csc304 lecture 8nisarg/teaching/304f19/...ย ยท mathematical setup csc304 - nisarg shah 9 โขour goal...
TRANSCRIPT
CSC304 Lecture 8
Mechanism Design with Money:
VCG mechanism
CSC304 - Nisarg Shah 1
RECAP: Game Theory
CSC304 - Nisarg Shah 2
โข Simultaneous-move Games
โข Nash equilibria
โข Prices of anarchy and stability
โข Cost-sharing games, congestion games, Braessโ paradox
โข Zero-sum games and the minimax theorem
โข Stackelberg games
Mechanism Design with Money
CSC304 - Nisarg Shah 3
โข Design the game structure in order to induce the desired behavior from the agents
โข Desired behavior?โข We will mostly focus on incentivizing agents to truthfully
reveal their private information
โข With moneyโข Can pay agents or ask agents for money depending on
what the agents report
Mathematical Setup
CSC304 - Nisarg Shah 4
โข A set of outcomes ๐ดโข ๐ด might depend on which agents are participating.
โข Each agent ๐ has a private valuation ๐ฃ๐ โถ ๐ด โ โ
โข Auctions:โข ๐ด has a nice structure.o Selling one item to ๐ buyers = ๐ outcomes (โgive to ๐โ)
o Selling ๐ items to ๐ buyers = ๐๐ outcomes
โข Agents only care about which items they receiveo ๐ด๐ = bundle of items allocated to agent ๐
o Use ๐ฃ๐ ๐ด๐ instead of ๐ฃ๐(๐ด) for notational simplicity
โข But for now, weโll look at the general setup.
Mathematical Setup
CSC304 - Nisarg Shah 5
โข Agent ๐ might misreport: report ๐ฃ๐ instead of ๐ฃ๐
โข Mechanism: (๐, ๐)โข Input: reported valuations ๐ฃ = ( ๐ฃ1, โฆ , ๐ฃ๐)
โข ๐ ๐ฃ โ ๐ด decides what outcome is implemented
โข ๐ ๐ฃ = (๐1, โฆ , ๐๐) decides how much each agent payso Note that each ๐๐ is a function of all reported valuations
โข Utility to agent ๐ : ๐ข๐ ๐ฃ = ๐ฃ๐ ๐ ๐ฃ โ ๐๐ ๐ฃโข โQuasi-linear utilitiesโ
Mathematical Setup
CSC304 - Nisarg Shah 6
โข Our goal is to design the mechanism (๐, ๐)โข ๐ is called the social choice function
โข ๐ is called the payment scheme
โข We want to several things from our mechanism
โข Truthfulness/strategyproofnessโข For all agents ๐, all ๐ฃ๐, and all ๐ฃ,
๐ข๐ ๐ฃ๐ , ๐ฃโ๐ โฅ ๐ข๐( ๐ฃ๐ , ๐ฃโ๐)
โข An agent is at least as happy reporting the truth as telling any lie, irrespective of what other agents report
Mathematical Setup
CSC304 - Nisarg Shah 7
โข Our goal is to design the mechanism (๐, ๐)โข ๐ is called the social choice function
โข ๐ is called the payment scheme
โข We want to several things from our mechanism
โข Individual rationalityโข For all agents ๐ and for all ๐ฃโ๐,
๐ข๐ ๐ฃ๐ , ๐ฃโ๐ โฅ 0
โข An agent doesnโt regret participating if she tells the truth.
Mathematical Setup
CSC304 - Nisarg Shah 8
โข Our goal is to design the mechanism (๐, ๐)โข ๐ is called the social choice function
โข ๐ is called the payment scheme
โข We want to several things from our mechanism
โข No payments to agentsโข For all agents ๐ and for all ๐ฃ,
๐๐ ๐ฃ โฅ 0
โข Agents pay the center. Not the other way around.
Mathematical Setup
CSC304 - Nisarg Shah 9
โข Our goal is to design the mechanism (๐, ๐)โข ๐ is called the social choice function
โข ๐ is called the payment scheme
โข We want to several things from our mechanism
โข Welfare maximizationโข Maximize ฯ๐ ๐ฃ๐ ๐ ๐ฃ
o In many contexts, payments are less important (e.g. ad auctions)
o Or think of the auctioneer as another agent with utility ฯ๐ ๐๐ ๐ฃ
โข Then, the total utility of all agents (including the auctioneer) is precisely the objective written above
Single-Item Auction
CSC304 - Nisarg Shah 10
Image Courtesy: Freepik
Rule 1: Each would tell me his/her value. Iโll give it to the one with the higher value.
