Compulsory license threats in a signaling game of drug procurement

Abstract

This paper presents a signaling game to formalize the interaction between a developing country and a pharmaceutical firm negotiating the supply of an essential medicine. During these negotiations, the government may threaten the firm with the issue of a compulsory license to force price reductions. However, the threat may be a poor signaling device of the government’s willingness to issue a compulsory license. Our model shows that, for a government, the threat may be used in various ways to fool the pharmaceutical firm about its real objectives. This result is consistent with stylized facts showing that threat strategies are used to obtain very opposite outcomes.

Appendix

This appendix studies the feasibility of the various equilibria considered in the model.

1.1 Separating equilibrium

Let us first consider the separating equilibrium in which a Strong government always waves the CL threat while a Weak one never threatens the Lab. In this equilibrium, a Lab which is threatened by the Government considers that it is matched with a Strong government and, in the opposite case, the Lab assumes that the Government is Weak. Equilibrium beliefs must respect:

$$\begin{aligned} \left\{ \begin{array}{l} P[\tau =S|s =T]=1 \\ P[\tau =S|s =N]=0. \end{array} \right. \end{aligned}$$

(14)

• Lab’s optimal behavior

In this equilibrium, the Government’s decision perfectly reveals its type. Hence, if the Government uses the threat, the Lab expecting to be faced with a Strong government will issue a VL and will make a profit $\pi _{L}(VL)=\alpha P_{VL}$ (if the Lab sets the monopoly price, the government will grant a CL and the Lab’s profit would be $\pi _{L}(CL)=\beta P_{CL}$ with $\pi _{L}(CL)<\pi _{L}(VL)$). In the other case, if the Government does not threaten the Lab, it will be considered as Weak and the Lab will set the monopoly price $P_{M}^{H}(N),$ with higher profits than those under a VL: $\pi _{L}(P_{M}^{H}(N))>\pi _{L}(VL)$.

• Government’s strategy

If, according to the equilibrium strategy, the Strong government waves the CL threat, the Lab will answer by the issuance of a VL and government’s surplus will be $U_{G}^{S}(VL|T)=P_{\max }-P_{VL}-c.$ If the Government deviates and refuses to threaten the Lab, it will be considered as Weak and the Lab will set the monopoly price $P_{M}^{H}(N).$ In turn, the Strong government will be able to grant a CL and will reach the surplus: $U_{G}^{S}(CL)=P_{\max }-P_{CL}-C^{S}.$ This deviation is optimal for this government as, $P_{CL}+C^{S}<$ $P_{VL}+c$. The separating equilibrium is impossible.

Note that a deviation is also optimal for a Weak government. If such a government does not use the CL threat, the Lab will set the monopoly price $P_{M}^{H}(N)$ and the government’s surplus will thus be: $U_{G}^{W}(P_{M}^{H}(N))=P_{\max }-P_{M}^{H}(N).$ If the Government decides to deviate and to threaten the Lab, it will be considered as Strong and the Lab will issue a VL. The government surplus will therefore be $U_{G}^{W}(VL|T)=P_{\max }-P_{VL}-c$. Deviation is optimal as $P_{VL}<P_{M}^{H}(N)-c=P_{M}^{H}(T)$.

1.2 Pooling equilibria

Let us now consider the possibility of pooling equilibria.

1.2.1 Pooling equilibrium with a generalized CL threat

In this equilibrium, both types of governments find it optimal to use the CL threat. Moreover, the Lab considers that a government which refuses to use this threat can only be Weak. Equilibrium beliefs must be:

$$\begin{aligned} \left\{ \begin{array}{l} P[\tau =S|s =T]=q \\ P[\tau =S|s =N]=0. \end{array} \right. \end{aligned}$$

(15)

• Lab’s optimal strategy

Consider first the case where the Government plays the equilibrium strategy and decides to threaten the Lab. In response, a Lab that decides to sell the drug directly and sets the monopoly price $P_{M}^{H}(T)$ expects that the Government will accept the status quo with probability $(1-q)$ (if the Government is Weak) or will impose a CL with probability q (if the Government is Strong). The expected profit of the Lab is:

$$\begin{aligned} E[\pi _{L}(P_{M}^{H}(T))]=q(\beta P_{CL})+(1-q)P_{M}^{H}(T). \end{aligned}$$

(16)

If the Lab decides to issue a VL, its profit will be $\pi _{L}(VL)=\alpha P_{VL}$. This last strategy is the optimal answer to the CL threat if $E[\pi _{L}(P_{M}^{H}(T))|T]<\pi _{L}(VL),$ i.e. if:

$$\begin{aligned} q>q_{1}=\frac{P_{M}^{H}(T)-\alpha P_{VL}}{P_{M}^{H}(T)-\beta P_{CL}}, \end{aligned}$$

(17)

with $q_{1}<1$ as $\beta P_{CL}<\alpha P_{VL}<$ $P_{M}^{H}(T)$ (cf. Equations 1 and 8).

