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

2

Les Mines de Bruoux, dug circa 1885

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]

3

Oliver Hart Bengt Holmström

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

No visitor !

" = 0

General visitor !% = 3 Targeted visitor !' = 7 Both visitors !

) = 10

Low effort +" = 0 0.72 0.18 0.08 0.02 Medium effort +% = 1 0.12 0.48 0.08 0.32 High effort +' = 2 0.4 0.6

11

Contract: 1" = 0 1% = 1 1' = 2 1) = 5

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

No visitor !

" = 0

General visitor !% = 3 Targeted visitor !' = 7 Both visitors !

) = 10

Low effort +" = 0 0.72 0.18 0.08 0.02 Medium effort +% = 1 0.12 0.48 0.08 0.32 High effort +' = 2 0.4 0.6

13

Contract: 1" = 0 1% = 1 1' = 2 1) = 5

Expected payments: (0.44, 2.24, 3.4) for (low, medium, high)

Example: Principal’s Perspective

No visitor !

" = 0

General visitor !% = 3 Targeted visitor !' = 7 Both visitors !

) = 10

Low effort +" = 0 0.72 0.18 0.08 0.02 Medium effort +% = 1 0.12 0.48 0.08 0.32 High effort +' = 2 0.4 0.6

14

Contract: 1" = 0 1% = 1 1' = 2 1) = 5

3' - expected payment = 7.2 - 3.4 = 3.8 3'= 7.2 3%= 5.2 3"= 1.3

Naïve Approach to Optimal Contracts

• Observation: Can compute optimal contract (maximizing principal’s

expected payoff) by solving one LP per action !" minimize (

)

*

",),)

s.t. (

)

*

",),) − 1" ≥ ( )

*"3,),) − 1"3 ∀56 ≠ 5 (IC) ,) ≥ 0 ∀=

• Caveats: Requires perfect knowledge of distributions; Running time

15

Representation Size of a Contract Setting

16

• 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

Our Example is Succinct

No visitor !

" = 0

General visitor !% = 3 Targeted visitor !' = 7 Both visitors !

) = 10

Low effort +" = 0 0.72 0.18 0.08 0.02 Medium effort +% = 1 0.12 0.48 0.08 0.32 High effort +' = 2 0.4 0.6

17

Additive Product

Our Example is Succinct

No visitor !

" = 0

General visitor !% = 3 Targeted visitor !' = 7 Both visitors !

) = 10

Low effort +" = 0 0.72 0.18 0.08 0.02 Medium effort +% = 1 0.12 0.48 0.08 0.32 High effort +' = 2 0.4 0.6

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Additive Product

Main Result 1

Robust optimization approach to simple linear contracts

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-
• Principal’s expected payoff is (1 − !).-

Notice: No dependence on details of distribution!

19

Expected welfare pie .- − 0-

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

• ver compatible distributions.
• Robustness to distribution details, only 1st 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 !

" = 0

General visitor !% = 3 Targeted visitor !' = 7 Both visitors !

) = 10

Low effort +" = 0 0.72 0.18 0.08 0.02 Medium effort +% = 1 0.12 0.48 0.08 0.32 High effort +' = 2 0.4 0.6

22

1'= 7.2 1%= 5.2 1"= 1.3

Example: Known Expected Rewards

No visitor !

" = 0

General visitor !% = 3 Targeted visitor !' = 7 Both visitors !

) = 10

Low effort +" = 0 ? ? ? ? Medium effort +% = 1 ? ? ? ? High effort +' = 2 ? ? ? ?

23

.'= 7.2 .%= 5.2 ."= 1.3

Example: Known Expected Rewards

No visitor !

" = 0

General visitor !% = 3 Targeted visitor !' = 7 Both visitors !

) = 10

Low effort +" = 0 0.72 0.18 0.08 0.02 Medium effort +% = 1 0.12 0.48 0.08 0.32 High effort +' = 2 0.4 0.6

24

1'= 7.2 1%= 5.2 1"= 1.3

Intuition / Take-Away

• If you don’t know enough about the setting to make contractual

transfers dependent on anything but the actions’ expected rewards,

• then optimize transfers given the actions’ expected rewards.

24

Example: Known Expected Rewards

No visitor ,

# = 0

General visitor ,3 = 3 Targeted visitor ,5 = 7 Both visitors ,

7 = 10

Low effort )# = 0 0.72 0.18 0.08 0.02 Medium effort )3 = 1 0.12 0.48 0.08 0.32 High effort )5 = 2 0.4 0.6

22

05= = 7.2 03= = 5.2 0#= = 1.3

Example: Known Expected Rewards

No visitor ,

# = 0

General visitor ,3 = 3 Targeted visitor ,5 = 7 Both visitors ,

7 = 10

Low effort )# = 0 ? ? ? ? Medium effort )3 = 1 ? ? ? ? High effort )5 = 2 ? ? ? ?

