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Estimating Approximate Incentive CompatibilityΒ Β· 2020. 7. 30.Β Β· 𝑀,π‘˜ 𝐿 β‡’ 𝑀...

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Page 1: Estimating Approximate Incentive CompatibilityΒ Β· 2020. 7. 30.Β Β· 𝑀,π‘˜ 𝐿 β‡’ 𝑀 π›Ύΰ·œβˆ’π›Ύ= ΰ·¨ 𝐿𝑀+ 𝑁 + 𝑑 𝑁 𝑑 π‘³π’˜π’Œ 𝑡 β€’ β‰€π‘³π’˜+π’Œ
Page 2: Estimating Approximate Incentive CompatibilityΒ Β· 2020. 7. 30.Β Β· 𝑀,π‘˜ 𝐿 β‡’ 𝑀 π›Ύΰ·œβˆ’π›Ύ= ΰ·¨ 𝐿𝑀+ 𝑁 + 𝑑 𝑁 𝑑 π‘³π’˜π’Œ 𝑡 β€’ β‰€π‘³π’˜+π’Œ
Page 3: Estimating Approximate Incentive CompatibilityΒ Β· 2020. 7. 30.Β Β· 𝑀,π‘˜ 𝐿 β‡’ 𝑀 π›Ύΰ·œβˆ’π›Ύ= ΰ·¨ 𝐿𝑀+ 𝑁 + 𝑑 𝑁 𝑑 π‘³π’˜π’Œ 𝑡 β€’ β‰€π‘³π’˜+π’Œ
Page 4: Estimating Approximate Incentive CompatibilityΒ Β· 2020. 7. 30.Β Β· 𝑀,π‘˜ 𝐿 β‡’ 𝑀 π›Ύΰ·œβˆ’π›Ύ= ΰ·¨ 𝐿𝑀+ 𝑁 + 𝑑 𝑁 𝑑 π‘³π’˜π’Œ 𝑡 β€’ β‰€π‘³π’˜+π’Œ
Page 5: Estimating Approximate Incentive CompatibilityΒ Β· 2020. 7. 30.Β Β· 𝑀,π‘˜ 𝐿 β‡’ 𝑀 π›Ύΰ·œβˆ’π›Ύ= ΰ·¨ 𝐿𝑀+ 𝑁 + 𝑑 𝑁 𝑑 π‘³π’˜π’Œ 𝑡 β€’ β‰€π‘³π’˜+π’Œ
Page 6: Estimating Approximate Incentive CompatibilityΒ Β· 2020. 7. 30.Β Β· 𝑀,π‘˜ 𝐿 β‡’ 𝑀 π›Ύΰ·œβˆ’π›Ύ= ΰ·¨ 𝐿𝑀+ 𝑁 + 𝑑 𝑁 𝑑 π‘³π’˜π’Œ 𝑡 β€’ β‰€π‘³π’˜+π’Œ
Page 7: Estimating Approximate Incentive CompatibilityΒ Β· 2020. 7. 30.Β Β· 𝑀,π‘˜ 𝐿 β‡’ 𝑀 π›Ύΰ·œβˆ’π›Ύ= ΰ·¨ 𝐿𝑀+ 𝑁 + 𝑑 𝑁 𝑑 π‘³π’˜π’Œ 𝑡 β€’ β‰€π‘³π’˜+π’Œ
Page 8: Estimating Approximate Incentive CompatibilityΒ Β· 2020. 7. 30.Β Β· 𝑀,π‘˜ 𝐿 β‡’ 𝑀 π›Ύΰ·œβˆ’π›Ύ= ΰ·¨ 𝐿𝑀+ 𝑁 + 𝑑 𝑁 𝑑 π‘³π’˜π’Œ 𝑡 β€’ β‰€π‘³π’˜+π’Œ
Page 9: Estimating Approximate Incentive CompatibilityΒ Β· 2020. 7. 30.Β Β· 𝑀,π‘˜ 𝐿 β‡’ 𝑀 π›Ύΰ·œβˆ’π›Ύ= ΰ·¨ 𝐿𝑀+ 𝑁 + 𝑑 𝑁 𝑑 π‘³π’˜π’Œ 𝑡 β€’ β‰€π‘³π’˜+π’Œ
Page 10: Estimating Approximate Incentive CompatibilityΒ Β· 2020. 7. 30.Β Β· 𝑀,π‘˜ 𝐿 β‡’ 𝑀 π›Ύΰ·œβˆ’π›Ύ= ΰ·¨ 𝐿𝑀+ 𝑁 + 𝑑 𝑁 𝑑 π‘³π’˜π’Œ 𝑡 β€’ β‰€π‘³π’˜+π’Œ
Page 11: Estimating Approximate Incentive CompatibilityΒ Β· 2020. 7. 30.Β Β· 𝑀,π‘˜ 𝐿 β‡’ 𝑀 π›Ύΰ·œβˆ’π›Ύ= ΰ·¨ 𝐿𝑀+ 𝑁 + 𝑑 𝑁 𝑑 π‘³π’˜π’Œ 𝑡 β€’ β‰€π‘³π’˜+π’Œ

