Occurrence of Deception Under the Oversight of a Regulator Having Reputation Concerns

Abstract

This paper studies deceptions conducted by agents in the presence of a regulator. The regulator is supposed to detect deviations from the “rightful” behavior through costly monitoring; thus she may not choose to be diligent in her job because of the associated costs. The goal is to understand the occurrence of deceptions when the interaction of the parties is not contractible, their behavior is not observable and the regulator has reputation concern for being perceived as diligent in a repeated incomplete-information setting. It is found that when the regulator faces a sequence of myopic agents, her payoff at any Nash equilibrium converges to the maximum payoff as the discount factor approaches to one for any prior belief on the regulator’s type. This suggests that, contrary to the well-known disappearance of reputation results in the literature, the reputation of the regulator for being diligent persists in the long-run in any equilibrium. These findings imply that socially undesirable behavior of the agents could be prevented through reputation concerns in this repeated setting.

Appendices

Appendix

A.1 The Proof of Lemma 1

The utility of agent by being truthful is u _A (T, σ _R ) = 0. And, her expected utility by being untruthful is

$$\displaystyle{ u_{A}(U,\sigma _{R}) =\gamma [(1-\beta )g -\beta l] + (1-\gamma )\sigma _{R}[(1-\beta )g -\beta l] + (1-\gamma )(1 -\sigma _{R})g. }$$

The agent’s best response correspondence against the strategic regulator’s strategy (which implies the cutoff detection probability) is given by:

$$ \displaystyle\begin{array}{rcl} \sigma _{A} \equiv BR_{A}(\sigma _{R}) = \left \{\begin{array}{@{}l@{\quad }l@{}} 1 \quad &\quad \text{if}\quad \sigma _{R} > \frac{g-\gamma \beta (g+l)} {(1-\gamma )\beta (g+l)} \\{} [0,1]\quad &\quad \text{if}\quad \sigma _{R} = \frac{g-\gamma \beta (g+l)} {(1-\gamma )\beta (g+l)} \\ 0 \quad &\quad \text{if}\quad \sigma _{R} < \frac{g-\gamma \beta (g+l)} {(1-\gamma )\beta (g+l)}. \end{array} \right.& &{}\end{array}$$

(A.1)

From this, we can deduce the cutoff prior beliefs. The strategy of the regulator that makes the agent indifferent between being truthful and untruthful, $\sigma _{R} = \frac{g-\gamma \beta (g+l)} {(1-\gamma )\beta (g+l)}$, is greater than 0 if $\gamma <\gamma ^{{\ast}} = \frac{g} {\beta (g+l)}$ and equals to 0 if γ = γ ^∗. If γ > γ ^∗, then BR _A (σ _R ) = 1 for any value of σ _R , i.e., even if the strategic regulator is lazy for sure.

The expected utility of the regulator by choosing to be diligent is u _R (σ _A , D) = (1 −σ _A )[β d − (1 −β)f] − c, whereas his expected utility by choosing to be lazy is u _R (σ _A , L) = −(1 −σ _A )f. Thus, regulator’s best response is given by:

$$\displaystyle\begin{array}{rcl} \sigma _{R} \equiv BR_{R}(\sigma _{A}) = \left \{\begin{array}{@{}l@{\quad }l@{}} 1 \quad &\quad \text{if}\quad \sigma _{A} < 1 - \frac{c} {\beta (d+f)} \\{} [0,1]\quad &\quad \text{if}\quad \sigma _{A} = 1 - \frac{c} {\beta (d+f)} \\ 0 \quad &\quad \text{if}\quad \sigma _{A} > 1 - \frac{c} {\beta (d+f)}. \end{array} \right.& &{}\end{array}$$

(A.2)

The strategy of agent that makes regulator indifferent between choosing to be diligent and lazy, $\sigma _{A} = 1 - \frac{c} {\beta (d+f)}$, is greater than 0 if $\beta > \frac{c} {f+d}$ (which holds by assumption).

