Announcements HW2 is out, due 10/15 before class 1 CS6501: Topics - - PowerPoint PPT Presentation

1

Announcements

ØHW2 is out, due 10/15 before class

CS6501: Topics in Learning and Game Theory (Fall 2019) Optimal Auction Design for Single-Item Allocation

(Part I)

Instructor: Haifeng Xu

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Outline

Ø Mechanism Design for Single-Item Allocation Ø Revelation Principle and Incentive Compatibility Ø The Revenue-Optimal Auction

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Single-Item Allocation

ØA single and indivisible item, 𝑜 buyers 1, ⋯ , 𝑜 = [𝑜] ØBuyer 𝑗 has a (private) value 𝑤* ∈ 𝑊

* about the item

ØOutcome: choice of the winner of the item, and payment 𝑞* from

each buyer 𝑗

ØObjectives: maximize revenue

• Last lecture: VCG auction maximizes welfare even for multiple items

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The Mechanism Design Problem

Objective: maximize revenue ∑*∈[/] 𝑞* Mechanism Design for Single-Item Allocation Described by ⟨𝑜, 𝑊, 𝑌, 𝑣⟩ where:

Ø 𝑜 = {1, ⋯ , 𝑜} is the set of 𝑜 buyers Ø𝑊 = 𝑊

6× ⋯× 𝑊 / is the set of all possible value profiles

Ø𝑌 = {𝑓9, 𝑓6, ⋯ , 𝑓/} is the set of all possible allocation outcomes Ø𝑣 = (𝑣6, ⋯ , 𝑣/) where 𝑣* = 𝑤*𝑦* − 𝑞* is the utility function of 𝑗

for any outcome 𝑦 ∈ 𝑌 and payment 𝑞* required from 𝑗

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The Mechanism Design Problem

Objective: maximize revenue ∑*∈[/] 𝑞*

ØCannot have any guarantee without additional assumptions ØWill assume public prior knowledge on buyer values. For

convenience, think of 𝑤* ∼ 𝑔

* independently

• Most results of this lecture hold for correlated 𝑤*’s, but easier to think

for independent cases

Mechanism Design for Single-Item Allocation Described by ⟨𝑜, 𝑊, 𝑌, 𝑣⟩ where:

Ø 𝑜 = {1, ⋯ , 𝑜} is the set of 𝑜 buyers Ø𝑊 = 𝑊

6× ⋯× 𝑊 / is the set of all possible value profiles

Ø𝑌 = {𝑓9, 𝑓6, ⋯ , 𝑓/} is the set of all possible allocation outcomes Ø𝑣 = (𝑣6, ⋯ , 𝑣/) where 𝑣* = 𝑤*𝑦* − 𝑞* is the utility function of 𝑗

for any outcome 𝑦 ∈ 𝑌 and payment 𝑞* required from 𝑗

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The Mechanism Design Problem

Mechanism Design for Single-Item Allocation Described by ⟨𝑜, 𝑊, 𝑌, 𝑣⟩ where:

Ø 𝑜 = {1, ⋯ , 𝑜} is the set of 𝑜 buyers Ø𝑊 = 𝑊

6× ⋯× 𝑊 / is the set of all possible value profiles

Ø𝑌 = {𝑓9, 𝑓6, ⋯ , 𝑓/} is the set of all possible allocation outcomes Ø𝑣 = (𝑣6, ⋯ , 𝑣/) where 𝑣* = 𝑤*𝑦* − 𝑞* is the utility function of 𝑗

for any outcome 𝑦 ∈ 𝑌 and payment 𝑞* required from 𝑗 Remarks:

ØGeneral mechanism design problem can be defined similarly Ø 𝑣* = 𝑤*𝑦* − 𝑞* is called quasi-linear utility function

• Not the only form of utility functions, but widely adopted

ØTypically, 𝑊

6 = ℝA, but can also be intervals like [𝑏, 𝑐]

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The Mechanism Design Problem

Mechanism Design for Single-Item Allocation Described by ⟨𝑜, 𝑊, 𝑌, 𝑣⟩ where:

