CS6501: T opics in Learning and Game Theory (Fall 2019) How Can - - PowerPoint PPT Presentation

CS6501: T

• pics in Learning and Game Theory

(Fall 2019) How Can Classifiers Induce Right Efforts?

Instructor: Haifeng Xu

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Outline

Ø Motivations and Model Ø Examples and Results

3

Decisions and Incentives

Often today, ML is used to assist decisions about human beings

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Decisions and Incentives

ØEducation

Often today, ML is used to assist decisions about human beings

5

Decisions and Incentives

ØEducation ØWhen a measure becomes a target, gaming behaviors happen

(Goodhart’s Law) Often today, ML is used to assist decisions about human beings

6

Decisions and Incentives

ØEducation ØWhen a measure becomes a target, gaming behaviors happen

(Goodhart’s Law)

ØMany other applications: recommender systems, hiring, finance…

• E.g., restaurants can game Yelp’s ranking metric by pay for positive

reviews or checkins

Often today, ML is used to assist decisions about human beings

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Decisions and Incentives

ØEducation ØWhen a measure becomes a target, gaming behaviors happen

(Goodhart’s Law)

ØMany other applications: recommender systems, hiring, finance…

• E.g., restaurants can game Yelp’s ranking metric by pay for positive

reviews or checkins

ØParticularly an issue when transparency is required

Often today, ML is used to assist decisions about human beings

Chief scientist of Obama 2012 Campaign

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Education as a Running Example

Strategic Behaviors Goal/score (determined by some measure)

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Education as a Running Example

Strategic Behaviors Goal/score (determined by some measure)

Desirable behavior

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Education as a Running Example

Strategic Behaviors

Undesirable behavior

Goal/score (determined by some measure)

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Education as a Running Example

ØSome strategic behaviors are desirable, and some are not

I think it’s best to. . . distinguish between seven different types of test preparation: Working more effectively; Teaching more; Working harder; Reallocation; Alignment; Coaching; Cheating. The first three are what proponents of high-stakes testing want to see

• - Daniel M. Koretz, Measuring up

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Education as a Running Example

ØSome strategic behaviors are desirable, and some are not

The Main Question How to design decision rules to induce desirable strategic behaviors?

ØUsually not possible to keep the rule confidential ØShould not simply use a rule that cannot be affected at all ØSo, this requires careful design

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The Mathematical Model

Ø𝑛 available actions (e.g., study hard, cheating) Ø𝑜 different features (e.g., HW grade, midterm grade) ØEach unit effort on action 𝑘 results in 𝛽%&(≥ 0) increase in feature 𝑗

1 𝑘 𝑛

. . . . . .

𝐺

.

𝐺& 𝐺

/

. . . . . .

𝛽.. 𝛽%. 𝛽%& 𝛽0%

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A Game between Agent and Principal

ØAgent’s action: allocation (𝑦., ⋯ , 𝑦0) of 1 unit of effort to actions

1 𝑘 𝑛

. . . . . .

𝐺

.

𝐺& 𝐺

/

. . . . . .

𝛽.. 𝛽%. 𝛽%& 𝛽0%

15

A Game between Agent and Principal

ØAgent’s action: allocation (𝑦., ⋯ , 𝑦0) of 1 unit of effort to actions

• Effort profile 𝑦(> 0) decides feature values

𝐺& = 𝑔

&(∑% 𝑦%𝛽%&)

(an increasing concave fnc)

𝑦. 𝑦% 𝑦0

. . . . . .

𝐺

.

𝐺& 𝐺

/

. . . . . .

𝛽.. 𝛽%. 𝛽%& 𝛽0%

∑% 𝑦% ≤ 1

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A Game between Agent and Principal

ØAgent’s action: allocation (𝑦., ⋯ , 𝑦0) of 1 unit of effort to actions

• Effort profile 𝑦(> 0) decides feature values

𝐺& = 𝑔

&(∑% 𝑦%𝛽%&)

(an increasing concave fnc)

ØPrincipal’s action: design the evaluation rule 𝐼(𝐺

., ⋯ , 𝐺 /)

• 𝐼 is increasing in every feature, and publicly known (e.g., a grading rule)

𝑦. 𝑦% 𝑦0

. . . . . .

