Problem of points
The division problem is a mathematical problem, which goes back to Luca Pacioli ( 1494). Blaise Pascal and Pierre de Fermat wrote each other letters about this issue.
- 2.1 Paccioli
- 2.2 Tartaglia
- 2.3 Cardano
- 2.4 Fermat and Pascal
Formulation
Two players A and B each define an equal money using E in a pot. To obtain the amount lying in the pot G = 2E they play a game of luck, which is composed of several rounds. In each round, a fair coin is tossed. For the game, they have agreed to the following rules:
Due to a higher power, however, the game must unexpectedly when a score before the decision: will b canceled. The first rule is violated it. The game can not be continued or repeated, and the money allocation must be the same.
Now Put yourself in the position of a judge, who will distribute the winning amount G in the pot on the two players 'fair'. Note that the word here has a legal "fair" as a mathematical meaning.
Proposal
The past players argued that the game was stopped foul. He wants to use his E to get reimbursed again, say half of G. He had, after all, can also catch up and win.
Counter-proposal
The leading player claimed the full amount of money for themselves. He insists on the "all or nothing " rule. Just when he is clearly in the lead, is to be expected that he also wins.
The two compromise proposals are neither " wrong " nor " right". It rather depends on the sense of justice of the viewer whether it evaluates one of the proposals as " wrong" or "right." How serious is the second rule even when but the first one was already broken?
Accessible appear the following two views:
- If the match is abandoned at equal points, so each gets half, so its use.
- Is there a leader, he shall not receive less than the Past.
Classical compromise solutions
Paccioli
A gets and B gets.
The split ratio is the Score a: b.
Tartaglia
A gets and B gets.
The split ratio is.
Cardano
A gets and B gets
The split ratio is.
Fermat and Pascal
A gets and B gets
The split ratio is.
Comments
In the chain
Increases monotonically from left to right, the preference of the leader.
The solution of Fermat and Pascal seems ultimately the " fairest " or " most correct " to be, because it divides the profit calculated in accordance with the individual probabilities of winning in a fictional game sequel.