𝜸 𝑖𝑖

Ξ³

Page 12: Estimating Approximate Incentive CompatibilityΒ Β· 2020. 7. 30.Β Β· 𝑀,π‘˜ 𝐿 β‡’ 𝑀 π›Ύΰ·œβˆ’π›Ύ= ΰ·¨ 𝐿𝑀+ 𝑁 + 𝑑 𝑁 𝑑 π‘³π’˜π’Œ 𝑡 β€’ β‰€π‘³π’˜+π’Œ

𝜸 𝑖𝑖

Ξ³

Page 13: Estimating Approximate Incentive CompatibilityΒ Β· 2020. 7. 30.Β Β· 𝑀,π‘˜ 𝐿 β‡’ 𝑀 π›Ύΰ·œβˆ’π›Ύ= ΰ·¨ 𝐿𝑀+ 𝑁 + 𝑑 𝑁 𝑑 π‘³π’˜π’Œ 𝑡 β€’ β‰€π‘³π’˜+π’Œ

𝛾

Page 14: Estimating Approximate Incentive CompatibilityΒ Β· 2020. 7. 30.Β Β· 𝑀,π‘˜ 𝐿 β‡’ 𝑀 π›Ύΰ·œβˆ’π›Ύ= ΰ·¨ 𝐿𝑀+ 𝑁 + 𝑑 𝑁 𝑑 π‘³π’˜π’Œ 𝑡 β€’ β‰€π‘³π’˜+π’Œ

𝛾

𝑖

π’•βˆ’π‘–(1), … , π’•βˆ’π‘–

(𝑁)

max𝑑𝑖,π‘‘π‘–β€²βˆˆβ„

𝐷

1

𝑁

𝑗=1

𝑁

𝑒 𝑑𝑖 , 𝑑𝑖′, π’•βˆ’π‘–π‘—

βˆ’ 𝑒 𝑑𝑖 , 𝑑𝑖 , π’•βˆ’π‘–π‘—

Page 15: Estimating Approximate Incentive CompatibilityΒ Β· 2020. 7. 30.Β Β· 𝑀,π‘˜ 𝐿 β‡’ 𝑀 π›Ύΰ·œβˆ’π›Ύ= ΰ·¨ 𝐿𝑀+ 𝑁 + 𝑑 𝑁 𝑑 π‘³π’˜π’Œ 𝑡 β€’ β‰€π‘³π’˜+π’Œ

𝛾

𝑖

π’•βˆ’π‘–(1), … , π’•βˆ’π‘–

(𝑁)

max𝑑𝑖,π‘‘π‘–β€²βˆˆβ„

𝐷

1

𝑁

𝑗=1

𝑁

𝑒 𝑑𝑖 , 𝑑𝑖′, π’•βˆ’π‘–π‘—

βˆ’ 𝑒 𝑑𝑖 , 𝑑𝑖 , π’•βˆ’π‘–π‘—

Page 16: Estimating Approximate Incentive CompatibilityΒ Β· 2020. 7. 30.Β Β· 𝑀,π‘˜ 𝐿 β‡’ 𝑀 π›Ύΰ·œβˆ’π›Ύ= ΰ·¨ 𝐿𝑀+ 𝑁 + 𝑑 𝑁 𝑑 π‘³π’˜π’Œ 𝑡 β€’ β‰€π‘³π’˜+π’Œ

𝛾

ෝ𝜸

𝑖

π’•βˆ’π‘–(1), … , π’•βˆ’π‘–

(𝑁)

