# Difference between revisions of "Austensmith Feddersen (2008) - Public Disclosure Private Revelation Or Silence"

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+ | {{Article | ||

+ | |Has page=Austensmith Feddersen (2008) - Public Disclosure Private Revelation Or Silence | ||

+ | |Has bibtex key= | ||

+ | |Has article title=Public Disclosure Private Revelation Or Silence | ||

+ | |Has author=Austensmith Feddersen | ||

+ | |Has year=2008 | ||

+ | |In journal= | ||

+ | |In volume= | ||

+ | |In number= | ||

+ | |Has pages= | ||

+ | |Has publisher= | ||

+ | }} | ||

*This page is referenced in [[BPP Field Exam Papers]] | *This page is referenced in [[BPP Field Exam Papers]] | ||

+ | |||

==Reference(s)== | ==Reference(s)== | ||

− | Austen-Smith, D. and T. Feddersen (2008), Public Disclosure, Private Revelation or Silence: Whistleblowing Incentives and Managerial Policy, Kellogg School of Management Working Paper. [http://www.edegan.com/pdfs/AustenSmith%20Feddersen%20(2008)%20-%20Public%20Disclosure%20Private%20Revelation%20or%20Silence.pdf pdf] ([http://www.edegan.com/repository/AustinSmith%20Feddersen%20(2008)%20-%20Whistleblowing%20--%20Handout.pdf Class Handout] and [http://www.edegan.com/repository/AustinSmith%20Feddersen%20(2008)%20-%20Whistleblowing%20--%20Slides.ppt Class Slides] | + | Austen-Smith, D. and T. Feddersen (2008), Public Disclosure, Private Revelation or Silence: Whistleblowing Incentives and Managerial Policy, Kellogg School of Management Working Paper. [http://www.edegan.com/pdfs/AustenSmith%20Feddersen%20(2008)%20-%20Public%20Disclosure%20Private%20Revelation%20or%20Silence.pdf pdf] ([http://www.edegan.com/repository/AustinSmith%20Feddersen%20(2008)%20-%20Whistleblowing%20--%20Handout.pdf Class Handout] and [http://www.edegan.com/repository/AustinSmith%20Feddersen%20(2008)%20-%20Whistleblowing%20--%20Slides.ppt Class Slides]) |

==Abstract== | ==Abstract== | ||

− | The public revelation of organizational wrongdoing by insiders, whistleblowing, is widely reported, economically | + | The public revelation of organizational wrongdoing by insiders, whistleblowing, is widely reported, economically significant and can be extremely costly to the whistleblowers. We develop a model of whistleblowing involving a manager and an employee. Each has a privately known type that specifes the relative weight placed on social rather than personal payoffs. The manager chooses a whistleblowing policy consisting of conditional penalties for various employee actions; the employee observes the policy and chooses between saying nothing, revealing a (privately observed) socially costly violation to the manager, or whistleblowing. Given common knowledge of manager types we characterize equilibrium whistleblowing policies and employee behavior. We show that there may be a nonmonotonic relationship between the severity of the violation and the likelihood of whistleblowing. When manager types are private information we provide suffcient conditions for a separating equilibrium. Managerial choice of whistleblowing policies thus serves a dual purpose: providing incentives for reporting violations and providing information to employees regarding the willingness of the manager to fix violations that are privately reported. |

+ | |||

+ | ==The Model== | ||

+ | |||

+ | There are two active players: | ||

+ | *Employees | ||

+ | *Managers | ||

+ | |||

+ | |||

+ | And two passive players (the firm and society) represented by nature. Game payoffs are to the firm and society and employees and managers payoffs are a function of these, depending on their types. High types (of both employees and managers) value society's payoffs more highly. | ||

+ | |||

+ | |||

+ | Violations are: | ||

+ | *Observed privately by employees | ||

+ | *<math>v \in [0,1]\,</math> where <math>v=0\,</math> represents no violation | ||

