A Talmudic constrained voting majority rule

For generalizations of Condorcet’s theorem under relaxations of Condorcet’s restrictive assumptions, see, for example, Grofman et al. (1983), Ladha (1992), Berend and Paroush (1998) and Ben-Yashar and Nitzan (2017a, b). For application of the Condorcet Theorem in a different field, see Ben-Yashar et al. (2018). For a comprehensive literature survey, see Nitzan and Paroush (2017).

The same view was expressed, for example, by the Israeli Supreme Court justice Yoram Danziger in the 16th annual conference of the Israeli Bar Association in Eilat (April 2016).

In the United States, unanimity is required for conviction in major felonies in 48 states (Reichelt 2006), except Oregon and Louisiana, in which conviction is allowed with a supermajority of 10 jurors against 2 (Johnson v. Louisiana, 406 U.S. 356 (1972). Apodaca v. Oregon, 406 U.S. 404 (1972)), but unanimity is required if the jury is small (5 or 6 jurors: Burch v. Louisiana, 441 U.S. 130 (1979). A supermajority is also required in many countries for approval of constitutional amendments.

Babylonian Talmud, Sanhedrin 17b, translated by Rabbi Jacob Shachter, (Epstein 1994). The Talmud (from the Hebrew תַּלְמוּד “study”) contains collections of rabbinical traditions, mainly interpretations of the Mishnah. The earlier version of the Talmud, the Jerusalem Talmud, was redacted around 400 CE. The prevailing version, the Babylonian Talmud, was redacted around 500 CE. The Mishnah is written in Hebrew and the Talmudim in Aramaic.

Sanhedrin (from the Greek συνέδριον, “sitting together” or “assembly”) is a tribunal of 23 jurors appointed in every city in the Land of Israel (Deuteronomy 16, 18). In addition, the Great Sanhedrin of 71 justices convened in the Hall of Hewn Stones located beside the Temple in Jerusalem.

Shachter’s translation of פּוֹטְרִין אוֹתוֹ as “acquitted”, is compatible with the mainstream commentators’ understanding. However, other translations are possible (see below footnotes 10 and 22).

For a survey of rabbinical explanations, see Glatt (2013). Glatt indicated that although the R. Kahana ruling (according to mainstream interpretation) seems as a “technicality” stemming from harsh legal formalism, no Talmudic commentator suggested such an explanation. That claim is, however, inaccurate. See, for example, Halevi (1963, Ch. 9).

E.g., Abulafia (1992, Sanhedrin 17a) quoted in Tosfot Rosh, (Sanhedrin 17a) and Safrai (1964).

Literally, פּוֹטְרִין אוֹתוֹ can be translated in this vein, but this translation is inconsistent with many other Talmudic topics. See Rabinovitch (2011, Laws of Sanhedrin Ch. 9 §1) and Hallevi (1966).

See Perla (1973, positive commandment 97, our translation): “Why it cannot be anticipated that the jury will find a new point in favor of the defendant? Not because it is definitely impossible to acquit him according to the law, but because each juror sees that none of his colleagues finds any point for the defendant, and thus he concludes that the defense really has no hope and consequently does not pay attention to search for such a point, although it is still possible that with careful and cautious examination the defendant may be found innocent”.

To ensure free votes and deliberations, Jewish law provides that the King and the High Priest cannot be members of the Sanhedrin (bSanhedrin 18b), and “…in capital charges we commence [deliberation and votes] with [the opinion of] those on the side [benches]” (Mishnah, Sanhedrin 4 §2).

See for example Zvi Hirsch Hayout’s gloss on Sanhedrin (17a).

See Kings I (21, 13).

See for example, Exodus (23, 6), Deuteronomy (16, 19), Kethuboth (101a), Sota (21a).

For a discussion of the Jewish law philosophy see Sect. 5.

We assume that \(p < 1\), since if there existed a juror with \(p = 1\), that divine juror would be enough to ensure that justice is done.

For a related discussion see also https://​risingentropy.​com/​a-talmudic-probability-puzzle/​.

Assuming “contamination” would strengthen our results.

