Contract Theory: A New Frontier for AGT
Transcript of Contract Theory: A New Frontier for AGT
Paul DΓΌtting β London School of Economics
Inbal Talgam-Cohen β Technion
ACM ECβ19 Tutorial
June 2019
Contract Theory: A New Frontier for AGT
Part I: Classic Theory
Plan
β’ Part I (Inbal): Classic Theoryβ’ Model
β’ Optimal Contracts
β’ Key Results
β’ Break (5-10 mins)
β’ Part II (Paul): Modern Approachesβ’ Robustness
β’ Approximation
β’ 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?
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An Old Idea
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Les Mines de Bruoux, dug circa 1885
Purpose of Contracts
β’ Contracts align interests to enable exploiting gains from cooperation
β’ βWhat are the common wages of labour, depends everywhere upon the contract usually made between those two parties, whose interests are not the same.β [Adam Smith 1776]
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Classic Contract Theory
βModern economies are held together by innumerable contractsβ
[2016 Nobel Prize Announcement]
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Laureates Oliver Hart and Bengt HolmstrΓΆm
Classic Applications
β’ Employment contracts
β’ Venture capital (VC) investment contracts
β’ Insurance contracts
β’ Freelance (e.g. book) contracts
β’ Government procurement contracts
β’ β¦
β Contracts are indeed everywhere
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New Applications
Classic applications are moving online and/or increasing in complexity
β’ Crowdsourcing platforms
β’ Platforms for hiring freelancers
β’ Online marketing and affiliation
β’ Complex supply chains
β’ Pay-for-performance medicare
β Algorithmic approach becoming more relevant
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Basic Contract Setting [HolmstrΓΆmβ79]
β’ 2 players: principal and agent
β’ Familiar ingredients: private information and incentives
β’ Letβs see an exampleβ¦
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Example
β’ Website owner (principal) hires marketing agent to attract visitors
β’ Two defining features:1. Agentβs actions are hidden - βmoral hazardβ
2. Principal never charges (only pays) agent - βlimited liabilityβ
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Moral Hazard
βWell then, says I, whatβs the use of you learning to do right when itβs troublesome to do right and ainβt no trouble to do wrong, and the wages is just the same?β
Mark Twain, Adventures of Huckleberry Finn
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Limited Liability
Typical example: an entrepreneur and a VC
β’ The entrepreneur builds the company
β’ The VC diversifies the risks and has deep pockets
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Timing
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Principal offers agent a contract
(parties have symmetric info)
Agent accepts
(or refuses)
Agent takes costly,hiddenaction
Actionβsoutcome
rewards the principal
Principal pays agent according
to contract
Time
2. Connection to AGT
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Relation to Other Incentive Problems [Salanie]
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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 -
New Frontier
β’ Economics and computation β lively interaction over past 2 decades
β’ Especially true for mechanism design and signaling
Can we recreate the success stories of AGT in the context of contracts?
β’ Are insights from CS useful for contracts? Is contract theory useful for AGT applications? In Part II: A preliminary YES to both
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Already Building Momentum
β’ Pioneering works:β’ Combinatorial agency [Babaioff Feldman and Nisanβ12,β¦]β’ Contract complexity [Babaioff and Winterβ14,β¦]β’ Incentivizing exploration [Frazier Kempe Kleinberg and Kleinbergβ14]β’ Robustness [Carrollβ15,β¦]β’ Adaptive design [Ho Slivkins and Vaughanβ16,...]