Case 1:

γ > γ ^∗: In this case, BR _A (σ _R ) = 1 for any σ _R . The unique fixed point of the best response correspondences is σ _A  = 1 and σ _R  = 0.

Case 2:

γ = γ ^∗: The strategy that makes the agent indifferent between telling the truth and lying is σ _R  = 0. For σ _R  > 0, BR _A (σ _R ) = 1. But, against σ _A  = 1, σ _R  > 0 cannot be a best response. Thus, the equilibrium strategies are $\sigma _{A} \in [1 - \frac{c} {\beta (d+f)},1]$ and σ _R (D) = 0. As we have assumed that the agent is truthful for sure when she is indifferent σ _A  = 1 and σ _R  = 0 in this case.

Case 3:

γ < γ ^∗: The unique intersection of the best response correspondences in this case is when $\sigma _{A} = 1 - \frac{c} {\beta (d+f)}$ and $\sigma _{R} = \frac{g-\gamma \beta (g+l)} {(1-\gamma )\beta (g+l)}$.

A.2 The Proofs of Lemmas 2–5

Proof (Proof of Lemma 2).

Given that agent chooses σ _A, t (h ^t) = 1 at h ^t, choosing diligent or lazy generates the same distribution of public signals so that the continuation payoff v(h ^t, i _d ) will be the same. Since the one-period utility u _R (T, L) = 0 > u _R (T, L) = −c, we conclude that σ _R, t (h ^t) = 0 complying with the one-shot deviation principle.

Proof (Proof of Lemma 3).

Suppose that there exists a Nash equilibrium with a positive probability history h ^t at which σ _R, t (h ^t) = 1. If agent is truthful with probability one at h ^t, then the regulator would have one-shot deviation gain by switching to lazy since the distribution of public signals, the posterior belief, and thus the continuation payoff would not change regardless of regulator’s action choice when the agent is truthful. And, the agent chooses to be untruthful with some probability on a Nash equilibrium only when the belief γ _t−1(h ^t) at t − 1 is less than γ ^∗ and the regulator chooses to be diligent by less than $\bar{\sigma }_{R} = \frac{\pi ^{{\ast}}-\beta \gamma } {\beta (1-\gamma )} < 1$.

Proof (Proof of Lemma 4).

Consider an arbitrary public history h ^t that is reached with positive probability with respect to some Nash equilibrium. Suppose for a contradiction that for every τ ≥ t and every h ^τ that comes after h ^t, σ _R, τ (h ^τ) = 0. Given γ _t−1(h ^t) < γ ^∗ and there will not be any detections after h ^t for every τ ≥ t and history since σ _R, τ (h ^τ) = 0 by hypothesis, and thus the expected probability of detection is going to be less than π ^∗ for every τ ≥ t and h ^τ. Thus, all the myopic agents are untruthful at every date and history starting h ^t. Thus, the regulator’s expected continuation payoff becomes v(h ^t) = −f which is less than the minmax payoff − e, providing us the desired contradiction.

Proof (Proof of Lemma 5).

First, note that in order to observe a detection at time t, the agent must have been untruthful. For the agent to have chosen untruthfulness with some positive probability at h ^t, the expected probability of detection must be lower than π ^∗. Given that the belief at the beginning of time t at h ^t is γ ≡ γ _t−1 (h ^t) < γ ^∗, this requires that $\sigma _{R,t}(h^{t}) \leq \frac{\pi ^{{\ast}}-\beta \gamma } {\beta (1-\gamma )}$, which is derived from (2). And, σ _R, t (h ^t) must be greater than zero because otherwise there would not be any detections. Lastly, it is easy to see from expression (3) that the smallest posterior is obtained when $\sigma _{R,t}(h^{t}) =\bar{\sigma } _{R}$ and equals to $\varGamma (\gamma ) = \frac{\gamma \beta } {\pi ^{{\ast}}}$.