See also Duran (1910, pp. 20–21) and Povarsky (2000, Sanhedrin Volume A, 17b). Literally, פּוֹטְרִין אוֹתוֹ bears HaCohen’s translation as well.

More accurately, there is no continuous tradition of rabbinical rulings in Jewish criminal law. According to the Talmudic tradition, the Sanhedrin ceased to discuss criminal cases at 30 AC (Rosh-Hashana 31a), and was dismantled at 425 AC by a Roman decree. Post-Talmudic Jewish law authorities (except Maimonides) concentrated mainly on ritual and civil laws that remained relevant for Jewish religious and communal life in the Diaspora. The mainstream translation of פּוֹטְרִין אוֹתוֹ as “acquitted” was suggested by commentators, not decisors, and the Talmudic literature distinguishes strictly between these two functions. It should also be emphasized that HaCohen suggested his adjudication in his commentary on Maimonides who quoted R. Kahana’s ruling literally, arguing that by פּוֹטְרִין אוֹתוֹ both R. Kahana and Maimonides meant “dismissed”. In the absence of a continuous Jewish criminal law ruling tradition, HaCohen’s interpretation is valid as any alternative interpretation.

See Sanhedrin (34a), Maimonides Laws of Sanhedrin Ch. 11 §1, Babad (1924, commandment 77 §3), Uziel (1964 Vol. 4, Hoshen Mishpat Ch. 11) and more.

In Sect. 4.4 we show that their claim is inaccurate.

Geradi and Yariv (2007) indicated that previous literature almost ignored the possibility of a pre-vote deliberation. They showed that, on the one hand, if pre-vote communication among jurors is precluded, different voting-rules may generate different equilibria, and on the other hand with pre-vote deliberation all decision rules except the two types of unanimity rule (requiring a unanimous consensus to acquit or requiring a unanimous consensus to convict) generate the same set of equilibria regardless of the information and preferences structure or the deliberation protocol. In fact, they showed that with deliberation and non-unanimous voting rules the outcomes are implemented unanimously, but this result is expected from Aumann’s (1976) Agreement Theorem. (See also footnote 14).

“Do not say at voting ‘it is sufficient that I behave as my mentor’ but say what you really think”. (Tosefta, Sanhedrin Ch. 3 §8). The Tosefta (from the Aramaic תּוֹסֶפְתָּא, “addition”) is a collection of traditional rulings that were omitted from the Mishnah. According to the rabbinic tradition the Tosefta was redacted by R. Oshaya and R. Hiya (pupils of R. Yehuda HaNassi) in the third century CE.

Especially in light of Gunn et al. (2016), as explained above.

A natural interpretation of \(p\) and \(q_{j}\) is measures of jurors’ competence and standard of proof, respectively (Coughlan 2000). That is, juror \(j\) prefers to convict whenever \(\beta > q_{j}\), and in this sense \(\left( {1 - q_{j} } \right)\) measures juror \(j\)’s extent of “reasonable doubt”.

We follow Coughlan (2000) in endogenizing \( {{m}}_{{j}}^{{G}}\) and \( {{m}}_{{j}}^{{I}}\) by specifying a juror’s perceptions about the consequences of mistrial. Coughlan explained that “These perceptions may incorporate many different factors, but it seems reasonable to model the utility of mistrial as simply the expected utility of an additional trial before a new jury”. In a footnote he added: “I also could discount the expected utility of future trials or apply a fixed cost/disutility to each new trial. Mistrial utilities incorporating these factors still allow calculation of the refined necessary and sufficient conditions for sincere voting, but analysis of such utility structures significantly increases the complexity of the presentation without providing much additional insight”. Recall that in this model there is no time dimension and consequently no discount factor. Thus, all what a juror has to consider is the possibility that after a number of repeated mistrials, the defendant will finally be either convicted or acquitted.

Throughout the paper we simulate the results under the assumption that \(q \in \left( {\frac{1}{2},1} \right)\) since we assume that a juror prefers to acquit a guilty than to convict an innocent. This assumption is well established in the Western judicial code (Blackstone 1830) and also in the Jewish judicial code as explained in Sect. 5.

