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

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

VCG Recap • 𝑔 𝑤 = 𝑏 ∗ = argmax 𝑏∈𝐵 σ 𝑗 𝑤 𝑗 (𝑏) − σ 𝑘≠𝑗 𝑤 𝑘 𝑏 ∗ • 𝑞 𝑗 𝑤 = max σ 𝑘≠𝑗 𝑤 𝑘 𝑏 𝑏 • Procedure ➢ Step 1: Choose the allocation to maximize social welfare ➢ Step 2: Payment charged to each agent 𝑗 is the externality that 𝑗 imposes on others o [Max welfare of others | 𝑗 absent] – [welfare of others | 𝑗 present] Under 𝑏 ∗ CSC304 - Nisarg Shah 2

VCG Recap • Four properties ➢ Maximize social welfare ➢ Dominant strategy incentive compatibility (DSIC) ➢ No payments to agents ➢ Individual rationality (IR) • Vickrey auction satisfies the first two • VCG adds Clarke’s pivot rule to satisfy all four CSC304 - Nisarg Shah 3

VCG Example • In the last lecture, we saw… ➢ Additive valuations: agent has value 𝑤 𝑗 𝑏 for each 𝑏 , 𝑤 𝑗 𝑇 = σ 𝑏∈𝑇 𝑤 𝑗 𝑏 ➢ Unit-demand valuations: Still have 𝑤 𝑗 𝑏 for each 𝑏 , 𝑤 𝑗 𝑇 = max 𝑏∈𝑇 𝑤 𝑗 𝑏 o Goods are “substitutes” • Another example… ➢ Complementary goods: value of the whole exceeds the sum of values of its parts CSC304 - Nisarg Shah 4

VCG Example • A chair ( 𝑑 ) and a table ( 𝑢 ) 𝑤 1 𝑑 = 3 𝑤 2 𝑢 = 4 • Allocation? • Payment? 𝑤 3 {𝑑, 𝑢} = 6 CSC304 - Nisarg Shah 5

VCG Example • A chair ( 𝑑 ) and a table ( 𝑢 ) 𝑤 1 𝑑 = 3 𝑤 2 𝑢 = 4 • Allocation? • Payment? 𝑤 3 {𝑑, 𝑢} = 8 CSC304 - Nisarg Shah 6

VCG Example: Seller as Agent • Seller ( 𝑇 ) wants to sell his car ( 𝑑 ) to buyer ( 𝐶 ) • Seller has a value for his own car: 𝑤 𝑇 𝑑 ➢ Individual rationality for the seller mandates that seller must get revenue at least 𝑤 𝑇 𝑑 • Idea: Add seller as another agent, and make his values part of the welfare calculations! CSC304 - Nisarg Shah 7

VCG Example: Seller as Agent 𝑤 𝑇 𝑑 = 3 𝑤 𝐶 𝑑 = 5 • What if… ➢ We give the car to buyer when 𝑤 𝐶 𝑑 > 𝑤 𝑇 (𝑑) and ➢ Buyer pays seller 𝑤 𝐶 𝑑 : Not DSIC for buyer! ➢ Buyer pays seller 𝑤 𝑇 (𝑑) : Not DSIC for seller! CSC304 - Nisarg Shah 8

VCG Example: Seller as Agent 𝑤 𝑇 𝑑 = 3 𝑤 𝐶 𝑑 = 5 • Allocation? Mechanism takes $3 from buyer, and gives ➢ Buyer gets the car (welfare = 5 ) $5 to the seller! • Payment? • Need external subsidy ➢ Buyer pays: 3 − 0 = 3 ➢ Seller pays: 0 − 5 = −5 CSC304 - Nisarg Shah 9

Problems with VCG • Difficult to understand in complex settings ➢ Need to reason about what allocation would maximize welfare if agent 𝑗 was absent • Only cares about welfare, not revenue ➢ Though, as we will see in a few lectures, gets pretty good revenue • With sellers and buyers, need external subsidy ➢ Actually, cannot get individual rationality, DSIC, no subsidy, and constant approximation of welfare • Might be computationally difficult to implement ➢ Computing welfare maximizing allocation may be hard CSC304 - Nisarg Shah 10

