bushindo

Interesting puzzle. I'm getting some non-intuitive results...

Thanks for the code. I think I see what is causing the discrepancy between the simulation results. I made a mistake in transcribing the game probabilities...

Excellent insight, bonanova. The idea of using log is brilliant. However, my coding (and theoretical results) disagree with the simulation shown you and prime. winning.pdf

You rang?

Here's my try

I like your answer. It is interesting how in America this problem is solved differently from my home country. I'm intrigued. How do people from your country solve this problem?

How many days does each person contribute to the total project days? Individuals don't contribute to the project's time. A can do a project, alone, in a days, but a does not increase the time of a project. Rather, 1/a contributes to the reciprocal of the project's time. 1/a +1/b+1/c = 1/10, etc. Allow me to rephrase cause I disagree: Including someone on a project and excluding someone else directly affects the amount of days a project takes. So in terms of making a three man team we can determine the day load attributed to each person (as well as rank them by productivity). In terms of proportional pay for effectiveness this is very important as we in manufacturing projects pay individuals incentives based on their individual contribution to group projects. So, by your later question, are you asking for how long each individual would take to complete the project working on his/her own? [edit from here] This would then, of course, make that proportional payment easier, using [actual time]/[individual time] to determine the proportion of total payment each individual should receive. essentially, i am asking how much of the duration is caused by each individual on the three man team if we assume that a person is just as productive no matter who they work with Let a, b, c, and d stand for the fraction of a project that A, B, C, and D can do within each day, respectively.

Correct. The star awaits the formula. Recursive formula

Here's my answer

I think I see where we agree and where we diverge now. This two-envelope paradox has two variants, A) There are two envelopes, both of which are unopened. We reach the same conclusion on this one. B) One of the envelope is opened and has $1000. This is where we disagree

Can you clarify the part highlighted in red? Do you mean specifically to generate N random numbers from the uniform distribution between, say, 0 and L? If I'm writing code for this experiment, I can't generate a random number without telling the computer precisely which probability distribution to use (and the corresponding distribution parameters). Most computer programs, for instance, will allow one to generate a random number uniformly between [0, L], but then you will need to supply the value for the upper limit L. (Reverend Bayes, is that you?) Randomness comes in many forms (e.g., normal, uniform, exponential, etc. ) and I don't think it is possible to generate random numbers without specifying which probability distribution we are working with.

If we want to maximize the minimum perceived percentage plus making sure that no one thinks someone else is receiving more money, then here's an approximate strategy

I'd love to hear about this experiment that does not depend on the probability distribution of how much money is in A and B. My feeling is that Reverends Bayes is hiding somewhere, possibly heavily disguised, in the set-up. But I may be wrong, I often am.

Can you elaborate on the bolded part? I'm not sure that I can parse that correctly. There are many ways to determine a 'best' answer in this question. Both you and Pickett found effective answer and there are in fact more answers that would work in providing everyone at least 25% of the fair share of the goods and money. I am now seeking the answer that provides everyone the most profit. I do not want the average percentage of perceived benefit from the will, I want to award the 'best' solution to the one who can give the most to the person who received the least. For example: (I am making these percentages up by the way) Remember each person expects to receive 1/4 of the value of the old man's wealth strategy 1's allocation strategy gave the following outcomes person 1=27% person 2=36% person 3=40% person 4=30% strategy 2's allocation strategy gave the following outcomes person 1= 30% person 2= 29% person 3= 28% person 4= 29% though the average gain in strategy 1 is 33.25% which is higher than strategy 2, person 1 only made 2% more than expected so this strategy is not preferred when compared to strategy 2 since its lowest recipient got 28% of the perceived wealth or 3% more than expected. So in these hypothetical cases, solution number 2 is perceived as better since the minimum beneficiary is better than the other cases minimum beneficiary. **and of course these percentages are perceived percentages of value they placed on the old man's wealth. If you are trying to maximize the minimum perceived percentage of the value, then you can divide the property in the following manner

Can you elaborate on the bolded part? I'm not sure that I can parse that correctly.

I think when you code this problem as plasmid suggested, you have to be careful to specify what kind of random distribution you are drawing from since that essentially will define your prior assumptions about A and B. Most random number generator have an explicit upper range, so it might be a problem to sample uniformly from a infinite real line. I think the discussion should be about what prior distribution is more representative of the puzzle conditions

If one of the people got the house which in Sonya's case means the house is worth 2.6 million should she also get 1/4 of the value back? Sounds like she got an incredible discount. I don't think that would be a problem

Love this problem. Here's how I would approach it.

Can you gain as much by switching back? Using the same reasoning? There is a paradox: A gain for switching can be anticipated. Yet, there is a preferred envelope, and if we initially chose it we should not switch. Not sure what you mean by "switching back". The 50/50 comes from randomly picking one of 2 envelopes from which we know one has double the money of the other. I don't see a paradox. The paradox comes from the fact that $1000 is arbitrary. Suppose I were to point to an envelope before any were opened, and I said "That envelope contains some amount of money; call it $X. The other envelope therefore must contain $2X or $X/2." You could now argue that the expected earnings from picking the other envelope are $5X/4, so you should choose the other envelope. But in reality, have I actually given you any more information than you already had when you only knew that one envelope contains twice as much money as the other? I think k-man is right. There is no infinite switching paradox in this formulation. That is because being able to see the envelope amount ($1000) nails down the value of one envelope. In other words, the amount $1000 is not arbitrary. The moment we see it, Reverend Bayes has already entered the room. Let me try to disentangle the paradox using Bayesian statistics

In that case

If each player will get the full amount if they happen to choose the same envelope, then

Answer for Question A

if they are equivalent then why don't they produce the same expected values as Bushindo pointed out. I misread the equation. Yours was

Nevermind, they are equivalent. See witzar's