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 Dtting London School of Economics Inbal Talgam-Cohen Technion ACM EC19 Tutorial June 2019 Plan Part I (Inbal): Classic Theory Model Optimal Contracts
Paul Dütting – London School of Economics Inbal Talgam-Cohen – Technion ACM EC’19 Tutorial June 2019
Part I: Classic Theory
*We thank Tim Roughgarden for feedback on an early version and Gabriel Carroll for helpful conversations; any mistakes are our own
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Les Mines de Bruoux, dug circa 1885
the contract usually made between those two parties, whose interests are not the same.” [Adam Smith 1776]
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“Modern economies are held together by innumerable contracts” [2016 Nobel Prize Announcement]
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Laureates Oliver Hart and Bengt Holmström
→ Contracts are indeed everywhere
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Classic applications are moving online and/or increasing in complexity
→ Algorithmic approach becoming more relevant
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“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
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Typical example: an entrepreneur and a VC
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Principal offers agent a contract (parties have symmetric info) Agent accepts (or refuses) Agent takes costly, hidden action Action’s
rewards the principal Principal pays agent according to contract Time
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Uninformed player has the initiative Informed player has the initiative Private information is hidden type Mechanism design (screening) Signaling (persuasion) Private information is hidden action Contract design
AGT applications? In Part II: A preliminary YES to both
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C.’19a]
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solutions are inappropriate
emphasize robust solutions to economic design problems More on this in Part II But first, let’s cover the basics
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1 ≤ ⋯ ≤ 𝑠 𝑛
𝑢 = 𝑢1, … , 𝑢𝑛 ≥ 0
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Recall two defining features
No visitor 𝑠
1 = 0
General visitor 𝑠
2 = 3
Targeted visitor 𝑠
3 = 7
Both visitors 𝑠
4 = 10
Low effort 𝑑1 = 0 0.72 0.18 0.08 0.02 Medium effort 𝑑2 = 1 0.12 0.48 0.08 0.32 High effort 𝑑3 = 2 0.4 0.6
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Contract: 𝑢1 = 0 𝑢2 = 1 𝑢3 = 2 𝑢4 = 5
𝑆3= 7. 7.2 𝑆2= 5 5.2 𝑆1= 1. 1.3
Fix action 𝑏𝑗. Agent
Principal
Utilities sum up to 𝑆𝑗 − 𝑑𝑗, action 𝑏𝑗’s expected welfare
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Contract setting:
𝑘
with expectations 𝑆𝑗
Payoff ≠ payment/transfer
No visitor 𝑠
1 = 0
General visitor 𝑠
2 = 3
Targeted visitor 𝑠
3 = 7
Both visitors 𝑠
4 = 10
Low effort 𝑑1 = 0 0.72 0.18 0.08 0.02 Medium effort 𝑑2 = 1 0.12 0.48 0.08 0.32 High effort 𝑑3 = 2 0.4 0.6
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Contract: 𝑢1 = 0 𝑢2 = 1 𝑢3 = 2 𝑢4 = 5
Exp xpect ected ed tran ansf sfers rs: (0. 0.44, , 2.24, 4, 3.4) for (low, medi dium, m, high) h) 1. 1.4 1. 1.24 0. 0.44
No visitor 𝑠
1 = 0
General visitor 𝑠
2 = 3
Targeted visitor 𝑠
3 = 7
Both visitors 𝑠
4 = 10
Low effort 𝑑1 = 0 0.72 0.18 0.08 0.02 Medium effort 𝑑2 = 1 0.12 0.48 0.08 0.32 High effort 𝑑3 = 2 0.4 0.6
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Contract: 𝑢1 = 0 𝑢2 = 1 𝑢3 = 2 𝑢4 = 5
Exp xpect ected ed tran ansf sfers rs: (0. 0.44, , 2.24, 4, 3.4) for (low, medi dium, m, high) h)
No visitor 𝑠
1 = 0
General visitor 𝑠
2 = 3
Targeted visitor 𝑠
3 = 7
Both visitors 𝑠
4 = 10
Low effort 𝑑1 = 0 0.72 0.18 0.08 0.02 Medium effort 𝑑2 = 1 0.12 0.48 0.08 0.32 High effort 𝑑3 = 2 0.4 0.6
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Contract: 𝑢1 = 0 𝑢2 = 1 𝑢3 = 2 𝑢4 = 5
𝑆3 - exp xpecte cted d tran ansf sfer er = 7 7.2 - 3.4 = 3 3.8 𝑆3= 7. 7.2 𝑆2= 5 5.2 𝑆1= 1. 1.3
𝑢 ≥ 0) [Innes’90]
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chooses the one that maximizes the principal’s expected payoff
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Goal: Design contract that maximizes principal’s payoff Optimization s.t. incentive compatibility (IC) constraints:
Related Problems: Implementability of action 𝑏𝑗; min pay for action 𝑏𝑗 Can all be solved using LPs!