Objective: The one who really needs it more should have it.
?
Single-Item Auction
CSC304 - Nisarg Shah 11
Image Courtesy: Freepik
Rule 2: Each would tell me his/her value. Iโll give it to the one with the higher value, but they have to pay me that value.
Objective: The one who really needs it more should have it.
?
Single-Item Auction
CSC304 - Nisarg Shah 12
Image Courtesy: Freepik
Implements the desired outcome. But not truthfully.
Objective: The one who really needs it more should have it.
?
Single-Item Auction
CSC304 - Nisarg Shah 13
Image Courtesy: Freepik
Rule 3: Each would tell me his/her value. Iโll give it to the one with the highest value, and charge them the second highest value.
Objective: The one who really needs it more should have it.
?
Single-item Vickrey Auction
CSC304 - Nisarg Shah 14
โข Simplifying notation: ๐ฃ๐ = value of agent ๐ for the item
โข ๐ ๐ฃ : give the item to agent ๐โ โ argmax๐ ๐ฃ๐
โข ๐ ๐ฃ : ๐๐โ = max๐โ ๐โ
๐ฃ๐, other agents pay nothing
Theorem:Single-item Vickrey auction is strategyproof.
Proof sketch:
Highest reported value among other agents
Case 1:๐ฃ๐ < ๐
True value of agent ๐
Case 2๐ฃ๐ = ๐
Case 3๐ฃ๐ > ๐
Increasingvalue
๐
Vickrey Auction: Identical Items
CSC304 - Nisarg Shah 15
โข Two identical xboxes
โข Each agent ๐ only wants one, has value ๐ฃ๐
โข Goal: give to the agents with the two highest values
โข Attempt 1
โข To agent with highest value, charge 2nd highest value.
โข To agent with 2nd highest value, charge 3rd highest value.
โข Attempt 2
โข To agents with highest and 2nd highest values, charge the 3rd
highest value.
โข Question: Which attempt(s) would be strategyproof?
โข Both, 1, 2, None?
VCG Auction
CSC304 - Nisarg Shah 16
โข Recall the general setup:โข ๐ด = set of outcomes, ๐ฃ๐ = valuation of agent ๐, ๐ฃ๐ = what
agent ๐ reports, ๐ chooses the outcome, ๐ decides payments
โข VCG (Vickrey-Clarke-Groves Auction)โข ๐ ๐ฃ = ๐โ โ argmax๐โ๐ด ฯ๐ ๐ฃ๐ ๐
โข ๐๐ ๐ฃ = max๐
ฯ๐โ ๐ ๐ฃ๐ ๐ โ ฯ๐โ ๐ ๐ฃ๐ ๐โ
Maximize welfare
๐โs payment = welfare that others lost due to presence of ๐
A Note About Payments
CSC304 - Nisarg Shah 17
โข ๐๐ ๐ฃ = max๐
ฯ๐โ ๐ ๐ฃ๐ ๐ โ ฯ๐โ ๐ ๐ฃ๐ ๐โ
โข In the first termโฆโข Maximum is taken over alternatives that are feasible
when ๐ does not participate.
โข Agent ๐ cannot affect this term, so can ignore in calculating incentives.
โข Could be replaced with any function โ๐ ๐ฃโ๐
o This specific function has advantages (weโll see)
Properties of VCG Auction
CSC304 - Nisarg Shah 18
โข Strategyproofness:โข Suppose agents other than ๐ report ๐ฃโ๐.