Ø 𝑜 = {1, ⋯ , 𝑜} is the set of 𝑜 buyers Ø𝑊 = 𝑊

6× ⋯× 𝑊 / is the set of all possible value profiles

Ø𝑌 = {𝑓9, 𝑓6, ⋯ , 𝑓/} is the set of all possible allocation outcomes Ø𝑣 = (𝑣6, ⋯ , 𝑣/) where 𝑣* = 𝑤*𝑦* − 𝑞* is the utility function of 𝑗

for any outcome 𝑦 ∈ 𝑌 and payment 𝑞* required from 𝑗 Remarks:

ØAssume risk neural players – i.e., all players maximize expected

utilities

ØWill guarantee 𝔽[𝑣*] ≥ 0 (a.k.a., individually rational or IR)

• Otherwise, players would not even bother coming to your auction

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The Design Space – Mechanisms

A mechanism (i.e., the game) is specified by ⟨𝐵, 𝑕⟩ where:

ØThat is, we will design ⟨𝐵, 𝑕⟩ ØA = 𝐵6× ⋯× 𝐵/ where 𝐵* is allowable actions for buyer 𝑗 Ø𝑕: 𝐵 → [𝑦, 𝑞] maps an action profile to [an allocation outcome

𝑦(𝑏) + a vector of payments 𝑞(𝑏)] for any 𝑏 = (𝑏6, ⋯ , 𝑏/) ∈ 𝐵

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The Design Space – Mechanisms

A mechanism (i.e., the game) is specified by ⟨𝐵, 𝑕⟩ where:

ØThat is, we will design ⟨𝐵, 𝑕⟩ ØPlayers’ utility function will be fully determined by ⟨𝐵, 𝑕⟩ ØThis is a game with incomplete information – 𝑤* is privately known

to player 𝑗; all other players only know its prior distribution

ØA = 𝐵6× ⋯× 𝐵/ where 𝐵* is allowable actions for buyer 𝑗 Ø𝑕: 𝐵 → [𝑦, 𝑞] maps an action profile to [an allocation outcome

𝑦(𝑏) + a vector of payments 𝑞(𝑏)] for any 𝑏 = (𝑏6, ⋯ , 𝑏/) ∈ 𝐵

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The Design Space – Mechanisms

Example 1: first-price auction Ø 𝐵* = ℝA for all 𝑗 Ø 𝑕 𝑏 allocates the item to the buyer 𝑗∗ = arg max

*∈[/] 𝑏* and asks

𝑗∗ to pay 𝑏*∗, and all other buyers pay 0 A mechanism (i.e., the game) is specified by ⟨𝐵, 𝑕⟩ where:

ØA = 𝐵6× ⋯× 𝐵/ where 𝐵* is allowable actions for buyer 𝑗 Ø𝑕: 𝐵 → [𝑦, 𝑞] maps an action profile to [an allocation outcome

𝑦(𝑏) + a vector of payments 𝑞(𝑏)] for any 𝑏 = (𝑏6, ⋯ , 𝑏/) ∈ 𝐵

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The Design Space – Mechanisms

Example 2: second-price auction Ø 𝐵* = ℝA for all 𝑗 Ø 𝑕 𝑏 allocates the item to the buyer 𝑗∗ = arg max

*∈[/] 𝑏* and asks 𝑗∗

to pay max2* 𝑏*, and all other buyers pay 0 A mechanism (i.e., the game) is specified by ⟨𝐵, 𝑕⟩ where:

ØA = 𝐵6× ⋯× 𝐵/ where 𝐵* is allowable actions for buyer 𝑗 Ø𝑕: 𝐵 → [𝑦, 𝑞] maps an action profile to [an allocation outcome

𝑦(𝑏) + a vector of payments 𝑞(𝑏)] for any 𝑏 = (𝑏6, ⋯ , 𝑏/) ∈ 𝐵

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The Design Space – Mechanisms