The optimal answer of the Lab to the CL threat thus relies on the frequency of Strong and Weak governments. When q is high, the probability of being faced with a Strong government leads the firm to issue a VL, in the other case, the firm will try to impose a monopoly price.

When a Government deviates and does not use the threat, the Lab considers the Government as Weak and sets the monopoly price, $P_{M}^{H}(N)$.

• Government’s optimal strategy

We must now study the conditions under which a Government deems it optimal to use the threat.

Case 1. $q>q_{1}$

In this first case, a Strong government has no incentive to use the threat.

If the Government threatens the Lab, the issue of a VL is optimal for the pharmaceutical firm and the Government’s surplus will be (whatever its type): $U_{G}^{\tau }(VL|T)=P_{\max }-P_{VL}-c$. If the Government decides not to use the threat, the Lab will set the monopoly price. In response, a Strong government will issue a CL and its surplus will be $U_{G}^{S}(CL)=P_{\max }-P_{CL}-C^{S}$. As $P_{CL}+C^{S}-c<P_{VL}$ (cf. Eq. 9). A Strong government strictly prefers not to use the threat. In Case 1, the equilibrium is impossible.

Case 2. $q\le q_{1}$

In this second case, whatever the Government’s threat policy, the Lab will set a monopoly price. It will set $P_{M}^{H}(T)=P_{CL}+C^{W}-\ c$ if the Government uses the CL threat and $P_{M}^{H}(N)=P_{CL}+C^{W}$ in the opposite case.

As a Weak government may accept the monopoly price, its surplus is: $U_{G}^{W}(P_{M}^{H}(T))=P_{\max }-P_{M}^{H}(T)-c$ in the threat case and $U_{G}^{W}(P_{M}^{H}(N))=$ $P_{\max }-P_{M}^{H}(N)$ in the other case. Given the definition of the two prices, we have $U_{G}^{W}(P_{M}^{H}(T))=U_{G}^{W}(P_{M}^{H}(N)),$ and whatever the threat strategy, the Weak government reaches the same surplus. The Strong government responds to the monopoly price by issuing a CL. In this case, whatever its threat policy, its surplus will be $U_{G}^{S}(CL)=P_{\max }-P_{CL}-C^{S}.$ For such a Government, the choice to use the threat or not does not matter. There is no reason to deviate from the equilibrium strategy, the pooling equilibrium may appear under the necessary condition $q\le q_{1}.$

1.2.2 Pooling equilibrium without CL threat

In this equilibrium, a Government never threatens the Lab with a CL and, from the Lab’s point of view, the threat may only be used by a Strong government. Equilibrium beliefs must be:

$$\begin{aligned} \left\{ \begin{array}{l} P[\tau =S|s =T]=1 \\ P[\tau =S|s =N]=q. \end{array} \right. \end{aligned}$$

• Lab’s optimal strategy

We know that, in response to the monopoly price, a Strong government will impose a CL and a Weak one will accept the status quo. Given the equilibrium beliefs, the expected profit for a Lab that receives no threat and chooses to set the monopoly price is: $E[\pi _{L}(P_{M}^{H}(N))]=q(\beta P_{CL})+(1-q)P_{M}^{H}(N).$ If the Lab grants a VL, it will obtain the certain profit $\pi _{L}(VL)=\alpha P_{VL}.$ Setting a monopoly price is thus optimal in the no-threat case if:

$$\begin{aligned} E[\pi _{L}(P_{M}^{H}(N))]> & {} \pi _{L}(VL) \end{aligned}$$

(18)

$$\begin{aligned}\Leftrightarrow & {} q(\beta P_{CL})+(1-q)P_{M}^{H}(N)>\alpha P_{VL} \end{aligned}$$

(19)

$$\begin{aligned} q< & {} q_{2}=\frac{\left( P_{M}^{H}(N)-\alpha P_{VL}\right) }{\left( P_{M}^{H}(N)-\beta P_{CL}\right) }<1. \end{aligned}$$

(20)

If the Government decides to use the CL threat, the Lab will consider this threat as a signal of strength and will grant a VL.