β’ Recent works:β’ Delegated search [Kleinberg and Kleinbergβ18,β¦]β’ Information acquisition [Azar and Micaliβ18,β¦]β’ Succinct models [DΓΌtting Roughgarden and T.-C.β19b,β¦]
β’ ECβ19 papers: β’ [Kleinberg and Raghavanβ19, Lavi and Shamashβ19, DΓΌtting Roughgarden and T.-
C.β19a]
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The Algorithmic Lens
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β’ Offers a language to discuss complexityβ’ Has popularized the use of approximation guarantees when optimal
solutions are inappropriate β’ Puts forth alternatives to average-case / Bayesian analysis that
emphasize robust solutions to economic design problems
More on this in Part IIBut first, letβs cover the basics
3. Formal Model
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Contract Setting
β’ Parameters π,π
β’ Agent has actions π1, β¦ , ππβ’ with costs 0 = π1 β€ β― β€ ππ (can always choose action with 0 cost)
β’ Principal has rewards 0 β€ π1 β€ β― β€ ππ
β’ Action ππ induces distribution πΉπ over rewards (βtechnologyβ)β’ with expectation π πβ’ Assumption: π 1 β€ β― β€ π π
β’ Contract = vector of transfers Τ¦π‘ = π‘1, β¦ , π‘π β₯ 0
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Recall two defining features
Example
No visitor π1 = 0
General visitor π2 = 3
Targeted visitor π3 = 7
Both visitorsπ4 = 10
Low effort π1 = 0
0.72 0.18 0.08 0.02
Medium effort π2 = 1
0.12 0.48 0.08 0.32
High effortπ3 = 2
0 0.4 0 0.6
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Contract: π‘1 = 0 π‘2 = 1 π‘3 = 2 π‘4 = 5
π 3= 7.2
π 2= 5.2
π 1= 1.3
Expected Utilities
Fix action ππ.
Agent
β’ πΌ[utility] = expected transfer Οπβ[π]πΉπ,ππ‘π minus cost ππ
Principal
β’ πΌ[payoff] = expected reward π π minus expected transfer Οπ πΉπ,ππ‘π
Utilities sum up to π π β ππ, action ππβs expected welfare
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Contract setting:β’ π actions {ππ}, costs ππβ’ π rewards ππβ’ π Γ π matrix πΉ of distributions
with expectations π π
Payoff β payment/transfer
Example: Agentβs Perspective
No visitor π1 = 0
General visitor π2 = 3
Targeted visitor π3 = 7
Both visitorsπ4 = 10
Low effort π1 = 0
0.72 0.18 0.08 0.02
Medium effort π2 = 1
0.12 0.48 0.08 0.32
High effortπ3 = 2
0 0.4 0 0.6
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Contract: π‘1 = 0 π‘2 = 1 π‘3 = 2 π‘4 = 5
Expected transfers: (0.44, 2.24, 3.4) for (low, medium, high)
1.4
1.24
0.44
Example: Agentβs Perspective
No visitor π1 = 0
General visitor π2 = 3
Targeted visitor π3 = 7
Both visitorsπ4 = 10
Low effort π1 = 0
0.72 0.18 0.08 0.02
Medium effort π2 = 1
0.12 0.48 0.08 0.32
High effortπ3 = 2
0 0.4 0 0.6
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Contract: π‘1 = 0 π‘2 = 1 π‘3 = 2 π‘4 = 5
Expected transfers: (0.44, 2.24, 3.4) for (low, medium, high)
Example: Principalβs Perspective
No visitor π1 = 0
General visitor π2 = 3
Targeted visitor π3 = 7
Both visitorsπ4 = 10
Low effort π1 = 0
0.72 0.18 0.08 0.02
Medium effort π2 = 1
0.12 0.48 0.08 0.32
High effortπ3 = 2
0 0.4 0 0.6
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Contract: π‘1 = 0 π‘2 = 1 π‘3 = 2 π‘4 = 5
π 3 - expected transfer = 7.2 - 3.4 = 3.8
π 3= 7.2
π 2= 5.2
π 1= 1.3
A Remark on Risk Averseness
β’ Recall 2nd defining feature: agent has limited liability (Τ¦π‘ β₯ 0) [Innesβ90]
β’ Popular alternative to risk-aversenessβ’ Utility from transfer π‘π is π’(π‘π) where π’ strictly concave
β’ Both assumptions justify why the agent enters the contractβ’ Rather than βbuying the projectβ and being her own boss
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A Remark on Tie-Breaking
β’ Standard assumption: If the agent is indifferent among actions, he chooses the one that maximizes the principalβs expected payoff
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4. Computing Optimal Contracts
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Contract Design
Goal: Design contract that maximizes principalβs payoff
Optimization s.t. incentive compatibility (IC) constraints:
β’ Maximize πΌ[payoff] from action ππβ’ Subject to ππ maximizing πΌ[utility] for agent
Related Problems: Implementability of action ππ; min pay for action ππCan all be solved using LPs!
