Switch perspectives ... 

Text

idea: admit those who score above 90

max score: 75%

max score: 60%

max score: 93%

max score: 88%

Why not?

High

Low

High

Low

Budget

Skill

Recall: 

• low score \(\neq\) low skill
• deterministic admissions policy cannot differentiate 

Comparison 

allocation

score

0

1

0.9

allocation

score

0

1

75

0.8

60

1

Given deterministic policy:

• low-skill students, regardless of budget, won't participate 
• Only the high-skill high-budget student can achieve positive utility 

allocation

0

1

0.9

1

utility

high skill

low skill

Given probabilistic policy:

• low-skill students, regardless of budget, won't participate 
• Only the high-skill high-budget student can achieve positive utility 

high skill

low skill

Given probabilistic policy:

• low-skill students, regardless of budget, won't participate 
• Only the high-skill high-budget student can achieve positive utility 

Three options: 

A. Score above 90 and admitted with certainty

B. Score between 70 and 90 and admitted w.p. 0.7

C. Don't apply. 

Suppose all students face the same budget \(\color{red}{b = 0.25}\) 

Suppose all students face the same budget \(\color{red}{b = 0.25}\) 

Compare this to the deterministic threshold: 

Observe that we are admitting low-skilled students with less probability, compared to deterministic threshold  

• Introducing subsidies ...

🪦 Slide Graveyard 

Alice.

• ... is preparing for some exam 
• ... has limited time to prepare
•  

skill: ??? 

skill: ??? 

skill: ??? 

skill: ??? 

score: 75%

score: 60%

score: 93%

score: 88%

Why not?

budget:
⌛

budget:⌛⌛⌛

budget:
⌛

budget:⌛⌛⌛

students may face constraints in budgets (on time) 

Why not?

students may face constraints in budgets (on time) 

Introduction

• Given privately constrained public signals, how do we optimally allocate to agents of different types? 
• Key idea: stochastic allocation enables higher allocation to a high type than to a low type, even when their observed signal is approximately equal due to budget constraints. 
• Multi-dimensional, monopolistic screening problem 

Menu 

2-skill 2-budget agents pick from the menu 

Explain why agents pick what they pick 

Model

• Principal \(P\) observes a public signal of quality: 

• \(P\) wants to hire agents with skills above threshold \(\tau \in \mathbb{R}_+\)

• \(P\) needs to design an allocation rule \(y:\mathbb{R}_+ \to [0, 1]\) 

Main results 

• A deterministic threshold mechanism sets a quality threshold \(\underline{q} \in \mathbb{R}_+\) and hires an agent if and only if the agent exhibits public quality \(q \geq \underline{q}\) 
• When agents have different budgets, we can prove via counterexample that a deterministic threshold mechanism is not optimal. 

• A lottery menu mechanism is a (weakly) monotone allocation rule \(y\) with menu options \((\underline q, x)\) where \(x\) is the probability of allocation resulting from an agent exhibiting quality \(\underline q\). 
• A menu option \((\underline q, x)\) is feasible for an agent \(A=(s,b)\) iff 
  • \((\underline q, x)\) is affordable: \(\underline q / s \leq b\)
  • \((\underline q, x)\) is a rational choice: \(x - \underline q / s \geq 0\)

• A lottery menu mechanism is a (weakly) monotone allocation rule \(y\) with menu options \((\underline q, x)\) where \(x\) is the probability of allocation resulting from an agent exhibiting quality \(\underline q\). 
• A menu option \((\underline q, x)\) is feasible for an agent \(A=(s,b)\) iff 
  • \((\underline q, x)\) is affordable: \(\underline q / s \leq b\)
  • \((\underline q, x)\) is a rational choice: \(x - \underline q / s \geq 0\)

Fixing \(\underline q\), if we decrease \(x\): 

• For larger budgets \(b\), upperbound on \(\underline q / s\) will be set sooner by decreasing \(x \implies\) discriminates more against agents with larger budgets  
•  

Future directions 

• What happens when we introduce a uniform subsidy (to budgets) to all agents? 
• Extending the model from 2-skill \(n\)-budget to \(m\)-skill \(n\)-budget
• Extending the model from discrete type space to continuous type space 

Thank you for listening!

