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

CS6501: T opics in Learning and Game Theory (Fall 2019) How Can Classifiers Induce Right Efforts? Instructor: Haifeng Xu

Outline Ø Motivations and Model Ø Examples and Results 2

Decisions and Incentives Often today, ML is used to assist decisions about human beings 3

Decisions and Incentives Often today, ML is used to assist decisions about human beings Ø Education 4

Decisions and Incentives Often today, ML is used to assist decisions about human beings Ø Education Ø When a measure becomes a target, gaming behaviors happen (Goodhart’s Law) 5

Decisions and Incentives Often today, ML is used to assist decisions about human beings Ø Education Ø When a measure becomes a target, gaming behaviors happen (Goodhart’s Law) Ø Many other applications: recommender systems, hiring, finance… • E.g., restaurants can game Yelp’s ranking metric by pay for positive reviews or checkins 6

Decisions and Incentives Often today, ML is used to assist decisions about human beings Ø Education Ø When a measure becomes a target, gaming behaviors happen (Goodhart’s Law) Ø Many other applications: recommender systems, hiring, finance… • E.g., restaurants can game Yelp’s ranking metric by pay for positive reviews or checkins Ø Particularly an issue when transparency is required Chief scientist of Obama 2012 Campaign 7

Education as a Running Example Goal/score Strategic Behaviors (determined by some measure) 8

Education as a Running Example Goal/score Strategic Behaviors (determined by some measure) Desirable behavior 9

Education as a Running Example Goal/score Strategic Behaviors (determined by some measure) Undesirable behavior 10

Education as a Running Example Ø Some 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 11

Education as a Running Example Ø Some strategic behaviors are desirable, and some are not The Main Question How to design decision rules to induce desirable strategic behaviors? Ø Usually not possible to keep the rule confidential Ø Should not simply use a rule that cannot be affected at all Ø So, this requires careful design 12

The Mathematical Model Ø 𝑛 available actions (e.g., study hard, cheating) Ø 𝑜 different features (e.g., HW grade, midterm grade) Ø Each unit effort on action 𝑘 results in 𝛽 %& (≥ 0) increase in feature 𝑗 𝛽 .. 1 𝐺 . 𝛽 %. . . . . . . 𝛽 %& 𝑘 𝐺 & 𝛽 0% . . . . . . 𝑛 𝐺 / 13

A Game between Agent and Principal Ø Agent’s action: allocation (𝑦 . , ⋯ , 𝑦 0 ) of 1 unit of effort to actions 𝛽 .. 1 𝐺 . 𝛽 %. . . . . . . 𝛽 %& 𝑘 𝐺 & 𝛽 0% . . . . . . 𝑛 𝐺 / 14

A Game between Agent and Principal Ø Agent’s action: allocation (𝑦 . , ⋯ , 𝑦 0 ) of 1 unit of effort to actions • Effort profile 𝑦(> 0) decides feature values 𝐺 & = 𝑔 & (∑ % 𝑦 % 𝛽 %& ) (an increasing concave fnc) 𝛽 .. 𝑦 . 𝐺 . 𝛽 %. . . . . . . ∑ % 𝑦 % ≤ 1 𝛽 %& 𝑦 % 𝐺 & 𝛽 0% . . . . . . 𝑦 0 𝐺 / 15

A Game between Agent and Principal Ø Agent’s action: allocation (𝑦 . , ⋯ , 𝑦 0 ) of 1 unit of effort to actions • Effort profile 𝑦(> 0) decides feature values 𝐺 & = 𝑔 & (∑ % 𝑦 % 𝛽 %& ) (an increasing concave fnc) Ø Principal’s action: design the evaluation rule 𝐼(𝐺 . , ⋯ , 𝐺 / ) • 𝐼 is increasing in every feature, and publicly known (e.g., a grading rule) 𝛽 .. 𝑦 . 𝐺 . 𝛽 %. . . . . . . ∑ % 𝑦 % ≤ 1 𝛽 %& 𝑦 % 𝐼 𝐺 & 𝛽 0% . . . . . . Evaluation rule 𝐼(𝐺 . , ⋯ , 𝐺 / ) 𝑦 0 𝐺 / 16

A Game between Agent and Principal Ø Agent’s action: allocation (𝑦 . , ⋯ , 𝑦 0 ) of 1 unit of effort to actions • Effort profile 𝑦(> 0) decides feature values 𝐺 & = 𝑔 & (∑ % 𝑦 % 𝛽 %& ) (an increasing concave fnc) Ø Principal’s action: design the evaluation rule 𝐼(𝐺 . , ⋯ , 𝐺 / ) • 𝐼 is increasing in every feature, and publicly known (e.g., a grading rule) Ø Principal has a desirable effort profile 𝑦 ∗ (e.g., 𝑦 ∗ = “work hard”) Ø Agent goal: choose 𝑦 to maximize 𝐼 Q : Can the principal design 𝐼 to induce her desirable 𝑦 ∗ ? 17

A Game between Agent and Principal Q : Can the principal design 𝐼 to induce her desirable 𝑦 ∗ ? Relation to problems we studied before Ø This is a Stackelberg game • First, principal announces the evaluation rule 𝐼 • Second, agent best responds to 𝐼 by picking effort profile 𝑦 Ø This is a mechanism design problem • Want to design evaluation rule 𝐼 to induce desirable response 𝑦 ∗ Ø More generally, this a principal-agent mechanism design problem • Rich literature in economics, explosive recent interest in EconCS 18

