Contract Theory: A New Frontier for AGT Paul Dtting Google and LSE - PowerPoint PPT Presentation

Contract Theory: A New Frontier for AGT Paul Dütting – Google and LSE SAGT’20 Keynote Based on joint work with: Tim Roughgarden (Columbia) and Inbal Talgam-Cohen (Technion)

An Old Idea Les Mines de Bruoux, dug circa 1885 2

Contract Theory • Contracts align interests to enable exploiting gains from cooperation • “Modern economies are held together by innumerable contracts” [From 2016 Nobel Prize in Economics Announcement] Oliver Hart Bengt Holmström 3

Modern Applications Classic applications of contract theory are moving online Optimization / computational approaches becoming more relevant • Crowdsourcing platforms [Ho et al.’16] • Platforms for hiring specialists • Online marketing • Sponsored influencer content • Health and other insurances, e.g., pay-for-performance medicare [Bastani’18] • Our goal: Apply the algorithmic lens to contract theory 4

Why Algorithmic Approach Successfully applied to other branches of microeconomic theory with incentives and asymmetric information • Mechanism design (screening) [Myerson’81, …] • Agents have private types • Algorithmic view in [Cai et al.’12, …] • Signaling (Bayesian persuasion) [Kamenica-Gentzkow’11, …] • Principal has private information • Algorithmic view in [Dughmi’14, …] • In classic contracts [Holmström’79], no types, principal less informed 5

Example of Contract Setting • Website owner (principal) hires marketing agent to attract visitors • 2 defining features: principal doesn’t directly observe agent’s actions (“moral hazard”); never charges agent (only pays) 6

Overview of Results In the classic contract model of [Holmström’79]: 1. A new explanation for ubiquity of simple linear contracts • Building upon [Carroll’15] • Robust optimization approach • Characterization of pathological cases where simple far from optimal 2. Tractable algorithm for finding nearly-optimal contract (up to ! ) • Utilizing natural structure of contract setting • Introducing relaxed incentive-compatibility to circumvent hardness 7

Additional Related Work • Pioneering work in a different model [Babaioff et al.’12, Babaioff- Winter’14] • Delegation / no money models [Kleinberg-Kleinberg’18, Kleinberg- Raghavan’18] 8

Model & Notation 9

Contract Setting • A principal and an agent • Actions ! " , … , ! % with costs 0 = ( " ≤ ⋯ ≤ ( % for agent • Rewards 0 = + " ≤ ⋯ ≤ + , for principal • Action ! - induces distribution . - over rewards • Expectations / " ≤ ⋯ ≤ / % • A contract is a vector of transfers 0 = 0 " , … , 0 , ≥ 0 • [Recall 2 defining features] 10

Example Contract: 1 " = 0 1 % = 1 1 ' = 2 1 ) = 5 No visitor General visitor Targeted visitor Both visitors ! " = 0 ! % = 3 ! ' = 7 ! ) = 10 Low effort 0.02 0.72 0.18 0.08 + " = 0 Medium effort 0.32 0.12 0.48 0.08 + % = 1 High effort 0.6 0 0.4 0 + ' = 2 11

Contract Design Problem An optimization problem with incentive compatibility (IC) constraints: • Maximize principal’s expected payoff from action ! " • Subject to action ! " maximizing expected utility for agent • # [payoff] = expected reward $ " minus expected payment ∑ & ' ",& ) & • # [utility] = expected payment ∑ & ' ",& ) & minus cost * " [All expectations over distributions {' " } mapping actions to rewards] 12

Example: Agent’s Perspective Contract: 1 " = 0 1 % = 1 1 ' = 2 1 ) = 5 No visitor General visitor Targeted visitor Both visitors ! " = 0 ! % = 3 ! ' = 7 ! ) = 10 Low effort 0.02 0.72 0.18 0.08 + " = 0 Medium effort 0.32 0.12 0.48 0.08 + % = 1 High effort 0.6 0 0.4 0 + ' = 2 Expected payments: (0.44, 2.24, 3.4) for (low, medium, high) 13

Example: Principal’s Perspective Contract: 1 " = 0 1 % = 1 1 ' = 2 1 ) = 5 No visitor General visitor Targeted visitor Both visitors ! " = 0 ! % = 3 ! ' = 7 ! ) = 10 Low effort 3 " = 1.3 0.02 0.72 0.18 0.08 + " = 0 Medium effort 3 % = 5.2 0.32 0.12 0.48 0.08 + % = 1 High effort 3 ' = 7.2 0.6 0 0.4 0 + ' = 2 3 ' - expected payment = 7.2 - 3.4 = 3.8 14

Naïve Approach to Optimal Contracts • Observation: Can compute optimal contract (maximizing principal’s expected payoff) by solving one LP per action ! " minimize ( * ",) , ) ) * " 3 ,) , ) − 1 " 3 ∀5 6 ≠ 5 (IC) s.t. ( * ",) , ) − 1 " ≥ ( ) ) , ) ≥ 0 ∀= • Caveats: Requires perfect knowledge of distributions; Running time 15

