Conference rusure::math

1229.0. "The Good, The Bad, The Ugly &..." by HERON::BUCHANAN (combinatorial bomb squad) Thu Apr 26 1990 11:30

	An old chestnut is the problem of the 3 way duel.   A,B & C are
fighting for honour or love or something.   They stand equidistant from
one another, and each takes it in turn to shoot.   Each can choose what
to shoot at.   Suppose that A is a perfect shot, B hits his target 80%
of the time, and C hits the target 50%.   If C is set to shoot first,
who should he shoot at?   What are the different players' probability of
winning the duel?

*******************************************************************************

	A variant of this generalizes cleanly.   Suppose that they don't
take it turns to shoot, but rather after each shoot, one of the survivors
is picked at random to fire the next.   Now let n be the number of duellists.

	Make two simplifying assumptions.

	(1) The probabilities of success, p_j, for j=1,..,n are algebraically
independent over the rationals.   (This means that they don't satisfy any
ponynomial equation with rationals as coefficients.    In particular, they
are irrational.   In particularly particular, p_j is not 0 or 1.)

	(2) No speaking or conniving or in any way dealing with other players 
is permitted, and each player attempts to maximize his chance of winning, and 
assumes that his opponents will do the same, etc.

	The winning gimmick that existed in the first version of the game is
no longer advantageous, but the tactics are still interesting.   I think I
have figured out n=4, but for larger n, perhaps some MAPLE or REDUCE support is
useful.

*******************************************************************************

	Finally, forget about the analysis.   Either version of the duel can
be made the basis of a pub game, say to decide who buys the next round of
drinks.   (British cultural bias here.)   It requires percentile dice, or a
lot of patience with coins.   Before the duel, each person decides 
independently how good a shot he wants to be!   He can pick anything from 1%
to 100%.   Everyone announces his p_j, and the shooting begins.   [Perhaps it's
an idea to normalize all the p_j to add to 1, if they are all ludicrously
tiny.]

	It is not at all obvious what to do.

Regards,
Andrew.

    Lets take the "standard' problem first (I've not heard it before):
    
>    If C is set to shoot first,
>who should he shoot at?   

    Who shoots next?  A or B?
    
    Doesn't matter really, either way C should just shoot into the air.
    
    --  Barry