• Government’s optimal strategy

Let us check the condition under which a Government prefers not to make the threat.

Case 1. $q<q_{2}$:

When q is low and no threat occurred at the beginning of the game, the firm’s optimal strategy is to set the monopoly price. The Weak government that accepts this price will obtain the surplus: $U_{G}^{W}(P_{M}^{H}(N))=P_{\max }-P_{CL}-C^{W}$. If the Weak government threatens the Lab, the latter will grant a VL and the Government’s surplus will be $U_{G}^{W}(VL|T)=P_{\max }-P_{VL}-c$. As $P_{CL}+C^{W}-c>P_{VL}$ (cf. Equation 9), the Weak government prefers to deviate from the equilibrium strategy. The equilibrium is impossible in this first case.

Case 2. $q>q_{2}$:

For high values of q,  the Lab will grant a VL whatever the threat strategy of the Government. Whatever its type, the Government gains no advantage by making the threat as this would induce a retaliation cost c and increase the procurement costs. The pooling equilibrium may occur under the necessary condition: $q>q_{2}.$

1.3 Hybrid equilibria

This appendix studies the existence of hybrid equilibria under our main assumption about the relative price values (8). In a general form, we will denote by $s(\tau )=\{ \alpha T+(1-\alpha )N|\alpha \in [0,1]\}$ a mixed strategy of a $\tau -$type government where the threat (T) is waved with probability $\alpha$ and is left aside (N) with probability $(1-\alpha )$. In a hybrid equilibrium, a Government uses a strategy $\sigma$ with:

$$\begin{aligned} \sigma =\left\{ \begin{array}{l} s(S)=\{ \mu T+(1- \mu )N| \mu \in [0,1]\} \\ s(W)=\{ \nu T+(1-\nu )N|\nu \in [0,1]\} \end{array} \right. \end{aligned}$$

A Strong government thus uses the threat with a probability $\mu$ and waives the threat opportunity with probability $(1- \mu ).$ In the same way, a Weak government will use the threat with a probability $\nu$ and will not with probability $(1-\nu ).$

In a general form, the Lab responds to a CL threat T by granting a VL with probability $\lambda$ and by setting the monopoly price with probability ($1-\lambda ).$ If the Government does not use the threat opportunity, the Lab will grant a VL with probability l and will set the monopoly price with probability ($1-l).$ The Lab’s strategy is therefore:

$$\begin{aligned} L(s)=\left\{ \begin{array}{l} L(T)=\{ \lambda VL+(1-\lambda )P_{M}^{H}(T)|\lambda \in [0,1]\} \\ L(N)=\{lVL+(1-l)P_{M}^{H}(T)|l\in [0,1]\}. \end{array} \right. \end{aligned}$$

(21)

Under our assumption (Eqs. 8 and 9), five hybrid equilibria are feasible according to the probabilities $\mu$ and $\nu$ defining the government’s strategy:

$$\begin{aligned} \begin{array}{ll} \text {Hybrid 1} &{} \text {Hybrid 2} \\ \sigma _{1}:\left\{ \begin{array}{l} \mu \in [0,1] \\ \nu =1 \end{array} \right. &{} \sigma _{2}:\left\{ \begin{array}{l} \mu \in [0,1] \\ \nu =0 \end{array} \right. \end{array} \begin{array}{l} \text {Hybrid 3} \\ \sigma _{3}:\left\{ \begin{array}{l} \mu \in [0,1] \\ \nu \in [0,1] \end{array} \right. \end{array} \begin{array}{ll} \text {Hybrid 4} &{} \text {Hybrid 5} \\ \sigma _{4}:\left\{ \begin{array}{l} \mu =1 \\ \nu \in [0,1] \end{array} \right. &{} \sigma _{5}: \left\{ \begin{array}{l} \mu =0 \\ \nu \in [0,1] \end{array} \right. \end{array} \end{aligned}$$

We study first the four semi-separating equilibria 1–2 and 4–5. The study of the Hybrid 3 equilibrium closes the section.