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First-Best Benchmark
β’ First-best = solution ignoring IC constraints
β’ What principal could extract if actions werenβt hiddenβ’ I.e., if could pick action and pay its cost
First-best = maxπ{π π β ππ}
β’ OPT β first-best due to IC constraints
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Implementability Problem
Given: Contract setting; action ππ
Determine: Is ππ implementable (exists contract Τ¦π‘ for which ππ is IC)
LP duality gives a simple characterization!
Proposition: Action ππ is implementable (up to tie-breaking) β
no convex combination of the other actions has same distribution over rewards at lower cost
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Implementability LP
ππ implementable βΊ LP feasible
π variables π‘π (transfers); π β 1 IC constraints
minimize 0
s.t.
π
πΉπ,ππ‘π β ππ β₯
π
πΉπβ²,ππ‘π β ππβ² βπβ² β π (IC)
π‘π β₯ 0 (LL)
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Agentβs expected utility from ππ
given contract Τ¦π‘
Dual* for Action ππ
Primal infeasible βΊβ feasible dual solution with objective > 0
π β 1 variables ππβ² (weights); π constraints
maximize ππ β
πβ²β π
ππβ²ππβ²
s.t.
πβ²β π
ππβ²πΉπβ²,π β€ πΉπ,π βπ β π
ππβ² β₯ 0;
πβ²β π
ππβ² = 1
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Convex combination of actions
Combined cost
Min Pay Problem
β’ Find minimum total transfer of a contract implementing action ππβ’ Same LP with updated objective:
minimize
π
πΉπ,ππ‘π
s.t.
π
πΉπ,ππ‘π β ππ β₯
π
πΉπβ²,ππ‘π β ππβ² βπβ² β π (IC)
π‘π β₯ 0 (LL)
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Optimal Contract Problem
Key observation:
β’ Can compute optimal contract by solving π LPs, one per action
Run-time per LP:
β’ Polynomial in π β 1 (constraints), π (variables)
Corollary:
β’ β optimal contract with β€ π β 1 nonzero transfers
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Criticism of LP-Based Approach
βMore normative than positiveβ:
1. Requires perfect knowledge of distribution matrix πΉ
2. What if polytime in π,π is too slow? β’ Recall example: π is exponential in number of visitor types to website
3. The contract that comes out of the LP may seem arbitrary
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Now
Part II
Part II
5. Structure of Optimal Contracts
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Let π > π
Optimal Contract for 2 Actions, 2 Rewards
βFailureβ reward π1
βSuccessβ reward π2 > π1
βShirkingβcost = 0
1 β π π
βWorkingβ cost = π > 0
1 β π π
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Optimal Contract for π = π = 2
Q: What does the optimal contract look like?
β’ Principal can always extract π 1 = 1 β π π1 + ππ2β Question interesting when optimal contract incentivizes work
β’ In this case:
first-best = π 2 β π = 1 β π π1 + ππ2 β π
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1 β π π
1 β π π
Optimal Contract for π = π = 2
β’ Notation: Contract pays π‘1 for failure, π‘2 for success
β’ IC constraint for working is:π‘1 1 β π + π‘2π β π β₯ π‘1 1 β π + π‘2π
βΊ πβ π π‘2 β π‘1 β₯ π (β)
β’ (β) binds at the optimal contract
β Optimal contract is: π‘1 = 0; π‘2 =π
πβπ
β Principal extracts: π 2 β ππ
πβπ
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Compare to first-best = π 2 β π
1 β π π
1 β π π
Optimal Contract for π = π = 2
Q: Structural properties of the optimal contract π‘1 = 0; π‘2 =π
πβπ?