𝐺

.

𝐺& 𝐺

/

. . . . . .

𝛽.. 𝛽%. 𝛽%& 𝛽0% 𝐼

Evaluation rule 𝐼(𝐺

., ⋯ , 𝐺 /)

∑% 𝑦% ≤ 1

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A Game between Agent and Principal

ØAgent’s action: allocation (𝑦., ⋯ , 𝑦0) of 1 unit of effort to actions

• Effort profile 𝑦(> 0) decides feature values

𝐺& = 𝑔

&(∑% 𝑦%𝛽%&)

(an increasing concave fnc)

ØPrincipal’s action: design the evaluation rule 𝐼(𝐺

., ⋯ , 𝐺 /)

• 𝐼 is increasing in every feature, and publicly known (e.g., a grading rule)

ØPrincipal has a desirable effort profile 𝑦∗ (e.g., 𝑦∗ = “work hard”) ØAgent goal: choose 𝑦 to maximize 𝐼

Q: Can the principal design 𝐼 to induce her desirable 𝑦∗?

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A Game between Agent and Principal

Relation to problems we studied before

ØThis is a Stackelberg game

• First, principal announces the evaluation rule 𝐼
• Second, agent best responds to 𝐼 by picking effort profile 𝑦

ØThis is a mechanism design problem

• Want to design evaluation rule 𝐼 to induce desirable response 𝑦∗

ØMore generally, this a principal-agent mechanism design problem

• Rich literature in economics, explosive recent interest in EconCS

Q: Can the principal design 𝐼 to induce her desirable 𝑦∗?

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Outline

Ø Motivations and Model Ø Examples and Results

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Example: Classroom Setting

𝑦. 𝑦; 𝑦< 𝐺= 𝐺> 1 2 2 1 𝐼

𝑦∗ = (0, 1, 0) Q: Can the principal induce the desirable 𝑦∗ = (0,1,0)?

cheating studying copying

𝐼 = 0.6 𝐺= + 0.4𝐺

D

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Example: Classroom Setting

𝑦. 𝑦; 𝑦< 𝐺= 𝐺> 1 2 2 1 𝐼

𝑦∗ = (0, 1, 0) Q: Can the principal induce the desirable 𝑦∗ = (0,1,0)?

ØAns: Yes

• For any unit of effort on cheating or copying, agent would rather

spend it on studying

cheating studying copying

𝐼 = 0.6 𝐺= + 0.4𝐺

D

22

Example: Classroom Setting

𝑦. 𝑦; 𝑦< 𝐺= 𝐺> 2 1 1 1.5 𝐼

𝐼 = 0.6 𝐺= + 0.4𝐺

D

Q: What about this setting?

cheating studying copying

𝑦∗ = (0, 1, 0)

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Example: Classroom Setting

𝑦. 𝑦; 𝑦< 𝐺= 𝐺> 2 1 1 1.5 𝐼

𝐼 = 0.6 𝐺= + 0.4𝐺

D

Q: What about this setting?

ØAns: No

• Spending 1 unit studying à H = 1
• Spending 1 unit on cheating à H = 1.2
• Problem: weight of exam is to large

cheating studying copying

𝑦∗ = (0, 1, 0)

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Example: Classroom Setting

𝑦. 𝑦; 𝑦< 𝐺= 𝐺> 2 1 1 1.5 𝐼

𝐼 = 0.4 𝐺= + 0.6𝐺

D

Q: What about changing 𝐼 to our class’s rule?

cheating studying copying

𝑦∗ = (0, 1, 0)

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Example: Classroom Setting

𝑦. 𝑦; 𝑦< 𝐺= 𝐺> 2 1 1 1.5 𝐼

𝐼 = 0.4 𝐺= + 0.6𝐺

D

Q: What about changing 𝐼 to our class’s rule?

