You go first!: coordination problems and the burden of proof in inquisitorial prosecution

Abstract

Prosecution of criminals is costly and subject to errors. In contrast to adversarial court procedures, in inquisitorial systems the prosecutor is regarded as an impartial investigator and an aide to the judge. We show in a sequential prosecution game of a Bayesian court that a strategic interaction between these two impartial agents exists where each player may hope to free ride on the other one´s investigative effort. This gives rise to inefficient equilibria. The model demonstrates that the effectiveness of some policy measures that intend to curb the free-riding problem critically depends on the assumed benevolence of the prosecutor. We find that, if policy makers are unable to infer the true preferences of the prosecutorial body, the high burden of proof in criminal law may reduce the probability of court errors. Our analysis, therefore, substantiates claims made in the literature that inquisitorial procedures are introduced to avoid wrongful acquittals.

Appendix

1.1 Proofs

1.1.1 Proof Proposition 1

(i) The strategies $\left\{ {(n,dr);(n,ac)} \right\}$ are always a Nash-equilibrium: if J acquits, (n, dr) is always best for P. If P drops, all J´s strategies show the same outcome. However, [(n, dr); (n, ac)] is a PBE only if out-of-equilibrium beliefs apply with $\mu <\frac{{c}_{J}}{h}$ and$\mu <\frac{\alpha }{\alpha +1}$: If J´s information set is reached, (n, ac) is only preferable to (inv) for J if$-\mu h>-{c}_{J}$, which yields $\mu <\frac{{c}_{J}}{h}$, if and $-\mu h>-\left(1-\mu \right)\alpha h$ hold, which yields$\mu <\frac{\alpha }{\alpha +1}$. This fully constitutes the PBE. (ii) It appears implausible, however, that (n, ac) is an equilibrium strategy for high values of$\gamma$. Given the setup of the game, it is easy to see that the condition $\mu \ge \gamma$ must apply when the information set of the judge is reached, as the prosecutor either charged blindly or investigated the case. Given the requirements for the judicial belief of (i), we restrict our focus on the equilibrium strategies $\left\{ {(n,dr);(n,ac)} \right\}$ on cases when $\gamma \le \frac{{c}_{J}}{h}$ applies. This reasoning about the condition $\mu \ge \gamma$ becomes even more pronounced if we applied the tradition of Kreps and Wilson (16) to analyze off-equilibrium strategies: Consider that the players ‘tremble’ in their strategies with a small probability $\varepsilon$, implying that at each information set, the equilibrium strategy by the player is actually played with probability $1 - \varepsilon$, and each of the other two off-equilibrium strategies with probability$\varepsilon /2$. This yields the judicial belief $\mu \left(g|ch\right)=\frac{\gamma \varepsilon }{\gamma \varepsilon +(1-\gamma )(\varepsilon /2)}\ge \gamma$ whenever the information set of the judge is reached. Thus, the strategies $\left\{ {(n,dr);(n,ac)} \right\}$ cannot be sequentially rational if $\gamma >\frac{{c}_{J}}{h}$ applies.

1.1.2 Proof Proposition 2

(i) For investigations to be a best response by J to blind charges by P, this must be more favorable than blind acquittals,$-{c}_{J}>-\gamma h\iff \gamma >\frac{{c}_{J}}{h}$, and better than blind convictions, $- c_{J} > - (1 - \gamma )\alpha h \Leftrightarrow \gamma < 1 - \frac{{c_{J} }}{\alpha h}$. Note that $\gamma >\frac{{c}_{J}}{h}$ implies that $\gamma >\frac{{c}_{P}}{h}$ holds for${c}_{J}>{c}_{P}$. For a blind charge to be optimal for P, this must be preferable to dropping the case,$-(1-\gamma )L>-\gamma h\iff \gamma h>(1-\gamma )L$, and also preferable to one’s own investigations,$\left(1-\gamma \right)L<{c}_{P}$. The former inequality always holds for ${c}_{P}<\gamma h$ and$\left(1-\gamma \right)L<{c}_{P}$, as we already established$\frac{{c}_{P}}{h}<\gamma$. (ii) As P charges all defendants in equilibrium, J has to form her beliefs as $\mu = \gamma$, which was considered in (i).