Ø In general, 𝐵, 𝑕 can be really arbitrary, up to your design Ø E.g, the following is a valid – though bad – mechanism Ø 𝐵* = {𝑘𝑣𝑛𝑞 𝑢𝑥𝑗𝑑𝑓 (𝐾), 𝑚𝑝𝑝𝑙 45° 𝑣𝑞 (𝑀)} Ø 𝑦(𝑏) gives the item to anyone of 𝑀 uniformly at random Ø 𝑞(𝑏) asks everyone to pay $0 A mechanism (i.e., the game) is specified by ⟨𝐵, 𝑕⟩ where:

ØA = 𝐵6× ⋯× 𝐵/ where 𝐵* is allowable actions for buyer 𝑗 Ø𝑕: 𝐵 → [𝑦, 𝑞] maps an action profile to [an allocation outcome

𝑦(𝑏) + a vector of payments 𝑞(𝑏)] for any 𝑏 = (𝑏6, ⋯ , 𝑏/) ∈ 𝐵

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Revenue at Equilibrium

ØHow to predict/estimate how much revenue we achieve? ØRevenue = expected revenue at (Bayesian) Nash equilibrium ØDue to incomplete information, player 𝑗’s strategy is 𝑡*: 𝑊

* → Δ(𝐵*)

where 𝑡*(𝑤*) is the mixed strategy of 𝑗 with private value 𝑤*

ØExpected utility of 𝑗 with value 𝑤* in mechanism ⟨𝐵, 𝑕⟩ is

𝔽(cd,ced)∼(fd gd ,fed ged ) 𝑤*𝑦* 𝑏*, 𝑏h* − 𝑞*(𝑏*, 𝑏h*)

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Revenue at Equilibrium

ØHow to predict/estimate how much revenue we achieve? ØRevenue = expected revenue at (Bayesian) Nash equilibrium ØDue to incomplete information, player 𝑗’s strategy is 𝑡*: 𝑊

* → Δ(𝐵*)

where 𝑡*(𝑤*) is the mixed strategy of 𝑗 with private value 𝑤*

ØExpected utility of 𝑗 with value 𝑤* in mechanism ⟨𝐵, 𝑕⟩ is

𝔽(cd,ced)∼(fd gd ,fed ged ) 𝑤*𝑦* 𝑏*, 𝑏h* − 𝑞*(𝑏*, 𝑏h*) 𝔽ged∼ied = 𝑉* 𝑡* 𝑤* 𝑤*, 𝑡h*

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Revenue at Equilibrium

ØHow to predict/estimate how much revenue we achieve? ØRevenue = expected revenue at (Bayesian) Nash equilibrium ØDue to incomplete information, player 𝑗’s strategy is 𝑡*: 𝑊

* → Δ(𝐵*)

where 𝑡*(𝑤*) is the mixed strategy of 𝑗 with private value 𝑤*

ØExpected utility of 𝑗 with value 𝑤* in mechanism ⟨𝐵, 𝑕⟩ is

𝔽(cd,ced)∼(fd gd ,fed ged ) 𝑤*𝑦* 𝑏*, 𝑏h* − 𝑞*(𝑏*, 𝑏h*) Strategy profile 𝑡∗ = (𝑡6

∗, ⋯ , 𝑡/ ∗) is a Bayes Nash Equilibrium (BNE)

for mechanism ⟨𝐵, 𝑕⟩ if for any player 𝑗 and value 𝑤* 𝑉* 𝑡*

∗ 𝑤*

𝑤*, 𝑡h*

∗

≥ 𝑉* 𝑏* 𝑤*, 𝑡h*

∗

, ∀ 𝑏* ∈ 𝐵* That is, 𝑡*

∗(𝑤*) is a best response to 𝑡h* ∗ for any 𝑗 and 𝑤*.

𝔽ged∼ied = 𝑉* 𝑡* 𝑤* 𝑤*, 𝑡h*

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Revenue at Equilibrium

Strategy profile 𝑡∗ = (𝑡6

∗, ⋯ , 𝑡/ ∗) is a Bayes Nash Equilibrium (BNE)

for mechanism ⟨𝐵, 𝑕⟩ if for any player 𝑗 and value 𝑤* 𝑉* 𝑡*

∗ 𝑤*

𝑤*, 𝑡h*

∗

≥ 𝑉* 𝑏* 𝑤*, 𝑡h*

∗

, ∀ 𝑏* ∈ 𝐵* That is, 𝑡*

∗(𝑤*) is a best response to 𝑡h* ∗ for any 𝑗 and 𝑤*.