ØAns: Yes

• Spending 1 unit studying à H = 1
• Shifting any amount of effort to copying or cheating only decreases H
• Whether we can induce 𝑦∗ does depends on our design of 𝐼

cheating studying copying

𝑦∗ = (0, 1, 0)

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Example: Classroom Setting

𝑦. 𝑦; 𝑦< 𝐺= 𝐺> 3 1 1 3 𝐼

𝐼 = 0.4 𝐺= + 0.6𝐺

D

Q: What about these effort transition values?

cheating studying copying

𝑦∗ = (0, 1, 0)

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Example: Classroom Setting

𝑦. 𝑦; 𝑦< 𝐺= 𝐺> 3 1 1 3 𝐼

𝐼 = 0.4 𝐺= + 0.6𝐺

D

Q: What about these effort transition values?

ØAns: No, regardless of what 𝐼 you choose

• For whatever (𝑦., 𝑦;, 𝑦<), (𝑦. +

GH ; , 0, 𝑦< + IH ; ) is better for agent

• There are cases where 𝑦∗ just cannot be induced regardless of 𝐼

cheating studying copying

𝑦∗ = (0, 1, 0)

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Example: Classroom Setting

𝑦. 𝑦; 𝑦< 𝐺= 𝐺> 𝐼

Q: In general, when would it be impossible to induce 𝑦∗?

cheating studying copying

𝑦∗ = (0, 1, 0)

𝛽.= 𝛽;= 𝛽;> 𝛽<>

𝐼 = 𝛾=𝐺= + 𝛾>𝐺

D

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Example: Classroom Setting

𝑦. 𝑦; 𝑦< 𝐺= 𝐺> 𝐼

Q: In general, when would it be impossible to induce 𝑦∗?

ØWith 𝐶 = 1 effort on studying, we get 𝐺=, 𝐺> = (𝛽;=, 𝛽;>) ØIf ∃ (𝑦., 𝑦;, 𝑦<) such that: (1) 𝑦. + 𝑦; + 𝑦< < 1; but (2) 𝑦.𝛽.= + 𝑦;𝛽;= ≥

𝛽;= and 𝑦;𝛽;> + 𝑦<𝛽<> ≥ 𝛽;>, then cannot induce effort on studying

• This condition does not depend on 𝐼

cheating studying copying

𝑦∗ = (0, 1, 0)

𝛽.= 𝛽;= 𝛽;> 𝛽<>

𝐼 = 𝛾=𝐺= + 𝛾>𝐺

D

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ØLet’s focus on the special case 𝑦∗ = 𝑓

% for some 𝑘

ØPrevious argument shows a necessary condition

There is no 𝑦., ⋯ , 𝑦0 ≥ 0 such that: 1. ∑% 𝑦% < 1 2. 𝑦 ⋅ 𝛽 ≥ 𝛽(𝑘,⋅) Note: 𝑦 here is a row vector

Which Effort Profile Can Be Incentivized, and How?

31

ØLet’s focus on the special case 𝑦∗ = 𝑓

% for some 𝑘

ØPrevious argument shows a necessary condition

There is no 𝑦., ⋯ , 𝑦0 ≥ 0 such that: 1. ∑% 𝑦% < 1 2. 𝑦 ⋅ 𝛽 ≥ 𝛽(𝑘,⋅) Note: 𝑦 here is a row vector Define 𝜆% ≔ min

I

∑% 𝑦% subject to (1) 𝑦 ⋅ 𝛽 ≥ 𝛽(𝑘,⋅); (2) 𝑦 ≥ 0. A necessary condition is 𝜆% ≥ 1.

Which Effort Profile Can Be Incentivized, and How?

