CSC304 Lecture 9 Mechanism Design w/ Money: More examples of VCG, - - PowerPoint PPT Presentation

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CSC304 Lecture 9 Mechanism Design w/ Money: More examples of VCG, winner determination and truthful approximation CSC304 - Nisarg Shah 1 VCG Recap = = argmax ()

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SLIDE 1

CSC304 Lecture 9

Mechanism Design w/ Money: More examples of VCG, winner determination and truthful approximation

CSC304 - Nisarg Shah 1

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VCG Recap

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  • 𝑔 𝑀 = π‘βˆ— = argmaxπ‘βˆˆπ΅ σ𝑗 𝑀𝑗(𝑏)
  • π‘žπ‘— 𝑀 = max

𝑏

Οƒπ‘˜β‰ π‘— π‘€π‘˜ 𝑏 βˆ’ Οƒπ‘˜β‰ π‘— π‘€π‘˜ π‘βˆ—

  • Procedure

➒ Step 1: Choose the allocation to maximize social welfare ➒ Step 2: Payment charged to each agent 𝑗 is the externality

that 𝑗 imposes on others

  • [Max welfare of others | 𝑗 absent] – [welfare of others | 𝑗 present]

Under π‘βˆ—

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SLIDE 3

VCG Recap

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  • Four properties

➒ Maximize social welfare ➒ Dominant strategy incentive compatibility (DSIC) ➒ No payments to agents ➒ Individual rationality (IR)

  • Vickrey auction satisfies the first two
  • VCG adds Clarke’s pivot rule to satisfy all four
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VCG Example

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  • In the last lecture, we saw…

➒ Additive valuations: agent has value 𝑀𝑗

𝑏 for each 𝑏, 𝑀𝑗 𝑇 = Οƒπ‘βˆˆπ‘‡ 𝑀𝑗 𝑏

➒ Unit-demand valuations: Still have 𝑀𝑗

𝑏 for each 𝑏, 𝑀𝑗 𝑇 = max

π‘βˆˆπ‘‡ 𝑀𝑗

𝑏

  • Goods are β€œsubstitutes”
  • Another example…

➒ Complementary goods: value of the whole exceeds the

sum of values of its parts

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VCG Example

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  • A chair (𝑑) and a table (𝑒)

𝑀1 𝑑 = 3 𝑀2 𝑒 = 4 𝑀3 {𝑑, 𝑒} = 6

  • Allocation?
  • Payment?
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SLIDE 6

VCG Example

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  • A chair (𝑑) and a table (𝑒)

𝑀1 𝑑 = 3 𝑀2 𝑒 = 4 𝑀3 {𝑑, 𝑒} = 8

  • Allocation?
  • Payment?
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SLIDE 7

VCG Example: Seller as Agent

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  • Seller (𝑇) wants to sell his car (𝑑) to buyer (𝐢)
  • Seller has a value for his own car: 𝑀𝑇 𝑑

➒ Individual rationality for the seller mandates that seller

must get revenue at least 𝑀𝑇 𝑑

  • Idea: Add seller as another agent, and make his

values part of the welfare calculations!

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VCG Example: Seller as Agent

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𝑀𝑇 𝑑 = 3

  • What if…

➒ We give the car to buyer when 𝑀𝐢 𝑑 > 𝑀𝑇(𝑑) and ➒ Buyer pays seller 𝑀𝐢 𝑑 : Not DSIC for buyer! ➒ Buyer pays seller 𝑀𝑇(𝑑) : Not DSIC for seller!

𝑀𝐢 𝑑 = 5

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VCG Example: Seller as Agent

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𝑀𝑇 𝑑 = 3

  • Allocation?

➒ Buyer gets the car (welfare = 5)

  • Payment?

➒ Buyer pays: 3 βˆ’ 0 = 3 ➒ Seller pays: 0 βˆ’ 5 = βˆ’5

𝑀𝐢 𝑑 = 5

Mechanism takes $3 from buyer, and gives $5 to the seller!

