A legal trial is a test of the hypothesis of Guilt. A judge examines evidence to evaluate the probability that the defendant is guilty and decides to convict him if the probability is high enough, or to acquit it if it isn’t. How high the probability of Guilt needs to be for a conviction depends on the standard of proof, which is proportional to the gravity of the allegation and the corresponding severity of the punishment.

But what determines the standard of proof? Let’s see. The judge has a utility function, defined over two possible states: Guilt or Innocence, and two possible decisions: Convict or Acquit.

The judge draws positive utility U(CG) from convicting a guilty defendant and negative utility U(CI) from convicting an innocent one. And he draws positive utility U(AI) from acquitting an innocent defendant and negative utility U(AG) from acquitting a guilty one. Based on these preferences, the probability of Guilt that leaves the judge indifferent between conviction and acquittal is given by:

(1)

Hence:

(2)

(This comes from a paper by Terry Connolly in this collection).

The judge will convict if the probability of Guilt is higher than P and acquit if it is lower. In order to examine the properties of P, we define BB=U(CI)/U(AG), CB=U(AI)/U(CG) and DB=-U(AG)/U(CG). We call BB the Blackstone Bias, after Sir William Blackstone’s Principle: “It is better that ten guilty persons escape than that one innocent suffer”. B>1 means that the pain of a wrongful conviction – a False Positive – is higher than the pain of a wrongful acquittal – a False Negative. Similarly, we call CB the Compassion Bias, where C>1 means that the judge draws more pleasure from a rightful acquittal – a True Negative – than from a rightful conviction – a True Positive. Finally, we can call DB the Distress Bias, where D>1 means that the pain of a wrongful acquittal – a False Negative – is higher than the pleasure of a rightful conviction – a True Positive. Using these definitions, (2) can be rewritten as:

(3)

where P is a function of the three biases and is independent of the utility function’s metric.

Assume first that the judge has no biases: BB=CB=DB=1. In this case, P=50%: conviction requires a Preponderance of evidence. An unbiased judge convicts if the defendant is more likely to be guilty than innocent. This may be an acceptable verdict for minor charges, where the limited size of the penalty renders the judge indifferent between False Positives and False Negatives and between True Positives and True Negatives. As the severity of the punishment increases, however, a conscientious judge will start caring more about avoiding a wrongful conviction than a wrongful acquittal. In this case, assuming for example the Blackstone Principle (BB=10), P increases to 85%: in order to convict, the judge will require Clear and convincing evidence. The same happens if we increase CB to 10, i.e. the judge cares more about reaching a rightful acquittal than a rightful conviction. If both BB and CB are increased to 10, P increases to 91%, thus entering the Beyond reasonable doubt zone. Notice that, if BB=CB, then (3) reduces to P=BB/(1+BB), which is 50% for BB=1, 91% for BB=10 and tends to 1 as BB grows further (P=99% requires BB=99). Hence, if BB=CB, increasing DB has no effect on P: as long as the judge is indifferent between the two ways of being wrong and the two ways of being right, his attitude towards guilt does not matter. DB affects P only if BB≠CB. If, for example, BB=10 and CB=1, then increasing DB from 1 to 10 increases P from 85% to 90%. If, on the other hand, BB=1 and CB=10, then the DB increase brings P down to 65%. This makes sense: a higher DB increases the sensitivity to wrongful acquittals and decreases the sensitivity to rightful convictions.

What happens with ‘perverse’ biases, i.e. lower than 1? For example, we can call BB=0.1 the Bismarck Bias: “It is better that ten innocent persons suffer than that one guilty escapes”. In this case, unsurprisingly, P decreases to 35%: the judge requires just about one-third probability of Guilt in order to convict. The same happens if CB=0.1 – which can be called the Callousness Bias. And if BB and CB are both 0.1, P goes all the way down to 9% – unpleasant news for defendants.

Notice that perversion is not about the signs of the utility function: U(CI) and U(AG) are still negative and U(CG) and U(AI) still positive. A perverse judge is not one who draws pleasure from wrongful verdicts and pain from rightful ones. Perversion is about relative utilities: U(CI)<U(AG) – the judge would rather convict an innocent person than acquit a guilty one – and U(AI)<U(CG) – he prefers to convict a guilty person to acquitting an innocent one. Compared to sign reversals, these may appear as secondary perversions. But they are all it is needed to bring havoc to the standard of proof.

In civilised legal systems, the standard of proof is inspired by worthy principles, aimed at safeguarding the rights of the innocent, especially as the severity of punishment increases. Uncivilised systems are characterized by the opposite tendency: a higher focus of hitting the guilty, combined with a lower concern for ‘collateral damage’. In reality, however, the distinction is not as neat: perverse utility functions exist also in advanced democracies, especially where judges have strong incentives to convict and collateral damage is less manifest.

A perverse verdict is the result of a bad decision process where, helped by hindsight, the judge imposes a high cost of False Positives on others, in order to avoid the cost of a Miss on himself. Do you want to catch the thief? Shoot everyone in sight. Do you want a culprit to blame for a negative outcome? Accuse him of negligence – how could he possibly ignore it? He should have known better.

Not as bloody in practice, but as uncivilised in principle.

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