CSC304 Lecture 12 Mechanism Design w/ Money: Revenue maximization - - PowerPoint PPT Presentation

CSC304 Lecture 12

Mechanism Design w/ Money: Revenue maximization Myerson’s Auction

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Revenue Maximization

Welfare vs Revenue

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• In welfare maximization, we want to maximize σ𝑗 𝑤𝑗 𝑏

➢ VCG = strategyproof + maximizes welfare on every single

instance

➢ Beautiful!

• In revenue maximization, we want to maximize σ𝑗 𝑞𝑗

➢ We can still use strategyproof mechanisms (revelation

principle).

➢ BUT…

Welfare vs Revenue

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• Different strategyproof mechanisms are better for

different instances.

• Example: 1 item, 1 bidder, unknown value 𝑤

➢ strategyproof = fix a price 𝑠, let the agent decide to

“take it” (𝑤 ≥ 𝑠) or “leave it” (𝑤 < 𝑠)

➢ Maximize welfare → set 𝑠 = 0

• Must allocate item as long as the agent has a positive value

➢ Maximize revenue → 𝑠 = ?

• Different 𝑠 are better for different 𝑤

Welfare vs Revenue

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• We cannot optimize revenue on every instance

➢ Need to optimize the expected revenue in the Bayesian

framework

• If we want to achieve higher expected revenue

than VCG, we cannot always allocate the item

➢ Revenue equivalence principle!

Single Item + Single Bidder

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• Value 𝑤 is drawn from distribution with CDF 𝐺
• Goal: post the optimal price 𝑠 on the item
• Revenue from price 𝑠 = 𝑠 ⋅ 1 − 𝐺 𝑠

(Why?)

• Optimal 𝑠∗ = argmax𝑠 𝑠 ⋅ 1 − 𝐺 𝑠

➢ “Monopoly price” ➢ Note: 𝑠∗ depends on 𝐺, but not on 𝑤, so the mechanism

is strategyproof.

Example

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• Suppose 𝐺 is the CDF of the uniform distribution
• ver [0,1] (denote by 𝑉 0,1 ).

➢ CDF is given by 𝐺 𝑦 = 𝑦 for all 𝑦 ∈ [0,1].

• Recall: E[Revenue] from price 𝑠 is 𝑠 ⋅ 1 − 𝐺 𝑠

➢ Q: What is the optimal posted price? ➢ Q: What is the corresponding optimal revenue?

• Compare this to the revenue of VCG, which is 0

➢ This is because if the value is less than 𝑠∗, we are willing

to risk not selling the item.

Single Item + Two Bidders

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• 𝑤1, 𝑤2 ∼ 𝑉[0,1]
• VCG revenue = 2nd highest bid = min(𝑤1, 𝑤2)

➢ 𝐹 min 𝑤1, 𝑤2

= 1/3 (Exercise!)

• Improvement: “VCG with reserve price”

➢ Reserve price 𝑠 ➢ Highest bidder gets the item if bid more than 𝑠 ➢ Pays max(𝑠, 2nd highest bid)

• “Critical payment” : Pay the least value you could have bid and still

won the item

Single Item + Two Bidders

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• Reserve prices are ubiquitous

➢ E.g., opening bids in eBay auctions ➢ Guarantee a certain revenue to auctioneer if item is sold

• 𝐹 revenue = 𝐹 max 𝑠, min 𝑤1, 𝑤2

➢ Maximize over 𝑠? Hard to think about.

• What about a strategyproof mechanism that is not

VCG + reserve price?

➢ What about just BNIC mechanisms?

Single-Parameter Environments

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• Roger B. Myerson solved

revenue optimal auctions in “single-parameter environments”

• Proposed a simple auction

that maximizes expected revenue

Single-Parameter Environments

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• Each agent 𝑗…

➢ has a private value 𝑤𝑗 drawn from a distribution with CDF

𝐺

𝑗 and PDF 𝑔 𝑗

➢ is “satisfied” at some level 𝑦𝑗 ∈ [0,1], which gives the

agent value 𝑦𝑗 ⋅ 𝑤𝑗

➢ is asked to pay 𝑞𝑗

• Examples

➢ Single divisible item ➢ Single indivisible item (𝑦𝑗 ∈ {0,1} – this is okay too!) ➢ Many items, single-minded bidders (again 𝑦𝑗 ∈ {0,1})

Myerson’s Lemma

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• Myerson’s Lemma:

For a single-parameter environment, a mechanism is strategyproof if and only if for all 𝑗

• 1. 𝑦𝑗 is monotone non-decreasing in 𝑤𝑗
• 2. 𝑞𝑗 = 𝑤𝑗 ⋅ 𝑦𝑗 𝑤𝑗 − ׬

𝑤𝑗 𝑦𝑗 𝑨 𝑒𝑨 + 𝑞𝑗(0)