𝐹

ΰ·œπ›Ύ = max𝑑𝑖,π‘‘π‘–β€²βˆˆπΉ

1

𝑁

𝑗=1

𝑁

𝑒 𝑑𝑖 , 𝑑𝑖′, π’•βˆ’π‘–π‘—

βˆ’ 𝑒 𝑑𝑖 , 𝑑𝑖 , π’•βˆ’π‘–π‘—

Page 17: Estimating Approximate Incentive CompatibilityΒ Β· 2020. 7. 30.Β Β· 𝑀,π‘˜ 𝐿 β‡’ 𝑀 π›Ύΰ·œβˆ’π›Ύ= ΰ·¨ 𝐿𝑀+ 𝑁 + 𝑑 𝑁 𝑑 π‘³π’˜π’Œ 𝑡 β€’ β‰€π‘³π’˜+π’Œ

𝛾

ෝ𝜸

𝑖

π’•βˆ’π‘–(1), … , π’•βˆ’π‘–

(𝑁)

𝐹

Page 18: Estimating Approximate Incentive CompatibilityΒ Β· 2020. 7. 30.Β Β· 𝑀,π‘˜ 𝐿 β‡’ 𝑀 π›Ύΰ·œβˆ’π›Ύ= ΰ·¨ 𝐿𝑀+ 𝑁 + 𝑑 𝑁 𝑑 π‘³π’˜π’Œ 𝑡 β€’ β‰€π‘³π’˜+π’Œ

𝛾

ෝ𝜸

𝑖

π’•βˆ’π‘–(1), … , π’•βˆ’π‘–

(𝑁)

𝐹

Page 19: Estimating Approximate Incentive CompatibilityΒ Β· 2020. 7. 30.Β Β· 𝑀,π‘˜ 𝐿 β‡’ 𝑀 π›Ύΰ·œβˆ’π›Ύ= ΰ·¨ 𝐿𝑀+ 𝑁 + 𝑑 𝑁 𝑑 π‘³π’˜π’Œ 𝑡 β€’ β‰€π‘³π’˜+π’Œ

𝛾

ෝ𝜸

𝑖

π’•βˆ’π‘–(1), … , π’•βˆ’π‘–

(𝑁)

𝐹

ΰ·œπ›Ύ βˆ’ 𝛾 ≀

Page 20: Estimating Approximate Incentive CompatibilityΒ Β· 2020. 7. 30.Β Β· 𝑀,π‘˜ 𝐿 β‡’ 𝑀 π›Ύΰ·œβˆ’π›Ύ= ΰ·¨ 𝐿𝑀+ 𝑁 + 𝑑 𝑁 𝑑 π‘³π’˜π’Œ 𝑡 β€’ β‰€π‘³π’˜+π’Œ
Page 21: Estimating Approximate Incentive CompatibilityΒ Β· 2020. 7. 30.Β Β· 𝑀,π‘˜ 𝐿 β‡’ 𝑀 π›Ύΰ·œβˆ’π›Ύ= ΰ·¨ 𝐿𝑀+ 𝑁 + 𝑑 𝑁 𝑑 π‘³π’˜π’Œ 𝑡 β€’ β‰€π‘³π’˜+π’Œ

𝑑𝑖′

𝑒 𝑑𝑖 ,βˆ™, π’•βˆ’π‘–

Page 22: Estimating Approximate Incentive CompatibilityΒ Β· 2020. 7. 30.Β Β· 𝑀,π‘˜ 𝐿 β‡’ 𝑀 π›Ύΰ·œβˆ’π›Ύ= ΰ·¨ 𝐿𝑀+ 𝑁 + 𝑑 𝑁 𝑑 π‘³π’˜π’Œ 𝑡 β€’ β‰€π‘³π’˜+π’Œ

𝑑𝑖′

1

𝑁σ𝑗=1𝑁 𝑒 𝑑𝑖 ,βˆ™, π’•βˆ’π‘–

𝑗

Page 23: Estimating Approximate Incentive CompatibilityΒ Β· 2020. 7. 30.Β Β· 𝑀,π‘˜ 𝐿 β‡’ 𝑀 π›Ύΰ·œβˆ’π›Ύ= ΰ·¨ 𝐿𝑀+ 𝑁 + 𝑑 𝑁 𝑑 π‘³π’˜π’Œ 𝑡 β€’ β‰€π‘³π’˜+π’Œ