• Theorem. Any finite Bayesian game admits a mixed BNE.

Ø Can be proved by Nash’s theorem Ø It so happens that in many natural Bayesian games we look at, there will be a pure BNE

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Revenue at Equilibrium

Q: what is the BNE for second-price auction? Truthful bidding is a dominant strategy equilibrium (thus also BNE)

ØTruthful bidding is a dominant strategy. That is, for any 𝑗 and 𝑤*,

for any 𝑏h*, we have 𝑤*𝑦* 𝑤*, 𝑏h* − 𝑞* 𝑤*, 𝑏h* ≥ 𝑤*𝑦* 𝑏*′, 𝑏h* − 𝑞*(𝑏*

m, 𝑏h*)

ØBidding 𝑤* remains optimal after expectation over 𝑏h* and 𝑤h*

Strategy profile 𝑡∗ = (𝑡6

∗, ⋯ , 𝑡/ ∗) is a Bayes Nash Equilibrium (BNE)

for mechanism ⟨𝐵, 𝑕⟩ if for any player 𝑗 and value 𝑤* 𝑉* 𝑡*

∗ 𝑤*

𝑤*, 𝑡h*

∗

≥ 𝑉* 𝑏* 𝑤*, 𝑡h*

∗

, ∀ 𝑏* ∈ 𝐵* That is, 𝑡*

∗(𝑤*) is a best response to 𝑡h* ∗ for any 𝑗 and 𝑤*.

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BNE for First-Price Auction

ØIn general, still an open question in economics and CS ØCan be computed for simple cases

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BNE for First-Price Auction

Example: Two bidders, 𝑤6, 𝑤n ∼ 𝑉([0,1]) independently

• Claim. 𝑐* 𝑤* = 𝑤*/2 forms a Bayes Nash Equilibrium.

Proof

ØBy symmetry, w.l.o.g., focus on bidder 1 ØAssume bidder 2 uses 𝑐n = 𝑤n/2; ℙ 𝑐n ≤ 𝑐 = min(2𝑐, 1) , ∀𝑐 ∈ [0,1] ØUtility of bidder 1 with value 𝑤6 and any bid 𝑐6 is

ℙ 𝑐6 ≥ 𝑐n) ×(𝑤6 − 𝑐6) = min 2𝑐6, 1 ×(𝑤6 − 𝑐6)

ØWhich 𝑐6 maximizes this utility?

• If 𝑐6 ≥ 1/2, it decreases in 𝑐6, so should bid at most 1/2
• Thus, utility is 2𝑐6(𝑤6 − 𝑐6), which is maximized at 𝑐6 = 𝑤6/2

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The Main Points . . .

Optimal Mechanism Design Design mechanism ⟨𝐵, 𝑕⟩ to maximize revenue at the BNE Ø A mechanism ⟨𝐵, 𝑕⟩ specifies action space 𝐵 and a mapping from action profiles to [an allocation outcome + payments] Ø Any mechanism describes a Bayesian game Ø We compute the revenue at some Bayes Nash equilibrium

• Since this is what we predict the players will behave
• Will design mechanisms that are very easy for players to play

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The Main Points . . .

