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.

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