1.3.1 Hybrid 1: Government plays strategy $s_{1}$

In this equilibrium, Strong governments are indifferent between the two threat possibilities while Weak always threaten the Lab. We have:

$$\begin{aligned} \sigma _{1}=\left\{ \begin{array}{l} s(S)=\{ \mu T+(1- \mu )N| \mu \in [0,1]\} \\ s(W)=T \end{array} \right. \end{aligned}$$

(22)

With strategy $\sigma _{1},$ the N-strategy may only be played by Strong governments, the optimal Lab’s response in the no-threat case is thus to grant a VL. This leaves only one feasible strategy in the set of the Lab’s strategies.

$$\begin{aligned} L_{1}=\left\{ \begin{array}{l} L(T)=\{ \lambda VL+(1-\lambda )P_{M}^{H}(T)|\lambda \in [0,1]\} \\ L(N)=VL \end{array} \right. \end{aligned}$$

(23)

According to strategy $L_{1}$, if, a Weak government uses the CL threat, the Lab will answer by choosing randomly between its two options and the Government expected utility will be:

$$\begin{aligned} E[U_{G}^{W}()|T]=\lambda \left[ P_{\max }-(P_{VL}+c)\right] +(1-\lambda ) \left[ P_{\max }-(P_{M}^{H}(T)+c)\right] \end{aligned}$$

(24)

If the Government does not threaten the Lab, the firm identifies the Government as Strong and grants a VL. The Government surplus will be: $U_{G}^{W}(VL|N)=P_{\max }-P_{VL}.$ For a Weak Government the threat is optimal if:

$$\begin{aligned} E[U_{G}^{W}()|T]\ge & {} U_{G}^{W}(VL|N) \end{aligned}$$

(25)

$$\begin{aligned}\Leftrightarrow & {} \lambda \left[ P_{\max }-(P_{VL}+c)\right] +(1-\lambda ) \left[ P_{\max }-(P_{M}^{H}(T)+c)\right] \ge P_{\max }-P_{VL} \end{aligned}$$

(26)

$$\begin{aligned}\Leftrightarrow & {} (1-\lambda )(P_{M}^{H}(T)-P_{VL})\le -c \end{aligned}$$

(27)

which is impossible as $P_{M}^{H}(T)>P_{VL}$.

1.3.2 Hybrid 2: Government plays strategy $\sigma _{2}$

$$\begin{aligned} \sigma _{2}=\left\{ \begin{array}{l} s(S)=\{ \mu T+(1- \mu )N| \mu \in [0,1]\} \\ s(W)=N. \end{array} \right. \end{aligned}$$

(28)

In this equilibrium, Strong governments are indifferent between the two threat options while Weak ignore the threat possibility. The T-strategy may only be played by Strong governments and the optimal Lab’s answer in the threat case is thus to issue a VL. The set of the Lab’s strategies restricts to $L_{2}$.

$$\begin{aligned} L_{2}()=\left\{ \begin{array}{l} L(T)=VL \\ L(N)=\{lVL+(1-l)P_{M}^{H}(N)|l\in [0,1]\}. \end{array} \right. \end{aligned}$$

(29)

As a threat signals a Strong government, the optimal Lab’s answer to the threat is to grant a VL. The strong government surplus in this case is $U_{G}^{S}(VL|T)=P_{\max }-P_{VL}-c.$ If the threat is not used, the Lab will grant a VL with probability l and will set the high price $P_{M}^{H}(N)=P_{\max }-P_{CL}-C^{S}$ with probability $(1-l).$ In the latter case, the Strong government will issue a CL. The S-type government expected surplus is thus:

$$\begin{aligned} E[U_{G}^{S}()|N]=l(P_{\max }-P_{VL})+(1-l)(P_{\max }-P_{CL}-C^{S}) \end{aligned}$$

(30)

In equilibrium, the Strong government must be indifferent between the two threat policies, this gives the equilibrium value of the probability l:

$$\begin{aligned} E[U_{G}^{S}()|N]=U_{G}^{S}(T)\Leftrightarrow -(1-l)(P_{VL}-\left( P_{CL}+C^{S}\right) )=c. \end{aligned}$$

(31)

As $P_{VL}>P_{CL}+C^{S},$ the last equation implies $l>1$ which precludes the equilibrium.