β’ Monotonicity property = transfer increases w/ rewardβ’ Generalizes to any π as long as π = 2
β’ As π, π draw closer, harder to distinguish work from shirk, so π‘2 grows
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1 β π π
1 β π π
Optimal Contract for 2 Actions, π Rewards
β’ π = 2,π > 2
β’ Recall: Thereβs an optimal contract with π β 1 = 1 nonzero transfers
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πΉ1,1 πΉ1,2 πΉ1,3 πΉ1,4 πΉ1,5
πΉ2,1 πΉ2,2 πΉ2,3 πΉ2,4 πΉ2,5
Optimal Contract for π = 2,π > 2
Q: Which reward ππ gets the nonzero transfer π‘π in optimal contract?
β’ Binding IC constraint for working is π‘ππΉ2,π β π = π‘ππΉ1,π
β Optimal contract is π‘π =π
πΉ2,πβπΉ1,π; principal extracts π 2 β π
πΉ2,π
πΉ2,πβπΉ1,π
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Optimal Contract for π = 2,π > 2
β’ Principal extracts π 2 β π1
1βπΉ1,π
πΉ2,π
β To maximize over all π, choose πβ that minimizes πΉ1,π
πΉ2,π
β’πΉ1,π
πΉ2,πis called the likelihood ratio of actions π1, π2
β’ Numerator (denominator) is likelihood of shirk (work) given reward ππ
Takeaway: Optimal contract pays for reward with min likelihood ratio
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Optimal Contract for π = 2,π > 2
Recap
Q: Which reward ππ gets the nonzero transfer π‘π in optimal contract?
A: Pay for ππ with min likelihood ratio πΉ1,π
πΉ2,π
Statistical inference intuition (holds for general π):
Principal is inferring agentβs action from the reward
β Pays more for rewards from which can infer agent is working
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An Extreme Example
β’ Assume reward ππβ has nonzero probability π only if agent worksβ’ I.e. if ππβ occurs, βgives awayβ agentβs action
β’ Optimal contract has single nonzero transfer π‘πβ =π
π
β’ The good: Principal extracts first-best = π 2 β π
β’ The bad: Contract non-monotoneβ’ (Recall: monotone = transfer increases with reward)
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Example with π > 2
Optimal contract incentivizes action π3
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π1 = 1 π2 = 1.1 π3 = 4.9 π4 = 5 π5 = 5.1 π6 = 5.2
π1 = 0 3/8 3/8 2/8 0 0 0
π2 = 1 0 3/8 3/8 2/8 0 0
π3 = 2 0 0 3/8 3/8 2/8 0
π4 = 2.2 0 0 0 3/8 3/8 2/8
Contract: π‘1 = 0 π‘2 = 0 π‘3 β .15 π‘4 β 3.9 π‘5 β 2 π‘6 = 0
Recap
Role of rewards in the model is two-fold:
1. Represent surplus to be shared
2. Signal to principal the agentβs action
β’ The optimal contract is shaped by (2)
β’ Can be mismatched with (1)
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6. Results on Monotonicity and Informativeness
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Regularity Conditions [Mirrleesβ99]
Q: Natural conditions for optimal contract monotonicity?
For π = 2 actions, πth reward must have min likelihood ratio
Definition: A contract setting satisfies MLRP (monotone likelihood ratio property) if
β actions ππ , ππβ² , π < πβ²: πΉπ,π
πΉπβ²,πdecreasing in π
Intuition: The higher the reward, the more likely the higher-cost action
Note: MLRP implies FOSD (πΉπβ² first-order stochastically dominates πΉπ)
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Regularity Conditions [Mirrleesβ99]
β’ MLRP insufficient for monotonicity with π > 2 (recall example)
β’ Sufficient with βCDF Propertyβ or if actions have increasing welfareβ’ βCDFP really has no clear economic interpretation, and its validity is much
more doubtful than that of MLRPβ [Salanieβ05]
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π1 = 1 π1 = 1.1 π1 = 4.9 π1 = 5 π1 = 5.1 π1 = 5.2
π1 = 0 3/8 3/8 2/8 0 0 0
π2 = 1 0 3/8 3/8 2/8 0 0
π3 = 2 0 0 3/8 3/8 2/8 0
π4 = 2.2 0 0 0 3/8 3/8 2/8
Informativeness [Grossman-Hartβ83]
Fix π actions, π Γπ stochastic matrix Ξ
Consider 2 contract settings (πΉ, π), (πΉβ², πβ²) s.t. β action ππ:
β’ π πβ² = π π
β’ πΉπβ² obtained from πΉπ as follows: Draw reward-index πβ² by drawing π from πΉπ,
then drawing from πth column of Ξ
β Settings have same expected rewards but πΉβ² is a coarsening of πΉ
Proposition: Min pay for action ππ is higher in coarser setting
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Informativeness [Holmstromβ79]
Suppose principal can observe additional signals indicating action, e.g., a report from agentβs direct supervisor
Statistical model connection:β’ Action = underlying parameterβ’ Reward + report = observed data
Given reward, does report give further info on action? If so β use it!