Optimal Mechanism Design Design mechanism ⟨𝐵, 𝑕⟩ to maximize revenue at the BNE Ø A mechanism ⟨𝐵, 𝑕⟩ specifies action space 𝐵 and a mapping from action profiles to [an allocation outcome + payments] Ø Any mechanism describes a Bayesian game Ø We compute the revenue at some Bayes Nash equilibrium

• Since this is what we predict the players will behave
• Will design mechanisms that are very easy for players to play

First major challenge: with so many possible actions in this world, what should I use? Ø Revelation principle says that you only need them to report their value 𝑤*

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Outline

Ø Mechanism Design for Single-Item Allocation Ø Revelation Principle and Incentive Compatibility Ø The Revenue-Optimal Auction

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Direct Revelation Mechanisms

ØThat is, the action for each player is to “report” their value (but

they don’t have to be honest…yet)

ØExamples: second-price auction, first-price auction ØNote: this restriction limits our design space as it limits our choice

• f 𝐵*’s
• Not clear yet whether this restriction will reduce our best achievable

revenue

• Will show that it indeed does not!
• Definition. A mechanism ⟨𝐵, 𝑕⟩ is a direct revelation mechanism

if 𝐵* = 𝑊

* for all 𝑗. In this case, the mechanism is described by 𝑕.

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Incentive-Compatibility

ØA similar but stronger IC requirement

• Definition. A direct revelation mechanism 𝑕

is Bayesian incentive-compatible (a.k.a., truthful or BIC) if truthful bidding forms a Bayes Nash equilibrium in the resulting game

• Definition. A direct revelation mechanism 𝑕

is Dominant- Strategy incentive-compatible (a.k.a., truthful or DIC) if truthful bidding is a dominant-strategy equilibrium in the resulting game

ØA DIC mechanism is also BIC

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Incentive-Compatibility: Examples

Second-price auction is dominant-strategy incentive-compatible, and thus also Bayesian incentive-compatible. First-price auction is not Bayesian incentive-compatible.

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Incentive-Compatibility: Examples

ØNot exactly a direct revelation mechanism as buyer only chooses

to accept or not accept, while not report their value

ØBut can be trivially modified to a direct revelation mechanism by

asking buyers to report their value and 𝑤* ≥ 𝑞 leads to an accept

Ø Both DIC and BIC

Second-price auction is dominant-strategy incentive-compatible, and thus also Bayesian incentive-compatible. First-price auction is not Bayesian incentive-compatible. Definition (Posted price). The auctioneer simply posts a fixed price 𝑞 to players in sequence until one buyer accepts.

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Incentive-Compatibility: Examples

ØConsider the following mechanism for the case with two bidders

and 𝑤6, 𝑤n ∼ 𝑉([0,1]) independently Modified First-Price Auction. Solicit bid 𝑐6, 𝑐n; highest bid wins and pays half its bid, i.e., max(𝑐6, 𝑐n)/2.

ØEquivalently, simulate first price auction where bidders bid 𝑐6/2, 𝑐n/2

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Incentive-Compatibility: Examples

ØConsider the following mechanism for the case with two bidders

and 𝑤6, 𝑤n ∼ 𝑉([0,1]) independently Modified First-Price Auction. Solicit bid 𝑐6, 𝑐n; highest bid wins and pays half its bid, i.e., max(𝑐6, 𝑐n)/2.

ØEquivalently, simulate first price auction where bidders bid 𝑐6/2, 𝑐n/2

• Claim. Modified first-price auction is BIC in the above example

ØAssuming bidder 2 truthfully bids 𝑤n. This is as if bidder 1 faces a

first price auction where bidder 2 bid 𝑐n = 𝑤n/2 and his bid is 𝑐6 = 𝑐/2 if he bids 𝑐 in the modified version

ØSince 𝑐* 𝑤* = 𝑤*/2 is a BNE of the first-price auction, thus 𝑐/2 =

𝑤*/2 (i.e., 𝑐 = 𝑤*) must be a best response

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Incentive-Compatibility: Examples

ØConsider the following mechanism for the case with two bidders

and 𝑤6, 𝑤n ∼ 𝑉([0,1]) independently Modified First-Price Auction. Solicit bid 𝑐6, 𝑐n; highest bid wins and pays half its bid, i.e., max(𝑐6, 𝑐n)/2.