1.3.3 Hybrid 4: Government plays strategy $\sigma _{4}$

$$\begin{aligned} \sigma _{4}=\left\{ \begin{array}{l} s(S)=T \\ s(W)=\{ \nu T+(1-\nu )N|\nu \in [0,1]\}. \end{array} \right. \end{aligned}$$

(32)

In this equilibrium, Strong governments always use the CL threat and Weak governments are indifferent between the two threat options. As Weak governments only may refuse to threaten the Lab, the optimal Lab’s answer in the no-threat case is thus to set the monopoly price $P_{M}^{H}(N).$ Therefore, the Lab’s set of feasible strategies is restricted to $L_{4}()$.

$$\begin{aligned} L_{4}()=\left\{ \begin{array}{l} L(T)=\{ \lambda VL+(1-\lambda )P_{M}^{H}(T)|\lambda \in [0,1]\} \\ L(N)=P_{M}^{H}(N). \end{array} \right. \end{aligned}$$

(33)

Given the threat strategy played by the Government, ex post beliefs are:

$$\begin{aligned} \left\{ \begin{array}{l} P[W|N]=1 \\ P[W|T]=\frac{P[T|W]P[W]}{P[T|W]P[W]+P[T|S]P[S]}=\frac{\nu (1-q)}{\nu (1-q)+q} <1-q \end{array} \right. \end{aligned}$$

(34)

• Lab’s strategy

According to strategy $L_{4}(),$ in case of threat, the Lab must be indifferent between setting the high price $P_{M}^{H}(T)$ and the granting of a VL. In the last case, the Lab’s profit is certain and equal to $\pi (P_{VL})=\alpha P_{VL}$ and, in the first case, the expected profit is $E[\pi (P_{M}^{H}(T))]=\frac{\nu (1-q)}{\nu (1-q)+q}P_{M}^{H}(T)+\frac{q}{\nu (1-q)+q}\beta P_{CL}.$ The indifference between the two strategies implies: $\pi (P_{VL})=E[\pi (P_{M}^{H}(T))],$ i.e.:

$$\begin{aligned} \nu =\frac{q\left( \alpha P_{VL}-\beta P_{CL}\right) }{(1-q)\left( P_{M}^{H}(T)-\alpha P_{VL}\right) }>0. \end{aligned}$$

It may easily be checked that $\nu \le 1$ implies $q\le q_{1}.$

• Government strategy

Strategy $\sigma _{4}$ describes a Strong government which systematically threatens the Lab with a possible CL. In answer, according to strategy $L_{4}()$ the Lab grants a VL with probability $\lambda$ and sets the price $P_{M}^{H}(T)$ with probability $(1-\lambda ).$ In the last case, the Government will issue a CL. The Strong government’s expected utility is

$$\begin{aligned} E[U_{G}^{S}()|T]=\lambda \left( P_{\max }-P_{VL}-c\right) +(1-\lambda )\left[ P_{\max }-(P_{CL}+C^{S})\right] . \end{aligned}$$

(35)

If the Government was not to use the threat, it would be considered as Weak and the Lab would set the high monopoly price. As the Strong government would thus issue a CL, its surplus would therefore be: $U_{G}^{S}(CL)=P_{ \max }-(P_{CL}+C^{S})$.

A Strong government find optimal to threaten the Lab if:

$$\begin{aligned} E[U_{G}^{S}()|T]\ge U_{G}^{S}(CL)\Leftrightarrow \lambda (P_{VL}-\left( P_{CL}+C^{S}-c\right) )\le 0. \end{aligned}$$

(36)

As $P_{VL}>(P_{CL}+C^{S}-c),$ the previous inequality implies $\lambda =0,$ ie the Lab must set the high price whatever the threat strategy: $L_{4}()=\left\{ L(T)=P_{M}^{H}(T),L(N)=P_{M}^{H}(N)\right\}$. In this case, the Weak government is also indifferent between the two strategies $U_{G}^{W}(P_{M}^{H}(T))=U_{G}^{W}(P_{M}^{H}(N))$ and may randomly choose to use or not the CL threat. The equilibrium may exist for $q\le q_{1}.$

