Backward induction and expected value calculations in an anonymous XVth century Italian manuscript

Abstract

Blaise Pascal famously calculated the expected value of a risky lottery in the 1654 correspondence with Pierre de Fermat in the context of the so-called points problem. Pascal solved this problem by backward induction, whereas Fermat—by counting combinations. This paper analyzes a more complex version of the points problem from an anonymous XVth century Italian manuscript stored in the Vatican Apostolic Library. In this manuscript, the problem of points is solved by backward induction in a similar way to Pascal’s train of thought. In this light, Pascal’s pioneering contribution may be not as novel as it is traditionally believed.

Keywords:

• Expected value
• backward induction
• points problem
• decision theory
• XVth century commercial arithmetics

JEL classification code:

• B11

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 Franklin (2015, 294–295) published an English translation.

2 In medieval Italy it was a common practice to denote unknown variables with c, the first letter of an Italian word cosa, which could be translated as something.

3 The value of winning the second game must be the same as that of the first. To see this, notice that if the second player wins the second game, the situation becomes symmetric (each player won one round) and the split must be 50%-50%. Thus, whatever the first player wins in the first game, the second player can recover after winning the second game. On July 29th, 1654, Pascal in his letter to Fermat discovers by calculation that “the value of the first game is equal to that of the second”.

4 A scanned copy is available at https://digi.vatlib.it/view/MSS_Urb.lat.291, a published version can be found in Franci (2002, 260) and Meusnier (2007).

5 Manuscript Urb.lat.291 contains several handwritings pointing to a co-authorship. Yet, most of the manuscript is written by one hand.

6 A footnote on p. 61r (in a different handwriting from the main text) indicates that Jo. Petrus de Penis finished this Italian translation of Leonardo’s work on June 18th, 1313.

7 “… don't throw this into the mind of everybody because, as they say, the one who shows everything is not prudent/pious.” Urb.lat.291 p. 94v.

8 aleatorum lucrum usura fortunae (Pascasius Justus 1561, 119).

9 In the context of dicing, this happens routinely when a die is lost after a toss and players cannot continue playing.

10 Thus, the author is clearly unaware of the solution of the points problem by forward induction described in the medieval manuscript Codice Magliabechiano Cl. XI, 120.

11 se quello che un giuoco vincesse un altro giuoco sarebbe al pari con quelli altri due e arebbe lo terzo di tutta la possta.

12 che perché sono 3 omini ne viene lo terzo per omo.

13 sicché adunqua quello 1/9 toccha per terzo a ciascuno, che è 1/27.

14 This is crucial in the context of gambling on dice (commonly among many players).

15 after the XIVth century Codice Magliabechiano Cl. XI, 120.

16 Similarly, an earlier example of the points problem in Codice Magliabechiano Cl. XI, 120 also does not employ any notion of chance, or luck, or fortune.

17 Ratio autem demonstrative super hoc est quod si facta divisione iterum ludus esset inchoandus, partes haberent deponere idem quod receperunt stante conditione.

18 Considerations of fairness and justice also play an important role in Pascal’s reasoning, see Jallais and Pradier (1997).

Additional information

Funding

Pavlo Blavatskyy is a member of the Entrepreneurship and Innovation Chair, which is part of LabEx Entrepreneurship (University of Montpellier, France) and funded by the French government [Labex Entreprendre, ANR-10-Labex-11-01].