Rudolf Taschner - Game Changers: Stories of the Revolutionary Minds behind Game Theory

EXERCISES

1. Mziriac's think-of-a-number game

In his book Problmes plaisants et dlectables qui se font par les nombres, published in the seventeenth century, Bachet de Mziriac presents the following game: Two players sit opposite one another. The first person names a number between 1 and 10. Then the second person thinks of a number between 1 and 10, adds it to the one just named and calls out the total. The first person then adds, in his turn, a number between 1 and 10 and calls out the new total. The two players alternate in this way until one of them is able to call out a number greater than 100. The one who can say a number greater than 100 has won the game.

Question: How does a conman proceed when he takes his opponent to the cleaner's in this game?

A) He tries to name one of the numbers 10, 20, 30, 40, 50, 60, 70, 80, or 90. As soon as he manages this, he proceeds in multiples of ten.

B) He tries to name one of the numbers 2, 13, 24, 35, 46, 57, 68, 79, or 90. As soon as he manages this, he proceeds in multiples of eleven.

C) He keeps naming a number that is 1 more than his opponent's, until his opponent names a number between 80 and 89. Then he responds with 90 and, because his opponent can only say a number between 91 and 100, the conman has won.

2. A force-majeure exercise

Two people playing a game of chance, in which the chances of each player winning a round are exactly fifty-fifty, agree that the first person to win five rounds will receive the entire stake. After the first player has won three rounds and the second player has won one, the game is interrupted by a force majeure, a superior force.

Question: Based on the rounds played so far, how should the stake be fairly divided up after the game has been interrupted?

A) The stake should be divided up at a ratio of 3:1 in favor of the first player.

B) The stake should be divided up at a ratio of 4:2 (i.e., 2:1), in favor of the first player.

C) The stake should be divided up at a ratio of 13:3 in favor of the first player.

3. A simple game of craps

Two dice are thrown. Beforehand, a player wagers a hundred dollars that the sum of the numbers thrown will be either seven or eleven. If he wins, he receives 300 dollars on top of his stake; if he loses, his stake goes to the gambling house.

Question: What winnings can the gambling house expect after 7,200 such games?

A) The house can expect a profit of about 80,000 dollars (before tax).

B) The house can expect a profit of about 40,000 dollars (before tax).

C) The house cannot expect any profit.

4. The Chevalier de Mr's mistake

The avid gambler Antoine Gombaud, known as the Chevalier de Mr, is said to have come to the following erroneous conclusion: he thought that, when throwing three dice, there was an equal probability of throwing a sum of eleven or a sum of twelve, since the sum of eleven could only be achieved by 1 + 4 + 6 = 1 + 5 + 5 = 2 + 3 + 6 = 2 + 4 + 5 = 3 + 3 + 5 = 3 + 4 + 4 and the sum of twelve could only be achieved by 1 + 5 + 6 = 2 + 4 + 6 = 2 + 5 + 5 = 3 + 3 + 6 = 3 + 4 + 5 = 4 + 4 + 4. That makes six different groups of numbers to get each of eleven and twelve, and, consequently, de Mr's view was that there is an equal probability of throwing a sum of eleven or twelve.

Question: How does this error arise, and which of the two total sums is more probable?

A) The Chevalier de Mr wasn't wrong at all, since the two sums are indeed equally probable.

B) The Chevalier de Mr overlooked the fact that, in the case of the total sum of eleven, three of the possible groups of numbers contain 2 identical summands, whereas in the case of the total sum of twelve, only two of the possible groups of numbers contain 2 identical summands, but another one of the possible groups of numbers is made up of 3 identical summands. The total sum of twelve is more probable than the total sum of eleven.