Problem 1. Fair four-way division of an inheritance.

We are going to work through an example of fair division in an inheritance, involving four people, Janice, Scott, Andrea and Eric.

The fair division procedure concerns three big items, the house in the small town in which they all grew up, a very nice cabin in the mountains where they spent many vacations when they were children, and a boat in which they all remember going on fishing trips.

Each of the four is asked to assign their own subjective value to the house. All of the different items need some work, and they are probably worth more to them, effectively, than the market value if they just simply got them appraised.

They all have different preferences as well; Eric has remained in the same area of the country and he has a lot of affection for the house in which he grew up. His own family is growing, and they need more space. He really would like to move into his parents' house. Janice doesn't care much about the house - she certainly doesn't want to live in it, and she knows it would need a lot of fixing up before they could get a good price for it. On the other hand, she has fond memories of the mountain cabin, and if she could get it, she would use it often for hiking get-aways from her city job. Scott wouldn't mind taking over the house, but he does not feel about it very strongly; he certainly doesn't intend to take vacations in the mountain cabin - his job doesn't leave him enough time to go there often for a few days, and he prefers to spend longer vacations elsewhere. But he is very much interested in the boat, with which he could go on day-outings during the weekend. Finally, Andrea is not so determined to take over the boat or the house or the cabin, and she just tries to gauge what she thinks would be a reasonable price for them.

When they are asked to put a monetary value on each item, taking into account not only what they think it is worth, but also what it is worth to them, they come up with the following:

They have to make these evaluations separately and simultaneously - no jockeying for position or lying about one's own interest after listening to what the others said.

a) We want to divide up this estate according to the Knaster procedure that we saw in class.

(i) Assign each item to the person who wanted it most. In this case, who gets the house, who gets the cabin and who gets the boat?

(ii) Imagine that each of them now pays into a common "pot" the amount corresponding to what they got "over" their share. For instance, Eric gets the house, worth in his estimate 200,000, of which his share would normally have been 25%; so he has to pay the excess, 75% of $200,000, into the pot. Work out how much each individual person pays into the pot.

(iii) Next, each person withdraws from the pot the money equivalent, according to their own evaluation, of the shares in the items that they didn't get. Work out how much each individual person withdraws from the pot.

(iv) What remains in the pot after this stage gets divided up evenly between all the heirs. Work out how much each individual person gets from this step.

(v) Finally, write down for each person what they got, in items (if they got any) and the monetary value that they assigned to the items. Also write down the actual money that they paid or received. Use these two answers to work out how much each person effectively received in monetary terms based on their valuations of the items.

(vi) For each person, now compare the value of what they felt they got (from part (v)) with what each person thought the whole estate was worth. Based on this result, do you feel that they will feel happy with this allocation or not? Why?

b) Imagine now a slightly different situation. We still have the same three items, and the same evaluations by each of the four heirs. But, because Eric spent much more time in the final years of their parents' life taking care of many things for them, all the children have agreed that this entitles him to 40% of the estate, while the other three get 20% each. Work through the same division process as outlined in (a) with these unequal weights. How much 'better' does each person feel they have done, in monetary terms, compared with what they expected to receive based on their original evaluation?

c) Do you think this process is a good method for dividing assets? Do you think the scheme is envy-free?

Problem 2. Wisdom of the Math Alive class.

Problem 3. Fair and envy-free division.

Problem 4. Binary Numbers.

• (a) Convert 177 and 327 to binary.
• (b) Add the resulting binary numbers.
• (c) Convert the resulting binary sum to decimal notation. How does the answer compare with what you would get if you just added 177 and 327 in decimal notation?

Problem 5. Properties of binary addition

Now let's examine what happens when we add two identical numbers. Make the following additions in binary:

1010011 +   1010011

11001 +   11001

100110111010 +   100110111010

Do you notice something special? Can you generalize this to a statement about what happens when you add any number in binary to itself (Hint: there is a similarity between the outcome and the two identical numbers that you add), and explain why this would be so? In other words, rephrase the special feature that you observe in the three sums above, so that it applies when you add ANY binary number (not just one of these examples) to itself, and explain why this holds. (Hint: what happens when you add up ten copies of the same number in the standard decimal system?)