Below (Sect. 3.3.4) we compare this result with Coughlan’s.

Feddersen and Pessendorfer (1998) calculated the (unique) symmetric responsive equilibrium strategies for a general\({\text{k}}^{*}\): \(\sigma \left( {q_{j} ,i} \right) = \left\{ {\begin{array}{*{20}c} 0 & {\beta < q_{j} } \\ {\frac{{p\left( {1 + \sqrt[{k - 1}]{{\left( {\frac{{1 - q_{j} }}{{q_{j} }}} \right)\left( {\frac{1 - p}{p}} \right)^{n - k + 1} }}} \right)}}{{p - \left( {1 - p} \right)\sqrt[{k - 1}]{{\left( {\frac{{1 - q_{j} }}{{q_{j} }}} \right)\left( {\frac{1 - p}{p}} \right)^{n - k + 1} }}}}} & {\beta \ge q_{j} } \\ \end{array} } \right.\)\(\sigma \left( {q_{j} ,g} \right) = \left\{ {\begin{array}{*{20}c} {\frac{{\sqrt[{n - k}]{{\left( {\frac{{1 - q_{j} }}{{q_{j} }}} \right)\left( {\frac{p}{1 - p}} \right)^{k} }} - 1}}{{p\left( {1 + \sqrt[{n - k}]{{\left( {\frac{{1 - q_{j} }}{{q_{j} }}} \right)\left( {\frac{1 - p}{p}} \right)^{k} }}} \right) - 1}}} & {\beta < q_{j} } \\ 1 & {\beta \ge q_{j} } \\ \end{array} } \right.\)

Since \(\lim_{n \to \infty } \beta \left( {r,n - 1, n} \right) = 1\), it follows that \(\exists {\overline{\text{n}}}\) such that \( {\beta }\left( {{{r,n}} - {{1, n}}} \right) > {{q}}_{{{j}}}\) for every \( {{n}} > {\overline{{n}}}\). Feddersen and Pessendorfer (1998) showed that in this case, \(\sigma \left( {q_{j} ,i} \right) = \left\{ {\begin{array}{*{20}c} 0 & {\beta < q_{j} } \\ {\frac{{p\sqrt[{n - 1}]{{\frac{{\left( {1 - p} \right)\left( {1 - q_{j} } \right)}}{{pq_{j} }}}} - \left( {1 - p} \right)}}{{p - \left( {1 - p} \right)\sqrt[{n - 1}]{{\frac{{\left( {1 - p} \right)\left( {1 - q_{j} } \right)}}{{pq_{j} }}}}}}} & {\beta \ge q_{j} } \\ \end{array} } \right.\) and \(\sigma \left( {q_{j} ,g} \right) = 1\).

More accurately, Coughlan showed that (under certain assumptions) informative-sincere voting is compatible with Nash equilibrium in this model if \(\left( {\frac{r}{1 - r}} \right)\left( {\frac{1 - p}{p}} \right) \le \frac{{q_{j} }}{{1 - q_{j} }} \le \left( {\frac{r}{1 - r}} \right)\left( {\frac{p}{1 - p}} \right)\).

Nevertheless, keep in mind that odd number of jurors is important only if conviction is not conditional upon unanimous approval.

If \( {{n}}\) is odd, the condition coincides with condition quoted in footnote 36. If \( {{n}}\) is even, the necessary and sufficient condition is \(\left( {\frac{r}{1 - r}} \right)\left( {\frac{1 - p}{p}} \right)^{2} \le \frac{{q_{j} }}{{1 - q_{j} }} \le \left( {\frac{r}{1 - r}} \right)\) or \(\left( {\frac{r}{1 - r}} \right) \le \frac{{q_{j} }}{{1 - q_{j} }} \le \left( {\frac{r}{1 - r}} \right)\left( {\frac{1 - p}{p}} \right)^{2}\).

