Information, Incentives, and Mechanism Design Nick Gravin Course Info @ 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

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

Games of Complete Information

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

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.

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

Games of Incomplete Information

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.

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.

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 .

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.

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 2 1 wins = Pr 𝑏 2 b 2 < b 1 = Pr 𝑣 2 𝑣 2 2 < 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.

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

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

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]

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.

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.

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.

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 +

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

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.

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.

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.

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.

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.

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.

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 1 2 = 2 3 ⋅ 1 2 = 1 3 . Revenue of the 2nd price auction is: 𝐄 𝑣 (2) = 1 3 .

The Revelation Principle