 1st price auction:
(a little tricky) may have many equilibria
(reasonable to assume) exists symmetric & monotone BNE
monotone: higher types bid higher values.
then highest value win.

40. Revenue Equivalence: application
Corollary When agents’ values are i.i.d. according to a continuous distribution, the 2nd price and 1st price auction have the same expected revenue.
Recall The 1st price auction with n=2 bidders, 𝑣 𝑖 ∼𝑈 0,1 .
An equilibrium: bid half your value.
Then the revenue is: 𝐄 𝑣 = 2 3 ⋅ 1 2 =
Revenue of the 2nd price auction is: 𝐄 𝑣 (2) =

41. The Revelation Principle