Review

• Agent \(i\) has skill \(s_i\), but principal only observes quality = skill \(\times\) effort 
• Principal wants agents above a certain quality threshold \(\hat{q}\) 
• Effort required for agent to reach said skill threshold \(\hat{q}\): \[e_i = \frac{\hat{q}}{s_i}\]
• Agent \(i\) is constrained by her budget on effort: \[e_i \in [0, b_i] \]
• Agent \(i\)'s utility \[u_i = x_i - e_i\]
  • where \(x_i\) is the probability that she gets allocated 

Review

• Agent \(i\) has
  • skill \(s_i\)
  • budget \(b_i\)
  • effort \(e_i \in [0, b_i]\)
• Agent gets allocated with probability \(x_i\) 
• Agent's utility \[u^A_i = x_i - e_i\]

• Principal observes agents' quality:

\(q_i = s_i \times e_i\) 

• Principal wants agents above a certain skill threshold \(\hat{q}\) 
• Principal's utility \[u^P =\sum_i (s_i - \hat{q})\]

What will the agent do?

• Suppose we offer agents a menu of options to choose from 
• Each option is a (quality threshold, allocation probability) tuple, \((\hat{q}, x)\) 
• Agent with skill \(s\) will consider option \((\hat{q}, x)\) if
  • effort required to reach \(\hat{q}\) is not going to exceed her budget, and 
  • utility is non-negative when exerting effort (i.e. individual rationality) 

Putting this into a graph...

Option \((\hat{q}_j, x_j)\) is feasible for agent if \(\hat{q}_j/s \leq \min \{b, x_j\}\)

Putting this into a graph...

i.e. visualizing agent's effort 

option \((\hat{q}_j, x_j)\) is feasible if \(\hat{q}_j/s \leq \min \{b, x_j\}\)

\(x\)

\(q\)

Consider a single agent with skill \(s\); 

When \(b=1\):

option \((\hat{q}_j, x_j)\) is feasible if \(\hat{q}_j/s \leq \min \{b, x_j\}\)

\(x\)

\(q\)

Consider a single agent with skill \(s\); 

When \(b=1\):

\(\implies\) Question: which is a feasible option? 

\(k = 1/s\)

option \((\hat{q}_j, x_j)\) is feasible if \(\hat{q}_j/s \leq \min \{b, x_j\}\)

\(x\)

\(q\)

Consider a single agent with skill \(s\); 

When \(b=1\):

\(\implies\) Question: which is a feasible option? 

\(k = 1/s\)

\(x\)

\(q\)

Consider a single agent with skill \(s\); 

When \(b=1\):

\(\implies\) Question: which is a feasible option? 

\(k = 1/s\)

option \((\hat{q}_j, x_j)\) is feasible if \(\hat{q}_j/s \leq \min \{b, x_j\}\)

\(x\)

\(q\)

Consider a single agent with skill \(s\); 

When \(b=1\):

\(\implies\) Question: which is a feasible option? 

\(k = 1/s\)

option \((\hat{q}_j, x_j)\) is feasible if \(\hat{q}_j/s \leq \min \{b, x_j\}\)

\(x\)

\(q\)

Consider a single agent with skill \(s\); 

When \(b=1\):

\(\implies\) Question: which is a feasible option? 

\(k = 1/s\)

option \((\hat{q}_j, x_j)\) is feasible if \(\hat{q}_j/s \leq \min \{b, x_j\}\)

\(x\)

\(q\)

Consider a single agent with skill \(s\); 

When \(b=1\):

\(\implies\) Question: which is a feasible option? 

\(k = 1/s\)

option \((\hat{q}_j, x_j)\) is feasible if \(\hat{q}_j/s \leq \min \{b, x_j\}\)

\(x\)

\(q\)

Consider a single agent with skill \(s\); 

When \(b=1\):

\(\implies\) Question: What about utility? 

\(k = 1/s\)

option \((\hat{q}_j, x_j)\) is feasible if \(\hat{q}_j/s \leq \min \{b, x_j\}\)

\(x\)

\(q\)

Consider a single agent with skill \(s\); 

When \(b=1\):

\(\implies\) Question: What about zero-utility curve? 

\(k = 1/s\)

option \((\hat{q}_j, x_j)\) is feasible if \(\hat{q}_j/s \leq \min \{b, x_j\}\)

\(x\)

\(q\)

Generalizing the feasible region

\(k = 1/s\)

When \(b=1\), 

\(x\)

\(q\)

Generalizing the feasible region

\(k = 1/s\)

When \(b<1\), 

feasible region

\(x\)

\(q\)

Generalizing the feasible region

\(k = 1/s\)

When \(b>1\), 

\(\implies\) budget is no longer limiting for agent 

feasible region

\(x\)

\(q\)

\(k = 1/s\)

What about agents with different budgets?