ØEquivalently, simulate first price auction where bidders bid 𝑐6/2, 𝑐n/2

Key insights:

ØWhatever manipulations bidders do at equilibrium, the auctioneer

can directly implement it on behalf of the bidders, thus in the modified mechanism being truthful becomes optimal for bidders

ØThis ideas turns out to generalize

• Claim. Modified first-price auction is BIC in the above example

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The Revelation Principle

Remarks

ØCan be stated more generally, but this version is sufficient for our

purpose of optimal auction design

• The same proof idea

ØCan thus focus on BIC mechanisms henceforth; Often omit word

“direction revelation” as we almost always design DR mechanisms

• Theorem. If there is a mechanism that achieves revenue 𝑆 at a

Bayes Nash equilibrium [resp. dominant-strategy equilibrium], then there is a direct revelation, Bayesian incentive-compatible [resp. DIC] mechanism achieving revenue 𝑆.

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The Revelation Principle

This simplifies our mechanism design task

• Theorem. If there is a mechanism that achieves revenue 𝑆 at a

Bayes Nash equilibrium [resp. dominant-strategy equilibrium], then there is a direct revelation, Bayesian incentive-compatible [resp. DIC] mechanism achieving revenue 𝑆. Optimal Mechanism Design for Single-Item Allocation Given instance ⟨𝑜, 𝑊, 𝑌, 𝑣⟩, supplemented with prior 𝑔

* *∈[/], design

the allocation function 𝑦: 𝑊 → 𝑌 and payment 𝑞: 𝑊 → ℝ/ such that truthful bidding is a BNE in the following Bayesian game: 1. Solicit bid 𝑐6 ∈ 𝑊

6, ⋯ , 𝑐/ ∈ 𝑊 /

2. Select allocation 𝑦 𝑐6, ⋯ , 𝑐/ ∈ 𝑌 and payment 𝑞(𝑐6, ⋯ , 𝑐/) Design goal: maximize expected revenue

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Proof (Bayesian Setting)

ØConsider any mechanism ⟨𝐵, 𝑕⟩ with BNE strategies 𝑡*: 𝑊

* → 𝐵*

ØDefine a new mechanism that simulates the BNE on behalf of players

Modified Mechanism. 1. Solicit reported value (as bid) 𝑐6 ∈ 𝑊

6, ⋯ , 𝑐/ ∈ 𝑊 /

2. Choose allocation outcome ̅ 𝑕 𝑐6, ⋯ , 𝑐/ = 𝑕(𝑡6 𝑐6 , ⋯ , 𝑡/(𝑐/)) and payment vector ̅ 𝑞 𝑐6, ⋯ , 𝑐/ = 𝑞(𝑡6 𝑐6 , ⋯ , 𝑡/(𝑐/))

• (If 𝑡*’s are mixed strategies, add expectation signs)

Argue that truthful bidding is a BNE in the modified mechanism

ØFocus on 𝑗 with value 𝑤*, and assume all other bidders bid truthfully ØThis is as if all other bidders play 𝑡h*(𝑤h*) in original mechanism ØThen, 𝑡*(𝑤*) must be bidder 𝑗’th optimal bid by definition of BNE ØSince auctioneer will apply function 𝑡* to 𝑗’s bid in the modified

mechanism, he should just bid 𝑤*

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Outline

Ø Mechanism Design for Single-Item Allocation Ø Revelation Principle and Incentive Compatibility Ø The Revenue-Optimal Mechanism

35

Optimal (Bayesian) Mechanism Design

ØPrevious formulation and simplification leads to the following

• ptimization problem

max

v,w

𝔽g∼i ∑*x6

/

𝑞*(𝑤6, ⋯ , 𝑤/)

• s. t.

𝔽ged∼ied 𝑤*𝑦* 𝑤*, 𝑤h* − 𝑞* 𝑤*, 𝑤h* ≥ 𝔽ged∼ied 𝑤*𝑦* 𝑐*, 𝑤h* − 𝑞* 𝑐*, 𝑤h* , ∀𝑗 ∈ 𝑜 , 𝑤*, 𝑐* ∈ 𝑊

*

𝑦(𝑤) ∈ 𝑌, ∀𝑤 ∈ 𝑊 𝔽ged∼ied 𝑤*𝑦* 𝑤*, 𝑤h* − 𝑞* 𝑤*, 𝑤h* ≥ 0, ∀𝑗 ∈ 𝑜 , 𝑤* ∈ 𝑊

*

36

Optimal (Bayesian) Mechanism Design

ØPrevious formulation and simplification leads to the following

• ptimization problem

max

v,w

𝔽g∼i ∑*x6

/

𝑞*(𝑤6, ⋯ , 𝑤/)

• s. t.

