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Computer-aided mechanism design Ye Fang, Swarat Chaudhuri, Moshe Vardi 1

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$150 $100$200 $130$175 $210$225 $140 $150 Private info: Winner = … Price = … A B C D E 2 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! 3

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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 4

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Second-Price Auction Rule: Winner highest bidder Payment second highest bid How will you bid under this rule? 5

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b rest = highest bid of rest bidders winning region b my > b rest lose region b my < b rest If the camera worth $200 to me, profit = ($200 – Price) or 0. $200 >= b rest, bid $200 $200 < b rest, bid $200 6

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Second-Price Auction Bidding truthfully is the best strategy. 7

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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 8

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Decision making mechanism Online Auction System 9 Voting System Reputation 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. 10

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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 11

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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? 12 easy!

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How to formalize this problem? Outcome Property System Setting Mechanism Magic Box 13 Agent model rule procedure

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Our Solution Outcome Property System Setting Mechanism Our System 14 Agent model rule procedure Language Synthesis Compiler

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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 15

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$150 $100$200 $130$175 $210$225 $140 $150 Private info: Winner = … Price = … A B C D E 16 Utility function = value -price

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Model Truth-telling Auction Setting Mechanism Our System 17 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 bid real value function utility(result){ If(bid = winningbid) { ut = value – price } else { ut = 0 } return ut } 18

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Mechanism function Rule(real[] B){ real winningbid = ?? real price = ?? return (winningbid, price) } 19

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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); } 20

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Specify the Desired Behavior main (){ … real[] B = [a_1.b, a_2.b, a_3.b] return result = 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! 21

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How to replace the question mark? function Rule(real[] B){ real winningbid = ?? real price = ?? return (winningbid, price) } 22

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Sort inputs sorted(B) function Rule(real[] B){ real winningbid = ?? real price = ?? return (winningbid, price) } 23

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Linear 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) } 24

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Put sorted(B) function Rule(real[] B) {real winningbid = … real price = … return (winningbid, price) } main(){ … return result = forall a_i, Let B’ = … a_i.ut(result) <= a_i.ut(Rule(B’)) } 25

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An Easier Problem Find an implementation of x < y Foo(int x, int y){ x = ? * x y = ? * y x > y 27

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Replace ? with x < y Foo(int x, int y){ x = c0 * x y = c1 * y x > y 28

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Weakest x < y Foo(int x, int y){ x/c0 > y/c1 s0: x = c0* x x>y/c1 s1: y = c1* y x>y x > y 29 x

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Generated x < y Foo(int x, int y){ x/c0 > y/c1 s0: x = c0* x x>y/c1 s1: y = c1* y x>y x > y Exists (c0, c1), ForAll(x, y), (x y/c1) 30

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Solve Generated Formula Exsits(c0, c1), ForAll(x, y), (x y/c1) SMT Solver (Satisfiability Modulo Theories) values for c0, c1 31

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Original sorted(B) function Rule(real[] B) {real winningbid = … real price = … return (winningbid, price) } main(){ … return result = forall a_i, Let B’ = … a_i.ut(result) >= a_i.ut(Rule(B’)) } 33

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Replace ? with 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) } 34

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Original 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, forall a_i, Let B’ = swap(B, I, v[i]) a_i.ut(result) >= a_i.ut(Rule(B’)) 35

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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) 36

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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] 37

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Our Contribution Outcome Property System Setting Mechanism Our System 38 Agent model rule procedure Language 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 properties Voting System no absolute fair mechanism 39

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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. 40

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