  • Need external subsidy
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Problems with VCG

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  • Difficult to understand in complex settings

➒ Need to reason about what allocation would maximize

welfare if agent 𝑗 was absent

  • Only cares about welfare, not revenue

➒ Though, as we will see in a few lectures, gets pretty good

revenue

  • With sellers and buyers, need external subsidy

➒ Actually, cannot get individual rationality, DSIC, no

subsidy, and constant approximation of welfare

  • Might be computationally difficult to implement

➒ Computing welfare maximizing allocation may be hard

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SLIDE 11

Single-Minded Bidders

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  • Combinatorial auction for a set of 𝑛 items 𝑇
  • Each agent 𝑗 has

➒ Value 𝑀𝑗 if receives a subset 𝑇𝑗 βŠ† 𝑇 ➒ Value 0 if doesn’t get a superset of 𝑇𝑗 ➒ β€œSingle-minded”

  • Welfare-maximizing allocation:

➒ Find a subset of players 𝑗 with the highest total value such

that their sets 𝑇𝑗 are disjoint

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SLIDE 12

Single-Minded Bidders

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  • Reduction to the Weighted Independent Set (WIS)

problem in a graph

➒ NP-hard to find the welfare-maximizing allocation ➒ Note: not even thinking about computing payments yet ➒ In fact, hard to approximately optimize welfare

  • No O(𝑛

1 2βˆ’πœ—) approximation (unless 𝑂𝑄 βŠ† π‘Žπ‘„π‘„)

  • Luckily, a simple greedy algorithm gives

𝑛-approximation (i.e., OPT/GREEDY ≀ 𝑛 )

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Greedy Algorithm

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  • Input: (𝑀𝑗, 𝑇𝑗) for each agent 𝑗
  • Output: Agents with mutually independent 𝑇𝑗
  • Greedy Algorithm:

➒ Sort the agents. Go over them one-by-one. Accept each

bid if no requested item is previously allocated.

  • Sort by what?

➒ 𝑀1 β‰₯ 𝑀2 β‰₯ β‹― β‰₯ π‘€π‘œ? 𝑛-approximation ➒

𝑀1 𝑇1 β‰₯ 𝑀2 𝑇2 β‰₯ β‹― π‘€π‘œ π‘‡π‘œ ? 𝑛-approximation

➒

𝑀1 𝑇1 β‰₯ 𝑀2 𝑇2 β‰₯ β‹― π‘€π‘œ π‘‡π‘œ ? 𝑛-approximation [Lehmann et al. 2011]

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Greedy Algorithm

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  • (allocation rule, payments) truthful if and only if

➒ Allocation is monotonic: If agent 𝑗 wins with (𝑀𝑗, 𝑇𝑗), it

must win with (𝑀𝑗

β€², 𝑇𝑗 β€²) where 𝑀𝑗 β€² β‰₯ 𝑀𝑗 and 𝑇𝑗 β€² βŠ† 𝑇𝑗

➒ Payments are critical prices: Agent 𝑗 pays the least value

(s)he could have reported and still won.

  • π‘žπ‘— = π‘€π‘˜βˆ— β‹…

|𝑇𝑗| π‘‡π‘˜βˆ—

➒ π‘˜βˆ— is the smallest index π‘˜ such that 𝑇

π‘˜ ∩ 𝑇𝑗 β‰  βˆ… and 𝑇 π‘˜ ∩

𝑇𝑙 = βˆ… for all 𝑙 < π‘˜, 𝑙 β‰  𝑗

➒ If agent 𝑗 reports less than this value, agent π‘˜ gets 𝑇

π‘˜ first,

and 𝑗 loses.

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Moral

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  • VCG can sometimes be too difficult to implement

➒ May look into approximately maximizing welfare ➒ Can set the payments right if the allocation rule is

monotone

  • Need for approximation is due to computational

considerations

  • Later in mechanism design without money…

➒ Can’t use payments to ensure truthfulness ➒ Will need to approximate welfare just to get truthfulness,

even without computational considerations

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SLIDE 16

Sponsored Search Auctions

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Sponsored Search Auctions

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  • Suppose the search engine receives a search query
  • 𝑙 advertisement slots

➒ β€œClickthrough rates” : 𝑑1 β‰₯ 𝑑2 β‰₯ β‹― β‰₯ 𝑑𝑙 β‰₯ 𝑑𝑙+1 = 0

  • π‘œ advertisers (bidders)

➒ Bidder 𝑗 derives value 𝑀𝑗 *per click* ➒ Final value to bidder 𝑗 for receiving slot π‘˜ = 𝑀𝑗 β‹… 𝑑

π‘˜

➒ Without loss of generality, 𝑀1 β‰₯ 𝑀2 β‰₯ β‹― β‰₯ π‘€π‘œ

  • Age-old question:

➒ Who gets which slot, and how much should they pay?