23

05= = 7.2 03= = 5.2 0#= = 1.3

Intuition / Take-Away

• If you don’t know enough about the setting to make contractual

transfers dependent on anything but the actions’ expected rewards,

• then optimize transfers given the actions’ expected rewards.

24

Example: Known Expected Rewards

No visitor !

" = 0

General visitor !% = 3 Targeted visitor !' = 7 Both visitors !

) = 10

Low effort +" = 0 ? ? ? ? Medium effort +% = 1 ? ? ? ? High effort +' = 2 ? ? ? ?

25

.'= 7.2 .%= 5.2 ."= 1.3

The Geometry of Linear Contracts

25

Intuition / Take-Away

• If you don’t know enough about the setting to make contractual

transfers dependent on anything but the actions’ expected rewards,

• then optimize transfers given the actions’ expected rewards.

26

The Geometry of Linear Contracts

25

Plot of Agent’s Utility

Agent’s expected utility `0. − ). as a function of ` for every action ".:

26

in indif ifference poin ints

Upper Envelope

• Observation: A linear contract with parameter ! implements the

action on the upper envelope at !

27

Upper Envelope

• Observation: A linear contract with parameter ! implements the

action on the upper envelope at !

28

Implemented Actions

• Observation: As ` increases, the implemented actions have

increasingly high welfare 0. − ).

28

Take Away

• Conclusion: Expectations !", … , !% completely determine upper

envelope and thus implemented actions

• Note: Everything holds for affine contracts too

29

Proof Sketch

30

Recall

Theorem: For every contract setting with known expected rewards, a linear contract maximizes the principal’s expected payoff in the worst-case

• ver compatible distributions.

31

Recall

Theorem: For every contract setting with known expected rewards, a linear contract maximizes the principal’s expected payoff in the worst-case

• ver compatible distributions.

32

Max-Min Visualization

• Fix a contract setting with known expected rewards

Co Contract ct Co Compatibl ble d distribu butions Pr Princi cipal’s ’s ex expec ected ed payof

• ff

Min Min o

• ver c

colu lumns Ma Max

• ver
• ver

rows ws

32

Linear Contracts are Robust

• What’s special about linear (and affine) contracts:

Linear/affine contract Compatible distributions Same expected payoff

33

Key Lemma

Lemma: For every contract ! there exist compatible distributions and an affine contract with "# ≥ 0 and better expected payoff Contract ! Compatible distributions Affine contract

34

Immediate Implication

à For every contract ! there exists an affine contract with "# ≥ 0 and better worst-case expected payoff Contract ! Compatible distributions Affine contract Min over columns

35

Additional Ingredient

Observation: In an affine contract, setting !" = 0 increases expected payoff

36

Proof of Theorem

à Optimal linear contract has best worst-case expected payoff QED Contract ! Compatible distributions Affine contract Min over columns Linear contract

37

Back to Key Lemma

Lemma: For every contract ! there exist compatible distributions and an affine contract with "# ≥ 0 and better expected payoff

• Consider non-affine contract ! that maps &

' ≤ ⋯ ≤ & * to !', … , !*

38

Key Lemma Proof Idea

• Compare contract ! to the affine contract defined by "#
• Let $%∗ be the action "# implements
• ∃ compatible distributions {)

%} s.t. ! implements $%∗ at higher agent

utility

39

Compatible Distributions

• For each ! ≠ !∗ define two-point distribution $

% on & ' and & ( s.t.

expected reward is )%

• For !∗ define two-point distribution $%∗ on &

' and & * or & * and & ( s.t.

expected reward is )%∗

Why Distributions Work

Observation: Consider arbitrary contract !. Consider line " between ($, !($)) and ($’, !($’)). If ) is distribution over $, $’ with expectation *. Then expected payment of ! is "(*).

*+ *+∗

à On {)

+} non-affine contract ! has same payment as "/ for i ≠ 2∗,

higher payment for 2∗ QED

Additional Results for Linear Contracts

• An alternative to max-min robustness: approximation
• Fraction of the optimal expected payoff achievable by best simple contract
• Result (informal): Linear contracts achieve good approximation except

in pathological settings with simultaneously:

• many actions;
• big spread among actions of expected rewards;
• big spread among actions of costs

42

Example of Pathological Setting

• Let ! → 0

(%&, %(, %), … ) = (1, 1 ! , 1 !( , … ) (.&, .(, .), … ) = (0, 1 ! − 2 + !, 1 !( − 3 + 2!, … )

43

Formally

Theorem: ! = worst-case ratio of optimal contract and best linear contract

• with " actions, ! = ";
• with ratio $ of highest to lowest $%, ! = Θ(log $);
• with ratio , of highest to lowest -%, ! = Θ(log ,)

Tight (even for best monotone contract)!