+ | *distributed according to <math>G(\cdot)\,</math> which is exogenous and has strictly positive density | ||

+ | |||

+ | |||

+ | Managers have: | ||

+ | *Types <math>s \in [0,1]\,</math> distributed according to the prior pdf <math>\beta\,</math> which is common knowledge | ||

+ | *Announce a whistleblowing policy <math>C(h,s) \ge 0\,</math> where <math>h\,</math> is the history of event, and <math>C(\cdot,\cdot)\,</math> is a schedule of penalities that will be imposed on the employee. These penalties are bounded <math>\overline(c) \ge C(h,s) \ge 0\,</math>, and costless to implement for the manager. | ||

+ | *The ability to make credible threats - they can commit to the penalty schedule above. | ||

+ | *A history consists of the action by an employee, the response by the manager, and <math>v\,</math> if it becomes known to the manager. | ||

+ | |||

+ | |||

+ | Employees have: | ||

+ | *Types <math>t \in [0,1]\,</math> distributed according to the prior pdf <math>\eta\,</math> which is common knowledge | ||

+ | *Responses: | ||

+ | **<math>\phi\,</math> - do nothing | ||

+ | **<math>p\,</math> - privately report | ||

+ | **<math>w\,</math> - whistleblow: report the violation publicly | ||

+ | |||

+ | |||

+ | The sequence of the model is: | ||

+ | *<math>v, t, s\,</math> are drawn by nature | ||

+ | *<math>v,t\,</math> are revealed to the employee, <math>s\,</math> is revealed to the manager | ||

+ | *The manager announces <math>C(h,s)\,</math> | ||

+ | *A type <math>t\,</math> employee chooses from <math>a_e(v,t) \in \{\phi,p.w\}\,</math> | ||

+ | **If <math>\phi\,</math> then nature reveals the violation with probability <math>q_{\phi}v\,</math>, where <math>q_{\phi} \in \left [0,1\right)\,</math>. The revelation of the information by nature is modelled as <math>\Omega_{\phi} \in \{0,1\}\,</math> | ||

+ | **If <math>p\,</math> then the manager chooses whether or not to fix the violation. Fixing costs the firm <math>\alpha v\,</math>, <math>\alpha >0\,</math>. Let <math>a_m(v,s) \in \{f,\sim f\}\,</math>. If not fix (<math>\sim f\,</math>), then nature chooses to reveal the violation with probability <math>q_pv\,</math>, where <math>q_p \in (q_{\phi},1)\,</math>, and again the revelation is modelled as <math>\Omega_{p} \in \{0,1\}\,</math>. If the violation is fixed it is never revealed. | ||

+ | **If <math>w\,</math>, then <math>v\,</math> becomes common knowledge (denoted <math>\Omega_{w} \equiv 1\,</math>). | ||

+ | |||

+ | There are the following crucial assumptions: | ||

+ | *The probability of a violation becoming known is increasing in the size of the violation | ||

+ | *The probabilities are ordered: <math>q_w = 1 > q_p v > q_{\phi} v > 0\,</math> | ||

+ | |||

+ | |||

+ | The payoffs to society are: | ||

+ | |||

+ | :<math> | ||

+ | \pi_S(h,v) = | ||

+ | \begin{cases} | ||

+ | 0 & \mbox{ if } a_e=p,a_m=f \\ | ||

+ | -\delta v & \mbox{ if } | ||

+ | a_e=\phi,\Omega_{\phi} =1 \mbox{ or } a_e=p,a_m=\sim f, \Omega_{\sim f} =1 \mbox{ or } a_e=w \\ | ||

