LAS TIC EN LA ESTRUCTURA DE LAS ORGANIZACIONES

We now turn to the Bayesian Nash equilibria of the game. We will start by assessing the baseline case of public signals, in order to compare it to the solution of the game with incomplete information.

4.2.1 Baseline case with public signals

As with our previous model, it is optimal for firm i to apply for leniency if firm j does.⁵¹ Therefore, either both or none of the firms apply for leniency in equilibrium. If firm j does not apply for leniency, it is optimal for firm i to do the same if and only if the following incentive compatibility constraint (ICC) holds:

θF ≥E[ρ|s_i, s_j]F θ ≥E[ρ|si, sj]

E[ρ|s_i, s_j] −θ ≤0 (6)

The ICC in (6) states that the expected punishment from not applying is smaller than the expected punishment from applying (and receiving leniency for sure).

We therefore get the following set of Bayesian Nash equilibria of the game:

ψ(si, sj) =







Apply (A) ,(si, sj) /∈Ω Not Apply (NA) ,(s_i, s_j) ∈Ω

51By also applying, firm i will lower its expected penalty from F (for sure) to 1+θ 2 F.

where

E[ρ|si, sj] −θ ≤0 ,∀(si, sj) ∈Ω

Ωis the set of signals that satisfy the ICC in equation (6). In words; for all combinations of signals, where the expected probability of conviction is lower than the degree of leniency (θ), the firms do not apply for leniency. For the remaining combinations of signals, both firms apply.

When E[ρ|si, sj] = θ, the firms do not apply for leniency. However, with a slight increase in one of the signals, both firms apply. As with our previous model, this causes a jump in expected penalties; now from E[ρ|_{si, sj}]F = θF to  1+θ

, both firms apply for leniency. At the same time, they have higher expected penalties than they would if they did not apply.⁵²

4.2.2 The case of incomplete information

In the situation with incomplete information, a firm bases its decision solely on its own signal. We must therefore find a cut-off value, x, for the signal. When the signal is larger than the cut-off, the probability of conviction is likely to be high. Therefore, the firm applies for leniency. If the signal is below the cut-off, the firm does not apply. We can express the firms’ symmetric strategy as:

φ(s_i) =

We can now find the set of x for which φ(si)constitutes a Bayesian Nash equilibrium.⁵³ As in 4.2.1, we need to assess the incentive compatibility constraint for NA, in order to determine the set of signals for which both firms choosing NA is an equilibrium:

Expected penalty from playing NA≤Expected penalty from playing A

52This is because it is not incentive compatible for them to choose NA, as each of them has an incentive to deviate. By applying for leniency, the deviating firm would get leniency for sure. As θF<E[ρ|si, sj]F, it could lower its expected punishment by deviating. Thus, it is not an equilibrium for the firms to choose NA.

53Note that as there is a set of x’s, there is also a set of Bayesian Nash equilibria. Harrington (2013) briefly discusses the Pareto ranking of the equilibria, and concludes that firms prefer the equilibrium with the highest cut-off x.

If firm i does not apply for leniency, one of three contingencies may occur. Firstly, firm j may apply for leniency, and firm i has to pay the fine F. Secondly, firm j may also choose not to apply, but the firms are discovered and prosecuted by the CA. They both have to pay F. Finally, firm j may choose not to apply, the firms may not be discovered and no fines are paid. We therefore get the following expected penalty from playing NA:

Prob. that firm j does not apply

·0 (7)

As H(x|si)is the probability that firm j receives a signal smaller than the cut-off x, it is also the probability that firm j does not apply for leniency. Note that this probability is conditional on firm i’s signal. The expected probability of conviction does not only depend on s_i, but it is also necessary that firm j does not apply for leniency. The expectation is therefore also conditional on sjbeing under the cut-off.

If firm i applies for leniency, conviction is certain. However, it is not certain whether or not firm j also applies. If firm i is the only one to apply, it receives leniency and must pay θF. If both firms apply, they face an expected penalty of 1+θ

2



F. We therefore get the following expected penalty from playing A:

H(x|si)

We can insert these expected penalties in the incentive compatibility constraint. If the expression in (7) is less than or equal to that in (8), firm i does not apply for leniency.

The ICC for NA is therefore:

[₁−H(x|s_i)]F+H(x|s_i)E[ρ|s_i,{s_j≤ x}]F≤ H(x|s_i)θF+ [₁−H(x|s_i)]^{ 1}+θ 2

 F (9)

Conversely, if the expression in (7) is strictly larger than that in (8), firm i applies for leniency. The ICC for A is therefore:

[1−H(x|s_i)]F+H(x|s_i)E[ρ|s_i,{s_j ≤x}]F >H(x|s_i)θF+ [1−H(x|s_i)]^{ 1}+θ 2

 F (10)

Rewriting either of the ICCs, we can define:

Let us now find the conditions for φ(s_i)to be a Bayesian Nash equilibrium:

1. It must be the case that firm i has no incentive to deviate (i.e. to play NA) when s_i > x. We must therefore have that ∆(s_i, x) > 0 for all s_i > x. As ∆(s_i, x) is strictly increasing in the firm’s signal,⁵⁴ this holds if ∆(x, x) ≥ 0.

2. It must be the case that firm i has no incentive to deviate (i.e. to play A) when s_i ≤ x. We must therefore have that ∆(s_i, x) ≤ 0 for all s_i ≤ x. As ∆(s_i, x) is strictly increasing in the firm’s signal, this holds if ∆(x, x) ≤ 0.

As both of these conditions must hold for φ(s_i) to be a Bayesian Nash equilibrium, we must have ∆(x, x) = 0 (such that x is an equilibrium cut-off). In other words, the ICC for not applying in (9), must hold with equality when the signal received, s_i, is the cut-off value x.

To summarize, we have the following possibilities for equilibrium cut-offs:

• If ∆(x, x) = 0, then x is an equilibrium cut-off.

When firm i receives the cut-off value x as a signal (i.e. when si =x), the expected penalty from choosing A must equal the expected penalty from choosing NA.

• x=s is an equilibrium cut-off.

It means that firms always apply for leniency, regardless of their signal.

Therefore, only condition 1 above must be satisfied, i.e. ∆(x, x) ≥ 0 or

previously discussed, it is always optimal to apply for leniency when the other firm does. Therefore, if one firm always applies for leniency, it is optimal for the other firm to do the same.

• If E[ρ|¯s] ≤θ, then x = ¯s is an equilibrium cut-off.

If the expectation of ρ when firm i receives the highest signal possible still is smaller than θ, never applying for leniency is an equilibrium for the firms.

First note that H(¯s|si) =1. The ICC for NA in (9) therefore yields:

0+E[ρ|s_i, s_j ≤ ¯s]F ≤θF+0⇒ E[ρ|s_i] ≤θ

Given that firm j never applies for leniency, it is optimal for firm i to do the same if (9) holds regardless of the signal firm i receives.

• Furthermore, if E[ρ|¯s] < θ (and θ < E[¯s), ¯s]), then ∆(¯s, ¯s) < 0. Note that, compared to the previous point, this is now a strict inequality. As ∆(s, s) > 0, by continuity there must therefore exist some x = s⁰, s⁰ ∈ [s, ¯s] , such that

∆(s⁰, s⁰) =0. Thus, there are at least three equilibrium cut-offs: x ∈ {s, s⁰, ¯s}.