32

ØLet’s focus on the special case 𝑦∗ = 𝑓

% for some 𝑘

ØPrevious argument shows a necessary condition

Note: 𝜆% ≤ 1 always because 𝑦 = 𝑓

% is feasible

Define 𝜆% ≔ min

I

∑% 𝑦% subject to (1) 𝑦 ⋅ 𝛽 ≥ 𝛽(𝑘,⋅); (2) 𝑦 ≥ 0. A necessary condition is 𝜆% ≥ 1.

Which Effort Profile Can Be Incentivized, and How?

33

ØLet’s focus on the special case 𝑦∗ = 𝑓

% for some 𝑘

ØPrevious argument shows a necessary condition

Note: 𝜆% ≤ 1 always because 𝑦 = 𝑓

% is feasible

Define 𝜆% ≔ min

I

∑% 𝑦% subject to (1) 𝑦 ⋅ 𝛽 ≥ 𝛽(𝑘,⋅); (2) 𝑦 ≥ 0. A necessary condition is 𝜆% = 1.

Which Effort Profile Can Be Incentivized, and How?

34

Which Effort Profile Can Be Incentivized, and How?

ØLet’s focus on the special case 𝑦∗ = 𝑓

% for some 𝑘

ØPrevious argument shows a necessary condition

Define 𝜆% ≔ min

I

∑% 𝑦% subject to (1) 𝑦 ⋅ 𝛽 ≥ 𝛽(𝑘,⋅); (2) 𝑦 ≥ 0. A necessary condition is 𝜆% = 1. Theorem: (1) There is a way to incentivize 𝑓

% if and only if

𝜆% = 1. (2) Whenever 𝑓

% can be incentivized, there is a linear

𝐼 of form 𝐼 = ∑& 𝛾& 𝐺& that incentivizes 𝑓

%.

35

Which Effort Profile Can Be Incentivized, and How?

ØLet’s focus on the special case 𝑦∗ = 𝑓

% for some 𝑘

ØPrevious argument shows a necessary condition

Define 𝜆% ≔ min

I

∑% 𝑦% subject to (1) 𝑦 ⋅ 𝛽 ≥ 𝛽(𝑘,⋅); (2) 𝑦 ≥ 0. A necessary condition is 𝜆% = 1. Theorem: (1) There is a way to incentivize 𝑓

% if and only if

𝜆% = 1. (2) Whenever 𝑓

% can be incentivized, there is a linear

𝐼 of form 𝐼 = ∑& 𝛾& 𝐺& that incentivizes 𝑓

%.

Proof

ØWe know if 𝜆% < 1, we cannot incentivize 𝑓

%, so 𝜆% = 1 is necessary

ØTo prove sufficiency, we construct a linear 𝐼 that indeed induce 𝑓

% when

𝜆% = 1

36

Linear 𝐼 That Induces 𝑓

%

ØConsider 𝐼 = ∑& 𝛾& 𝐺&, agent’s optimization problem

max

I∈XY 𝐼 = ∑& 𝛾& ⋅ 𝑔 & ∑Z 𝑦Z𝛽Z&

Value of feature 𝑗

37

Linear 𝐼 That Induces 𝑓

%

ØConsider 𝐼 = ∑& 𝛾& 𝐺&, agent’s optimization problem

max

I∈XY 𝐼 = ∑& 𝛾& ⋅ 𝑔 & ∑Z 𝑦Z𝛽Z&

ØWhen would the optimal solution be 𝑦∗ = 𝑓

%?