Outline Ø Motivations and Model Ø Examples and Results 19

Example: Classroom Setting 1 𝑦 . 𝐺 = cheating 2 𝐼 𝑦 ; 2 𝐺 > studying 𝐼 = 0.6 𝐺 = + 0.4𝐺 D 𝑦 < 1 𝑦 ∗ = (0, 1, 0) copying Q : Can the principal induce the desirable 𝑦 ∗ = (0,1,0) ? 20

Example: Classroom Setting 1 𝑦 . 𝐺 = cheating 2 𝐼 𝑦 ; 2 𝐺 > studying 𝐼 = 0.6 𝐺 = + 0.4𝐺 D 𝑦 < 1 𝑦 ∗ = (0, 1, 0) copying Q : Can the principal induce the desirable 𝑦 ∗ = (0,1,0) ? Ø Ans: Yes • For any unit of effort on cheating or copying, agent would rather spend it on studying 21

Example: Classroom Setting 2 𝑦 . 𝐺 = cheating 1 𝐼 𝑦 ; 1 𝐺 > studying 𝐼 = 0.6 𝐺 = + 0.4𝐺 D 𝑦 < 𝑦 ∗ = (0, 1, 0) 1.5 copying Q : What about this setting? 22

Example: Classroom Setting 2 𝑦 . 𝐺 = cheating 1 𝐼 𝑦 ; 1 𝐺 > studying 𝐼 = 0.6 𝐺 = + 0.4𝐺 D 𝑦 < 𝑦 ∗ = (0, 1, 0) 1.5 copying Q : What about this setting? Ø Ans: No • Spending 1 unit studying à H = 1 • Spending 1 unit on cheating à H = 1.2 • Problem: weight of exam is to large 23

Example: Classroom Setting 2 𝑦 . 𝐺 = cheating 1 𝐼 𝑦 ; 1 𝐺 > studying 𝐼 = 0.4 𝐺 = + 0.6𝐺 D 𝑦 < 𝑦 ∗ = (0, 1, 0) 1.5 copying Q : What about changing 𝐼 to our class’s rule? 24

Example: Classroom Setting 2 𝑦 . 𝐺 = cheating 1 𝐼 𝑦 ; 1 𝐺 > studying 𝐼 = 0.4 𝐺 = + 0.6𝐺 D 𝑦 < 𝑦 ∗ = (0, 1, 0) 1.5 copying Q : What about changing 𝐼 to our class’s rule? Ø Ans: 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 25

Example: Classroom Setting 3 𝑦 . 𝐺 = cheating 1 𝐼 𝑦 ; 1 𝐺 > studying 𝐼 = 0.4 𝐺 = + 0.6𝐺 D 𝑦 < 𝑦 ∗ = (0, 1, 0) 3 copying Q : What about these effort transition values? 26

Example: Classroom Setting 3 𝑦 . 𝐺 = cheating 1 𝐼 𝑦 ; 1 𝐺 > studying 𝐼 = 0.4 𝐺 = + 0.6𝐺 D 𝑦 < 𝑦 ∗ = (0, 1, 0) 3 copying Q : What about these effort transition values? Ø Ans: No, regardless of what 𝐼 you choose G H I H • For whatever (𝑦 . , 𝑦 ; , 𝑦 < ) , (𝑦 . + ; , 0, 𝑦 < + ; ) is better for agent • There are cases where 𝑦 ∗ just cannot be induced regardless of 𝐼 27

Example: Classroom Setting 𝛽 .= 𝑦 . 𝐺 = cheating 𝛽 ;= 𝐼 𝑦 ; 𝛽 ;> 𝐺 > studying 𝐼 = 𝛾 = 𝐺 = + 𝛾 > 𝐺 D 𝑦 < 𝑦 ∗ = (0, 1, 0) 𝛽 <> copying Q : In general, when would it be impossible to induce 𝑦 ∗ ? 28

Example: Classroom Setting 𝛽 .= 𝑦 . 𝐺 = cheating 𝛽 ;= 𝐼 𝑦 ; 𝛽 ;> 𝐺 > studying 𝐼 = 𝛾 = 𝐺 = + 𝛾 > 𝐺 D 𝑦 < 𝑦 ∗ = (0, 1, 0) 𝛽 <> copying Q : In general, when would it be impossible to induce 𝑦 ∗ ? Ø With 𝐶 = 1 effort on studying , we get 𝐺 = , 𝐺 > = (𝛽 ;= , 𝛽 ;> ) Ø If ∃ (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 29

Which Effort Profile Can Be Incentivized, and How? Ø Let’s focus on the special case 𝑦 ∗ = 𝑓 % for some 𝑘 Ø Previous argument shows a necessary condition There is no 𝑦 . , ⋯ , 𝑦 0 ≥ 0 such that: ∑ % 𝑦 % < 1 1. 2. 𝑦 ⋅ 𝛽 ≥ 𝛽(𝑘,⋅) Note: 𝑦 here is a row vector 30

Which Effort Profile Can Be Incentivized, and How? Ø Let’s focus on the special case 𝑦 ∗ = 𝑓 % for some 𝑘 Ø Previous argument shows a necessary condition ∑ % 𝑦 % subject to (1) 𝑦 ⋅ 𝛽 ≥ 𝛽(𝑘,⋅) ; (2) 𝑦 ≥ 0 . A Define 𝜆 % ≔ min I necessary condition is 𝜆 % ≥ 1 . There is no 𝑦 . , ⋯ , 𝑦 0 ≥ 0 such that: ∑ % 𝑦 % < 1 1. 2. 𝑦 ⋅ 𝛽 ≥ 𝛽(𝑘,⋅) Note: 𝑦 here is a row vector 31