Representation Size of a Contract Setting • For 2 visitor types (general & targeted), ! = 4 • For $ types, ! = 2 & • Alternative succinct representation: for every visitor type ' ∈ [$] , • reward + , • independent probability ℱ .,, to visit site given action 0 . • A contract setting is succinct if it has a succinct representation • i.e., has additive rewards and product distributions 16

Our Example is Succinct Additive No visitor General visitor Targeted visitor Both visitors ! " = 0 ! % = 3 ! ' = 7 ! ) = 10 Low effort 0.02 0.72 0.18 0.08 + " = 0 Medium effort Product 0.32 0.12 0.48 0.08 + % = 1 High effort 0.6 0 0.4 0 + ' = 2 17

Our Example is Succinct Additive No visitor General visitor Targeted visitor Both visitors ! " = 0 ! % = 3 ! ' = 7 ! ) = 10 Low effort 0.02 0.72 0.18 0.08 Main Result 1 + " = 0 Medium effort Product 0.32 0.12 0.48 0.08 + % = 1 Robust optimization approach to simple linear contracts High effort 0.6 0 0.4 0 + ' = 2 18 18

Linear Contracts • Determined by parameter ! ∈ [0,1] • For reward ( ) the principal pays the agent !( ) • Generalization to affine: !( ) + ! + • Agent’s expected utility from action , - is !. - − 0 - Expected welfare pie • Principal’s expected payoff is (1 − !). - . - − 0 - Notice: No dependence on details of distribution! 19

Max-Min Approach • “It is probably the great robustness of linear rules… that accounts for their popularity. That point is not made as effectively as we would like by our model” [Milgrom-Holmström’87] • Recent breakthrough: First sense in which linear contracts are max- min optimal [Carroll’15] 20

Our Robustness Result Theorem: For every contract setting with known expected rewards, a linear contract maximizes the principal’s expected payoff in the worst-case over compatible distributions. • Robustness to distribution details, only 1 st moments known [Scarf’58] • Used in auctions [Azar et al.’13, Bandi-Bertsimas’14, Carroll’17, Gravin-Lu’18] • In [Carroll’15] robustness is to unknown technologies of the agent 21

Example: Known Expected Rewards No visitor General visitor Targeted visitor Both visitors ! " = 0 ! % = 3 ! ' = 7 ! ) = 10 Low effort 1 " = 1.3 0.02 0.72 0.18 0.08 + " = 0 Medium effort 1 % = 5.2 0.32 0.12 0.48 0.08 + % = 1 High effort 1 ' = 7.2 0.6 0 0.4 0 + ' = 2 22

Example: Known Expected Rewards No visitor General visitor Targeted visitor Both visitors ! " = 0 ! % = 3 ! ' = 7 ! ) = 10 Low effort . " = 1.3 ? ? ? ? + " = 0 Medium effort . % = 5.2 ? ? ? ? + % = 1 High effort . ' = 7.2 ? ? ? ? + ' = 2 23

Example: Known Expected Rewards Intuition / Take-Away Example: Known Expected Rewards Intuition / Take-Away Example: Known Expected Rewards • If you don’t know enough about the setting to make contractual • If you don’t know enough about the setting to make contractual transfers dependent on anything but the actions’ expected rewards, transfers dependent on anything but the actions’ expected rewards, No visitor No visitor No visitor General visitor General visitor General visitor Targeted visitor Targeted visitor Targeted visitor Both visitors Both visitors Both visitors • then optimize transfers given the actions’ expected rewards. • then optimize transfers given the actions’ expected rewards. , ! , " = 0 # = 0 # = 0 ! % = 3 , 3 = 3 , 3 = 3 ! ' = 7 , 5 = 7 , 5 = 7 , , ! 7 = 10 ) = 10 7 = 10 Low effort Low effort Low effort 1 " = 1.3 0 # = 0 # = 0.02 0.02 ? 0.72 0.72 ? 0.18 0.18 ? 0.08 0.08 ? = 1.3 = 1.3 + " = 0 ) # = 0 ) # = 0 Medium effort Medium effort Medium effort 1 % = 5.2 0 3 = 0 3 = 0.32 0.32 ? 0.12 0.12 ? 0.48 0.48 ? 0.08 0.08 ? = 5.2 = 5.2 ) 3 = 1 + % = 1 ) 3 = 1 High effort High effort High effort 1 ' = 7.2 0 5 = 0 5 = 0.6 0.6 ? = 7.2 = 7.2 0 0 ? 0.4 0.4 ? 0 0 ? ) 5 = 2 ) 5 = 2 + ' = 2 23 22 24 24 24

Example: Known Expected Rewards No visitor General visitor Targeted visitor Both visitors ! " = 0 ! % = 3 ! ' = 7 ! ) = 10 Low effort . " = 1.3 ? ? ? ? The Geometry of Linear Contracts + " = 0 Medium effort . % = 5.2 ? ? ? ? + % = 1 High effort . ' = 7.2 ? ? ? ? + ' = 2 25 25

Intuition / Take-Away Plot of Agent’s Utility • If you don’t know enough about the setting to make contractual Agent’s expected utility `0 . − ) . as a function of ` for every action " . : transfers dependent on anything but the actions’ expected rewards, • then optimize transfers given the actions’ expected rewards. in indif ifference poin ints The Geometry of Linear Contracts 25 26 26