QuantyFrog | The Embezzler's Game

The game described in this paper has been adapted from the cash-flow diversion game in (Grenadier and Malenko, 2011). However, it discretizes the processes described in the paper to ask the research question, "Under what conditions would managers be incentivized to divert free cash flow from the shareholders and understate payoff projects?"

This is important especially in terms of corporate finance where managers may intentionally choose to delay a project to benefit themselves. It helps in understanding each part of a corporation as an individual agent with their own incentives and beliefs.

While "discretization" is an all-encompassing term, this project particularly differentiates itself from (Grenadier and Malenko, 2011) in a variety of ways: it changes the continuous time distribution structure of the game and turns it into a 2 period model, replaces the differential equations with simpler payoffs, and transforms the action space from a continuous path of diversion to one that has only two options (early and late).

As for the game itself, there are 2 players (a Manager and Shareholders) in which a manager can invest early (day 0) or late (day 1). The project's observable value P grows in such a manner that $P_0 < P_1$. The manager privately knows the sunk cost with $\text{Pr}(\theta_L) = p$. After investment, they own an equity fraction $\alpha \in (0,1)$ and can try to divert from the cash flow for personal consumption. Diversion is risky as the shareholders may audit at cost c recovering the theft. If the shareholders believe that the project is expensive ($\theta$) they are less likely to audit and the manager can pocket a larger amount. Thus, the manager would prefer to think $\theta = \theta_H$ and therefore, prefer to delay investment.

The Extensive Form

Nature | ┌───────────┴───────-────┐ │ │ θ_H (1-p) θ_L (p) │ │ ┌─────┴─────┐ ┌─────┴─────┐ Early │ │ Late Early │ │ Late │ │ │ │ ● ● ● ● Shareholders observe action and form posterior belief θ̃: θ̃ = θ_L if action = Early θ̃ = θ_H if action = Late Shareholders choose audit probability ψ*(θ̃) Manager chooses diversion amount d = D(ψ* + θ - θ̃)

Manager's Equity Value:

\text{Equity}_M(\theta) = \begin{cases} \alpha(P_0 - \theta), & t = 0 \\ \delta \alpha(P_1 - \theta), & t = 1 \end{cases}

Skim Function:

\text{skim}(\tilde{\theta}, \theta) = (1-\alpha) D(\tilde{\theta}, \theta)

Game Sequence

Sequence

Nature draws $\theta \in \{\theta_L, \theta_H\}$ and tells the manager only
Period 0 - Manager's move:
Action $a \in \{\text{Early}, \text{Late}\}$
Outsiders observe $a$ and form the posterior belief: $\tilde{\theta} = \begin{cases} \theta_L, & \text{if } a = \text{early} \\ \theta_H, & \text{if } a = \text{late} \end{cases}$
Audit decision: Given $\tilde{\theta}$, shareholders choose an audit cutoff $\psi^*(\tilde{\theta})$ that maximizes their expected payoff.
Cash flow realization: If the manager invested early, cash flow $P_0 - \theta$ is realized. If they wait, $P_1 - \theta$ realizes with discount factor $\delta$.
Diversion: Before flow is observed, manager picks an amount $d = D(\psi^* + \theta - \tilde{\theta})$ to skim.
Audit outcome: Shareholders verify with probability $q_T \psi(\psi^* + D(\psi^*))$ and recover $d$ if they catch the manager.

Key Payoff Components

$\lambda \in (0,1)$: Fraction of each diverted dollar the manager actually enjoys

Diversion rent:

B(\tilde{\theta}, \theta) = D(\psi^*, \tilde{\theta} + \theta)[1 - \psi(\psi^*, \tilde{\theta} + \theta + D(\cdot))] = D(1-q_T)

Analysis

Pooled Equilibriums

(i) Pooling on "Early":

If both types choose Early, then shareholders seeing early will form the belief that the manager is average on cost. Thus:

\tilde{\theta} = p\theta_H + (1-p)\theta_L

\psi^*(\tilde{\theta}) = (1-\alpha)B(\tilde{\theta}, \theta)

In this case, the high cost manager has a profitable deviation to late triggering $\tilde{\theta} = \theta_H$ which lowers $\psi^*$ and $D^*$.