(typically, 𝑞𝑗 0 = 0)

• Generalizes critical payments

➢ For every “𝜀” allocation,

pay the lowest value that would have won it

Myerson’s Lemma

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• Note: allocation determines unique payments

𝑞𝑗 = 𝑤𝑗 ⋅ 𝑦𝑗 𝑤𝑗 − න

𝑤𝑗

𝑦𝑗 𝑨 𝑒𝑨 + 𝑞𝑗(0)

• A corollary: revenue equivalence

➢If two mechanisms use the same allocation 𝑦𝑗, they

“essentially” have the same expected revenue

• Another corollary: optimal revenue auctions

➢Optimizing revenue = optimizing some function of

allocation (easier to analyze)

Myerson’s Theorem

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• “Expected Revenue = Expected Virtual Welfare”

➢ Recall: 𝑞𝑗 = 𝑤𝑗 ⋅ 𝑦𝑗 𝑤𝑗 − ׬

𝑤𝑗 𝑦𝑗 𝑨 𝑒𝑨 + 𝑞𝑗(0)

➢ Take expectation over draw of valuations + lots of calculus

𝐹{𝑤𝑗∼𝐺𝑗} Σ𝑗 𝑞𝑗 = 𝐹{𝑤𝑗∼𝐺𝑗} Σ𝑗 𝜒𝑗 ⋅ 𝑦𝑗

• 𝜒𝑗 = 𝑤𝑗 −

1−𝐺𝑗(𝑤𝑗) 𝑔𝑗(𝑤𝑗)

= virtual value of bidder 𝑗

• σ𝑗 𝜒𝑗 ⋅ 𝑦𝑗 = virtual welfare

Myerson’s Theorem

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• Myerson’s auction:

➢ A strategyproof auction maximizes the (expected)

revenue if its allocation rule maximizes the virtual welfare subject to monotonicity and it charges critical payments.

• Charging critical payments is easy.
• But maximizing virtual welfare subject to

monotonicity is tricky.

➢ Let’s get rid of the monotonicity requirement!

Myerson’s Theorem Simplified

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• Regular Distributions

➢ A distribution 𝐺 is regular if its virtual value function

𝜒 𝑤 = 𝑤 − (1 − 𝐺 𝑤 )/𝑔 𝑤 is non-decreasing in 𝑤.

➢ Many important distributions are regular, e.g., uniform,

exponential, Gaussian, power-law, …

• Lemma

➢ If all 𝐺

𝑗’s are regular, the allocation rule maximizing virtual

welfare is already monotone.

• Myerson’s Corollary:

➢ When all 𝐺

𝑗’s are regular, the strategyproof auction

maximizes virtual welfare and charges critical payments.

Single Item + Single Bidder

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• Setup:

➢ Single indivisible item, single bidder, value 𝑤 drawn from a

regular distribution with CDF 𝐺 and PDF 𝑔

• Goal:

➢ Maximize 𝜒 ⋅ 𝑦, where 𝜒 = 𝑤 −

1−𝐺 𝑤 𝑔 𝑤

and 𝑦 ∈ {0,1}

• Optimal auction:

➢ 𝑦 = 1 iff 𝜒 ≥ 0 ⇔ 𝑤 ≥

1−𝐺 𝑤 𝑔 𝑤

⇔ 𝑤 ≥ 𝑤∗ where 𝑤∗ =

1−𝐺 𝑤∗ 𝑔 𝑤∗

➢ Critical payment: 𝑤∗ ➢ This is VCG with a reserve price of 𝜒−1(0)!

Example

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• Optimal auction:

➢ 𝑦 = 1 iff 𝜒 ≥ 0 ⇔ 𝑤 ≥

1−𝐺 𝑤 𝑔 𝑤

➢ Critical payment: 𝑤∗ such that 𝑤∗ =

1−𝐺 𝑤∗ 𝑔 𝑤∗

• Distribution is 𝑉 0,1 :

➢ 𝑦 = 1 iff 𝑤 ≥

1−𝑤 1 ⇔ 𝑤 ≥ 1 2

➢ Critical payment =

1 2

➢ That is, we post the optimal price of 0.5

Single Item + 𝑜 Bidders

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• Setup:

➢ Single indivisible item, each bidder 𝑗 has value 𝑤𝑗 drawn

from a regular distribution with CDF 𝐺

𝑗 and PDF 𝑔 𝑗

• Goal:

➢ Maximize σ𝑗 𝜒𝑗 ⋅ 𝑦𝑗 where 𝜒𝑗 = 𝑤𝑗 −

1−𝐺𝑗 𝑤𝑗 𝑔𝑗 𝑤𝑗

and 𝑦𝑗 ∈ {0,1} such that σ𝑗 𝑦𝑗 ≤ 1

Single Item + 𝑜 Bidders

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• Optimal auction:

➢ If all 𝜒𝑗 < 0:

• Nobody gets the item, nobody pays anything
• For all 𝑗, 𝑦𝑗 = 𝑞𝑗 = 0

➢ If some 𝜒𝑗 ≥ 0:

• Agent with highest 𝜒𝑗 wins the item, pays critical payment
• 𝑗∗ ∈ 𝑏𝑠𝑕𝑛𝑏𝑦𝑗 𝜒𝑗 𝑤𝑗 , 𝑦𝑗∗ = 1, 𝑦𝑗 = 0 ∀𝑗 ≠ 𝑗∗
• 𝑞𝑗∗ = 𝜒𝑗∗

−1 max 0, max𝑘≠𝑗∗ 𝜒𝑘 𝑤𝑘

• Note: The item doesn’t necessarily go to the highest value

agent!

Special Case: iid Values

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• Suppose all distributions are identical (say CDF 𝐺 and

PDF 𝑔)

• Check that the auction simplifies to the following

➢ Allocation: item goes to bidder 𝑗∗ with highest value if her

value 𝑤𝑗∗ ≥ 𝜒−1 0

➢ Payment charged = max 𝜒−1(0), max

𝑘≠𝑗∗ 𝑤𝑘

• This is again VCG with a reserve price of 𝜒−1(0)

Example

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• Two bidders, both drawing iid values from 𝑉[0,1]

➢ 𝜒 𝑤 = 𝑤 −

1−𝑤 1 = 2𝑤 − 1

➢ 𝜒−1 0 = 1/2

• Auction:

➢ Give the item to the highest bidder if their value is at

least ½

➢ Charge them max(½, 2nd highest bid)

Example

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• Two bidders, one with value from 𝑉[0,1], one with

value from 𝑉[3,5]

➢ 𝜒1 𝑤1 = 2𝑤1 − 1 ➢ 𝜒2 𝑤2 = 𝑤2 −

1−𝐺

2 𝑤2

𝑔

2 𝑤2

= 𝑤2 −

1−𝑤2−3

2

Τ

1 2

= 2𝑤2 − 5

• Auction:

➢ If 𝑤1 < ½ and 𝑤2 < 5/2, the item remains unallocated. ➢ Otherwise…

• If 2𝑤1 − 1 > 2𝑤2 − 5, agent 1 gets it and pays max ½, 𝑤2 − 2
• If 2𝑤1 − 1 < 2𝑤2 − 5, agent 2 gets it and pays max Τ

5 2 , 𝑤1 + 2

Extensions

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• Irregular distributions:

➢ E.g., multi-modal or extremely heavy tail distributions ➢ Need to add the monotonicity constraint ➢ Turns out, we can “iron” irregular distributions to make

them regular and then use Myerson’s framework

• Relaxing DSIC to BNIC

➢ Myerson’s mechanism has optimal revenue among all

DSIC mechanisms

➢ Turns out, it also has optimal revenue among the much

larger class of BNIC mechanisms!

• Approx. Optimal Auctions

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• Optimal auctions become unintuitive and difficult

to understand with unequal distributions, even if they are regular

➢ Simpler auctions preferred in practice ➢ We still want approximately optimal revenue

• Theorem [Hartline & Roughgarden, 2009]:

➢ For iid values from regular distributions, VCG with bidder-

specific reserve prices gives a 2-approximation of the

• ptimal revenue.

Approximately Optimal

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• Still relies on knowing bidders’ distributions
• Theorem [Bulow and Klemperer, 1996]:

➢ For i.i.d. values,

𝐹[Revenue of VCG with 𝑜 + 1 bidders] ≥ 𝐹[Optimal revenue with 𝑜 bidders]

• “Spend that effort in getting one more bidder than

in figuring out the optimal auction”

Simple proof

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• One can show that VCG with 𝑜 + 1 bidders has the

max revenue among all 𝑜 + 1 bidder strategyproof auctions that always allocate the item

➢ Via revenue equivalence

• Consider the auction: “Run 𝑜-bidder Myerson on

the first 𝑜 bidders. If the item is unallocated, give it to agent 𝑜 + 1 for free.”

➢ 𝑜 + 1 bidder DSIC auction ➢ As much revenue as 𝑜-bidder Myerson auction

Optimizing Revenue is Hard

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• Slow progress beyond single-parameter setting

➢ Even with just two items and one bidder with i.i.d. values

for both items, the optimal auction DOES NOT run Myerson’s auction on individual items!

➢ “Take-it-or-leave-it” offers for the two items bundled

might increase revenue

• But nowadays, the focus is on simple,

approximately optimal auctions instead of complicated, optimal auctions.