<body><div><ul><li>What if, as in certain systems of law, a single vote of not guilty can end a defendant’s life? </li><li>Bernoulli’s statistics shows how judges can operate probabilistically. </li><li>It is tricky – though doable – for a person to act with probability p on a single occasion. </li></ul>In dubio pro reo. – a prudential principle of Roman lawThe courts of the ancient Hebrews refused to convict an accused unless all but one of the justices submitted a vote of guilty. If there was unanimity in the call for conviction, the defendant was let go. This is odd in light of the established legal view as well as the folk intuition, which says that unanimity is desirable and should be required for a conviction.Once the contra-unanimity rule appeared in the tractate Sanhedrin (from the Greek “synhedrion,” sitting together, assembly) of the Babylonian Talmud, the Jewish sages and legal scholars have struggled to rationalize their puzzlement. Why would the authoritative book of laws endorse such a patently irrational rule? Among the offered opinions is the idea that a transgressor who receives a unanimous verdict is so obviously guilty and beyond redemption that it is not for humans to mete out punishment. Only the deity can. Another idea is that unanimity raises the specters of collusion and groupthink. Perhaps the judges had it in for the defendant on grounds beyond the evidence, and this should count in his (or her) favor.The theological argument is not compelling psychologically or even on its own theological grounds. If it must be left to God to punish those who are guilty beyond human doubt, then why punish (or release) those where the humans are not entirely sure? The social psychological argument is not compelling either, for if the judges wish to collude and if they wish to see the accused executed, it is a trivial task for them to generate n – 1 guilty votes.Are we stuck? In my opinion, the linguistic interpretation of this dilemma is most compelling. Rabbi Meir Halevi Abulafia, the “RaMah,” suggested that the Talmudic expression does not refer to the defendant being released, but to being sent away for immediate execution (Glatt, 2013). In this view, there is no dilemma. Yet, as sages and scholars debate the issue to the present day, it should be taken seriously. To a psychologist, the question is how a panel of judges, jurors, or any group of voters, might achieve a desired aggregate outcome short of unanimity, especially when they are barred from deliberation.<figure><img src="https://img.particlenews.com/image.php?type=thumbnail_580x000&amp;url=1vccX3_0vwTxzpD00" alt="https://img.particlenews.com/image.php?url=1vccX3_0vwTxzpD00" width="639" height="514" ></figure>Suppose you sit on a panel and you learn that a desired action will only be taken if all but one panelist endorses it. With what probability would you vote for it? Following Jacob Bernoulli, a godfather of frequentist statistics (Clayton, 2022), we can apply binomial calculus to find the maximum probability of having k out of n votes of guilt, where n is the number of judges. This probability is highest if each independent judge votes “yes” with a probability of k/n. Figure 1 shows the proof as worked out by Dr. Peyam Tabrizi. ChatGPT produced the same result (personal communication, Dr. Daniel Villiger).Figure 2, generated by Dr. Johannes Ullrich, shows the distributions of the probability of a group’s guilty verdict for different values of k and n = 10. The curve that has its maximum to the far right of the graph shows the results for k = 9. We see that even in this case, the maximum probability of a group verdict is below .4.<figure><img src="https://img.particlenews.com/image.php?type=thumbnail_580x000&amp;url=1Rzk4o_0vwTxzpD00" alt="https://img.particlenews.com/image.php?url=1Rzk4o_0vwTxzpD00" width="639" height="651" ><figcaption>The probability of a collective guilty verdict as a function of the individual's probability of voting guilty for k from 1 to 9 and group size n = 10</figcaption></figure>Being human, judges may feel guilt when casting a vote of guilt. As the facts of a case become clearer, guilty votes (and a conviction) should become more probable, while the intensity of the guilt should become attenuated. A psychologically sophisticated and introspective judge would want to find an equilibrium where the amount of guilt is smallest relative to the likelihood of doing the right thing. This is a challenging task if it is not the case that any increase in guilt is perfectly cancelled out by a change in the likelihood of being right.The task is further complicated if an individual judge can only assume, but not know, that other judges will vote guilty with the same probability. In our scheme of independent voting, each individual’s probability of voting guilty may be low and still result in the aggregate outcome that condemns the defendant.Judges are, in fact, executioners, although they delegate the physical act of killing (or incarcerating) to specialists (Harrington, 2013). In wartime, firing squads are sometimes set up so that the shooting conscripts are told that one of the rifles holds a blank, or, alternatively, that all but one rifle carry blanks. Either way, each conscript has a psychological out. You can’t prove, they can claim, that they killed a person. Now, what would happen if all judges sought a guilty verdict, and then n – 1 guilty tickets and one not-guilty ticket were produced and randomly given to the judges who then throw them—unseen—into the ballot? Would they feel better, having achieved the collective goal without carrying the burden of individual responsibility? All told, the principle of in dubio and the requirement of unanimity seems the sanest solution. It is not only conservative by limiting wrongful convictions (type I errors), it also limits missed convictions (type II errors) in comparison with the n – 1 rule. Perhaps its greatest virtue is that it does not let the judges off the emotional hook.We have assumed that individuals can act on single occasions in a probabilistic manner. I sometimes confound students by asking them to raise their hand with a probability of 1/3. Frequentists like Bernoulli are stumped. They can only ask if 30 percent of the students raised their hands and then infer, without proof or rationale, that each individual raised a hand with p = .333. Bayesians, also without proof, might say that each individual carried a subjective probability, but hey, where’s that at? A pragmatic solution is to externalize the generation of probability and cast a die. If the number 1 or 2 comes up, raise your hand; otherwise, don’t (Krueger, 2010).Collective action is difficult to optimize on a good day. Statistics and the study of interpersonal dynamics can shed some light. Still, anyone sitting in judgment of the fate of another must be sensitive to and mature in the management of their own emotions.I thank my friends Peyam Tabrizi, Johannes Ullrich, and Daniel Villiger for contributing to this effort. I thank the algorithm used by Psychology Today Online for placing this essay under the label of "guilt." There is no shame in that.</div></body>

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Justice With Dissent

What if, as in certain systems of law, a single vote of not guilty can end a defendant’s life?. Bernoulli’s statistics shows how judges can operate probabilistically. It is tricky – though doable – for a person to act with probability p on a single occasion.

Justice With Dissent

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