𝔽ged∼ied 𝑤*𝑦* 𝑤*, 𝑤h* − 𝑞* 𝑤*, 𝑤h* ≥ 𝔽ged∼ied 𝑤*𝑦* 𝑐*, 𝑤h* − 𝑞* 𝑐*, 𝑤h* , ∀𝑗 ∈ 𝑜 , 𝑤*, 𝑐* ∈ 𝑊

*

𝑦(𝑤) ∈ 𝑌, ∀𝑤 ∈ 𝑊 𝔽ged∼ied 𝑤*𝑦* 𝑤*, 𝑤h* − 𝑞* 𝑤*, 𝑤h* ≥ 0, ∀𝑗 ∈ 𝑜 , 𝑤* ∈ 𝑊

*

BIC constraints Individually rational (IR) constraints

37

Optimal (Bayesian) Mechanism Design

ØPrevious formulation and simplification leads to the following

• ptimization problem

max

v,w

𝔽g∼i ∑*x6

/

𝑞*(𝑤6, ⋯ , 𝑤/)

• s. t.

𝔽ged∼ied 𝑤*𝑦* 𝑤*, 𝑤h* − 𝑞* 𝑤*, 𝑤h* ≥ 𝔽ged∼ied 𝑤*𝑦* 𝑐*, 𝑤h* − 𝑞* 𝑐*, 𝑤h* , ∀𝑗 ∈ 𝑜 , 𝑤*, 𝑐* ∈ 𝑊

*

𝑦(𝑤) ∈ 𝑌, ∀𝑤 ∈ 𝑊 𝔽ged∼ied 𝑤*𝑦* 𝑤*, 𝑤h* − 𝑞* 𝑤*, 𝑤h* ≥ 0, ∀𝑗 ∈ 𝑜 , 𝑤* ∈ 𝑊

*

ØThis problem is challenging because we are optimizing over

functions 𝑦: 𝑊 → 𝑌 and 𝑞: 𝑊 → ℝ/

38

Optimal DIC Mechanism Design

ØDesigning optimal dominant-strategy incentive compatible (DIC)

mechanism is a strictly more constrained optimization problem max

v,w

𝔽g∼i ∑*x6

/

𝑞*(𝑤6, ⋯ , 𝑤/)

• s. t.

𝔽ged∼ied 𝑤*𝑦* 𝑤*, 𝑤h* − 𝑞* 𝑤*, 𝑤h* ≥ 𝔽ged∼ied 𝑤*𝑦* 𝑐*, 𝑤h* − 𝑞* 𝑐*, 𝑤h* , ∀𝑗 ∈ 𝑜 , 𝑤*, 𝑐* ∈ 𝑊

*

𝑦(𝑤) ∈ 𝑌, ∀𝑤 ∈ 𝑊 𝔽ged∼ied 𝑤*𝑦* 𝑤*, 𝑤h* − 𝑞* 𝑤*, 𝑤h* ≥ 0, ∀𝑗 ∈ 𝑜 , 𝑤* ∈ 𝑊

*

39

Optimal DIC Mechanism Design

ØDesigning optimal dominant-strategy incentive compatible (DIC)

mechanism is a strictly more constrained optimization problem max

v,w

𝔽g∼i ∑*x6

/

𝑞*(𝑤6, ⋯ , 𝑤/)

• s. t.

