Division problem

The division problem is a mathematical problem that goes back to Luca Pacioli (1494). Blaise Pascal and Pierre de Fermat wrote letters on this problem.

formulation

Two players A and B each put an equal amount of money E into a pot. For the amount G = 2E in the pot, you play a game of chance, which is made up of several rounds. A fair coin is tossed in each round. For the game they have agreed the following rules:

The game must be played until one of the two players has won n times.
The first person to win n times receives the amount in the pot. Regardless of how close the lead was, the other one gets nothing.

However, due to force majeure, the game must unexpectedly be stopped at score a: b before the decision is made. The first rule is broken. Game cannot continue or replay and the money must be split immediately.

Now put yourself in the position of a judge who is supposed to distribute the prize amount G in the pot to the two players “fairly”. Note that the word “just” here has a legal rather than a mathematical meaning.

suggestion

The past player argues that the game ended illegally. He wants his use E again get reimbursed half speak of G . After all, he could have caught up and won.

Counter proposal

The leading player claims the full amount of money. He insists on the "all or nothing" rule. Especially when he is clearly in the lead, one can expect that he will also win.

The two uncompromising proposals are neither “wrong” nor “right”. Rather, it depends on the viewer's sense of justice whether he evaluates one of the suggestions as “wrong” or “right”. How heavy is the second rule if the first has already been broken?

The following two views appear fair:

If the game is aborted when there is a tie, everyone receives half, i.e. their stake.
If there is a leader, he must never get less than the one behind.

Classic compromise solutions

Pacioli

A gets and B gets . ${\ displaystyle {\ frac {a} {a + b}} \, G}$ ${\ displaystyle {\ frac {b} {a + b}} \, G}$

The division ratio is for score a: b . ${\ displaystyle a: b \,}$

Tartaglia

A gets and B gets . ${\ displaystyle \ left ({\ frac {ab} {n}} + 1 \ right) E}$ ${\ displaystyle \ left ({\ frac {ba} {n}} + 1 \ right) E}$

The division ratio is . ${\ displaystyle (a-b + n) :( b-a + n) \,}$

Cardano

A gets and B gets ${\ displaystyle {\ frac {\ sum \ limits _ {k = 1} ^ {nb} k} {\ sum \ limits _ {k = 1} ^ {na} k + \ sum \ limits _ {k = 1} ^ {nb} k}} \, G}$ ${\ displaystyle {\ frac {\ sum \ limits _ {k = 1} ^ {na} k} {\ sum \ limits _ {k = 1} ^ {na} k + \ sum \ limits _ {k = 1} ^ {nb} k}} \, G}$

The division ratio is . ${\ displaystyle {\ sum \ limits _ {k = 1} ^ {nb} k}: {\ sum \ limits _ {k = 1} ^ {na} k}}$

Fermat and Pascal

A gets and B gets ${\ displaystyle {\ frac {2 ^ {a + b}} {2 ^ {2n-1}}} \ sum _ {k = 0} ^ {nb-1} {2n-1- (a + b) \ choose k} \, G}$ ${\ displaystyle {\ frac {2 ^ {a + b}} {2 ^ {2n-1}}} \ sum _ {k = 0} ^ {na-1} {2n-1- (a + b) \ choose k} \, G}$

The division ratio is . ${\ displaystyle \ sum _ {k = 0} ^ {nb-1} {2n-1- (a + b) \ choose k}: \ sum _ {k = 0} ^ {na-1} {2n-1 - (a + b) \ choose k}}$

Remarks

In the chain

Proposal - Tartaglia - Cardano - Fermat / Pascal - counter-proposal

The preference for the leader increases monotonously from left to right.

The solution by Fermat and Pascal ultimately seems to be the "fairest" or "most correct" one, because it divides the win amount according to the individual winning probabilities in a fictitious game continuation. In order to solve the problem, both had assumed that the players competing against each other had the same skill level. This is understandable, because Pacioli formulated the division problem in 1494 in relation to an aborted ball game, it was only later that it was not quite comprehensibly related to an aborted game of chance.

literature

Andreas Büchter, Hans-Wolfgang Henn: Elementary Stochastics: An introduction to the mathematics of data and chance . Springer 2007, ISBN 9783540453819 , pp. 263-266

Web links

The development of the concept of probability from 1654 to 1718 (pdf) - Master's thesis with the problem of division

Individual evidence

↑ Thomas Bronder: Game, chance and commerce. Theory and practice of the game of money between math, law and reality . Springer-Verlag Berlin Heidelberg (2016), XXIII, 313 pp., Softcover ISBN 978-3-662-48828-7 , eBook ISBN 978-3-662-48829-4 , problem of division s. Pp. 12-15, doi : 10.1007 / 978-3-662-48829-4

[1] Thomas Bronder: Game, chance and commerce. Theory and practice of the game of money between math, law and reality . Springer-Verlag Berlin Heidelberg (2016), XXIII, 313 pp., Softcover ISBN 978-3-662-48828-7 , eBook ISBN 978-3-662-48829-4 , problem of division s. Pp. 12-15, doi : 10.1007 / 978-3-662-48829-4