Contract Theory: A New Frontier for AGT Part I: Classic Theory Paul - PowerPoint PPT Presentation

Contract Theory: A New Frontier for AGT Part I: Classic Theory Paul Dütting – London School of Economics Inbal Talgam-Cohen – Technion ACM EC’19 Tutorial June 2019

Plan • Part I (Inbal): Classic Theory • Model • Optimal Contracts • Key Results • Break (5-10 mins) • Part II (Paul): Modern Approaches • Robustness • Approximation • Computational Complexity *We thank Tim Roughgarden for feedback on an early version and Gabriel Carroll for helpful conversations; any mistakes are our own

1. What is a Contract? 3

An Old Idea Les Mines de Bruoux, dug circa 1885 4

Purpose of Contracts • Contracts align interests to enable exploiting gains from cooperation • “What are the common wages of labour, depends everywhere upon the contract usually made between those two parties, whose interests are not the same .” [Adam Smith 1776] 5

Classic Contract Theory “ Modern economies are held together by innumerable contracts ” [2016 Nobel Prize Announcement] Laureates Oliver Hart and Bengt Holmström 6

Classic Applications • Employment contracts • Venture capital (VC) investment contracts • Insurance contracts • Freelance (e.g. book) contracts • Government procurement contracts • … → Contracts are indeed everywhere 7

New 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 → Algorithmic approach becoming more relevant 8

Basic Contract Setting [Holmström’79] • 2 players: principal and agent • Familiar ingredients: private information and incentives • Let’s see an example… 9

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 ” 10

Moral Hazard “ Well then, says I, what ’ s the use of you learning to do right when it ’ s troublesome to do right and ain ’ t no trouble to do wrong, and the wages is just the same? ” Mark Twain, Adventures of Huckleberry Finn 11

Limited Liability Typical example: an entrepreneur and a VC • The entrepreneur builds the company • The VC diversifies the risks and has deep pockets 12

Timing Time Action’s Principal offers agent Agent takes Principal Agent outcome pays agent a contract costly, accepts (parties have rewards the according hidden (or refuses) symmetric info) principal to contract action 13

2. Connection to AGT 14

Relation to Other Incentive Problems [Salanie] Uninformed player Informed player has the initiative has the initiative Private Mechanism Signaling information is design (screening) (persuasion) hidden type Private information is Contract design - hidden action 15

New Frontier • Economics and computation – lively interaction over past 2 decades • Especially true for mechanism design and signaling Can we recreate the success stories of AGT in the context of contracts? • Are insights from CS useful for contracts? Is contract theory useful for AGT applications? In Part II: A preliminary YES to both 16

Already Building Momentum • 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,...] • 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] 17

The Algorithmic Lens • Offers a language to discuss complexity • Has popularized the use of approximation guarantees when optimal solutions are inappropriate • Puts forth alternatives to average-case / Bayesian analysis that emphasize robust solutions to economic design problems More on this in Part II But first, let’s cover the basics 18

3. Formal Model 19

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 20

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 21

Contract setting: • 𝑜 actions {𝑏 𝑗 } , costs 𝑑 𝑗 • Expected Utilities 𝑛 rewards 𝑠 𝑘 • 𝑜 × 𝑛 matrix 𝐺 of distributions with expectations 𝑆 𝑗 Fix action 𝑏 𝑗 . Agent • 𝔽 [utility] = expected transfer σ 𝑘∈[𝑛] 𝐺 𝑗,𝑘 𝑢 𝑘 minus cost 𝑑 𝑗 Payoff ≠ payment/transfer Principal • 𝔽 [payoff] = expected reward 𝑆 𝑗 minus expected transfer σ 𝑘 𝐺 𝑗,𝑘 𝑢 𝑘 Utilities sum up to 𝑆 𝑗 − 𝑑 𝑗 , action 𝑏 𝑗 ’s expected welfare 22

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) 23

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 0.72 0.18 0.08 0.02 𝑑 1 = 0 Medium effort 0.12 0.48 0.08 0.32 𝑑 2 = 1 High effort 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) 24

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 25

A Remark on Risk Averseness • Recall 2nd defining feature: agent has limited liability ( Ԧ 𝑢 ≥ 0 ) [Innes’90] • Popular alternative to risk-averseness • Utility from transfer 𝑢 𝑘 is 𝑣(𝑢 𝑘 ) where 𝑣 strictly concave • Both assumptions justify why the agent enters the contract • Rather than “ buying the project ” and being her own boss 26

A Remark on Tie-Breaking • Standard assumption: If the agent is indifferent among actions, he chooses the one that maximizes the principal’s expected payoff 27

4. Computing Optimal Contracts 28

Contract Design Goal : Design contract that maximizes principal’s payoff Optimization s.t. incentive compatibility (IC) constraints: • Maximize 𝔽 [payoff] from action 𝑏 𝑗 • Subject to 𝑏 𝑗 maximizing 𝔽 [utility] for agent Related Problems: Implementability of action 𝑏 𝑗 ; min pay for action 𝑏 𝑗 Can all be solved using LPs! 29

First-Best Benchmark • First-best = solution ignoring IC constraints • What principal could extract if actions weren’t hidden • I.e., if could pick action and pay its cost First-best = max 𝑗 {𝑆 𝑗 − 𝑑 𝑗 } • OPT ≠ first-best due to IC constraints 30

Implementability Problem Given: Contract setting; action 𝑏 𝑗 Determine: Is 𝑏 𝑗 implementable (exists contract Ԧ 𝑢 for which 𝑏 𝑗 is IC) LP duality gives a simple characterization! Proposition: Action 𝑏 𝑗 is implementable (up to tie-breaking) ⇔ no convex combination of the other actions has same distribution over rewards at lower cost 31

Implementability LP 𝑏 𝑗 implementable ⟺ LP feasible 𝑛 variables 𝑢 𝑘 (transfers); 𝑜 − 1 IC constraints minimize 0 𝐺 𝑗 ′ ,𝑘 𝑢 𝑘 − 𝑑 𝑗 ′ ∀𝑗 ′ ≠ 𝑗 (IC) s.t. ෍ 𝐺 𝑗,𝑘 𝑢 𝑘 − 𝑑 𝑗 ≥ ෍ 𝑘 𝑘 Agent’s expected 𝑢 𝑘 ≥ 0 (LL) utility from 𝑏 𝑗 given contract Ԧ 𝑢 32