+ | -v & \mbox{ otherwise} \\ | ||

+ | \end{cases} | ||

+ | \,</math> | ||

+ | |||

+ | The payoffs to the firm are: | ||

+ | |||

+ | :<math> | ||

+ | \pi_F(h,v) = | ||

+ | \begin{cases} | ||

+ | -\alpha v & \mbox{ if } a_e=p,a_m=f \\ | ||

+ | -(\alpha + \delta) v & \mbox{ if } | ||

+ | a_e=\phi,\Omega_{\phi} =1 \mbox{ or } a_e=p,a_m=\sim f, \Omega_{\sim f} =1 \mbox{ or } a_e=w \\ | ||

+ | 0 & \mbox{ otherwise} \\ | ||

+ | \end{cases} | ||

+ | \,</math> | ||

+ | |||

+ | Where <math>\delta > 0\,</math> represents some multiplier of the violation to represent damage done to the firm's reputation beyond the cost of the violation. | ||

+ | |||

+ | |||

+ | The payoffs to the manager are: | ||

+ | |||

+ | :<math>\pi_m(h,v,s) = s \pi_S(h,v) + (1-s) \pi_F(h,v)\,</math> | ||

+ | |||

+ | |||

+ | |||

+ | The payoffs to the employee are: | ||

+ | |||

+ | :<math>\pi_e(h,v,s,t) = t \pi_S(h,v) + (1-t) (\pi_F(h,v) - C(h,s))\,</math> | ||

+ | |||

+ | |||

+ | The solution concept is Perfect Bayesian equilibria in (weakly) undominated strategies. | ||

+ | |||

+ | ==Three Benchmark Cases== | ||

+ | |||

+ | ===No penalties=== | ||

+ | |||

+ | The following two conditions are arrived at by plugging into the managers utility function (which involves plugging into the firm's and society's utility functions): | ||

+ | |||

+ | |||

+ | Assuming an employee reports a violation to a manager, <math>a_e=p\,</math>, then the manager prefers to fix the violation iff <math>v > \hat{v}(s)\,</math> where: | ||

+ | |||

+ | :<math>\hat{v}(1) = 0\,</math> | ||

+ | |||

+ | :<math>\hat{v}(0) = \frac{\alpha}{q_p(\alpha+\delta)}\,</math> | ||

+ | |||

+ | |||

+ | A type <math>s\,</math> manager strictly prefers the employee to report a violation privately rather than stay silent iff <math>v > \dot{v}(s)\,</math> where: | ||

+ | |||

+ | :<math>\dot{v}(1) = 0\,</math> | ||

+ | |||

+ | :<math>\dot{v}(0) = \frac{\alpha}{q_{\phi}(\alpha+\delta)}\,</math> | ||

+ | |||

+ | |||

+ | Furthermore <math>\dot{v}(0) > \hat{v}(0)\,</math> | ||

+ | |||

+ | |||

+ | Essentially: | ||

+ | *A type 1 manager (who cares solely about society): | ||

+ | **Fixes everything reported to her | ||

+ | **Wants everything reported to her | ||

+ | *A type 0 manager (who cares solely about the firm) has the fixing and reporting cutoffs as above. | ||

+ | |||

+ | |||

+ | Thus, depending on the level of the violation and the manager's type there are two cutoffs, that are a function of the employees belief about a manager's type <math>\beta\,</math>: <math>T_{\phi p}\,</math> - the cutoff between keeping silent and reporting internally, and <math>T_{p w}\,</math> - the cutoff between reporting internally and whistleblowing. | ||

+ | |||

+ | |||

+ | Three types of violation: | ||

+ | *High violations: <math>v \in \left ( \dot{v}, 1 \right ]\,</math> | ||

+ | **Any type of manager will prefer that their employee report rather than stay silent | ||

+ | **Employees therefore prefer to report rather than stay silent | ||

+ | **By <math>\dot{v}(0) > \hat{v}(0)\,</math> any type of manager will fix a violation that they are aware of | ||

+ | **Employees therefore prefer to report internally than whistleblow | ||

+ | **<math>T_{\phi p}(v,\beta) = 0\,</math> and <math>T_{p w}(v,\beta) = 1\,</math> | ||