โข Agent ๐ reports ๐ฃ๐ โ outcome chosen is ๐ ๐ฃ = ๐
โข Utility to agent ๐ = ๐ฃ๐ ๐ โ โ โ ฯ๐โ ๐ ๐ฃ๐ ๐
โข Agent ๐ wants ๐ to maximize ๐ฃ๐ ๐ + ฯ๐โ ๐ ๐ฃ๐ ๐
โข ๐ chooses ๐ to maximize ๐ฃ๐ ๐ + ฯ๐โ ๐ ๐ฃ๐ ๐
โข Hence, agent ๐ is best off reporting ๐ฃ๐ = ๐ฃ๐
o ๐ chooses ๐ that maximizes the utility to agent ๐
Term that agent ๐ cannot affect
Properties of VCG Auction
CSC304 - Nisarg Shah 19
โข Individual rationality:โข ๐โ โ argmax๐โ๐ด ๐ฃ๐ ๐ + ฯ๐โ ๐ ๐ฃ๐ ๐
โข ๐ โ argmax๐โ๐ด ฯ๐โ ๐ ๐ฃ๐ ๐
๐ข๐ ๐ฃ๐ , ๐ฃโ๐
= ๐ฃ๐ ๐โ โ ๐โ ๐
๐ฃ๐ ๐ โ ๐โ ๐
๐ฃ๐ ๐โ
= ๐ฃ๐ ๐โ + ๐โ ๐
๐ฃ๐ ๐โ โ ๐โ ๐
๐ฃ๐ ๐
= Max welfare to all agentsโ max welfare to others when ๐ is absent
โฅ 0
Properties of VCG Auction
CSC304 - Nisarg Shah 20
โข No payments to agents:โข Suppose the agents report ๐ฃ
โข ๐โ โ argmax๐โ๐ด ฯ๐ ๐ฃ๐ ๐
โข ๐ โ argmax๐โ๐ด ฯ๐โ ๐ ๐ฃ๐ ๐
๐๐ ๐ฃ
= ๐โ ๐
๐ฃ๐ ๐ โ ๐โ ๐
๐ฃ๐ ๐โ
= Max welfare to others when ๐ is absentโ welfare to others when ๐ is present
โฅ 0
Properties of VCG Auction
CSC304 - Nisarg Shah 21
โข Welfare maximization:โข By definition, since ๐ chooses the outcome maximizing
the sum of reported values
โข Informal result:โข Under minimal assumptions, VCG is the unique auction
satisfying these properties.
VCG: Simple Example
CSC304 - Nisarg Shah 22
โข Suppose each agent has a value XBox and a value for PS4.
โข Their value for {๐๐ต๐๐ฅ, ๐๐4} is the max of their two values.
A1 A2 A3 A4
XBox 3 4 8 7
PS4 4 2 6 1
Q: Who gets the xbox and who gets the PS4?
Q: How much do they pay?
VCG: Simple Example
CSC304 - Nisarg Shah 23
A1 A2 A3 A4
XBox 3 4 8 7
PS4 4 2 6 1
Allocation:
โข A4 gets XBox, A3 gets PS4
โข Achieves maximum welfare of 7 + 6 = 13
VCG: Simple Example
CSC304 - Nisarg Shah 24
A1 A2 A3 A4
XBox 3 4 8 7
PS4 4 2 6 1
Payments:
โข Zero payments charged to A1 and A2โข โDeletingโ either does not change the outcome/payments for others
โข Can also be seen by individual rationality
VCG: Simple Example
CSC304 - Nisarg Shah 25
A1 A2 A3 A4
XBox 3 4 8 7
PS4 4 2 6 1
Payments:
โข Payment charged to A3 = 11 โ 7 = 4โข Max welfare to others if A3 absent: 7 + 4 = 11
o Give XBox to A4 and PS4 to A1
โข Welfare to others if A3 present: 7
VCG: Simple Example
CSC304 - Nisarg Shah 26
A1 A2 A3 A4
XBox 3 4 8 7
PS4 4 2 6 1
Payments:
โข Payment charged to A4 = 12 โ 6 = 6โข Max welfare to others if A4 absent: 8 + 4 = 12
o Give XBox to A3 and PS4 to A1
โข Welfare to others if A4 present: 6
VCG: Simple Example
CSC304 - Nisarg Shah 27
A1 A2 A3 A4
XBox 3 4 8 7
PS4 4 2 6 1
Final Outcome:
โข Allocation: A3 gets PS4, A4 gets XBox
โข Payments: A3 pays 4, A4 pays 6
โข Net utilities: A3 gets 6 โ 4 = 2, A4 gets 7 โ 6 = 1