44

Main Result 2

Tractable algorithm for nearly-optimal contracts in succinct settings

45

Recall Example

46

• Visitor type 1 is “general”, visitor type 2 is “targeted”
• Rewards from each type: #$ = 3, #( = 7
• Probabilities of each type: 0.2,0.1 given ,$, 0.8,0.4 given ,(,

1,0.6 given ,0

Recall: Succinct Contract Settings

• ! types, " = 2% possible outcomes
• Given action &', each type ( appears independently wp ℱ',+
• Principal’s reward = sum of rewards ,+ for each appearing type (
• Normalized contract setting: -' ∈ [0,1] for every action &'
• Succinct representation is simpler and arguably more realistic

47

Goal

Use succinct structure to exponentially speed-up finding the optimal contract in comparison to the naïve LP-based method

48

Recall: Naïve LP-based Approach

Based on solving ! instances of the “MIN-PAY” problem:

• Given action "#, find optimal contract that implements "#

minimize )

*

+

#,*-*

s.t. )

*

+

#,*-* − 2# ≥ ) *

+#4,*-* − 2#4 ∀67 ≠ 6 (IC) There are polynomial in =, exponential in > many variables, but only ! constraints – Ellipsoid to the rescue?

49

The Dual

maximize '

()*(

+()(-( − -()) s.t. '

()*(

+() − 1 ≤

∑6)76 86)96),; 96,;

∀= ∈ [@] A separation oracle boils down to finding a type subset with minimum likelihood in the combination distribution ∑()*( +()B() relative to B

(

50

Computational Hardness

• Solving the separation oracle exactly is NP-hard
• In fact computing the optimal expected payoff in succinct contract

settings in time polynomial in ! turns out to be NP-hard A solution from algorithmic mechanism design: Slightly relax the IC constraints

51

!-IC

Definition: Given a contract ", action #$ is !-IC if (1 + !) ∑* +

$,*"* − .$ ≥ ∑* +$0,*"* − .$0 ∀23 ≠ 2

In normalized settings, the agent loses ≤ ! by choosing a !-IC action Idea: Agent is willing to sacrifice ! utility to please principal

• possible formalization of a bias empirically known to exist [Akerlof’82]
• [By !-IC contract we mean a contract " and !-IC action #$ that pleases

principal]

52

Theorem

• Let OPT be the expected payoff of the optimal (IC) contract.

Theorem: There is an Ellipsoid-based algorithm that given a succinct normalized contract setting with $ types and a parameter % > 0, returns a %-IC contract with expected payoff ≥ OPT in time polynomial in $.

• (Recall: Running time of naïve method is exponential in $)

53

Ellipsoid-based Algorithm

• Strengthened dual:

maximize '

()*(

+()(-( − -()) s.t. 1 + 5 '

()*(

+() − 1 ≤

∑8)98 :8);8),= ;8,=

∀? ∈ [B]

• Run Ellipsoid calling an FPTAS for the separation oracle
• FPTAS runs in time polynomial in D and exponential in E

54

Additional Results

• Is the !-IC notion useful more widely? E.g., can !-IC linear contracts

perform better?

• The answer is yes – we show good approximation by !-IC linear

contracts even in pathological settings

• The computational landscape related to (approximately) optimal

contracts turns out to be quite rich

• More hardness + positive results in our paper

55

Summary

56

Recap 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

57

Discussion

• Contract theory is becoming more relevant for algorithm designers
• How relevant will the theory of algorithms be for contract designers?
• Caveat: Usual gap between model and applications
• More pronounced than in mechanism design and/or persuasion?
• Algorithmic approach can provide qualitative insights, e.g.:
• optimize the contract to available moment information
• slight relaxation of IC constraints can be significant
• Our take-away: Opportunities for new success stories

58

Many Open Problems

For example, in model of [Holmström’79]:

• Is there a “simple” contract that provides a constant factor

approximation to the optimal contract?

• We already know it must be non-monotone
• Are there natural regularity conditions under which linear/simple

contracts are near optimal?

• We already know that MLRP does not suffice

Thank you!

[1] Simple versus Optimal Contracts Paul Dütting, Tim Roughgarden, Inbal Talgam-Cohen ACM EC’19 [2] The Complexity of Optimal Contracts Paul Dütting, Tim Roughgarden, Inbal Talgam-Cohen ACM-SIAM SODA’20