+ | *Medium violations: <math>v \in \left [\hat{v}, \dot{v} \right )\,</math> | ||

+ | **Any type of manager will fix the violation | ||

+ | **Type zero managers would rather not know about the violation | ||

+ | **As no penalties, the employees utility is independent of the managers type | ||

+ | **Therefore all employee prefer to report privately rather than blowing the whistle | ||

+ | **The employee will report rather than staying silent if their type is above the threshold: <math>t > T_{\phi p}(v,\beta)\,</math> | ||

+ | **<math>0 < T_{\phi p}(v,\beta)\,</math> and <math>T_{p w}(v,\beta) = 1\,</math> | ||

+ | *Minor violations: <math>v \in \left [0, \hat{v} \right )\,</math> | ||

+ | **Type 1 managers fix everything, and type 0 fix nothing. Likewise a type 1 manager wants to hear about everything a type 0 about nothing. | ||

+ | **Again the following only holds under the assumption of no penalties, so that the employees actions are indenpendent of the managers type. | ||

+ | **<math>T_{\phi p}(v,\beta)\,</math> is decreasiong in <math>\beta\,</math> in this range. Only employee types <math>t\,</math> greater than the threshold report, and as their belief about the type of the manager increases more violations are reported. | ||

+ | **<math>T_{p w}(v,\beta)\,</math> is increasing in both <math>v\,</math> and <math>\beta\,</math>. Only employee types <math>t\,</math> greater than the threshold whistleblow, and as the pertinent consideration is whether the violation is fixed, and the employee prefers to avoid the reputational cost. | ||

+ | |||

+ | |||

+ | ===Whistleblowing Under a Type 1 Manager=== | ||

+ | |||

+ | Assume that the manager's type is common knowledge, and that managers can impose penalties up to a fixed limit. First we assume that <math>\beta =1\,</math>. | ||

+ | |||

+ | This gives the following results: | ||

+ | *When <math>c\,</math> is sufficiently large, <math>\overline{c} \ge \frac{\alpha \dot{v}}{4}\,</math>, all types report privately. | ||

+ | *The penalty exists only to prevent silence, as employees will rather report privately than whistleblow. | ||

+ | *When <math>c\,</math> is small, there exist low violations and low-types of employees that will not report | ||

+ | **They would rather bear <math>c\,</math> and risk that the violations are made public by nature | ||

+ | |||

+ | |||

+ | The manager's optimal punishment is: | ||

+ | |||

+ | :<math> | ||

+ | C(a_e,1) = | ||

+ | \begin{cases} | ||

+ | \overline{c} & \mbox{ if } a_e = \phi \\ | ||

+ | 0 & mbox{ otherwise} | ||

+ | \end{cases} | ||

+ | \,</math> | ||

+ | |||

+ | |||

+ | ===Whistleblowing Under a Type 0 Manager=== | ||

+ | |||

+ | The case for a low type manager is more complex. From proposition 4 in the paper, the manager's optimal punishment is: | ||

+ | |||

+ | :<math> | ||

+ | C(a_e,0) = | ||

+ | \begin{cases} | ||

+ | \overline{c} & \mbox{ if } a_e = p \mbox{ and } v \in \left [\hat{v},\dot{v} \right ) \\ | ||

+ | c(v) \in [\tilde{c}(v),\overline{c}] & \mbox{ if } a_e = p \mbox{ and } v < \hat{v} \\ | ||

+ | \overline{c} & \mbox{ if } a_e = w \mbox{ and } v < \dot{v} \\ | ||

+ | 0 & \mbox{ otherwise} | ||

+ | \end{cases} | ||

+ | \,</math> | ||

+ | |||

+ | where <math>\tilde{c}(v) = \frac{(q_p - q_{\phi})v}{(1 - q_{\phi}v)}\overline{c}\,</math> | ||

+ | |||

+ | This is somewhat complicated and best explained as follows: | ||

+ | *For all violations in <math>v \ge \dot{v}\,</math> are reported privately to the manager. | ||