Single-Minded Bidders • Combinatorial auction for a set of 𝑛 items 𝑇 • Each agent 𝑗 has ➢ Value 𝑤 𝑗 if receives a subset 𝑇 𝑗 ⊆ 𝑇 ➢ Value 0 if doesn’t get a superset of 𝑇 𝑗 ➢ “Single - minded” • Welfare-maximizing allocation: ➢ Find a subset of players 𝑗 with the highest total value such that their sets 𝑇 𝑗 are disjoint CSC304 - Nisarg Shah 11

Single-Minded Bidders • Reduction to the Weighted Independent Set (WIS) problem in a graph ➢ NP-hard to find the welfare-maximizing allocation ➢ Note: not even thinking about computing payments yet ➢ In fact, hard to approximately optimize welfare 1 2 −𝜗 ) approximation (unless 𝑂𝑄 ⊆ 𝑎𝑄𝑄 ) o No O(𝑛 • Luckily, a simple greedy algorithm gives 𝑛 -approximation (i.e., OPT/GREEDY ≤ 𝑛 ) CSC304 - Nisarg Shah 12

Greedy Algorithm • Input: (𝑤 𝑗 , 𝑇 𝑗 ) for each agent 𝑗 • Output: Agents with mutually independent 𝑇 𝑗 • Greedy Algorithm: ➢ Sort the agents. Go over them one-by-one. Accept each bid if no requested item is previously allocated. • Sort by what? ➢ 𝑤 1 ≥ 𝑤 2 ≥ ⋯ ≥ 𝑤 𝑜 ? 𝑛 -approximation 𝑤 1 𝑤 2 𝑤 𝑜 𝑇 1 ≥ 𝑇 2 ≥ ⋯ 𝑇 𝑜 ? 𝑛 -approximation ➢ 𝑤 1 𝑤 2 𝑤 𝑜 𝑇 1 ≥ 𝑇 2 ≥ ⋯ 𝑇 𝑜 ? 𝑛 -approximation [Lehmann et al. 2011] ➢ CSC304 - Nisarg Shah 13

Greedy Algorithm • (allocation rule, payments) truthful if and only if ➢ Allocation is monotonic: If agent 𝑗 wins with (𝑤 𝑗 , 𝑇 𝑗 ) , it ′ ≥ 𝑤 𝑗 and 𝑇 𝑗 ′ ⊆ 𝑇 𝑗 ′ , 𝑇 𝑗 ′ ) where 𝑤 𝑗 must win with (𝑤 𝑗 ➢ Payments are critical prices: Agent 𝑗 pays the least value (s)he could have reported and still won. |𝑇 𝑗 | • 𝑞 𝑗 = 𝑤 𝑘 ∗ ⋅ 𝑇 𝑘∗ ➢ 𝑘 ∗ is the smallest index 𝑘 such that 𝑇 𝑘 ∩ 𝑇 𝑗 ≠ ∅ and 𝑇 𝑘 ∩ 𝑇 𝑙 = ∅ for all 𝑙 < 𝑘, 𝑙 ≠ 𝑗 ➢ If agent 𝑗 reports less than this value, agent 𝑘 gets 𝑇 𝑘 first, and 𝑗 loses. CSC304 - Nisarg Shah 14

Moral • VCG can sometimes be too difficult to implement ➢ May look into approximately maximizing welfare ➢ 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… ➢ Can’t use payments to ensure truthfulness ➢ Will need to approximate welfare just to get truthfulness, even without computational considerations CSC304 - Nisarg Shah 15