1.1.3 Proof Proposition 3

(i) Given that P only charges the guilty defendants, for J it is always optimal to convict all charged defendants without further investigations. Given blind convictions by J, investigations are rational for P when $- c_{P} > - (1 - \gamma )\alpha h \Leftrightarrow \gamma < 1 - \frac{{c_{P} }}{\alpha h}$ and $- c_{P} > - \gamma h \Leftrightarrow \gamma > \frac{{c_{P} }}{h}$. (ii) As P charges only the guilty defendants, $\mu = 1$ applies, as considered in (i).

1.1.4 Proof Proposition 4

The judge is indifferent between investigation and blind convictions if $- c_{J} = - (1 - \mu )\alpha h$ holds, which yields the threshold $\mu = 1 - \frac{{c_{J} }}{\alpha h}$. We write the posterior belief of the judge as $\mu = \frac{\gamma }{{\gamma + (1 - \phi_{P} )(1 - \gamma )}}$. Thus, the judge is indifferent if $\mu = \frac{\gamma }{{\gamma + (1 - \phi_{P} )(1 - \gamma )}} = 1 - \frac{{c_{J} }}{\alpha h}$ holds, which gives $\phi *_{P} = \frac{{(1 - \gamma )\alpha h - c_{J} }}{{(1 - \gamma )\left( {\alpha h - c_{J} } \right)}}$. $\phi {*}_{P}\in (0;1)$ applies if$\gamma <1-\frac{{c}_{J}}{\alpha h}$. The prosecutor is indifferent between investigation and blind charges if $- c_{P} = - \phi_{J} (1 - \gamma )L - (1 - \phi_{J} )(1 - \gamma )\alpha h$ holds. This yields $\phi *_{J} = \frac{{(1 - \gamma )\alpha h - c_{P} }}{(1 - \gamma )\alpha h - (1 - \gamma )L}$. The numerator is positive for$\gamma <1-\frac{{c}_{P}}{\alpha h}$, and smaller than the denominator given$\left(1-\gamma \right)L<{c}_{P}$. These requirements are always met under Proposition 2 and 3.

1.1.5 Proof Proposition 5

(i) Given that J blindly convicts, P will respond with a blind charge if $- (1 - \gamma )\alpha h > - c_{P}$, which gives $\gamma > 1 - \frac{{c_{p} }}{\alpha h}$, and $- (1 - \gamma )\alpha h > - \gamma h \Leftrightarrow \gamma > \frac{\alpha }{1 + \alpha }$. J´s best response to a blind charge is a blind conviction if $- (1 - \gamma )\alpha h > - c_{J}$, which yields $\gamma > 1 - \frac{{c_{J} }}{\alpha h}$, and$-(1-\gamma )\alpha h>-\gamma h\iff \gamma >\frac{\alpha }{1+\alpha }$. Due to${c}_{J}>{c}_{P}$, the inequality $\gamma >1-\frac{{c}_{p}}{\alpha h}$ guarantees that $\gamma >1-\frac{{c}_{J}}{\alpha h}$ holds.

1.2 Mixed strategy outcomes

P may choose between the pure strategies (inv), (n, ch), and (n, dr). J may choose between the pure strategies (inv), (n, co), and (n, ac). Given the existence of the strategy (inv) and $\frac{{c}_{J}}{h}<\frac{\alpha }{1+\alpha }<1-\frac{{c}_{J}}{\alpha h}$, P cannot be made indifferent between (n, ch) and (n, dr), and J cannot be indifferent between (n, ac) and (n, co). In other words, if P (or J) would be indifferent between the mentioned two pure strategies for a given γ, they would strictly prefer to choose the pure strategy (inv). Consequently, 2 × 2 candidates for mixed strategy equilibria remain.