+ | *For any violation <math>v \in \left[\hat{v},\dot{v} \right)\,</math> the employee reports privately if his type is above a cutoff, and otherwise stays silent | ||

+ | *For any violation <math>v \in \left(0,\hat{v}\right)\,</math> and <math>c \ge \tilde{c}(v)\,</math> as a punishment for reporting privately, the employee stays silent if his type is below a cutoff, and otherwise blows the whistle. | ||

+ | |||

+ | |||

+ | ==Seperating Equilibria== | ||

+ | |||

+ | Let the punishment strategy for the type 1 manager above be denoted <math>C_1^*\,</math> and the punishment strategy for the type 0 manager above be denoted <math>C_0^*\,</math>. Supposing that both of these strategies were available in the set <math>C\,</math>, and that employees observed managers announcing these punishment strategies, then the employee could believe that contingent on observing a <math>C_1^*\,</math> strategy that he is facing a type 1 employee and that contingent on observing a <math>C_0^*\,</math> strategy that he is facing a type 0 employee. Proposition 5 in the paper, on p34, uses this to prove that there can exist a seperating equilibrium with these characteristics. | ||

+ | |||

+ | |||

+ | ==Conclusions from the Paper== | ||

+ | |||

+ | The following is a list of bullet point conclusions taken directly from the paper: | ||

+ | #Violations can be partitioned into three sets, of which only the "lowest" need be non- | ||

+ | empty | ||

+ | #The "highest" violations are invariably reported privately by all employee types to either type of manager who surely fixes the violation and does not penalize the employee | ||

+ | #The "moderate" violations are fixed by both types of manager conditional on being privately informed, but only the high type manager prefers to be told of such violations whereas the low type manager penalizes any private reporting of "moderate" or "low" violations | ||

+ | #Only the high type manager fixes "low" violations when informed but not all employee types are willing to report all such violations; the low type manager prefers not to be informed and does not fix the violation if the employee nevertheless reports it privately; and sufficiently high employee types blow the whistle on "low" violations. | ||

+ | #Only the low type manager penalizes whistleblowing or private reporting; the high type manager only penalizes remaining silent. Moreover, not all low or moderate violations are reported to either type of manager, even when there are no penalties for any action. |

## Latest revision as of 18:14, 29 September 2020

Article | |
---|---|

Has bibtex key | |

Has article title | Public Disclosure Private Revelation Or Silence |

Has author | Austensmith Feddersen |

Has year | 2008 |

In journal | |

In volume | |

In number | |

Has pages | |

Has publisher | |

© edegan.com, 2016 |

- This page is referenced in BPP Field Exam Papers

## Contents

## Reference(s)

Austen-Smith, D. and T. Feddersen (2008), Public Disclosure, Private Revelation or Silence: Whistleblowing Incentives and Managerial Policy, Kellogg School of Management Working Paper. pdf (Class Handout and Class Slides)

## Abstract

The public revelation of organizational wrongdoing by insiders, whistleblowing, is widely reported, economically significant and can be extremely costly to the whistleblowers. We develop a model of whistleblowing involving a manager and an employee. Each has a privately known type that specifes the relative weight placed on social rather than personal payoffs. The manager chooses a whistleblowing policy consisting of conditional penalties for various employee actions; the employee observes the policy and chooses between saying nothing, revealing a (privately observed) socially costly violation to the manager, or whistleblowing. Given common knowledge of manager types we characterize equilibrium whistleblowing policies and employee behavior. We show that there may be a nonmonotonic relationship between the severity of the violation and the likelihood of whistleblowing. When manager types are private information we provide suffcient conditions for a separating equilibrium. Managerial choice of whistleblowing policies thus serves a dual purpose: providing incentives for reporting violations and providing information to employees regarding the willingness of the manager to fix violations that are privately reported.