Sponsored Search Auctions CSC304 - Nisarg Shah 16

Sponsored Search Auctions • Suppose the search engine receives a search query • 𝑙 advertisement slots ➢ “ Clickthrough rates” : 𝑑 1 ≥ 𝑑 2 ≥ ⋯ ≥ 𝑑 𝑙 ≥ 𝑑 𝑙+1 = 0 • 𝑜 advertisers (bidders) For convenience ➢ Bidder 𝑗 derives value 𝑤 𝑗 *per click* ➢ Final value to bidder 𝑗 for receiving slot 𝑘 = 𝑤 𝑗 ⋅ 𝑑 𝑘 ➢ Without loss of generality, 𝑤 1 ≥ 𝑤 2 ≥ ⋯ ≥ 𝑤 𝑜 • Age-old question: ➢ Who gets which slot, and how much should they pay? CSC304 - Nisarg Shah 17

Sponsored Search : VCG • VCG ➢ Maximize welfare: 𝑘 th bidder gets 𝑘 th slot ( 1 ≤ 𝑘 ≤ 𝑙 ) ➢ Payment of 𝑘 th bidder? • Increase in social welfare to others if 𝑘 abstains ➢ Bidders 𝑘 + 1 through 𝑙 + 1 get “upgraded” by one slot 𝑙+1 ➢ Payment of bidder 𝑘 = σ 𝑗=𝑘+1 𝑤 𝑗 ⋅ (𝑑 𝑗−1 − 𝑑 𝑗 ) 𝑑 𝑗−1 −𝑑 𝑗 𝑙+1 ➢ Payment to bidder 𝑘 “per click” = σ 𝑗=𝑘+1 𝑤 𝑗 ⋅ 𝑑 𝑘 ➢ Not very intuitive… CSC304 - Nisarg Shah 18

Sponsored Search : VCG • What happens if all clickthrough rates are same? ➢ 𝑑 1 = 𝑑 2 = ⋯ = 𝑑 𝑙 > 𝑑 𝑙+1 = 0 • Payment of bidder 𝑘 per click 𝑑 𝑗−1 −𝑑 𝑗 𝑙+1 ➢ σ 𝑗=𝑘+1 𝑤 𝑗 ⋅ = 𝑤 𝑙+1 𝑑 𝑘 • Bidders 1 through 𝑙 pay the value of bidder 𝑙 + 1 ➢ Familiar? VCG for 𝑙 identical items CSC304 - Nisarg Shah 19

Sponsored Search : GSP • Generalized Second Price Auction (GSP) ➢ For 1 ≤ 𝑘 ≤ 𝑙 ➢ Bidder 𝑘 gets slot 𝑘 ➢ Bidder 𝑘 pays the bid of bidder 𝑘 + 1 • A natural extension of the second price auction ➢ We already saw that this is not truthful even with two identical slots ➢ Highest bidder paying 2 nd highest bid → wants to lower bid to become 2 nd highest bidder and pay 3 rd highest bid CSC304 - Nisarg Shah 20

Sponsored Search : GSP • 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 ( 𝑄𝑝𝐵 ≤ 1.282 ) and generates at least half the revenue of VCG [Caragiannis et al. 2011, Dutting et al. 2011, Lucier et al. 2012] CSC304 - Nisarg Shah 21

VCG vs GSP • VCG ➢ Truthful in dominant strategy → more confidence that players will bid truthfully ➢ Theoretical welfare/revenue guarantees will hold ➢ Though players might still misreport… ➢ Difficult to understand • GSP ➢ Need to rely on players reaching a Nash equilibrium ➢ Good welfare and revenue ➢ Easy to understand CSC304 - Nisarg Shah 22

VCG vs GSP • Google uses GSP • Facebook used GSP, but switched to VCG ➢ They argue that maximizing welfare has two benefits ➢ Advertisers are happy → attract more advertisers → more long-term revenue ➢ Users are happy (?!) → users use FB more → more slots to sell → more long-term revenue • No consensus CSC304 - Nisarg Shah 23

Sponsored Search Reality • Value is proportional to clickthrough rate ➢ Could it be that users clicking on the 2 nd slot are more likely buyers than those clicking on the 1 st slot? • Ad engines also want to produce quality results ➢ 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 ➢ Advertiser divide their budget among ad engines CSC304 - Nisarg Shah 24