## Decision

An archive of questions and answers that may be of interest to puzzle enthusiasts.
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Question 1 - allais:
The Allais Paradox involves the choice between two alternatives:

A. 89% chance of an unknown amount
10% chance of \$1 million
1% chance of \$1 million

B. 89% chance of an unknown amount (the same amount as in A)
10% chance of \$2.5 million
1% chance of nothing
What is the rational choice? Does this choice remain the same if the unknown amount is \$1 million? If it is nothing? Show Answer

Question 2 - division:
N-Person Fair Division

If two people want to divide a pie but do not trust each other, they can still ensure that each gets a fair share by using the technique that one person cuts and the other person chooses. Generalize this technique to more than two people. Take care to ensure that no one can be cheated by a coalition of the others. Show Answer

Question 3 - dowry:
Sultan's Dowry

A sultan has granted a commoner a chance to marry one of his hundred daughters. The commoner will be presented the daughters one at a time. When a daughter is presented, the commoner will be told the daughter's dowry. The commoner has only one chance to accept or reject each daughter; he cannot return to a previously rejected daughter. The sultan's catch is that the commoner may only marry the daughter with the highest dowry. What is the commoner's best strategy assuming he knows nothing about the distribution of dowries? Show Answer

Question 4 - envelope:
Someone has prepared two envelopes containing money. One contains twice as much money as the other. You have decided to pick one envelope, but then the following argument occurs to you: Suppose my chosen envelope contains \$X, then the other envelope either contains \$X/2 or \$2X. Both cases are equally likely, so my expectation if I take the other envelope is .5 * \$X/2 + .5 * \$2X = \$1.25X, which is higher than my current \$X, so I should change my mind and take the other envelope. But then I can apply the argument all over again. Something is wrong here! Where did I go wrong?

In a variant of this problem, you are allowed to peek into the envelope you chose before finally settling on it. Suppose that when you peek you see \$100. Should you switch now? Show Answer

Question 5 - exchange:

Who pays for the drinks? Show Answer

Question 6 - high.or.low:
I pick two numbers, randomly, and tell you one of them. You are supposed to guess whether this is the lower or higher one of the two numbers I picked. Can you come up with a method of guessing that does better than picking the response "low" or "high" randomly (i.e. probability to guess right > .5) ? Show Answer

Question 7 - monty.hall:
You are a participant on "Let's Make a Deal." Monty Hall shows you three closed doors. He tells you that two of the closed doors have a goat behind them and that one of the doors has a new car behind it. You pick one door, but before you open it, Monty opens one of the two remaining doors and shows that it hides a goat. He then offers you a chance to switch doors with the remaining closed door. Is it to your advantage to do so? Show Answer

Question 8 - newcomb:
Newcomb's Problem
A being put one thousand dollars in box A and either zero or one million dollars in box B and presents you with two choices:
(1) Open box B only.
(2) Open both box A and B.
The being put money in box B only if it predicted you will choose option (1).
The being put nothing in box B if it predicted you will do anything other than
choose option (1) (including choosing option (2), flipping a coin, etc.).

Assuming that you have never known the being to be wrong in predicting your actions, which option should you choose to maximize the amount of money you get? Show Answer

Question 9 - prisoners:
Three prisoners on death row are told that one of them has been chosen at random for execution the next day, but the other two are to be freed. One privately begs the warden to at least tell him the name of one other prisoner who will be freed. The warden relents: 'Susie will go free.' Horrified, the first prisoner says that because he is now one of only two remaining prisoners at risk, his chances of execution have risen from one-third to one-half! Should the warden have kept his mouth shut? Show Answer

Question 10 - red:
I show you a shuffled deck of standard playing cards, one card at a time. At any point before I run out of cards, you must say "RED!". If the next card I show is red (i.e. diamonds or hearts), you win. We assume I the "dealer" don't have any control over what the order of cards is.

The question is, what's the best strategy, and what is your probability of winning ? Show Answer

Question 11 - rotating.table:
Four glasses are placed upside down in the four corners of a square rotating table. You wish to turn them all in the same direction, either all up or all down. You may do so by grasping any two glasses and, optionally, turning either over. There are two catches: you are blindfolded and the table is spun after each time you touch the glasses. Assuming that a bell rings when you have all the glasses up, how do you do it? pt type="text/javascript"> function reveal(a){ var e=document.getElementById(a); if(!e)return true; if(e.style.display=="none"){ e.style.display="block" } else { e.style.display="none" } return true; } Show Answer

Question 12 - stpetersburg:
What should you be willing to pay to play a game in which the payoff is calculated as follows: a coin is flipped until it comes up heads on the nth toss and the payoff is set at 2^n dollars? Show Answer

Question 13 - truel:
A, B, and C are to fight a three-cornered pistol duel. All know that A's chance of hitting his target is 0.3, C's is 0.5, and B never misses. They are to fire at their choice of target in succession in the order A, B, C, cyclically (but a hit man loses further turns and is no longer shot at) until only one man is left. What should A's strategy be? Show Answer