\(b=1\)

\(x\)

\(q\)

\(k = 1/s\)

What about agents with different budgets?

\(b=1\)

\(x\)

\(q\)

\(k = 1/s\)

What about agents with different skills?

\(x\)

\(q\)

\(1/s_H\)

What about agents with different skills?

Suppose we have agents with low skills \(s_L\) and high skills \(s_H\) but they face the same budget 

\(1/s_L\)

What is the optimal menu? 

• Recall: principal's utility is \(\sum_i (s_i - \hat{q})\). 
• Ideally, we want all agents with high skills. 
• However, we can't tell high-skilled agents from low-skilled agents because all we observe is quality. 

• Start with the simple case of two kinds of skill types and two kinds of budget types
  • An agent can either be high-skilled \(s_H\) or low-skilled \(s_L\)  
  • An agent can either have a high budget \(b_H\) or a low budget \(b_L\)
    • WLOG, suppose \(b_L < 1\)

What is the optimal menu? 

• If \(s_L\cdot b_H \leq s_H\cdot b_L\): 
  • Posting a single threshold of \(\hat{q} = s_L\cdot b_H + \epsilon\) is optimal because the threshold allows us to filter out all agents with low skills 
• The more interesting case -- \(s_L\cdot b_H > s_H\cdot b_L\):
  • ​Just looking at quality won't help us differ high-skilled agents from low-skilled agents

Principal's utility is \(\sum_i (s_i - \hat{q})\). 

Simple menu 1 ("keep all high-skilled")

\(x\)

\(q\)

Simple menu 1 

\(x\)

\(q\)

low skill people who reached \(\hat{q}\)

(because they have high budgets) 

Simple menu 2 ("no low skill")

\(x\)

\(q\)

Simple menu 2 ("no low skill")

\(x\)

\(q\)

Menu with jumps 

\(x\)

\(q\)

Menu with jumps 

\(x\)

\(q\)

Simple menu 2

\(x\)

\(q\)

What is the optimal menu? 

\(x\)

\(q\)

two skills, same limiting budget

What is the optimal menu? 

\(x\)

\(q\)

expressing optimal menu in math: ...

What is the optimal menu? 

\(x\)

\(q\)

What is the optimal menu? 

\(x\)

\(q\)

More specifically, 

• Principal's first-best is to hire everyone who is above a certain threshold 
• Principal's utility is   \[ u^P(x,e) = \sum_i x_i \cdot (s_i - t) \] where \(t \geq 0\) is some threshold 
  • For a utility maximizing principal, they will only pick agents with \(s_i \geq t\) 

Public Budget 

• Suppose we have a single public budget \(b\) for all agents; i.e. \(F^b\) contains one element
  • all agents face the same budget \(b\) and we know it  
• Suppose that allocation rule \(g\) has a deterministic quality threshold \(\tau\)
• We argue that there always exists a threshold \(\tau = t\) that achieves the first-best
  • i.e. Principal awards agents with \(s_i \geq t\) 

Proof.

• To achieve threshold \(\tau\), agent \(i\) will need to put in effort at least \(\tau / s_i\) - think of this as min effort required to achieve \(\tau\)  
• Recall that agent \(i\)'s utility is \(\underbrace{g(q_i)}_{\in [0, 1]} - e_i\) 
  • \(\implies\) Agent \(i\)'s max possible utility is \(1 - \tau / s_i\) if she wants to reach quality threshold \(\tau\) 
• To reach threshold \(t\), agent \(i\)'s max possible utility is \(1 - t/s_i\) and she will only put in effort if \(1 - t/s_i \geq 0\)  

Proof cont'd.

• To reach threshold \(t\), agent \(i\)'s max possible utility is \(1 - t/s_i\) and she will only put in effort if \(1 - t/s_i \geq 0\)  
• \(\implies s_i \geq t\) - i.e. only agents with skill \(s_i \geq t\) will put in effort  
• Therefore, Principal's utility \(\sum_i x_i \cdot (s_i - t)\) is maximized because we're only summing up non-negative quantities 

Formally, 

• Agent \(i = 1, ... , n\) 
• For each agent \(i\), she has a 
  • skill \(s_i \sim F^s\)
  • budget \(b_i \sim F^b\)
  • effort \(e_i \in [0, b_i] \) 
  • \(F^s ,F^b\) are known 
• We observe each agent's quality: \(q_i = \underbrace{s_i}_{\text{skill}} \times \underbrace{e_i}_{\text{effort}}\) 

Formally, 

Agent's effort 

• To achieve quality threshold \(\tau\), we only need effort \(\tau / s_i\)

\(\to\) When will an agent put in effort? 

