A.1 - C - This is one of our fundamental assumptions. Only the person themselves
can decide the value of a good, because that value is subjective.

A.2 - A - This is one of our fundamental assumptions. People are not
extraordinarily greedy. If they pay $12, they expect $12 of value.

A.3 - A - Here we act as mind-readers and see that Alex thinks he
got the piece with 3 out of 12 chips, so 25% of the value, and
Bart got 1/4 of the cookie by volume, so 25% of the value. This
is equitable since everybody got the same amount, but not fair
because everybody wanted at least half!

A.4 - B - Notice the first answer claims the division is unfair.
In this case the division is equitable and fair, but boring. They
could have been happier, but oh well, they are happy enough.

A.5 - B - Notice the first answer claims the division is unfair,
but we count it as fair even if people get more than they deserve.
The important part is that nobody gets less than what they deserve.They each
get 75% which iis more than enough.

B.1 - E - You might notice that most of the other games are radically
unfair. In other words, in each of the other games, one player has
a strategy to take so much loot that the other player cannot get their
fair share. In E, the second player can get more than their fair share,
but never at the expense of the first player (who always gets his fair
share if he plays the winning strategy of Be Honest).

B.2 - E - Ah yes, the winning strategy is to be honest. But be honest
in whose eyes? Your own. So easy.

B.3 - C - Choose the best piece in your own eyes. So easy. Although
the formula is nice too...

B.4 - C - This is the only one with both pieces of equal value.
The cake is worth say $3, with the chocolate side $2 and the strawberry side $1.
A is wrong: you can be sure.
B makes one piece worth $1 (half chocolate) and the other worth $2 ($1+$1)
C makes one piece worth (3/4)*$2 = $1.50 and the other worth (1/4)*$2 + (1/1)*$1 = $1.50, even stevens.
D makes one piece worth $2 and the other worth $1
E is wrong. If it was right, she could have just made one slice and let Bob
have it all.

B.5 - A - However she wants to decide which is bigger is up to her, but
the strategy is simple: choose the bigger pile.

C.1 - C - Notice how A and B involve a lot shooting. D means whoever
goes first wins, and E doesn't even make sense (what if two people have
the same favorite piece? I guess the shooting starts early!)

C.2 - E - Be honest, in your own opinion.  It doesn't matter if anybody
else thinks the pieces are even. It's actually good for society if they
don't (the extra value comes from these differences).

C.3 - C - Be honest, in your own opinion. It doesn't matter which piece other
people like, your strategy is just to claim all the pieces you wouldn't want to
lose. If you only claim the best one, you might not get it (could be more than
one person who only want that piece), but you could lose all the others (if
only one person wanted the second best piece, then that piece is gone!)

C.4 - D - Worst case is actually pretty simple: Mario gets one of the pieces,
worst case is he gets the $4 piece, which is unfair (he thinks he deserves
$5 = ($4+$5+$6)/3 ).

C.5 - A - His fair share is ($2+$7+$9)/3 = $6. By letting the $7 piece go
unclaimed, he stands a chance of losing that extra $7-$6 = $1. Well, he
loses his share of it, which by the time this happens is $1/(2 players)
= $0.50 per player. And A works this out in detail: he might only get $4.50
instead of $5. Not a huge loss, but a loss none-the-less. Had he claimed the
$7 and the $9 he would have been guaranteed at least $7, more than his fair
share. Greedy in this case was just foolish.

D.1 - D - not a great description, but that is the game. No shooting, and
no just choosing how much you get without anyone else having input.

D.2 - B - You are supposed to take one piece form each player, so take the
best one, duh.

D.3 - B - You cannot be sure the chooser will take the "right" piece, so
just make all the pieces the same.

D.4 - E - All of these are basically true. It is especially good to use
if you ever find yourself in the A situation (say when trying to help
your 3yr old nephew divide up the party hats and stuffed animals at the
birthday party and someone comes late).

D.5 - C - This is actually an easy problem, though it took me a bit to
remember that. Cid chops his 40% into 4 pieces, 10% + 10% + 10% = 10%.
Deb chooses one of them. Cid has 10% + 10% + 10% = 30% left. Easy.