## The Model

There are two active players:

- Employees
- Managers

And two passive players (the firm and society) represented by nature. Game payoffs are to the firm and society and employees and managers payoffs are a function of these, depending on their types. High types (of both employees and managers) value society's payoffs more highly.

Violations are:

- Observed privately by employees
- [math]v \in [0,1]\,[/math] where [math]v=0\,[/math] represents no violation
- distributed according to [math]G(\cdot)\,[/math] which is exogenous and has strictly positive density

Managers have:

- Types [math]s \in [0,1]\,[/math] distributed according to the prior pdf [math]\beta\,[/math] which is common knowledge
- Announce a whistleblowing policy [math]C(h,s) \ge 0\,[/math] where [math]h\,[/math] is the history of event, and [math]C(\cdot,\cdot)\,[/math] is a schedule of penalities that will be imposed on the employee. These penalties are bounded [math]\overline(c) \ge C(h,s) \ge 0\,[/math], and costless to implement for the manager.
- The ability to make credible threats - they can commit to the penalty schedule above.
- A history consists of the action by an employee, the response by the manager, and [math]v\,[/math] if it becomes known to the manager.

Employees have:

- Types [math]t \in [0,1]\,[/math] distributed according to the prior pdf [math]\eta\,[/math] which is common knowledge
- Responses:
- [math]\phi\,[/math] - do nothing
- [math]p\,[/math] - privately report
- [math]w\,[/math] - whistleblow: report the violation publicly

The sequence of the model is:

- [math]v, t, s\,[/math] are drawn by nature
- [math]v,t\,[/math] are revealed to the employee, [math]s\,[/math] is revealed to the manager
- The manager announces [math]C(h,s)\,[/math]
- A type [math]t\,[/math] employee chooses from [math]a_e(v,t) \in \{\phi,p.w\}\,[/math]
- If [math]\phi\,[/math] then nature reveals the violation with probability [math]q_{\phi}v\,[/math], where [math]q_{\phi} \in \left [0,1\right)\,[/math]. The revelation of the information by nature is modelled as [math]\Omega_{\phi} \in \{0,1\}\,[/math]
- If [math]p\,[/math] then the manager chooses whether or not to fix the violation. Fixing costs the firm [math]\alpha v\,[/math], [math]\alpha \gt 0\,[/math]. Let [math]a_m(v,s) \in \{f,\sim f\}\,[/math]. If not fix ([math]\sim f\,[/math]), then nature chooses to reveal the violation with probability [math]q_pv\,[/math], where [math]q_p \in (q_{\phi},1)\,[/math], and again the revelation is modelled as [math]\Omega_{p} \in \{0,1\}\,[/math]. If the violation is fixed it is never revealed.
- If [math]w\,[/math], then [math]v\,[/math] becomes common knowledge (denoted [math]\Omega_{w} \equiv 1\,[/math]).

There are the following crucial assumptions:

- The probability of a violation becoming known is increasing in the size of the violation
- The probabilities are ordered: [math]q_w = 1 \gt q_p v \gt q_{\phi} v \gt 0\,[/math]

The payoffs to society are:

- [math] \pi_S(h,v) = \begin{cases} 0 & \mbox{ if } a_e=p,a_m=f \\ -\delta v & \mbox{ if } a_e=\phi,\Omega_{\phi} =1 \mbox{ or } a_e=p,a_m=\sim f, \Omega_{\sim f} =1 \mbox{ or } a_e=w \\ -v & \mbox{ otherwise} \\ \end{cases} \,[/math]

The payoffs to the firm are:

- [math] \pi_F(h,v) = \begin{cases} -\alpha v & \mbox{ if } a_e=p,a_m=f \\ -(\alpha + \delta) v & \mbox{ if } a_e=\phi,\Omega_{\phi} =1 \mbox{ or } a_e=p,a_m=\sim f, \Omega_{\sim f} =1 \mbox{ or } a_e=w \\ 0 & \mbox{ otherwise} \\ \end{cases} \,[/math]

Where [math]\delta \gt 0\,[/math] represents some multiplier of the violation to represent damage done to the firm's reputation beyond the cost of the violation.

