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CYCLIC MONOTONICITY, REVENUE EQUIVALENCE AND

BUDGETS

Ahuva Mu’alem HIT July 2016

Budgets

ALSO:

- Daily budgets in Google’s Ad Auctions - Billing alerts by Amazon CloudWatch

CHALLANGE: Bidders can over-report their private values while under-reporting their private budgets to avoid charges  Maximizing the social welfare is impossible

Classic Auctions: Building Blocks Prices

Allocation Rule

Strategic Behavior

Classical Auctions Single-item Auctions [Myerson ‘81]: ⇒ Optimal Auction Truthfulness ⇔ Monotonicity ⇔ Revenue Equivalence

⇒ Optimal Auction

Classical Auctions Single-item Auctions [Myerson ‘81]: ⇒ Optimal Auction Truthfulness ⇔ Monotonicity ⇔ Revenue Equivalence

⇒ Optimal Auction

Classical Auctions Single-item Auctions [Myerson ‘81]: ⇒ Optimal Auction Truthfulness ⇔ Monotonicity ⇔ Revenue Equivalence

⇒ Optimal Auction • Truthfulness ≈ rational players truthfully bid their values • Monotonicity ≈ a higher bid has a greater probability for receiving the item • Revenue Equivalence ≈ the payment is uniquely defined (up to an additive constant)

Classical Auctions Single-item Auctions [Myerson ‘81]: • Truthfulness ⇔

Monotonicity

⇔

Revenue Equivalence

⇒ Optimal Auction

Multiple-item Auctions: • [Rochet ‘87]: Truthfulness ⇔ Cyclic-Monotonicity • [Heydenreich, Müller, Uetz, and Vohra ‘09]: Cyclic-Monotonicity + Anti-Symmetry ⇔ Revenue Equivalence

Classical Auctions Single-item Auctions [Myerson ‘81]: • Truthfulness ⇔

Monotonicity

⇔

Revenue Equivalence

⇒ Optimal Auction

Multiple-item Auctions: • [Rochet ‘87]: Truthfulness ⇔ Cyclic-Monotonicity • [Heydenreich, Müller, Uetz, and Vohra ‘09]: Cyclic-Monotonicity + Anti-Symmetry ⇔ Revenue Equivalence

Auctions with Budgets

Recent Related Work: [Hatfield and Milgrom ’05], [Borgs, Chayes, Immorlica, Mahdian, and Saberi ’05], [Dobzinski, Lavi and Nisan ’08], [Malakhov and Vohra ’08], [Pai and Vohra ‘14], [Dobzinski and Leme ’14], [Daskalakis, Devanur and Weinberg ’15]

This Talk: • Part I:

Budgeted Cyclic-Monotonicity

• Part II:

Budgeted Revenue-Equivalence

The Model • Wlog, we assume a single player

• Wlog, a single player with private value 𝑣 ∈ 𝒱 and private budget B ∈ 𝓑. • A social choice function

𝑓 ∶ 𝒱×𝓑→𝒜

• A payment function

𝑝 ∶ 𝒱×𝓑→ℝ

• A mechanism

(𝑓, 𝑝)

The Model Def: The Mechanism 𝒇, 𝒑 is implementable if • (Truthfulness): if 𝑓(𝑣, 𝑩) = 𝑎, 𝑓(𝑣′, 𝑩′ ) = 𝑎′ and 𝒑 𝒗′ , 𝑩′ ≤ 𝑩 then 𝒗 𝑎 − 𝑝 𝑣, 𝐵 ≥ 𝒗 𝑎′ − 𝑝 𝑣 ′ , 𝐵′ . (“Any affordable deviation wrt value and/or budget is not profitable”) • (Payment-Feasibility): if 𝑓(𝑣, 𝐵) = 𝑎 then 0 ≤ 𝒑 𝑣, 𝐵 ≤ min 𝑣 𝑎 , 𝑩 . Def: A SCF 𝒇 is called implementable if there exists a payment 𝒑 s.t. the mechanism (𝒇, 𝒑) is implementable.