• Ans: when [\

[I] |I_I∗ ≥ [\ [I]` |I_I∗ for all 𝑘′ (verify it after class)

• Spell the derivatives out:

∑& 𝛾& ⋅ 𝛽%& ⋅ 𝑔

& b ∑Z 𝑦Z ∗𝛽Z& ≥ ∑& 𝛾& ⋅ 𝛽%`& ⋅ 𝑔 & b ∑Z 𝑦Z ∗𝛽Z& , ∀𝑘′

Eq.(1)

38

Linear 𝐼 That Induces 𝑓

%

ØConsider 𝐼 = ∑& 𝛾& 𝐺&, agent’s optimization problem

max

I∈XY 𝐼 = ∑& 𝛾& ⋅ 𝑔 & ∑Z 𝑦Z𝛽Z&

ØWhen would the optimal solution be 𝑦∗ = 𝑓

%?

• Ans: when [\

[I] |I_I∗ ≥ [\ [I]` |I_I∗ for all 𝑘′ (verify it after class)

• Spell the derivatives out:

∑& 𝛾& ⋅ 𝛽%& ⋅ 𝑔

& b ∑Z 𝑦Z ∗𝛽Z& ≥ ∑& 𝛾& ⋅ 𝛽%`& ⋅ 𝑔 & b ∑Z 𝑦Z ∗𝛽Z& , ∀𝑘′

Eq.(1)

Q: Given 𝜐% = 1, do there exist 𝛾 ≠ 0 so that Eq. (1) holds? Ø Eq (1) is also a set of linear constraints on 𝛾 Ø Ans: yes, through an elegant duality argument

39

Choosing the 𝛾

ØGoal:∑& 𝛾& ⋅ 𝛽%& ⋅ 𝑔

& b ∑Z 𝑦Z ∗𝛽Z& ≥ ∑& 𝛾& ⋅ 𝛽%`& ⋅ 𝑔 & b ∑Z 𝑦Z ∗𝛽Z& , ∀𝑘′

ØLet 𝐵%,& = 𝛽%& ⋅ 𝑔

& b ∑Z 𝑦Z ∗𝛽Z& which is a constant (𝑦∗ is given)

• Let 𝐵(𝑘,⋅) denotes the 𝑘’th row

ØNeed to check the linear system

𝐵 𝑘,⋅ ⋅ 𝛾= ≥ 𝐵 𝑘b,⋅ ⋅ 𝛾=, ∀𝑘′ 𝛾 ≥ 0 ∃𝛾 ≠ 0 such that

40

Choosing the 𝛾

ØGoal:∑& 𝛾& ⋅ 𝛽%& ⋅ 𝑔

& b ∑Z 𝑦Z ∗𝛽Z& ≥ ∑& 𝛾& ⋅ 𝛽%`& ⋅ 𝑔 & b ∑Z 𝑦Z ∗𝛽Z& , ∀𝑘′

ØLet 𝐵%,& = 𝛽%& ⋅ 𝑔

& b ∑Z 𝑦Z ∗𝛽Z& which is a constant (𝑦∗ is given)

• Let 𝐵(𝑘,⋅) denotes the 𝑘’th row

ØNeed to check the linear system

𝐵 𝑘,⋅ ⋅ 𝛾= ≥ 𝐵 𝑘b,⋅ ⋅ 𝛾=, ∀𝑘′ 𝛾 ≥ 0 ∃𝛾 ≠ 0 such that s.t. 𝟐 ≥ 𝐵 ⋅ 𝛾=, ∀𝑙 𝛾 ≥ 0 max

i

𝐵 𝑘,⋅ ⋅ 𝛾=

⇔

• btains opt ≥ 1

41

Choosing the 𝛾

ØGoal:∑& 𝛾& ⋅ 𝛽%& ⋅ 𝑔

& b ∑Z 𝑦Z ∗𝛽Z& ≥ ∑& 𝛾& ⋅ 𝛽%`& ⋅ 𝑔 & b ∑Z 𝑦Z ∗𝛽Z& , ∀𝑘′

ØLet 𝐵%,& = 𝛽%& ⋅ 𝑔

& b ∑Z 𝑦Z ∗𝛽Z& which is a constant (𝑦∗ is given)