The payoffs to the manager are:

- [math]\pi_m(h,v,s) = s \pi_S(h,v) + (1-s) \pi_F(h,v)\,[/math]

The payoffs to the employee are:

- [math]\pi_e(h,v,s,t) = t \pi_S(h,v) + (1-t) (\pi_F(h,v) - C(h,s))\,[/math]

The solution concept is Perfect Bayesian equilibria in (weakly) undominated strategies.

## Three Benchmark Cases

### No penalties

The following two conditions are arrived at by plugging into the managers utility function (which involves plugging into the firm's and society's utility functions):

Assuming an employee reports a violation to a manager, [math]a_e=p\,[/math], then the manager prefers to fix the violation iff [math]v \gt \hat{v}(s)\,[/math] where:

- [math]\hat{v}(1) = 0\,[/math]

- [math]\hat{v}(0) = \frac{\alpha}{q_p(\alpha+\delta)}\,[/math]

A type [math]s\,[/math] manager strictly prefers the employee to report a violation privately rather than stay silent iff [math]v \gt \dot{v}(s)\,[/math] where:

- [math]\dot{v}(1) = 0\,[/math]

- [math]\dot{v}(0) = \frac{\alpha}{q_{\phi}(\alpha+\delta)}\,[/math]

Furthermore [math]\dot{v}(0) \gt \hat{v}(0)\,[/math]

Essentially:

- A type 1 manager (who cares solely about society):
- Fixes everything reported to her
- Wants everything reported to her

- A type 0 manager (who cares solely about the firm) has the fixing and reporting cutoffs as above.

Thus, depending on the level of the violation and the manager's type there are two cutoffs, that are a function of the employees belief about a manager's type [math]\beta\,[/math]: [math]T_{\phi p}\,[/math] - the cutoff between keeping silent and reporting internally, and [math]T_{p w}\,[/math] - the cutoff between reporting internally and whistleblowing.

Three types of violation:

- High violations: [math]v \in \left ( \dot{v}, 1 \right ]\,[/math]
- Any type of manager will prefer that their employee report rather than stay silent
- Employees therefore prefer to report rather than stay silent
- By [math]\dot{v}(0) \gt \hat{v}(0)\,[/math] any type of manager will fix a violation that they are aware of
- Employees therefore prefer to report internally than whistleblow
- [math]T_{\phi p}(v,\beta) = 0\,[/math] and [math]T_{p w}(v,\beta) = 1\,[/math]

- Medium violations: [math]v \in \left [\hat{v}, \dot{v} \right )\,[/math]
- Any type of manager will fix the violation
- Type zero managers would rather not know about the violation
- As no penalties, the employees utility is independent of the managers type
- Therefore all employee prefer to report privately rather than blowing the whistle
- The employee will report rather than staying silent if their type is above the threshold: [math]t \gt T_{\phi p}(v,\beta)\,[/math]
- [math]0 \lt T_{\phi p}(v,\beta)\,[/math] and [math]T_{p w}(v,\beta) = 1\,[/math]

- Minor violations: [math]v \in \left [0, \hat{v} \right )\,[/math]
- Type 1 managers fix everything, and type 0 fix nothing. Likewise a type 1 manager wants to hear about everything a type 0 about nothing.
- Again the following only holds under the assumption of no penalties, so that the employees actions are indenpendent of the managers type.
- [math]T_{\phi p}(v,\beta)\,[/math] is decreasiong in [math]\beta\,[/math] in this range. Only employee types [math]t\,[/math] greater than the threshold report, and as their belief about the type of the manager increases more violations are reported.
- [math]T_{p w}(v,\beta)\,[/math] is increasing in both [math]v\,[/math] and [math]\beta\,[/math]. Only employee types [math]t\,[/math] greater than the threshold whistleblow, and as the pertinent consideration is whether the violation is fixed, and the employee prefers to avoid the reputational cost.

