Simple vs. Optimal Contracts Paul Dütting – London School of Economics (Math) Tim Roughgarden – Columbia (CS) Inbal Talgam-Cohen – Technion (CS) ACM EC @FCRC Phoenix, June 2019
Contract Theory • Contracts align interests to enable exploiting gains from cooperation • “ Modern economies are held together by innumerable contracts ” [2016 Nobel Prize Announcement] Oliver Hart Bengt Holmström 2
Classic Applications • Employment contracts • Venture capital (VC) investment contracts • Insurance contracts • Freelance (e.g. book) contracts • Government procurement contracts • … → Contracts are indeed everywhere 3
Modern Applications Classic applications are moving online and/or increasing in complexity: • Crowdsourcing platforms • Platforms for hiring freelancers • Online marketing and affiliation • Complex supply chains • Pay-for-performance medicare → Of interest to AGT; algorithmic approach becoming more relevant 4
The Algorithmic Lens Research agenda: What can we learn about contract design through the algorithmic lens? 1. Robust alternatives to average-case / Bayesian analysis 2. Approximation guarantees when optimal solutions inappropriate 3. (Complexity issues – in different work) For more information please see our EC’19 tutorial website 5
A Building Momentum • Some pioneering works: • Combinatorial agency [Babaioff Feldman and Nisan’12] • Contract complexity [Babaioff and Winter’14] • Incentivizing exploration [Frazier Kempe Kleinberg and Kleinberg’14] • Robustness [Carroll’15] • Adaptive design [Ho Slivkins and Vaughan’16] • Some recent works: • Delegated search [Kleinberg and Kleinberg’18] • Information acquisition [Azar and Micali’18] • Succinct models [Dütting Roughgarden and T.- C.’19b] • EC’19 papers: • [Kleinberg and Raghavan’19, Lavi and Shamash’19, Dütting Roughgarden and T.- C.’19a] 6
Basic Contract Setting: An Example • Website owner (principal) hires marketing agent to attract visitors • Two defining features: 1. Agent’s actions are hidden - “ moral hazard ” 2. Principal never charges (only pays) agent - “ limited liability ” 7
Relation to other Incentive Problems • Mechanism design • Agents have hidden types • Contracts [Holstrom’79] • No hidden types • Principal less informed • Signaling (Bayesian persuasion) • Principal more informed 8
Our Results • In the model of [Holmstrom ’ 79]: 1. New robustness (max-min) justification for simple, linear contracts • “ Standing on the shoulders ” of [Carroll ’ 15] 2. Approximation guarantees for linear contracts • Linear is far from optimal only in pathological cases • Approximation is tight even for monotone contracts 9
Model: Contract Setting • Parameters 𝑜, 𝑛 • Agent has actions 𝑏 1 , … , 𝑏 𝑜 • with costs 0 = 𝑑 1 ≤ ⋯ ≤ 𝑑 𝑜 (can always choose action with 0 cost) • Principal has rewards 0 ≤ 𝑠 1 ≤ ⋯ ≤ 𝑠 𝑛 • Action 𝑏 𝑗 induces distribution 𝐺 𝑗 over rewards (“technology”) • with expectation 𝑆 𝑗 Recall two • Assumption: 𝑆 1 ≤ ⋯ ≤ 𝑆 𝑜 defining features • Contract = vector of transfers Ԧ 𝑢 = 𝑢 1 , … , 𝑢 𝑛 ≥ 0 10
Example Contract: 𝑢 1 = 0 𝑢 2 = 1 𝑢 3 = 2 𝑢 4 = 5 No visitor General visitor Targeted visitor Both visitors 𝑠 1 = 0 𝑠 2 = 3 𝑠 3 = 7 𝑠 4 = 10 Low effort 𝑆 1 = 1. 1.3 0.72 0.18 0.08 0.02 𝑑 1 = 0 Medium effort 𝑆 2 = 5 5.2 0.12 0.48 0.08 0.32 𝑑 2 = 1 High effort 𝑆 3 = 7. 7.2 0 0.4 0 0.6 𝑑 3 = 2 11
Contract Design Problem An optimization problem with incentive compatibility (IC) constraints Maximize principal’s 𝔽 [payoff] from action 𝑏 𝑗 subject to action 𝑏 𝑗 maximizing 𝔽 [utility] for agent • 𝔽 [payoff] = expected reward 𝑆 𝑗 minus expected payment σ 𝑘 𝐺 𝑗,𝑘 𝑢 𝑘 • 𝔽 [utility] = expected payment σ 𝑘 𝐺 𝑗,𝑘 𝑢 𝑘 minus cost 𝑑 𝑗 12
Example: Agent’s Perspective Contract: 𝑢 1 = 0 𝑢 2 = 1 𝑢 3 = 2 𝑢 4 = 5 No visitor General visitor Targeted visitor Both visitors 𝑠 1 = 0 𝑠 2 = 3 𝑠 3 = 7 𝑠 4 = 10 Low effort 0. 0.44 0.72 0.18 0.08 0.02 𝑑 1 = 0 Medium effort 1.24 1. 0.12 0.48 0.08 0.32 𝑑 2 = 1 High effort 1.4 1. 0 0.4 0 0.6 𝑑 3 = 2 Exp xpect ected ed tran ansf sfers rs: (0. 0.44, , 2.24, 4, 3.4) for (low, medi dium, m, high) h) 13