At this point note that any strategy $\sigma _{4}$ leading the Lab to post a high price would lead to a perfect Bayesian equilibrium. As the lab will prefer to post a high price to the granting of a VL if: $E[\pi (P_{M}^{H}(T))]\ge \pi (P_{VL}),$ the second necessary condition for such an equilibrium is:

$$\begin{aligned} \nu \ge \frac{q\left( \alpha P_{VL}-\beta P_{CL}\right) }{(1-q)\left( P_{M}^{H}(T)-\alpha P_{VL}\right) }. \end{aligned}$$

1.3.4 Hybrid 5: Government plays strategy $\sigma _{5}$

$$\begin{aligned} \sigma _{5}=\left\{ \begin{array}{l} s(S)=N \\ s(W)=\{ \nu T+(1-\nu )N|\nu \in [0,1]\}. \end{array} \right. \end{aligned}$$

(37)

According to this strategy, only Weak governments may use the CL threat. In case of threat, the optimal Lab policy is thus to set the monopoly price $P_{M}^{H}(T)$. The Lab strategy is therefore given by strategy $L_{5}$:

$$\begin{aligned} L_{5}()=\left\{ \begin{array}{l} L(T)=P_{M}^{H}(T) \\ L(N)=\{lVL+(1-l)P_{M}^{H}(N)|l\in [0,1]\}. \end{array} \right. \end{aligned}$$

(38)

• Lab’s strategy

Given the government’s threat policy, the Lab updates its beliefs:

$$\begin{aligned} \left\{ \begin{array}{l} P[W|T]=1 \\ P[W|N]=\frac{P[N|W]P[W]}{P[N|W]P[W]+P[N|S]P[S]}=\frac{(1-\nu )(1-q)}{(1-\nu )(1-q)+q}<(1-q). \end{array} \right. \end{aligned}$$

(39)

If the Government uses the threat, the Lab considers that it is Weak and sets the monopoly price which leads to a higher profit than the grant of a VL.

In the no-threat case, the Lab must be indifferent between the granting of a VL and the setting of the monopoly price $P_{M}^{H}(N)$. In the first case, its profit will be $\pi _{L}(VL)=\alpha P_{VL}.$ In the second case, its profit relies on the actual type of the Government. If the Government is Weak, the Lab’s profit will be $\pi _{L}(P_{M}^{H}(N))=P_{M}^{H}(N);$ if it is Strong the Government will issue a CL and the Lab’s profit will be:  $\pi _{L}(P_{CL})=\beta P_{CL}.$The expected profit of the Lab which sets a monopoly price in the no-threat case is:

$$\begin{aligned} E[\pi _{L}(P_{M}^{H}(N))]=\frac{(1-\nu )(1-q)}{(1-\nu )(1-q)+q}P_{M}^{H}(N)+ \frac{q}{(1-\nu )\nu (1-q)+q}\beta P_{CL}. \end{aligned}$$

(40)

This leads to the indifference condition:

$$\begin{aligned} E[\pi _{L}(P_{M}^{H}(N))]=\pi _{L}(VL)\Leftrightarrow (1-\nu )=\frac{q\left[ \alpha P_{VL}-\beta P_{CL}\right] }{(1-q)\left( P_{M}^{H}(N)-\alpha P_{VL}\right) }>0. \end{aligned}$$

(41)

It is easy to check that $\nu \in [0,1]$ under the necessary condition $q\le q_{2}.$

• Government’s strategy

Consider first a Strong government. If following the equilibrium strategy this Government does not threaten the Lab, the firm will answer by granting a VL with probability l and by setting the monopoly price with probability $(1-l).$ In the latter case, the Government will issue a CL. The Strong government expected utility is: $E[U_{G}^{S}()|N]=l\left( P_{\max }-P_{VL}\right) +(1-l)\left[ P_{\max }-(P_{CL}+C^{S})\right] .$

On the contrary, if the Government threatens the firm, the Lab will consider that it is Weak and will set the monopoly price. Thus, the Strong government will have the opportunity to issue a CL and will reach the surplus: $U_{G}^{S}(CL)=P_{\max }-(P_{CL}+C^{S})$. Strategy $s_{5}$ may be optimal if $E[U_{G}^{S}()|N]\ge U_{G}^{S}(T),$ i.e. if:

$$\begin{aligned} l\left[ P_{VL}-(P_{CL}+C^{S})\right] \le 0. \end{aligned}$$

(42)