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First-best = max
𝑗 {𝑆𝑗 − 𝑑𝑗}
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Given: Contract setting; action 𝑏𝑗 Determine: Is 𝑏𝑗 implementable (exists contract Ԧ 𝑢 for which 𝑏𝑗 is IC) LP duality gives a simple characterization! Proposition: Action 𝑏𝑗 is implementable (up to tie-breaking) ⇔ no convex combination of the other actions has same distribution over rewards at lower cost
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𝑏𝑗 implementable ⟺ LP feasible 𝑛 variables 𝑢𝑘 (transfers); 𝑜 − 1 IC constraints minimize 0 s.t.
𝑘
𝐺𝑗,𝑘𝑢𝑘 − 𝑑𝑗 ≥
𝑘
𝐺𝑗′,𝑘𝑢𝑘 − 𝑑𝑗′ ∀𝑗′ ≠ 𝑗 (IC) 𝑢𝑘 ≥ 0 (LL)
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Agent’s expected utility from 𝑏𝑗 given contract Ԧ 𝑢
Primal infeasible ⟺ ∃ feasible dual solution with objective > 0 𝑜 − 1 variables 𝜇𝑗′ (weights); 𝑛 constraints maximize 𝑑𝑗 −
𝑗′≠𝑗
𝜇𝑗′𝑑𝑗′ s.t.
𝑗′≠𝑗
𝜇𝑗′𝐺𝑗′,𝑘 ≤ 𝐺
𝑗,𝑘 ∀𝑘 ∈ 𝑛
𝜇𝑗′ ≥ 0;
𝑗′≠𝑗
𝜇𝑗′ = 1
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Convex combination of actions Combined cost
minimize
𝑘
𝐺𝑗,𝑘𝑢𝑘 s.t.
𝑘
𝐺𝑗,𝑘𝑢𝑘 − 𝑑𝑗 ≥
𝑘
𝐺𝑗′,𝑘𝑢𝑘 − 𝑑𝑗′ ∀𝑗′ ≠ 𝑗 (IC) 𝑢𝑘 ≥ 0 (LL)
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Key observation:
Run-time per LP:
Corollary:
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“More normative than positive”:
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Now Part II Part II
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Let 𝜌 > 𝑞
“Failure” reward 𝑠
1
“Success” reward 𝑠
2 > 𝑠 1
“Shirking” cost = 0 1 − 𝑞 𝑞 “Working” cost = 𝑑 > 0 1 − 𝜌 𝜌
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Q: What does the optimal contract look like?
1 + 𝑞𝑠 2
→ Question interesting when optimal contract incentivizes work
first-best = 𝑆2 − 𝑑 = 1 − 𝜌 𝑠
1 + 𝜌𝑠 2 − 𝑑
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1 − 𝑞 𝑞 1 − 𝜌 𝜌
𝑢1 1 − 𝜌 + 𝑢2𝜌 − 𝑑 ≥ 𝑢1 1 − 𝑞 + 𝑢2𝑞 ⟺ 𝜌 − 𝑞 𝑢2 − 𝑢1 ≥ 𝑑 (∗)
→ Optimal contract is: 𝑢1 = 0; 𝑢2 =
𝑑 𝜌−𝑞
→ Principal extracts: 𝑆2 − 𝑑
𝜌 𝜌−𝑞
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Compare to first-best = 𝑆2 − 𝑑
1 − 𝑞 𝑞 1 − 𝜌 𝜌
Q: Structural properties of the optimal contract 𝑢1 = 0; 𝑢2 =
𝑑 𝜌−𝑞?
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1 − 𝑞 𝑞 1 − 𝜌 𝜌
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𝐺
1,1
𝐺
1,2
𝐺
1,3
𝐺
1,4
𝐺
1,5
𝐺2,1 𝐺2,2 𝐺2,3 𝐺2,4 𝐺2,5
Q: Which reward 𝑠
𝑘 gets the nonzero transfer 𝑢𝑘 in optimal contract?
1,𝑘
→ Optimal contract is 𝑢𝑘 =
𝑑 𝐺2,𝑘−𝐺1,𝑘; principal extracts 𝑆2 − 𝑑 𝐺2,𝑘 𝐺2,𝑘−𝐺1,𝑘
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1 1−
𝐺1,𝑘 𝐺2,𝑘
→ To maximize over all 𝑘, choose 𝑘∗ that minimizes
𝐺1,𝑘 𝐺2,𝑘
𝐺2,𝑘 is called the likelihood ratio of actions 𝑏1, 𝑏2
𝑘
Takeaway: Optimal contract pays for reward with min likelihood ratio
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Recap Q: Which reward 𝑠
𝑘 gets the nonzero transfer 𝑢𝑘 in optimal contract?