• Let 𝐵(𝑘,⋅) denotes the 𝑘’th row

ØNeed to check the linear system

s.t. 𝟐 ≥ 𝐵 ⋅ 𝛾=, ∀𝑙 𝛾 ≥ 0 max

i

𝐵 𝑘,⋅ ⋅ 𝛾=

• btains opt ≥ 1

s.t. 𝑧 ⋅ 𝐵 ≥ 𝐵(𝑘, : ) 𝑧 ≥ 0 min

m

𝟐 ⋅ 𝑧= Dual LP

42

Choosing the 𝛾

ØGoal:∑& 𝛾& ⋅ 𝛽%& ⋅ 𝑔

& b ∑Z 𝑦Z ∗𝛽Z& ≥ ∑& 𝛾& ⋅ 𝛽%`& ⋅ 𝑔 & b ∑Z 𝑦Z ∗𝛽Z& , ∀𝑘′

ØLet 𝐵%,& = 𝛽%& ⋅ 𝑔

& b ∑Z 𝑦Z ∗𝛽Z& which is a constant (𝑦∗ is given)

• Let 𝐵(𝑘,⋅) denotes the 𝑘’th row

ØNeed to check the linear system

s.t. 𝟐 ≥ 𝐵 ⋅ 𝛾=, ∀𝑙 𝛾 ≥ 0 max

i

𝐵 𝑘,⋅ ⋅ 𝛾=

• btains opt ≥ 1

s.t. 𝑧 ⋅ 𝐵 ≥ 𝐵(𝑘, : ) 𝑧 ≥ 0 min

m

𝟐 ⋅ 𝑧= Dual LP Ø The constraint is ∑Z 𝑧Z 𝛽Z& ⋅ 𝑔

& b ≥ 𝛽%& ⋅ 𝑔 & b, ∀𝑗

i.e., ∑Z 𝑧Z 𝛽Z& ≥ 𝛽%&, ∀𝑗

43

Choosing the 𝛾

ØGoal:∑& 𝛾& ⋅ 𝛽%& ⋅ 𝑔

& b ∑Z 𝑦Z ∗𝛽Z& ≥ ∑& 𝛾& ⋅ 𝛽%`& ⋅ 𝑔 & b ∑Z 𝑦Z ∗𝛽Z& , ∀𝑘′

ØLet 𝐵%,& = 𝛽%& ⋅ 𝑔

& b ∑Z 𝑦Z ∗𝛽Z& which is a constant (𝑦∗ is given)

• Let 𝐵(𝑘,⋅) denotes the 𝑘’th row

ØNeed to check the linear system

s.t. 𝟐 ≥ 𝐵 ⋅ 𝛾=, ∀𝑙 𝛾 ≥ 0 max

i

𝐵 𝑘,⋅ ⋅ 𝛾=

• btains opt ≥ 1

s.t. 𝑧 ⋅ 𝐵 ≥ 𝐵(𝑘, : ) 𝑧 ≥ 0 min

m

𝟐 ⋅ 𝑧= Dual LP Ø The constraint is ∑Z 𝑧Z 𝛽Z& ⋅ 𝑔

& b ≥ 𝛽%& ⋅ 𝑔 & b, ∀𝑗

i.e., ∑Z 𝑧Z 𝛽Z& ≥ 𝛽%&, ∀𝑗 Ø Dual opt is exactly the def of 𝜆%(= 1)

44

Choosing the 𝛾

ØGoal:∑& 𝛾& ⋅ 𝛽%& ⋅ 𝑔

& b ∑Z 𝑦Z ∗𝛽Z& ≥ ∑& 𝛾& ⋅ 𝛽%`& ⋅ 𝑔 & b ∑Z 𝑦Z ∗𝛽Z& , ∀𝑘′

ØLet 𝐵%,& = 𝛽%& ⋅ 𝑔

& b ∑Z 𝑦Z ∗𝛽Z& which is a constant (𝑦∗ is given)