The Model Def: The Mechanism 𝒇, 𝒑 is implementable if • (Truthfulness): if 𝑓(𝑣, 𝑩) = 𝑎, 𝑓(𝑣′, 𝑩′ ) = 𝑎′ and 𝒑 𝒗′ , 𝑩′ ≤ 𝑩 then 𝒗 𝑎 − 𝑝 𝑣, 𝐵 ≥ 𝒗 𝑎′ − 𝑝 𝑣 ′ , 𝐵′ . (“Any affordable deviation wrt value and/or budget is not profitable”) • (Payment-Feasibility): if 𝑓(𝑣, 𝐵) = 𝑎 then 0 ≤ 𝒑 𝑣, 𝐵 ≤ min 𝑣 𝑎 , 𝑩 . Def: A SCF 𝒇 is called implementable if there exists a payment 𝒑 s.t. the mechanism (𝒇, 𝒑) is implementable.

Notation:

𝜷 𝑎 =

𝐢𝐧𝐟

𝑣, 𝐵 ∈ 𝒱×ℬ

𝑩

𝑓 𝑣, 𝐵 = 𝑎}

i.e., the minimum reported budget required to obtain the outcome 𝑎 ∈ 𝐴. We also sometimes call it the “budget level of a” Fact: By Implementability: if 𝑓 𝑣, 𝐵 = 𝑎 then B ≥ 𝜷 𝒂 ≥ 𝑝𝑎

Notation:

𝜹 𝑎, 𝑎′ = 𝐢𝐧𝐟

𝑣∈ 𝒱

′

𝒗 𝒂 −𝒗 𝒂

𝑓 𝑣 = 𝑎}

i.e., the minimum happiness of receiving 𝑎 compared to 𝑎′ (assuming zero payments).

Fact: By Implemetability: if 𝑓 𝑣 = 𝑎 then 𝒗 𝒂 − 𝒗 𝒂′ ≥ 𝜹 𝒂, 𝒂′ ≥ 𝑝𝑎 − 𝑝𝑎′

Graph interpretation of Cyclic Monotonicity Theorem [Rochet ‘87]: A social choice function 𝒇: 𝒱 → 𝒜 is truthfully implementable if and only if the allocation graph 𝑮𝜹 has no negative cycles. Definition ]Allocation Graph ]: Let 𝑮𝜹 be a complete directed graph over the nodes {𝑎1 , 𝑎2 , … , 𝑎|𝐴| }, where the directed edge 𝑎𝑖 , 𝑎𝑘 has length 𝜹 𝑎𝑖 , 𝑎𝑘 . (𝑎1 , 𝑎2 ) 𝒂𝟐

𝒂𝟏 (𝑎2 , 𝑎1 )

Sufficiency First attempt to generalize 𝜹:

𝜹 𝑎, 𝑎′ =

inf

𝒗, 𝑩 ∈ 𝓥×𝓑

𝑣 𝑎 − 𝑣 𝑎′

𝑓 𝑣, 𝑩 = 𝑎}

Theorem: If the allocation graph 𝑮𝜹, 𝜷 has no negative cycles then 𝑓: 𝒱 × 𝓑 → 𝒜 is implementable with private budgets. Remark: The other direction is incorrect (as shown next).

Example: non-necessity 𝑩 =𝟎 𝒗

𝒂 10

𝒂′ 0

𝑩′ = 𝟓 𝒗

𝒂 10

𝒂′ 0

• Clearly, 𝑝𝑎′ = 0, 𝑝𝑎 = 5 is a feasible truthful payment • However, 𝑮𝜹, 𝜷 contains a negative cycle:

𝟓 = min{𝛽 𝑎 , 𝛿 (𝑎, 𝑎′ )} = min{5 , 10}

Theorem: If the 𝒂′ 𝒂 −𝟏𝟎 = min{𝛽 𝑎′ , 𝛿 (𝑎′, 𝑎)} = min{0, −10}

Necessity Second attempt to generalize 𝜹: 𝛿 𝑎, 𝑎′ =

inf

𝑣, 𝐵 ∈ 𝒱 × 𝓑 ∩ 𝛽 𝑎′ , ∞

𝑣 𝑎 − 𝑣 𝑎′

𝑓 𝑣, 𝐵 = 𝑎}

Theorem: If 𝑓: 𝒱 × 𝓑 → 𝒜 is implementable with private budgets then the allocation graph 𝑮𝜹, 𝜷 has no negative cycles. Remark: The other direction is incorrect (as shown next).

