After graduating from college, you have taken an important managing position in the prestigious financial...

induction/takeover Added material

Problem

After graduating from college, you have taken an important managing position
in the prestigious financial firm of "Mary and Lee".
You are responsible for all the decisions concerning take-over bids.
Your immediate concern is whether to take over "Financial Data".
There is no doubt that you will be successful if you are the first to
bid and that this will be profitable for the firm and you in the long
run.  However, you know that there exist another n financial firms,
similar to "Mary and Lee", that are also considering the possibility.
Although you are likely to be the first one to move, you know that
just after a take-over there is a lot of adjustment that needs to be
done.  In fact, for a period of time following any take-over the
successful firm becomes a prime candidate for a take-over which will
cost the job of whoever is responsible for take-overs.  Among all
financial firms it is common knowledge that the managers responsible
for take-overs are rational and intelligent.  What is your best response?

Show solution

Assume the takeover is wise for n.  The takeover is then unwise for
n+1, as the other companies now find themselves in the same situation
as you for n.  If the decision is unwise for n, by similar reasoning
it is wise to takeover FD for n+1.  Now note that for n=1 the takeover
decision is clearly unwise, hence by induction you should takeover
FD iff n is even.

Additional material

[ Editor: in 1995 the same puzzle was posted with 25 hungry lions in
place of the firms, asking what happens when extra food is thrown in. ]

This is in the puzzle archive under the name `induction/takeover', in
the guise of companies taking over other companies rather than lions
eating other lions.

It is easy to show by induction that it is safe to eat the food if and
only if there are an odd number of lions present.  So whichever lion is
nearest the food eats it.

-- Gareth Rees <gdr11@cl.cam.ac.uk>