𝔽ged∼ied 𝑤*𝑦* 𝑤*, 𝑤h* − 𝑞* 𝑤*, 𝑤h* ≥ 𝔽ged∼ied 𝑤*𝑦* 𝑐*, 𝑤h* − 𝑞* 𝑐*, 𝑤h* , ∀𝑗 ∈ 𝑜 , 𝑤*, 𝑐* ∈ 𝑊

*

𝑦(𝑤) ∈ 𝑌, ∀𝑤 ∈ 𝑊 𝔽ged∼ied 𝑤*𝑦* 𝑤*, 𝑤h* − 𝑞* 𝑤*, 𝑤h* ≥ 0, ∀𝑗 ∈ 𝑜 , 𝑤* ∈ 𝑊

*

∀ 𝑤h*

40

Optimal DIC Mechanism Design

ØDesigning optimal dominant-strategy incentive compatible (DIC)

mechanism is a strictly more constrained optimization problem max

v,w

𝔽g∼i ∑*x6

/

𝑞*(𝑤6, ⋯ , 𝑤/)

• s. t.

𝔽ged∼ied 𝑤*𝑦* 𝑤*, 𝑤h* − 𝑞* 𝑤*, 𝑤h* ≥ 𝔽ged∼ied 𝑤*𝑦* 𝑐*, 𝑤h* − 𝑞* 𝑐*, 𝑤h* , ∀𝑗 ∈ 𝑜 , 𝑤*, 𝑐* ∈ 𝑊

*

𝑦(𝑤) ∈ 𝑌, ∀𝑤 ∈ 𝑊 𝔽ged∼ied 𝑤*𝑦* 𝑤*, 𝑤h* − 𝑞* 𝑤*, 𝑤h* ≥ 0, ∀𝑗 ∈ 𝑜 , 𝑤* ∈ 𝑊

*

∀ 𝑤h*

• Corollary. Optimal DIC mechanism achieves revenue at most that
• f optimal BIC mechanism.

41

Myerson’s Optimal Auction

Theorem (informal). For single-item allocation with prior distribution 𝑤* ∼ 𝑔

* independently, the following auction is BIC and optimal:

1. Solicit buyer values 𝑤6, ⋯ , 𝑤/ 2. Transform 𝑤* to “virtual value” 𝜚*(𝑤*) where 𝜚* 𝑤* = 𝑤* − 6h}d(gd)

id(gd)

3. If there exists 𝜚* 𝑤* ≥ 0, allocate item to 𝑗∗ = arg max

*∈[/] 𝜚*(𝑤*)

and charge him the minimum bid needed to win, i.e., 𝜚*

h6 max max ~•*∗ 𝜚~(𝑤~) , 0

; Other bidders pay 0. 4. If 𝜚* 𝑤* < 0 for all 𝑗, keep the item and no payments

42

Myerson’s Optimal Auction

Theorem (informal). For single-item allocation with prior distribution 𝑤* ∼ 𝑔

* independently, the following auction is BIC and optimal:

1. Solicit buyer values 𝑤6, ⋯ , 𝑤/ 2. Transform 𝑤* to “virtual value” 𝜚*(𝑤*) where 𝜚* 𝑤* = 𝑤* − 6h}d(gd)

id(gd)

3. If there exists 𝜚* 𝑤* ≥ 0, allocate item to 𝑗∗ = arg max

*∈[/] 𝜚*(𝑤*)

and charge him the minimum bid needed to win, i.e., 𝜚*

h6 max max ~•*∗ 𝜚~(𝑤~) , 0

; Other bidders pay 0. 4. If 𝜚* 𝑤* < 0 for all 𝑗, keep the item and no payments

ØRecall second-price auction, we also charge the minimum bid to win,

but directly use the bid to determine winner

ØKey differences from second-price auction: (1) use virtual value to

determine winner; (2) added a “fake bidder” with virtual value 0

43

Remarks

Myerson’s optimal auction is noteworthy for many reasons

ØMatches practical experience: when buyer values are i.i.d,

• ptimal auction is a second price auction with reserve 𝜚h6(0).

ØApplies to “single parameter” problems more generally ØThe optimal BIC mechanism just so happens to be DIC and

deterministic!!

• Not true for multiple items – there exists revenue gap even when

selling two items to two bidders

Thank You

Haifeng Xu

University of Virginia hx4ad@virginia.edu