Two travelers who want to meet each come to a crossroad and must go left, right or straight ahead; which...

Problem

Two travelers who want to meet each come to a crossroad and must go
left, right or straight ahead; which road should X take?

Here is a little decision theory puzzle, invented by Wlodzimierz
Rabinowicz.  It was presented not as a puzzle, but as an example to
illustrate the superiority of causal decision theory over evidentiary
decision theory.  I'm not so sure that it succeeds.

Suppose two travelers want to meet each other.  They have started from
different directions and now each of them has come to a crossroad.
Each of them has to choose between going left, right, or straight
ahead.


           ___o___
         /         \
        /           \
       /             \
X-->--0-------o-------0---<---Y
       \             /
        \           /
         \____o____/

The zeros (0) are the current positions of the travelers, the lowercase
o's their potential meeting points.

Suppose neither traveler can just sit and wait for the other.  Since the
middle road is winding, the three roads take equally long; and there is
a hefty toll on the middle road.  Suppose also that each likes his left
road a bit better than the other two (maybe the scenery?) (though this
is not the same road for each, of course).  Here is the utility matrix
for X.  Y's is exactly analogous.

             Y goes...
          L    S    R
    L     1    1   11
X   S    -5    5   -5
    R    10    0    0

(They meet just in case one goes right and the other left, or they both
go straight, and this is the most important contributor to utility; the
S road is less desirable because of the toll; and X likes L a little
better than R.)

No communication between them is possible.

Now, suppose that X and Y are psychologically very similar to each
other, and that they know this, so that each takes his own decision to
be *evidence* of what the other will choose.  X takes the probability
that Y will go left, to be a bit higher than .5; and he takes the
probability that Y will go left given that X goes left, to be a bit
higher than .8 [and similarly for prob(Y goes right | X goes right)].

(Note that this problem is a bit Newcomby--the traveler takes his own
decision to be *evidence* for what someone else does, but does not
think his own decision will *cause* the other person to do anything.)

Which road should X take?

-- Jamie Dreier <pl436000@brownvm.brown.edu>  (December 1995)

[ Editor: the opening sentence is the editor's summary; Dreier's
posting, "The Travelers' Trilemma", follows it.  See also
decision/newcomb. ]