Example: insufficiency 𝑩 = 𝟏𝟎 𝒗 𝒗’ • • • •

𝒂 20 11

𝒂′ 10 0

𝑩′ = 𝟐𝟎 𝒗 𝒗’

𝒂 20 11

⇒ 𝒑𝒂′ = 𝟎 ⇒ 𝟏𝟎 ≥ v(a) − v(a′) ≥ 𝑝𝑎 − 𝑝𝑎′ ≥ 𝒑𝒂 ⇒ it is rational to deviate from 𝑣 ′ , 10 to 𝑣 ′ , 20 However, 𝑮, 𝜷 contains no negative cycle:

𝟏𝟎 = min{𝛽 𝑎 ,  (𝑎, 𝑎′ )} = min {20, 10}

Theorem: If the 𝒂′ 𝒂 𝟏𝟎 = min{𝛽 𝑎′ , (𝑎′, 𝑎)} = min {10, ∞}

𝒂′ 10 0

Revenue Equivalence Example 𝑩 = ∞

𝒂

𝒂′

𝒗

7

6

𝒗′

6

5

p = truthful payment ⇔

𝑝𝑎 − 𝑝𝑎′ = 1

To see why:

1 = 𝑣 𝑎 − 𝑣 𝑎′ ≥ 𝑝𝑎 − 𝑝𝑎′ ≥ 𝑣 ′ 𝑎 − 𝑣 ′ 𝑎′ = 1 Truthfulness wrt a

Truthfulness wrt a’

Revenue Equivalence Example 𝑩 = ∞

𝒂

𝒂′

𝒗

7

6

𝒗′

6

5

𝒑 = truthful payment ⇔

𝑝𝑎 − 𝑝𝑎′ = 1

To see why:

1 = 𝑣 𝑎 − 𝑣 𝑎′ ≥ 𝑝𝑎 − 𝑝𝑎′ ≥ 𝑣 ′ 𝑎 − 𝑣 ′ 𝑎′ = 1 Truthfulness wrt a

Truthfulness wrt a’

Revenue Equivalence Example B=10

𝒂

𝒂’

𝒄

𝒄′

B’=20

𝒂

𝒂’

𝒄

𝒄′

𝒗

7

6

17

15

𝒗

7

6

17

15

𝒗’

6

5

15

13

𝒗’

6

5

15

13

⇒ A (possibly distinct) constant for every budget level !

Budgeted Revenue Equivalence Definition: 𝑓: 𝒱 × ℬ → 𝐴 satisfies the revenue equivalence property if for any two implementable mechanisms (𝑓, 𝑝) and (𝑓, 𝑝’) we have that 𝜷 𝒂 = 𝜷 𝒂′ implies that 𝑝 𝑎 − 𝑝 𝑎′ = 𝑝′ 𝑎 − 𝑝′ 𝑎′ . Notation: Δ 𝑎, 𝑎′ denotes the length of the shortest path from 𝑎 to a’ in the allocation graph 𝐺𝛿, 𝛽 Theorem: A generically implementable 𝑓: 𝒱 × 𝓑 → 𝒜 satisfies the revenue equivalence property if and only if for every 𝒂, 𝒂′ ∈ 𝑨 s.t. 𝜷 𝒂 = 𝜷 𝒂′ we have that Δ 𝑎, 𝑎′ + Δ 𝑎, 𝑎′ = 0.

Thm: A generically implementable 𝑓 satisfies the revenue equivalence property IFF 𝛽 𝑎 = 𝛽 𝑎′ ⟹ Δ 𝑎, 𝑎′ + Δ 𝑎, 𝑎′ = 0. Proof Sketch of Necessity: • Suppose that the mechanism (f, p) is implementable. • Define a subgraph 𝓗𝛿 with an edge from 𝒂 to 𝒂′ ∈ 𝓐 if both (1) 𝜷 𝒂 = 𝜷 𝒂′ and (2) 𝒑𝒂 − 𝒑𝒂′ = 𝜹 𝒂, 𝒂′ . • Clearly, all directed cycles in 𝓗𝛿 have zero length. • Now, if 𝛽 𝑎 = 𝛽 𝑎′ but 𝚫 𝒂, 𝒂′ + 𝚫 𝒂′, 𝒂 > 𝟎, then a and a’ belong to distinct connected components of 𝓗𝛿 . We then can easily construct a feasible truthful payment p’ to contradict the revenue equivalence property. ∎

Thanks !