• Given \(\tau\), agent will put in effort
  • \(e_i = 0\) if \(1 - \tau/s_i < 0\) 
    • Agent puts in no effort if maximal attainable utility at \(\tau\) is negative 
  • \(e_i = \min \{b, \tau/s_i\}\) otherwise 

Deriving \(\tau\) 

• Recall that the principal will award agents with skill \(s_i \geq t \to \) awarded agent at the boundary has skill \(s_i = t\)  
• For an agent at the boundary, she either has a binding budget (\(e_i = \frac{\tau}{t} = b\)) or zero utility (\(1 - \frac{\tau}{t} = 0\)).

Case 1: \(e = \tau / t = b\)

• From \(\tau/t = b\), we have \(\tau = tb\) 
• Suppose \( b > 1\). 
• Utility is \( 1 - \tau / t = 1 - b < 0\) -- This cannot happen. Therefore, \(b \leq 1\). 
• \(\to \tau / t \leq 1 \to \tau \leq t\)
• In general, utility \(= 1 - \tau / t \geq 0 \to t \geq \tau\)
• Therefore, \(\tau = t\)

Case 2: \(1 - \tau / t = 0\)

• Solving for the above equation, we have that \(\tau = t\) 

\(t / s_i \in (b, \infty)\) 

• In words: For agent \(i\), the minimum effort required to achieve quality threshold \(t\) is greater than the budget for effort 
• Agent \(i\) with skill s.t. \(s_i \cdot b < t\) will not be selected by the principal \(\implies g(s_i \cdot b) = 0\) 
• \(\implies\) Agent \(i\) will not exert effort; \(e = 0\) 
• \(\implies\) Quality \(q_i = 0 < t\), Principal will not select such an agent  

\(t / s_i \in (0,  b]\) 

• In words: For agent \(i\), the minimum effort required to achieve quality threshold \(t\) is between 0 and budget
• To maximize agent's utility, she will put in effort \(e_i = t/s_i \leq b\), and her utility is \(g(s_i \cdot e_i) - e_i = g(t) - t/s_i \) 
• Recall that \(g(t) = x_i \in [0, 1]\), therefore agent \(i\)'s maximum possible utility is \(1 - t/s_i\)   
• Agent will not put in effort if \(1 - t/s_i < 0\) 
  • \(\implies 1 < t/s_i \leq b\) 
• Agent will put in effort if \(1 - t/s_i \geq 0\)
  • ​\(\implies b > 1 \geq t/s_i\)

Public budget 

• If \(b \leq 1\) (i.e. budget is binding): 
  • \(t\) is used when \( \tau / t \leq b \to \tau \leq tb \) 
  • Agent exerts effort and meets threshold \(\tau\) if her utility is positive, that is, \(1 - \tau / t \geq 0 \to t \geq \tau \)
• Since \( b \leq 1\), \( \tau \leq tb \) is binding, and suppose we set \(\tau = tb\) 
• This ensures that agents with \( s \geq t\) will choose to exert effort \(b\), meet the threshold, and are given rewards 

Public budget 

• Moreover, for agents with skills \(s < t\), they will put in effort \(e \leq b\) 

Variant 2 

• Principal's first-best is to hire the top \(k\) candidates 
• \(s_i \geq 0\) 
• \( C = k\) is now binding 
• Allocation rule \(g\) maps observed qualities, reported budgets \(\hat{b}\), reported skills \(\hat{s}\) to allocation \(x\) 

Variant 2 

• Suppose allocation rule \(g\) ignores reports for now and only uses observed qualities 
• Assume for now that the probability of a tie (in terms of quality?) is 0 
• Agents care about their probability of being in the top \(k\) 
• Denote the largest quality by \(q^{(1)}\) and the smallest quality by \(q^{(n)}\)

Variant 2 

• For agent \(i\), her interim utility is \[\Pr_F[q_i \geq q^{(k)}] - e_i\]