Download Computer-aided mechanism design

Computer-aided mechanism
design
Ye Fang, Swarat Chaudhuri,
Moshe Vardi
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Winner = …
Price = …
$200
$100
$150
$175
Private info:
$130
A
B
C
D
$210
$225
$140
$150
E
$150
Utility function = value -price
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First-Price Auction
Rule:
• Winner
• Payment
highest bidder
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
• Payment
highest bidder
second highest bid
How will you bid under this rule?
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If the camera worth $200 to me, profit = ($200 – Price) or 0.
$200 >= brest , bid $200
$200 < brest , bid $200
winning region
bmy > brest
brest = highest bid of rest bidders
lose region
bmy < brest
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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 mechanism
Online Auction System
Reputation System
……
Voting 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-telling
2) efficient allocation
But, what if you are not given a rule, and you
want the players to behave in certain way?
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How to formalize this problem?
System Setting
Agent model
procedure
Outcome Property
rule
Magic Box
Mechanism
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Our Solution
System Setting
Agent model
procedure
Outcome Property
rule
Our System
Language
Synthesis Compiler
Mechanism
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Our Solution
Language
• to encode the setting
• to encode the property
Synthesis Program
• reduce to the program to a first order logic
formula
• use SMT solver to search for missing
implementations
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Winner = …
Price = …
$200
$100
$150
$175
Private info:
$130
A
B
C
D
$210
$225
$140
$150
E
$150
Utility function = value -price
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Model
Auction Setting
Truth-telling
rule
Agent model
• bid
• Private value
• Utility function
• Partial
Our System
procedure
rule
• How the auction is conducted
Mechanism
• Complete
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Agent Model
Class Agent {
real bid
real value
function 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 Behavior
main (){
Bidding truthfully always yields more profit!
…
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’))
}
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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 < y
Foo(int x, int y){
x/c0 > y/c1
s0: x = c0* x
x>y/c1
s1: y = c1* y
x>y
}
@assert: x > y
x<y
x/c0<y/c1
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Generated Fornula
@assume: x < y
Foo(int x, int y){
x/c0 > y/c1
s0: x = c0* x
x>y/c1
s1: y = c1* y
x>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
System Setting
Agent model
procedure
Outcome Property
rule
Our System
Language
Synthesis Compiler
Mechanism
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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 properties
Voting 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.
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