This equation implies $l=0$ meaning that in equilibrium the Lab must play the degenerated Lab strategy: $L_{5}()=\left\{ L(T)=P_{M}^{H}(T),L(N)=P_{M}^{H}(N)\right\} .$ Consider now the Weak government, if this Government does not threaten the Lab, the firm will respond by setting the monopoly price $P_{M}^{H}(N)=P_{CL}+C^{W}$ and the Government’s expected surplus is: $U_{G}^{W}(N)=\left[ P_{\max }-(P_{CL}+C^{W})\right] .$ If the Government uses the threat, the Lab will set the price $P_{M}^{H}(T)=P_{CL}+C^{W}-c$ with the same Government surplus: $U_{G}^{W}(T)=P_{\max }-(P_{CL}+C^{W}-c)-c.$ The Weak government is thus indifferent between the two options.

As in the previous equilibrium, for both type of Governments there is no strict preference between the two threats options, no Government finds optimal to deviate from the equilibrium strategy. This hybrid equilibrium is, therefore, feasible under condition $q<q_{2}.$ At this point, note again that any strategy $\sigma _{5}$ leading the Lab to post a high price would lead to a perfect Bayesian equilibrium. As the lab will prefer to post a high price to the granting of a VL if: $E[\pi (P_{M}^{H}(T))]\ge \pi (P_{VL}),$ the second necessary condition for such an equilibrium is:

$$\begin{aligned} (1-\nu )\ge \frac{q\left[ \alpha P_{VL}-\beta P_{CL}\right] }{(1-q)\left[ P_{M}^{H}(N)-\alpha P_{VL}\right] }. \end{aligned}$$

1.3.5 Hybrid 3: Government plays strategy $\sigma _{3}$

In this equilibrium, we assume that both players use mixed strategies whatever their type or whatever the signal they observe.

$$\begin{aligned} \sigma _{3}=\left\{ \begin{array}{l} s(S)=\{ \mu T+(1- \mu )N| \mu \in [0,1]\} \\ s(W)=\{ \nu T+(1-\nu )N|\nu \in [0,1]\}. \end{array} \right. \end{aligned}$$

(43)

• The Lab’s indifference condition

Given the equilibrium strategy of the Government, the Lab revises its beliefs according to a Bayesian procedure. Posterior beliefs are

$$\begin{aligned} \left\{ \begin{array}{l} P[W|T]=\frac{P[T|W]P[W]}{P[T|W]P[W]+P[T|S]P[S]}=\frac{\nu (1-q)}{\nu (1-q)+q \mu } \\ P[W|N]=\frac{P[N|W]P[W]}{P[N|W]P[W]+P[N|S]P[S]}=\frac{(1-\nu )(1-q)}{(1-\nu )(1-q)+q(1- \mu )} \end{array} \right. . \end{aligned}$$

(44)

If the Lab decides to set a monopoly price, its expected profit is

$$\begin{aligned} \left\{ \begin{array}{ll} E[\pi _{L}(P_{M}^{H}(T)]=\frac{\nu (1-q)}{\nu (1-q)+q \mu }P_{M}^{H}(T)+\frac{q \mu }{\nu (1-q)+q \mu }\beta P_{CL} &{} \text {In the Threat case} \\ E[\pi _{L}(P_{M}^{H}(N))]=\frac{(1-\nu )(1-q)}{(1-\nu )(1-q)+q(1- \mu )}P_{M}^{H}(N)+\frac{q(1- \mu )}{(1-\nu )(1-q)+q(1- \mu )}\beta P_{CL} &{} \text {In the no Threat case.} \end{array} \right. \end{aligned}$$

and the profit is $\pi _{L}(P_{VL})=\alpha P_{VL}$ if it decides to grant a VL. If Eq. (43) properly describes the optimal strategy of the Lab, this one must be indifferent between granting a VL and setting the monopoly price, thus

$$\begin{aligned} \left\{ \begin{array}{l} E[\pi _{L}(P_{M}^{H}(T)]=\pi _{L}(P_{VL}) \\ E[\pi _{L}(P_{M}^{H}(N))]=\pi _{L}(P_{VL}). \end{array} \right. \end{aligned}$$

After some calculations, these two equalities require:

$$\begin{aligned} \left\{ \begin{array}{l} \nu =\frac{(1-q)P_{M}^{H}(N)+q\beta P_{CL}-\alpha P_{VL}}{(1-q)c} \\ \mu =\nu \frac{(1-q)}{q}\frac{\left[ P_{M}^{H}(T)-\alpha P_{VL}\right] }{\left[ \alpha P_{VL}-\beta P_{CL}\right] } \end{array} \right. , \end{aligned}$$

(45)

with $\nu \in [0,1]$ and $\mu \in [0,1]$ if and only if $q\in [q_{1},q_{2}]$ and $\nu > \mu$ if $q>q_{1},$ in equilibrium, the Weak government uses the threat more frequently than the Strong one. In turn, this implies: $P[W|N]<P[W]<P[W|T].$ Waving the threat is a signal of weakness.