• Let 𝐵(𝑘,⋅) denotes the 𝑘’th row

ØNeed to check the linear system

s.t. 𝟐 ≥ 𝐵 ⋅ 𝛾=, ∀𝑙 𝛾 ≥ 0 max

i

𝐵 𝑘,⋅ ⋅ 𝛾=

• btains opt ≥ 1

s.t. 𝑧 ⋅ 𝐵 ≥ 𝐵(𝑘, : ) 𝑧 ≥ 0 min

m

𝟐 ⋅ 𝑧= Dual LP Ø The constraint is ∑Z 𝑧Z 𝛽Z& ⋅ 𝑔

& b ≥ 𝛽%& ⋅ 𝑔 & b, ∀𝑗

i.e., ∑Z 𝑧Z 𝛽Z& ≥ 𝛽%&, ∀𝑗 Ø Dual opt is exactly the def of 𝜆%(= 1) Primal opt = 1 𝛾 can be easily constructed

45

Choosing the 𝛾

ØGoal:∑& 𝛾& ⋅ 𝛽%& ⋅ 𝑔

& b ∑Z 𝑦Z ∗𝛽Z& ≥ ∑& 𝛾& ⋅ 𝛽%`& ⋅ 𝑔 & b ∑Z 𝑦Z ∗𝛽Z& , ∀𝑘′

ØLet 𝐵%,& = 𝛽%& ⋅ 𝑔

& b ∑Z 𝑦Z ∗𝛽Z& which is a constant (𝑦∗ is given)

• Let 𝐵(𝑘,⋅) denotes the 𝑘’th row

ØNeed to check the linear system

s.t. 𝟐 ≥ 𝐵 ⋅ 𝛾=, ∀𝑙 𝛾 ≥ 0 max

i

𝐵 𝑘,⋅ ⋅ 𝛾=

• btains opt ≥ 1

s.t. 𝑧 ⋅ 𝐵 ≥ 𝐵(𝑘, : ) 𝑧 ≥ 0 min

m

𝟐 ⋅ 𝑧= Dual LP Ø The constraint is ∑Z 𝑧Z 𝛽Z& ⋅ 𝑔

& b ≥ 𝛽%& ⋅ 𝑔 & b, ∀𝑗

i.e., ∑Z 𝑧Z 𝛽Z& ≥ 𝛽%&, ∀𝑗 Ø Dual opt is exactly the def of 𝜆%(= 1) Primal opt = 1 𝛾 can be easily constructed

46

General 𝑦∗

ØSimilar conclusion holds with similar proof ØIt turns out that the condition depends on 𝑇∗, the support of 𝑦∗

Theorem: (1) There is a way to incentivize 𝑦∗ if and only if 𝜆o∗ = 1 for some suitably defined 𝜆o∗. (2) Whenever 𝑦∗ can be incentivized, there is a linear 𝐼 that incentivizes 𝑦∗.

47

Optimization Version of the Problem

ØPreviously, principal has a single 𝑦∗ to induce

• Some of 𝑦∗ can be incentivized, and some cannot

ØA natural optimization version of the problem

• Among all incentivizable 𝑦∗, how can principal incentivize the “best” one
• Assume a utility function 𝑕(𝑦) over 𝑦

48

Optimization Version of the Problem

ØPreviously, principal has a single 𝑦∗ to induce

• Some of 𝑦∗ can be incentivized, and some cannot

ØA natural optimization version of the problem

• Among all incentivizable 𝑦∗, how can principal incentivize the “best” one
• Assume a utility function 𝑕(𝑦) over 𝑦

ØProblem: maximize 𝑕(𝑦) subject to 𝑦 is incentivizable

Theorem: The above problem is NP-hard, even when 𝑕 is concave.

Open question: Ø What kind of 𝑕 can be optimized? Linear? Ø What kind effort transition graph makes the problem more tractable?

Thank You

Haifeng Xu

University of Virginia hx4ad@virginia.edu