E.1 - A - Notice how the other games are almost all unfair. D intrigues
me a little. Is it fair (as in, is there a winning strategy for everyone)?
If people are all honest, then it seems pretty good, but there seems to
be liitle motivation to bid honestly (you want OTHER people to bid honestly,
not you). At any rate, no bidding in last diminisher, just diminishing.

E.2 - D - This is be honest strategy with the "unless it is exactly fair"
exception built-in.

E.3 - D - Yup, same strategy, just with numbers.

E.4 - D - Yup, same strategy, just with numbers.

E.5 - D - Actually, this is a different strategy, a better one. Last player
has no need to shrink a piece that is too big.

F.1 - B - Well, it looks like "back to the loot" should be "back to the group".
Who writes these things? Notice the difference between A and B is Sweeney's
rule versus Knaster's rule. C and A are actually the same, except that Sweeney
is sneakier in C and just hopes no one notices the extra money. D is kind
of mean: all the extra money goes to the losers, none to the winner. This
encourages people to bid low, and lowers the expected happiness for everyone.
E is just completely unfair, though it basically means every bids 1/N as
much as they should, but like D, also encourages them to bid low.

F.2 - A - Honest, in your own opinion. D and E are flawed on two counts:
one is that sometimes you'd want to D and sometimes E, based on how your
own honest bid compared, but two is that they are probably lying to you!

F.3 - E - Yup, all of these are good explanations.

F.4 - D - With two players and one item, sealed bids becomes very simple: The
two players compromise on the price, ($36 + $24)/2 = $30 is the compromise
price. The high bidder pays the low bidder the fair share of the compromise
price, $30/2 = $15.  If you don't know that, then you can just do it the regular
way: high bidder pays the group $36 and gets the loot. Harley takes $18 back,
and Quinn takes $12. That leaves $6 in the middle, and Quinn gets $3 of it,
Harley the other $3. Since all the money was originally Harley's, we are really
just interested in what Quinn got: $12 + $3 is $15.

F.5 - C - Tom gets both piecs, because he is the high bidder both times. Tom
pays the group $20 + $12 for the two pieces of loot. Tom takes back $10 + $6,
and Wyatt takes $2.50 + $3.50. That leaves $20 + $12 - $10 - $6 - $2.50 - $3.50
which is $10 to be split evenly, so Tom takes back another $5 of his own money,
and Wyatt takes the other $5. Wyatt got a total of $2.50 + $3.50 + $5 = $11.

On the real exam, I'll probably have three people involved, but maybe only
one item. The fancy 3person+5item version can be the bonus question!

G: Bonus. This one you have to figure out for yourself. Here is wrong answer,
and an explanation of why it is wrong.

Give everyone the cheapest room!
Alex: Loft, $150/mo
Bart: Regular, $100/mo
Carl: Master, $200/mo
Dirk: Guest, $125/mo
That is a total of $150 + $100 + $200 + $125 = $575/mo! Oh, which is not
nearly enough to cover the rent, much less TV and internet.

Ok attempt number 2: Give everyone the cheapest room, but charge them extra!
Alex: Loft, $175/mo
Bart: Regular, $150/mo
Carl: Master, $225/mo
Dirk: Guest, $150/mo
That is a total of $800/mo, so we made the rent! Except EVERYONE feels
cheated! They are paying more than what they are getting in return!
Everyone is angry, no one does the chores, pretty soon the leftover
pizza in the fridge becomes sentient and devours the neighborhood.
Oh yeah, and no money for the TV or internet either. :-(

So you have to do better at assigning the rooms. You can charge people
extra (or less) but make sure they see the value: so for instance
if you charge them extra, it should be paying for the TV or internet,
and they should be willing to pay that much extra. For instance,
if we had managed to get internet, Alex would have been willing to
pay $150 + $5 = $155 for the loft, and with internet and TV, he'd
be willing to pay $150 + $5 + $20 = $175 for the loft. In the second
answer though, there was no money to pay for the internet and TV, so
he feels like he is getting charged for nothing!