• The Government’s indifference condition

Given the Lab’s strategy, the government’s expected surplus generated by its threat policy is (recall that a Strong government will issue a CL if the firm tries to set a monopoly price):

$$\begin{aligned} \left\{ \begin{array}{ll} E[U_{G}^{S}(T)]=\lambda \left( P_{\text {max}}-P_{VL}-c\right) +(1-\lambda )\left( P_{\text {max}}-P_{CL}-C^{S}\right) &{} \text {Threat case} \\ E[U_{G}^{S}(N)]=l\left( P_{\text {max}}-P_{VL}\right) +(1-l)\left( P_{\text { max}}-P_{CL}-C^{S}\right) &{} \text {Opposite case.} \end{array} \right. , \end{aligned}$$

(46)

for a Strong government and

$$\begin{aligned} \left\{ \begin{array}{ll} E[U_{G}^{W}(T)]=\lambda \left( P_{\text {max}}-P_{VL}-c\right) +(1-\lambda )\left( P_{\text {max}}-P_{CL}-C^{W}\right) &{} \text {Threat case} \\ E[U_{G}^{W}(N)]=l\left( P_{\text {max}}-P_{VL}\right) +(1-l)\left( P_{\text { max}}-P_{CL}-C^{W}\right) &{} \text {Opposite case} \end{array} \right. , \end{aligned}$$

(47)

for a Weak government. If Eq. (12) properly describes the Government’s strategy, whatever its type the Government must be indifferent between using the threat or not. We must have:

$$\begin{aligned} \left\{ \begin{array}{l} E[U_{G}^{S}(T)]=E[U_{G}^{S}(N)] \\ E[U_{G}^{W}(T)]=E[U_{G}^{W}(N)] \end{array} \right. , \end{aligned}$$

(48)

which imposes $\lambda =l=0.$ In other words, regardless of the Government’s threat policy, in the equilibrium, the Lab always sets the monopoly price and, consequently, the use of the threat has no influence on the Government’s surplus. Deviation from the equilibrium strategy is useless. This equilibrium exists for $q\in [q_{1},q_{2}].$ Again, any strategy $\sigma _{3}$ leading the Lab to post a high price would lead to this equilibrium. The necessary condition for Hybrid H3 is:

$$\begin{aligned} \left\{ \begin{array}{l} E[\pi _{L}(P_{M}^{H}(T)]\ge \pi _{L}(P_{VL}) \\ E[\pi _{L}(P_{M}^{H}(N))]\ge \pi _{L}(P_{VL}). \end{array} \right. , \end{aligned}$$

which can be rewritten as

$$\begin{aligned} \left\{ \begin{array}{l} \nu \ge A \mu \text { with }A\text {=}\frac{q\left[ \alpha P_{VL}-\beta P_{CL}\right] }{(1-q) \left[ P_{M}^{H}(T)-\alpha P_{VL}\right] }>1\qquad (1) \\ \nu \le \left( 1-B\right) + \mu B,\text { with }B=\frac{q\left[ \alpha P_{VL}-\beta P_{CL}\right] }{(1-q) \left[ P_{M}^{H}(N)-\alpha P_{VL}\right] }\in [0,1]\qquad (2). \end{array} \right. \end{aligned}$$

For a given value of $q\in [q_{1},q_{2}],$ the set of couples ($\mu ,\nu$) consistent with the Hybrid 3 equilibrium is plotted as the grey surface in Fig. 5. Note that as the slope of the line (1) is greater than 1, we have $\nu \ge \mu ,$ i.e. the Weak government uses more often the threat than the Strong one and the threat is a signal of weakness, $P[S|T]<P[S])$.

Fig. 5

Probabilities consistent with Hybrid 3