In addition to Proposition 4, a second mixed strategy equilibrium exists when P mixes between (inv) and (n, ch), and J mixes between (inv) and (n, ac.). J is indifferent when ${-\phi }_{P}\gamma {c}_{J}-\left(1{-\phi }_{P}\right){c}_{J}=-\gamma h$ holds, which gives ${\phi }_{P}=\frac{{c}_{J}-\gamma h}{(1-\gamma ){c}_{J}}$ and requires the condition $\gamma <\frac{{c}_{J}}{h}$ to hold. P becomes indifferent if ${-{c}_{P}-(1-\phi }_{J})\gamma \left(h+L\right)=-{\phi }_{J}\left(1-\gamma \right)L-(1-{\phi }_{J})(\gamma h+L)$. This only holds if $\left(1-\gamma \right)L={c}_{P}$, and allows ${\phi }_{J}\in (0;1)$.

No mixed strategy equilibrium can exist when P mixes between (inv) and (n, dr), and J mixes between (inv) and (n, co). For J to become indifferent, the following condition needs to hold: $-{\phi }_{P}\gamma {c}_{J}-\left(1-{\phi }_{P}\right)\gamma h=-(1-{\phi }_{P})\gamma h$, which cannot be fulfilled for ${\phi }_{P}>0$. Note that, given P plays (n, dr), any combination of J´s strategies would be a best response but provide identical outcomes to the pure strategy solution. We thus restrict the analysis of mixed strategy outcomes to cases where both players apply mixed strategies.

No mixed strategy equilibrium can exist when P mixes between (inv) and (n, dr), and J mixes between (inv) and (n, ac). For J to become indifferent, again the following condition needs to hold: $-{\phi }_{P}\gamma {c}_{J}-\left(1-{\phi }_{P}\right)\gamma h=-\gamma h\iff {\phi }_{P}\gamma \left(h-{c}_{J}\right)=0$, which cannot be fulfilled for ${\phi }_{P}>0$ and $h>{c}_{J}$.

1.3 Comparison of equilibria

Equilibrium	Impartial prosecutor	Opportunistic prosecutor
PBE No. 1	$(\gamma >\frac{{c}_{J}}{h}$ violates sequential rationality)	$(\gamma >\frac{{c}_{J}}{h}$ violates sequential rationality)
PBE No. 2	${c}_{P}>\left(1-\gamma \right)L;$ $\gamma >\frac{{c}_{J}}{h};\gamma <1-\frac{{c}_{J}}{\alpha h}$	$\gamma >\frac{L}{V+L};{c}_{P}>\left(1-\gamma \right)L;$ $\gamma >\frac{{c}_{J}}{h};\gamma <1-\frac{{c}_{J}}{\alpha h}$
PBE No. 3	$\gamma <1-\frac{{c}_{P}}{\alpha h}$ ; $\gamma >\frac{{c}_{P}}{h}$	(does not exist)
PBE No. 4	$\phi *_{P} = \frac{{(1 - \gamma )\alpha h - c_{J} }}{{(1 - \gamma )\left( {\alpha h - c_{J} } \right)}}$ $\phi *_{J} = \frac{{(1 - \gamma )\alpha h - c_{P} }}{(1 - \gamma )\alpha h - (1 - \gamma )L}$ ${c}_{P}>\left(1-\gamma \right)L$; $\gamma <1-\frac{{c}_{J}}{\alpha h}$	$\phi *_{P} = \frac{{(1 - \gamma )\alpha h - c_{J} }}{{(1 - \gamma )\left( {\alpha h - c_{J} } \right)}}$ $\phi {*}_{J}=\frac{\left(1-\gamma \right)V+{c}_{P}}{\left(1-\gamma \right)(V+L)}$ ${c}_{P}<\left(1-\gamma \right)L$ ; $\gamma <1-\frac{{c}_{J}}{\alpha h}$
PBE No. 5	$\gamma >1-\frac{{c}_{P}}{\alpha h}$ ; $\gamma >\frac{\alpha }{1+\alpha }$	$\gamma >1-\frac{{c}_{J}}{\alpha h}$ ; $\gamma >\frac{\alpha }{1+\alpha }$