𝑑𝑖′

1

𝑁σ𝑗=1𝑁 𝑒 𝑑𝑖 ,βˆ™, π’•βˆ’π‘–

𝑗

Page 24: Estimating Approximate Incentive CompatibilityΒ Β· 2020. 7. 30.Β Β· 𝑀,π‘˜ 𝐿 β‡’ 𝑀 π›Ύΰ·œβˆ’π›Ύ= ΰ·¨ 𝐿𝑀+ 𝑁 + 𝑑 𝑁 𝑑 π‘³π’˜π’Œ 𝑡 β€’ β‰€π‘³π’˜+π’Œ
Page 25: Estimating Approximate Incentive CompatibilityΒ Β· 2020. 7. 30.Β Β· 𝑀,π‘˜ 𝐿 β‡’ 𝑀 π›Ύΰ·œβˆ’π›Ύ= ΰ·¨ 𝐿𝑀+ 𝑁 + 𝑑 𝑁 𝑑 π‘³π’˜π’Œ 𝑡 β€’ β‰€π‘³π’˜+π’Œ

𝑒1, … , 𝑒𝑁 π’˜,π’Œ

𝑀 ≀ π‘˜

1

𝑁σ𝑒𝑖

Page 26: Estimating Approximate Incentive CompatibilityΒ Β· 2020. 7. 30.Β Β· 𝑀,π‘˜ 𝐿 β‡’ 𝑀 π›Ύΰ·œβˆ’π›Ύ= ΰ·¨ 𝐿𝑀+ 𝑁 + 𝑑 𝑁 𝑑 π‘³π’˜π’Œ 𝑡 β€’ β‰€π‘³π’˜+π’Œ

ෝ𝜸

𝑁𝑀, π‘˜ 𝐿

Page 27: Estimating Approximate Incentive CompatibilityΒ Β· 2020. 7. 30.Β Β· 𝑀,π‘˜ 𝐿 β‡’ 𝑀 π›Ύΰ·œβˆ’π›Ύ= ΰ·¨ 𝐿𝑀+ 𝑁 + 𝑑 𝑁 𝑑 π‘³π’˜π’Œ 𝑡 β€’ β‰€π‘³π’˜+π’Œ

ෝ𝜸

𝑁𝑀, π‘˜ 𝐿⇒ 𝑀

Page 28: Estimating Approximate Incentive CompatibilityΒ Β· 2020. 7. 30.Β Β· 𝑀,π‘˜ 𝐿 β‡’ 𝑀 π›Ύΰ·œβˆ’π›Ύ= ΰ·¨ 𝐿𝑀+ 𝑁 + 𝑑 𝑁 𝑑 π‘³π’˜π’Œ 𝑡 β€’ β‰€π‘³π’˜+π’Œ

𝑁𝑀, π‘˜ 𝐿⇒ 𝑀

ΰ·œπ›Ύ βˆ’ 𝛾 = ෨𝑂 𝐿𝑀 +π‘˜

𝑁+

𝑑

𝑁

𝑑

𝑁

𝑀, π‘˜ 𝐿

β‡’ 𝑀

Page 29: Estimating Approximate Incentive CompatibilityΒ Β· 2020. 7. 30.Β Β· 𝑀,π‘˜ 𝐿 β‡’ 𝑀 π›Ύΰ·œβˆ’π›Ύ= ΰ·¨ 𝐿𝑀+ 𝑁 + 𝑑 𝑁 𝑑 π‘³π’˜π’Œ 𝑡 β€’ β‰€π‘³π’˜+π’Œ

𝑁𝑀, π‘˜ 𝐿⇒ 𝑀

ΰ·œπ›Ύ βˆ’ 𝛾 = ෨𝑂 𝐿𝑀 +π‘˜

𝑁+

𝑑

𝑁

𝑑

ΰ·œπ›Ύ βˆ’ 𝛾 = ෨𝑂 π‘³π’˜ +π’Œ

𝑡+

𝑑

𝑁

β€’ ≀ π‘³π’˜ +π’Œ

𝑡

‒𝑑

𝑁

𝑁

𝑀, π‘˜ 𝐿

β‡’ 𝑀

Page 30: Estimating Approximate Incentive CompatibilityΒ Β· 2020. 7. 30.Β Β· 𝑀,π‘˜ 𝐿 β‡’ 𝑀 π›Ύΰ·œβˆ’π›Ύ= ΰ·¨ 𝐿𝑀+ 𝑁 + 𝑑 𝑁 𝑑 π‘³π’˜π’Œ 𝑡 β€’ β‰€π‘³π’˜+π’Œ