Example: Principal’s Perspective Contract: 𝑢 1 = 0 𝑢 2 = 1 𝑢 3 = 2 𝑢 4 = 5 No visitor General visitor Targeted visitor Both visitors 𝑠 1 = 0 𝑠 2 = 3 𝑠 3 = 7 𝑠 4 = 10 Low effort 𝑆 1 = 1. 1.3 0.72 0.18 0.08 0.02 𝑑 1 = 0 Medium effort 𝑆 2 = 5 5.2 0.12 0.48 0.08 0.32 𝑑 2 = 1 High effort 𝑆 3 = 7. 7.2 0 0.4 0 0.6 𝑑 3 = 2 𝑆 3 - exp xpecte cted d tran ansf sfer er = 7 7.2 - 3.4 = 3 3.8 14
LP-Based Solution Observation: Can compute optimal contract by solving 𝑜 LPs, one per action Expected transfer to minimize 𝐺 𝑗,𝑘 𝑢 𝑘 agent for action 𝑏 𝑗 𝑘 𝐺 𝑗 ′ ,𝑘 𝑢 𝑘 − 𝑑 𝑗 ′ ∀𝑗 ′ ≠ 𝑗 (IC) s.t. 𝐺 𝑗,𝑘 𝑢 𝑘 − 𝑑 𝑗 ≥ 𝑘 𝑘 𝑢 𝑘 ≥ 0 (LL) Agent’s expected utility from 𝑏 𝑗 given contract Ԧ 𝑢 • Caveats: (1) imperfect distribution knowledge (2) impractical contract 15
Result 1: Robust Optimality 16
Linear Contracts • Determined by parameter 𝛽 ∈ [0,1] : • Given reward 𝑠 𝑘 , principal transfers 𝛽𝑠 𝑘 to agent • Generalization to affine: 𝛽𝑠 𝑘 + 𝛽 0 • Agent’s expected utility from action 𝑏 𝑗 is 𝛽𝑆 𝑗 − 𝑑 𝑗 No dependence on • Principal’s expected payoff is (1 − 𝛽)𝑆 𝑗 details of distribution! • Really popular in practice 17
Robustness • “It is probably the great robustness of linear rules […] that accounts for their popularity” [Milgrom- Holmström’87] • Breakthrough formulation of [Carroll’15] : Linear contracts are optimal in the worst-case over unknown extra actions available to agent • Alternative formulations? • Standard CS formulation of uncertainty when input is stochastic: assume only first moments of the distribution are known [Scarf’58] 18
Example No visitor General visitor Targeted visitor Both visitors 𝑠 1 = 0 𝑠 2 = 3 𝑠 3 = 7 𝑠 4 = 10 Low effort 𝑆 1 = 1. 1.3 0.72 0.18 0.08 0.02 𝑑 1 = 0 Medium effort 𝑆 2 = 5 5.2 0.12 0.48 0.08 0.32 𝑑 2 = 1 High effort 𝑆 3 = 7. 7.2 0 0.4 0 0.6 𝑑 3 = 2 19
Example No visitor General visitor Targeted visitor Both visitors 𝑠 1 = 0 𝑠 2 = 3 𝑠 3 = 7 𝑠 4 = 10 Low effort 𝑆 1 = 1. 1.3 ? ? ? ? 𝑑 1 = 0 Medium effort 𝑆 2 = 5 5.2 ? ? ? ? 𝑑 2 = 1 High effort 𝑆 3 = 7. 7.2 ? ? ? ? 𝑑 3 = 2 20
New Robustness Result Theorem: • Given a contract setting with unknown distributions but known expectations, • a linear contract is optimal in the worst-case over all compatible distributions → Same conclusion as [Carroll’15] , under very different hypothesis! Intuition : If you don’t know enough to design a contract depending on anything but the expected rewards, optimize wrt what you know 21
Proof Overview: Max-Min Visualization • Fix a contract setting with known expected rewards Compatible atible di dist stribu butio tions ns Min over column umns Max over Contract ct rows Principal’s exp xpecte cted d pay ayoff ff 22
Proof Overview: Linear Contracts are Robust Compatible atible di dist stribu butio tions ns Linear ar/affine /affine Sam ame exp xpecte cted d contract act pay ayoff ff 23
Proof Overview: Key Lemma Lemma: For every contract 𝑢 there exist compatible distributions and an affine contract with 𝛽 0 ≥ 0 and better expected payoff Compatible atible di dist stributio butions ns Contract act 𝑢 Affin ine e contract act 24
Key Lemma Suffices → For every contract 𝑢 there exists an affine contract with 𝛽 0 ≥ 0 and better worst-case expected payoff Compatible atible di dist stributio butions ns Min over column umns Contract act 𝑢 Affin ine e contract act 25
Key Lemma Suffices In an affine contract, setting 𝛽 0 = 0 increases expected payoff → Optimal linear contract has best worst-case expected payoff Compatible atible di dist stributio butions ns QED Min over column umns Contract act 𝑢 Affin ine e contract act Linear ar contract act 26
Result 2: Approximation 27
Approximation What fraction of the optimal payoff is achievable by a simple contract? • Result (informal): Linear contracts achieve constant approximation except in pathological settings with simultaneously: • many actions; • big spread of expected rewards; • big spread of costs 28
Example of Pathological Setting • Let 𝜗 → 0 (𝑆 1 , 𝑆 2 , 𝑆 3 , … ) = (1, 1 𝜗 , 1 𝜗 2 , … ) (𝑑 1 , 𝑑 2 , 𝑑 3 , … ) = (0, 1 𝜗 − 2 + 𝜗, 1 𝜗 2 − 3 + 2𝜗, … ) 29
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