Payoff (Early):

U_E(\theta_H) = \alpha(P_0 - \theta_H) + (1-q_T(\xi, \theta_H))(1-\alpha)D(\tilde{\theta}, \theta_H)

Deviation (Late):

U_L(\theta_H) = \delta\alpha(P_1 - \theta_H) + (1-q_T(\theta_H, \theta_H))(1-\alpha)D(\theta_H, \theta_H)

Since deviating under the high cost belief such that $U_L(\theta_H) > U_E(\theta_H)$ exists, the high type would always deviate.

(ii) Pooling on "Late":

If both types delay, then the low cost type would prefer to deviate to early.

Deviation (Early):

U_E(\theta_L) = \alpha(P_0 - \theta_L) + (1-q_T(\theta_L, \theta_L))(1-\alpha)D(\theta_L, \theta_L)

Payoff (Late):

U_L(\theta_L) = \delta\alpha(P_1 - \theta_L) + (1-q_T(\tilde{\theta}, \theta_L))(1-\alpha)D(\tilde{\theta}, \theta_L)

Separating Equilibrium

Since $U_E(\theta_L) > U_L(\theta_L)$, there would be no pooling and the low cost would deviate. Thus, there are no pooled equilibriums.

We assume that $\theta_H$ chooses late and $\theta_L$ chooses early. If early, $\tilde{\theta} = \theta_L$ and if late $\tilde{\theta} = \theta_H$.

Audits are set at $\psi^*(\theta_L)$ and $\psi^*(\theta_H)$ optimally, which sets the diversion rents at:

B(\theta_L, \theta_L) = D(\theta_L, \theta_L)(1 - \psi(\psi^*(\theta_L) + D(\theta_L, \theta_L)))

= D(\theta_L, \theta_L)(1 - q_T(\theta_L, \theta_L))

Similarly:

B(\theta_H, \theta_L) = D(\theta_H, \theta_L)(1 - q_T(\theta_H, \theta_L))

B(\theta_H, \theta_H) = D(\theta_H, \theta_H)(1 - q_T(\theta_H, \theta_H))

Low type (no deviation to late):

U_E(\theta_L) > U_L(\theta_L)

\Leftrightarrow \alpha(P_0 - \theta_L) + (1-\alpha)B(\theta_L, \theta_L) > \delta\alpha(P_1 - \theta_L) + (1-\alpha)B(\theta_H, \theta_L)

Since the shareholders believe $\theta_L$, they think the diversion rent on the right side of the inequality is smaller.

High type (no deviation to early):

U_L(\theta_H) > U_E(\theta_H)

\Leftrightarrow \delta\alpha(P_1 - \theta_H) + (1-\alpha)B(\theta_H, \theta_H) > \alpha(P_0 - \theta_H) + (1-\alpha)B(\theta_L, \theta_H)

The same logic applies where shareholders believe $\theta_H$.

Equilibrium Result

Therefore, the low type would choose $U_E(\theta_L)$ and high type would choose $U_L(\theta_H)$ always. This causes a separating equilibrium.

Conclusion

In conclusion, I think the separated equilibrium analysis shows the conditions that would incentivize managers to embezzle money from the shareholders based on their own nature while choosing if they want to delay investment or not. Considering that every player is a rational individual choosing to maximize their utility in the given time period, I believe that they would always choose these options in a 2 period bayesian model.

References

Grenadier, S. R., and A. Malenko. "Real Options Signaling Games with Applications to Corporate Finance." Review of Financial Studies 24.12 (2011): 3993–4036. Web.
Version: Author's final manuscript.