# Auction

Six paintings, worth $1500, $2500, $3500, $4500, $5500, and $6500, are auctioned,
and their are only two bidders, Ann and Beth. The paintings are auctioned in a random
order unknown to the bidders. Starting bid is always $1000, and it is increases by
$1000 in each round. The two bidders alternate bidding. Thereby geting a picture
(since the other player said "no" is also considered a move, thus after that the
player who said "no" gets the $1000 offer.

Ann's and Beth's goal is to
maximize the sum of cash money and value in paintings. The bidders goal is **not**
to be better than the other bidder---they don't care about each other.

Depending on how much money both
bidders initially have, we get different variants.

Here is the applet

We have a two-player sequential game. If we assume that the ordering of the paintings
is done at the very beginning, but only the next non-auctioned painting is revealed to
the two players, then we have a formulation
with non-perfect information. However, the same game can also be formulated as a game of
perfect information, and of course we would prefer this formualation. Here we assume that
at the very beginning only the very first picture is chosen. After it has been auctioned, we
have another random move selecting the second painting with equal probability, and so on.
Therefore backwards induction should allow us to analyze the game, and find the solution.
The only problem with this game is that the number of positions is lareg,
too large for drawing a game tree or even a game digraph. ...

Let's assume throughout that all amounts are expressed in thousands of dollars.

## The last picture occurs

Let A(a,b,c) denote the position that the last picture is put on stage, and it is worth c,
and Ann has still an amount of a (in thousands of dollars), and Beth still b, and Ann is to move.
The corresponding position where B is to move is denoted by B(a,b,c).
...