Sufficient statistic theorem: The principal should condition transfers on a sufficient statistic for all available signals
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Recap
β’ Classic lit has made headway in making sense of optimal contracts
β’ E.g. through statistical inference connections
Limitations:
β’ Conditions like actions having increasing welfare are too strong
β’ βCoarseningβ relation is a very partial order on contract settings
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A Way Forward: Simple Contracts
β’ Linear contracts: Determined by parameter πΌ β [0,1]
β’ For reward ππ the principal pays the agent πΌππβ’ Generalization to affine: πΌππ + πΌ0
β’ Agentβs expected utility from action ππ is πΌπ π β ππβ’ Principalβs expected payoff is (1 β πΌ)π π
Notice: No dependence on details of distribution!
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7. Model Extensions & Summary
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Extensions
1. Continuum of actions: Studied in particular with 2 rewards [Mirrleesβ99]
2. Continuum of rewards: Functional analysis [Pageβ87]
3. Multiple agents: Teamwork, free-riding [Holmstromβ82]
4. Multiple principals: Agentβs success in a project benefits 2 principals [Bernheim-Whinstonβ86]
5. Multitasking: Actions can be substitutes or complements for agent [Holmstrom-Milgromβ91]
6. Adverse selection: Agents also have hidden types [E.g., Chiappori et al.β94]
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Dynamics
1. Multiple time periods, agent takes action at each period
β’ In this model [Holmstrom and Milgromβ87] give first robustnessexplanation for real-life contracts taking a simple, often linear form
2. Renegotiation after action is taken
β’ May prevent implementing costly actions
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Incomplete Contracts
Famous example from 1920s [Klein et al.β78]:
β’ Contract between GM and car-part manufacturer
β’ GM committed; manufacturer kept costs high (βheld upβ GM)
Problem caused by incomplete contract setting:
β’ Players can make specific investments
β’ Not all appear in contract due to transaction costs [Coaseβ37]
βLeads to underinvestment; here renegotiation can be socially useful
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Recap of Part I
Contracts incentivize someone to do something for us although we get the rewards and they incur the cost
β’ A model with familiar components of private info, incentives that βfit togetherβ in fundamentally different way than auctions
β’ Two defining features: (1) Hidden actions (2) Limited agent liability
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Recap of Part I: Main Results
β’ Implementability, Min Pay and Optimal Contract all solvable with LPs in poly(π,π) runtime if distributions known
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Optimal Contract 2 rewards π rewards
2 actions
Monotonicity&
Pay for reward with min likelihood ratio
Pay for reward with min likelihood ratio
π actions MonotonicityStrong assumptions
needed
Resources
1. Jean-Jacques Laffont and David Martimort, βThe Theory of Incentives: The Principal-Agent Modelβ, Princeton U. Press 2002
2. Patrick Bolton and Mathias Dewatripont, βContract Theoryβ, MIT Press 2005
3. Bernard Salanie, βThe Economics of Contracts: A Primerβ, MIT Press 2005 (see in particular Chapter 5)
*See Appendices of [DΓΌtting Roughgarden and T.-C.β19a] for more details on many of the basics covered in this tutorial
*For tutorial bibliography see tutorial website
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Questions?
β’ After the break: Algorithmic aspects
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