A: Pay for 𝑠
𝑘 with min likelihood ratio 𝐺1,𝑘 𝐺2,𝑘
Statistical inference intuition (holds for general 𝑜): Principal is inferring agent’s action from the reward → Pays more for rewards from which can infer agent is working
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𝑑 𝜗
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Optimal contract incentivizes action 𝑏3
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𝑠
1 = 1
𝑠
2 = 1.1 𝑠 3 = 4.9
𝑠
4 = 5
𝑠
5 = 5.1 𝑠 6 = 5.2
𝑑1 = 0 3/8 3/8 2/8 𝑑2 = 1 3/8 3/8 2/8 𝑑3 = 2 3/8 3/8 2/8 𝑑4 = 2.2 3/8 3/8 2/8 Contract: 𝑢1 = 0 𝑢2 = 0 𝑢3 ≈ .15 𝑢4 ≈ 3.9 𝑢5 ≈ 2 𝑢6 = 0
Role of rewards in the model is two-fold:
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Q: Natural conditions for optimal contract monotonicity? For 𝑜 = 2 actions, 𝑛th reward must have min likelihood ratio Definition: A contract setting satisfies MLRP (monotone likelihood ratio property) if ∀ actions 𝑏𝑗, 𝑏𝑗′, 𝑗 < 𝑗′:
𝐺𝑗,𝑘 𝐺𝑗′,𝑘 decreasing in 𝑘
Intuition: The higher the reward, the more likely the higher-cost action Note: MLRP implies FOSD (𝐺𝑗′ first-order stochastically dominates 𝐺𝑗)
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more doubtful than that of MLRP” [Salanie’05]
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𝑠
1 = 1
𝑠
1 = 1.1 𝑠 1 = 4.9
𝑠
1 = 5
𝑠
1 = 5.1 𝑠 1 = 5.2
𝑑1 = 0 3/8 3/8 2/8 𝑑2 = 1 3/8 3/8 2/8 𝑑3 = 2 3/8 3/8 2/8 𝑑4 = 2.2 3/8 3/8 2/8
Fix 𝑜 actions, 𝑛 × 𝑛 stochastic matrix Π Consider 2 contract settings (𝐺, 𝑠), (𝐺′, 𝑠′) s.t. ∀ action 𝑏𝑗:
′ = 𝑆𝑗
′ obtained from 𝐺𝑗 as follows: Draw reward-index 𝑘′ by drawing 𝑘 from 𝐺𝑗,
then drawing from 𝑘th column of Π → Settings have same expected rewards but 𝐺′ is a coarsening of 𝐺 Proposition: Min pay for action 𝑏𝑗 is higher in coarser setting
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Suppose principal can observe additional signals indicating action, e.g., a report from agent’s direct supervisor Statistical model connection:
Given reward, does report give further info on action? If so – use it! Sufficient statistic theorem: The principal should condition transfers on a sufficient statistic for all available signals
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Limitations:
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𝑘 the principal pays the agent 𝛽𝑠 𝑘
𝑘 + 𝛽0
Notice: No dependence on details of distribution!
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1. Continuum of actions: Studied in particular with 2 rewards [Mirrlees’99] 2. Continuum of rewards: Functional analysis [Page’87] 3. Multiple agents: Teamwork, free-riding [Holmstrom’82] 4. Multiple principals: Agent’s success in a project benefits 2 principals [Bernheim-Whinston’86] 5. Multitasking: Actions can be substitutes or complements for agent [Holmstrom-Milgrom’91] 6. Adverse selection: Agents also have hidden types [E.g., Chiappori et al.’94]
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explanation for real-life contracts taking a simple, often linear form
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Famous example from 1920s [Klein et al.’78]:
Problem caused by incomplete contract setting:
→Leads to underinvestment; here renegotiation can be socially useful
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Contracts incentivize someone to do something for us although we get the rewards and they incur the cost
together” in fundamentally different way than auctions
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in poly(𝑜, 𝑛) runtime if distributions known
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Optimal Contract 2 rewards 𝑛 rewards 2 actions Monotonicity & Pay for reward with min likelihood ratio Pay for reward with min likelihood ratio 𝑜 actions Monotonicity Strong assumptions needed
Incentives: The Principal-Agent Model”, Princeton U. Press 2002
Press 2005
2005 (see in particular Chapter 5) *See Appendices of [Dütting Roughgarden and T.-C.’19a] for more details on many of the basics covered in this tutorial *For tutorial bibliography see tutorial website
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