Post on 14-Jan-2016
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Computer-aided mechanism design
Ye Fang, Swarat Chaudhuri, Moshe Vardi
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$150$100$200
$130$175
$210 $225 $140 $150 $150Private info:
Winner = …Price = …
A B C D E
Utility function = value -price
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First-Price Auction
Rule:• Winner highest bidder• Payment highest bid
How much will you bid based on this rule?• Try to maximize my profit.
• If I am the bidder, I will UNDERBID!
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First-Price Auction
If everyone thinks like me:• Payment EQUALS highest bid• Highest bid LESS THAN true value• Profit LESS THAN highest true value
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Second-Price Auction
Rule:• Winner highest bidder• Payment second highest bid
How will you bid under this rule?
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brest = highest bid of rest bidders
winning regionbmy > brest
lose regionbmy < brest
If the camera worth $200 to me, profit = ($200 – Price) or 0.
$200 >= brest , bid $200
$200 < brest , bid $200
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Second-Price Auction
Bidding truthfully is the best strategy.
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Rules & Behaviors
First-Price – Bidders bid lower than how much they think the
camera worth to them
Second-Price – Bidders’ bids equal to how much they think the
camera worth to them
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Decision making mechanismOnline Auction System
Voting SystemReputation System
……
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What is common?
Multi-agents • private information• conflicting preferences
The decision-making entity• aiming to achieve a desirable outcome – In auction, reveal bidders private information or try
to maximize the seller’s profit – In public resource auction, achieve efficient
allocation of resources.
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How to achieve desirable outcome?
Decision maker has no control over their behaviors.Agents are self-interested. Answer:• Design mechanisms • Agents are better to behave “nicely”• Deter liars, cheaters
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If given rules, we can choose one by finding the best strategy of each player.
Second-Price Auction:1) truth-telling2) efficient allocation
But, what if you are not given a rule, and you want the players to behave in certain way?
easy!
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How to formalize this problem?
Outcome PropertySystem Setting
Mechanism
Magic Box
Agent model rule
procedure
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Our Solution
Outcome PropertySystem Setting
Mechanism
Our System
Agent model rule
procedureLanguage
Synthesis Compiler
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Our SolutionLanguage • to encode the setting • to encode the propertySynthesis Program • reduce to the program to a first order logic
formula• use SMT solver to search for missing
implementations
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$150$100$200
$130$175
$210 $225 $140 $150 $150Private info:
Winner = …Price = …
A B C D E
Utility function = value -price
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Model
Truth-tellingAuction Setting
Mechanism
Our System
Agent model rule
procedure
• bid• Private value• Utility function
• How the auction is conducted
• Partial
rule
• Complete
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Agent Model
Class Agent {real bidreal valuefunction utility(result){
If(bid = winningbid) { ut = value – price }
else { ut = 0 }return ut
}
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Mechanism
function Rule(real[] B){real winningbid = ??real price = ??return (winningbid, price)
}
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When auction starts
main (){Agent a_1 = new Agent(“1”)Agent a_2 = new Agent(“2”)Agent a_3 = new Agent(“3”)real[] B = [a_1.b, a_2.b, a_3.b]
return result = Rule(B);}
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Specify the Desired Behaviormain (){
…real[] B = [a_1.b, a_2.b, a_3.b]
return result = Rule(B);
@assert: forall a_i, Let B’ = swap(B, i, v[i])
a_i.ut(result) <= a_i.ut(Rule(B’))}
Bidding truthfully always yields more profit!
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How to replace the question mark?
function Rule(real[] B){real winningbid = ??real price = ??return (winningbid, price)
}
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Sort inputs first
@assume: sorted(B)function Rule(real[] B){
real winningbid = ??real price = ??return (winningbid, price)
}
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Linear Function
@assume: sorted(B)function Rule(real[] B){
real winningbid = ? * B[0] + … + ? * B[B.size-1]
real price = ? * B[0] + … + ? * B[B.size-1]
return (winningbid, price)}
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Put together
@assume: sorted(B)function Rule(real[] B){ real winningbid = …
real price = …return
(winningbid, price)}
main(){…
return result = Rule(B);@assert: forall a_i,
Let B’ = …a_i.ut(result)
<= a_i.ut(Rule(B’)) }
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An Easier Problem
Find an implementation of Foo: @assume: x < y Foo(int x, int y){
x = ? * x y = ? * y } @assert: x > y
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Replace ? with identifiers
@assume: x < y Foo(int x, int y){
x = c0 * x y = c1 * y } @assert: x > y
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Weakest Precondition
@assume: x < yFoo(int x, int y){
x/c0 > y/c1s0: x = c0* xx>y/c1
s1: y = c1* yx>y }
@assert: x > y
x<y
x/c0<y/c1
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Generated Fornula
@assume: x < yFoo(int x, int y){
x/c0 > y/c1s0: x = c0* xx>y/c1
s1: y = c1* yx>y }
@assert: x > y
Exists (c0, c1), ForAll(x, y),(x < y) implies (x/c0 > y/c1)
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Solve Generated Formula
Exsits(c0, c1), ForAll(x, y), (x < y) implies (x/c0 > y/c1)
SMT Solver(Satisfiability Modulo Theories)
values for c0, c1
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Original Problem
@assume: sorted(B)function Rule(real[] B){ real winningbid = …
real price = …return
(winningbid, price)}
main(){…
return result = Rule(B);@assert: forall a_i,
Let B’ = …a_i.ut(result)
>= a_i.ut(Rule(B’)) }
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Replace ? with Identifiers
@assume: sorted(B)function Rule(real[] B){
real winningbid = c[0] * B[0] + … + c[B.size-1] * B[B.size-1]
real price = d[0] * B[0] + … + d[B.size-1] * B[B.size-1]
return (winningbid, price)}
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Original Problem @assume: sorted(B)function Rule(real[] B){
real winningbid = c[0] * B[0] + … + c[B.size-1] * B[B.size-1]
real price = d[0] * B[0] + … + d[B.size-1] * B[B.size-1]
return (winningbid, price)}@assert: forall a_i,
Let B’ = swap(B, I, v[i])a_i.ut(result) >= a_i.ut(Rule(B’))
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Generated Formula
Compute weakest precondition given assertion• f(c[0], …, c[B.size-1], d[0], …, d[B.size-1])
Formula to solve:• Exists(C, D), ForAll(B), sorted(B) implies f(C, D)
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Solve Generated Formula
• Exists(C, D), ForAll(B), sorted(B) implies f(C, D)
SMT Solver(Satisfiability Modulo Theories)
values for c[0], …, c[B.size-1], d[0], …, d[B.size-1]
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Our Contribution
Outcome PropertySystem Setting
Mechanism
Our System
Agent model rule
procedureLanguage
Synthesis Compiler
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What we have achieved?We reconstructed a set of classical mechanisms• single-item auction• Google online ads auction New mechanisms • multistage auction• result in new propertiesVoting System• no absolute fair mechanism
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Future Work
Extend to model with arbitrarily large number of agents.
Enrich the kind of mechanism functions that can be handled.
Explore more complicated real-life preference aggregation systems.