Notice that even under Coughlan’s revelation mechanism, if \(\beta \left( {r,n - 1, n} \right) > q_{j}\) for any \(k < n,\) then as long as \(n_{c} < n\) in the preliminary straw vote, even if \(n_{c} = n\) in the final vote there is still a “reasonable doubt” according to the societal norms. This problem is aggravated in odd juries, because satisfaction Coughlan’s conditions in this case requires ceiling \(q\) but \(\beta \left( {r,n - 1, n} \right) \le q_{j}\) implies that \(q\) is close to 1.

Baharad et Al. (2012) suggested an algorithm for estimation of jurors’ competence, based on their previous decisions. However, the application of this algorithm faces two major problems. First, the algorithm implicitly assumes that jurors have always vote informatively. Second, we still have no way to know whether a past decision was, indeed, correct.

By Feddersen and Pessendorfer, assuming \(q_{j} = q\ \forall j \in N\), the probabilities of Type I and Type II errors under the unanimity rule are \(L_{I}^{\rm UN} = \left( {\gamma_{I} } \right)^{n} = \left( {\frac{{\left( {2p - 1} \right)\sqrt[{n - 1}]{{\frac{{\left( {1 - q} \right)\left( {1 - p} \right)}}{pq}}}}}{{p - \left( {1 - p} \right)\sqrt[{n - 1}]{{\frac{{\left( {1 - q} \right)\left( {1 - p} \right)}}{pq}}}}}} \right)^{n}\) and \(L_{G}^{\rm UN} = 1 - \left( {\gamma_{G} } \right)^{n} = 1 - \left( {\frac{2p - 1}{{p - \left( {1 - p} \right)\sqrt[{n - 1}]{{\frac{{\left( {1 - q} \right)\left( {1 - p} \right)}}{pq}}}}}} \right)^{n}\).

Taking the limits of \( {{L}}_{{I}}^{{UN}}\) and \(\it {\text{L}}_{\text{G}}^{\text{UN}}\) (see footnote 41) yields \(\lim_{n \to \infty } L_{I}^{{{\text{UN}}}} = \left( {\frac{{\left( {1 - q} \right)\left( {1 - p} \right)}}{pq}} \right)^{{\frac{p}{2p - 1}}}\) and \(\lim_{n \to \infty } L_{G}^{{{\text{UN}}}} = \left( {\frac{{\left( {1 - q} \right)\left( {1 - p} \right)}}{pq}} \right)^{{\frac{1 - p}{{2p - 1}}}}\).

Under Feddersen and Pessendorfer’s calibration \(\lim_{n \to \infty } L_{I}^{{{\text{UN}}}} = 2.27 \times 10^{ - 1}\) and \(\lim_{n \to \infty } L_{G}^{{{\text{UN}}}} = 0.47\), while under our alternative calibration \(\lim_{n \to \infty } L_{I}^{{{\text{UN}}}} = 1.87 \times 10^{ - 6}\) and \(\lim_{n \to \infty } L_{G}^{{{\text{UN}}}} = {0}{\text{.999}}\).

This result is unsurprising since in Coughlan’s model the optimal rule is \(k^{*} = n\).

Zechariah 8, §16.

Jerusalem Talmud, Sanhedrin 1,1. For further discussion, see Lipschütz and Schwarz (2020).

The only human attributed with perfect judicial competence in Judaism is the Messiah (bSanhedrin 93b). According to the Talmudic tradition (Rosh-Hashana 21b), King Solomon’s pretension of being able to judge intuitively without witnesses or evidence was refuted by God, and even that wisest man could not really identify the real mother based on his analysis of the mothers’ behavior in the Judgment of King Solomon (Kings I, 3:16–28). The Babylonian Talmud (Makkot 23b) concludes that the verdict “she is the mother thereof” was actually issued by a celestial voice.

For a comprehensive discussion of this issue, see Abraham (2016).

Deuteronomy (8, 6, and 28, 9), Babylonian Talmud Sota (14a) and more.

See Maimonides, Laws of De’ot (general proper behavior, Ch. 7 §1–2).

Exodus 23, 2.

See Baba-Batra (93a), Maimonides Laws of Torts (8, §14), Laws of Sanhedrin (20, §1) and more.

Maimonides, Book of Commandments, (negative commandment 290), Translated by Berel Bell.