For convenience

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SLIDE 18

Sponsored Search : VCG

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  • VCG

➒ Maximize welfare: π‘˜th bidder gets π‘˜th slot (1 ≀ π‘˜ ≀ 𝑙) ➒ Payment of π‘˜th bidder?

  • Increase in social welfare to others if π‘˜ abstains

➒ Bidders π‘˜ + 1 through 𝑙 + 1 get β€œupgraded” by one slot ➒ Payment of bidder π‘˜ = σ𝑗=π‘˜+1

𝑙+1

𝑀𝑗 β‹… (π‘‘π‘—βˆ’1 βˆ’ 𝑑𝑗)

➒ Payment to bidder π‘˜ β€œper click” = σ𝑗=π‘˜+1

𝑙+1

𝑀𝑗 β‹…

π‘‘π‘—βˆ’1βˆ’π‘‘π‘— π‘‘π‘˜

➒ Not very intuitive…

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SLIDE 19

Sponsored Search : VCG

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  • What happens if all clickthrough rates are same?

βž’π‘‘1 = 𝑑2 = β‹― = 𝑑𝑙 > 𝑑𝑙+1 = 0

  • Payment of bidder π‘˜ per click

βž’Οƒπ‘—=π‘˜+1

𝑙+1

𝑀𝑗 β‹…

π‘‘π‘—βˆ’1βˆ’π‘‘π‘— π‘‘π‘˜

= 𝑀𝑙+1

  • Bidders 1 through 𝑙 pay the value of bidder 𝑙 + 1

➒ Familiar? VCG for 𝑙 identical items

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SLIDE 20

Sponsored Search : GSP

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  • Generalized Second Price Auction (GSP)

➒ For 1 ≀ π‘˜ ≀ 𝑙 ➒ Bidder π‘˜ gets slot π‘˜ ➒ Bidder π‘˜ pays the bid of bidder π‘˜ + 1

  • A natural extension of the second price auction

➒ We already saw that this is not truthful even with two

identical slots

➒ Highest bidder paying 2nd highest bid β†’ wants to lower

bid to become 2nd highest bidder and pay 3rd highest bid

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SLIDE 21

Sponsored Search : GSP

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  • Truth-telling is not a Nash equilibrium 
  • But there is a good Nash equilibrium that realizes

the VCG outcome, i.e., maximizes welfare and generates as much revenue as VCG ☺

[Edelman et al. 2007]

  • Even the worst Nash equilibrium gives 1.282-

approximation to welfare (𝑄𝑝𝐡 ≀ 1.282) and generates at least half the revenue of VCG

[Caragiannis et al. 2011, Dutting et al. 2011, Lucier et al. 2012]

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SLIDE 22

VCG vs GSP

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  • VCG

➒ Truthful in dominant strategy β†’ more confidence that

players will bid truthfully

➒ Theoretical welfare/revenue guarantees will hold ➒ Though players might still misreport… ➒ Difficult to understand

  • GSP

➒ Need to rely on players reaching a Nash equilibrium ➒ Good welfare and revenue ➒ Easy to understand

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VCG vs GSP

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  • Google uses GSP
  • Facebook used GSP, but switched to VCG

➒ They argue that maximizing welfare has two benefits ➒ Advertisers are happy β†’ attract more advertisers β†’ more

long-term revenue

➒ Users are happy (?!) β†’ users use FB more β†’ more slots to

sell β†’ more long-term revenue

  • No consensus
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SLIDE 24

Sponsored Search Reality

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  • Value is proportional to clickthrough rate

➒ Could it be that users clicking on the 2nd slot are more

likely buyers than those clicking on the 1st slot?

  • Ad engines also want to produce quality results

➒ An advertiser having a high value for a slot does not

necessarily mean his ad is appropriate for the slot

  • Theoretical analysis does not take into account

market competition

➒ Advertiser divide their budget among ad engines