### Whistleblowing Under a Type 1 Manager

Assume that the manager's type is common knowledge, and that managers can impose penalties up to a fixed limit. First we assume that [math]\beta =1\,[/math].

This gives the following results:

- When [math]c\,[/math] is sufficiently large, [math]\overline{c} \ge \frac{\alpha \dot{v}}{4}\,[/math], all types report privately.
- The penalty exists only to prevent silence, as employees will rather report privately than whistleblow.
- When [math]c\,[/math] is small, there exist low violations and low-types of employees that will not report
- They would rather bear [math]c\,[/math] and risk that the violations are made public by nature

The manager's optimal punishment is:

- [math] C(a_e,1) = \begin{cases} \overline{c} & \mbox{ if } a_e = \phi \\ 0 & mbox{ otherwise} \end{cases} \,[/math]

### Whistleblowing Under a Type 0 Manager

The case for a low type manager is more complex. From proposition 4 in the paper, the manager's optimal punishment is:

- [math] C(a_e,0) = \begin{cases} \overline{c} & \mbox{ if } a_e = p \mbox{ and } v \in \left [\hat{v},\dot{v} \right ) \\ c(v) \in [\tilde{c}(v),\overline{c}] & \mbox{ if } a_e = p \mbox{ and } v \lt \hat{v} \\ \overline{c} & \mbox{ if } a_e = w \mbox{ and } v \lt \dot{v} \\ 0 & \mbox{ otherwise} \end{cases} \,[/math]

where [math]\tilde{c}(v) = \frac{(q_p - q_{\phi})v}{(1 - q_{\phi}v)}\overline{c}\,[/math]

This is somewhat complicated and best explained as follows:

- For all violations in [math]v \ge \dot{v}\,[/math] are reported privately to the manager.
- For any violation [math]v \in \left[\hat{v},\dot{v} \right)\,[/math] the employee reports privately if his type is above a cutoff, and otherwise stays silent
- For any violation [math]v \in \left(0,\hat{v}\right)\,[/math] and [math]c \ge \tilde{c}(v)\,[/math] as a punishment for reporting privately, the employee stays silent if his type is below a cutoff, and otherwise blows the whistle.

## Seperating Equilibria

Let the punishment strategy for the type 1 manager above be denoted [math]C_1^*\,[/math] and the punishment strategy for the type 0 manager above be denoted [math]C_0^*\,[/math]. Supposing that both of these strategies were available in the set [math]C\,[/math], and that employees observed managers announcing these punishment strategies, then the employee could believe that contingent on observing a [math]C_1^*\,[/math] strategy that he is facing a type 1 employee and that contingent on observing a [math]C_0^*\,[/math] strategy that he is facing a type 0 employee. Proposition 5 in the paper, on p34, uses this to prove that there can exist a seperating equilibrium with these characteristics.

## Conclusions from the Paper

The following is a list of bullet point conclusions taken directly from the paper:

- Violations can be partitioned into three sets, of which only the "lowest" need be non-

empty

- The "highest" violations are invariably reported privately by all employee types to either type of manager who surely fixes the violation and does not penalize the employee
- The "moderate" violations are fixed by both types of manager conditional on being privately informed, but only the high type manager prefers to be told of such violations whereas the low type manager penalizes any private reporting of "moderate" or "low" violations
- Only the high type manager fixes "low" violations when informed but not all employee types are willing to report all such violations; the low type manager prefers not to be informed and does not fix the violation if the employee nevertheless reports it privately; and sufficiently high employee types blow the whistle on "low" violations.
- Only the low type manager penalizes whistleblowing or private reporting; the high type manager only penalizes remaining silent. Moreover, not all low or moderate violations are reported to either type of manager, even when there are no penalties for any action.