𝑁𝑀, π‘˜ 𝐿⇒ 𝑀

ΰ·œπ›Ύ βˆ’ 𝛾 = ෨𝑂 𝐿𝑀 +π‘˜

𝑁+

𝑑

𝑁

𝑑

ΰ·œπ›Ύ βˆ’ 𝛾 = ෨𝑂 π‘³π’˜ +π’Œ

𝑡+

𝑑

𝑁

β€’ ≀ π‘³π’˜ +π’Œ

𝑡

‒𝑑

𝑁

ΰ·œπ›Ύ βˆ’ 𝛾 = ෨𝑂 𝐿𝑀 +π‘˜

𝑁+

𝒅

𝑡

β€’ ≀ 𝐿𝑀 +π‘˜

𝑁

‒𝒅

𝑡

𝑁

𝑀, π‘˜ 𝐿

β‡’ 𝑀

Page 31: Estimating Approximate Incentive CompatibilityΒ Β· 2020. 7. 30.Β Β· 𝑀,π‘˜ 𝐿 β‡’ 𝑀 π›Ύΰ·œβˆ’π›Ύ= ΰ·¨ 𝐿𝑀+ 𝑁 + 𝑑 𝑁 𝑑 π‘³π’˜π’Œ 𝑡 β€’ β‰€π‘³π’˜+π’Œ

𝑁𝑀, π‘˜ 𝐿⇒ 𝑀

ΰ·œπ›Ύ βˆ’ 𝛾 = ෨𝑂 𝐿𝑀 +π‘˜

𝑁+

𝑑

𝑁

𝑑

ΰ·œπ›Ύ βˆ’ 𝛾 = ෨𝑂 π‘³π’˜ +π’Œ

𝑡+

𝑑

𝑁

β€’ ≀ π‘³π’˜ +π’Œ

𝑡

‒𝑑

𝑁

ΰ·œπ›Ύ βˆ’ 𝛾 = ෨𝑂 𝐿𝑀 +π‘˜

𝑁+

𝒅

𝑡

β€’ ≀ 𝐿𝑀 +π‘˜

𝑁

‒𝒅

𝑡

ΰ·œπ›Ύ βˆ’ 𝛾 = ෨𝑂 𝐿𝑀 +π‘˜

𝑁+

𝑑

𝑁

𝑀 = 𝑂1

𝑁, π‘˜ = 𝑂 𝑁

(𝑀, π‘˜)

𝑁

𝑁

𝑀, π‘˜ 𝐿

β‡’ 𝑀

Page 32: Estimating Approximate Incentive CompatibilityΒ Β· 2020. 7. 30.Β Β· 𝑀,π‘˜ 𝐿 β‡’ 𝑀 π›Ύΰ·œβˆ’π›Ύ= ΰ·¨ 𝐿𝑀+ 𝑁 + 𝑑 𝑁 𝑑 π‘³π’˜π’Œ 𝑡 β€’ β‰€π‘³π’˜+π’Œ

[0, πœ…]

ΰ·œπ›Ύ βˆ’ 𝛾 = ෨𝑂#bidders + πœ…βˆ’1

(#samples)

Page 33: Estimating Approximate Incentive CompatibilityΒ Β· 2020. 7. 30.Β Β· 𝑀,π‘˜ 𝐿 β‡’ 𝑀 π›Ύΰ·œβˆ’π›Ύ= ΰ·¨ 𝐿𝑀+ 𝑁 + 𝑑 𝑁 𝑑 π‘³π’˜π’Œ 𝑡 β€’ β‰€π‘³π’˜+π’Œ

[0, πœ…]

ΰ·œπ›Ύ βˆ’ 𝛾 = ෨𝑂#bidders 3/2+ πœ…βˆ’1

(#samples)

Page 34: Estimating Approximate Incentive CompatibilityΒ Β· 2020. 7. 30.Β Β· 𝑀,π‘˜ 𝐿 β‡’ 𝑀 π›Ύΰ·œβˆ’π›Ύ= ΰ·¨ 𝐿𝑀+ 𝑁 + 𝑑 𝑁 𝑑 π‘³π’˜π’Œ 𝑡 β€’ β‰€π‘³π’˜+π’Œ

[0, πœ…]

ΰ·œπ›Ύ βˆ’ 𝛾 = ෨𝑂#bidders #units 2+ πœ…βˆ’1

(#samples)

Page 35: Estimating Approximate Incentive CompatibilityΒ Β· 2020. 7. 30.Β Β· 𝑀,π‘˜ 𝐿 β‡’ 𝑀 π›Ύΰ·œβˆ’π›Ύ= ΰ·¨ 𝐿𝑀+ 𝑁 + 𝑑 𝑁 𝑑 π‘³π’˜π’Œ 𝑡 β€’ β‰€π‘³π’˜+π’Œ

β€’

β€’

β€’

β€’β€’

β€’

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