junk mail analysis, part 2

[Wei Dai <weidai@eskimo.com>]

Last time I gave the equilibrium for the junk mail game.  Now I will look
at a modified game that allows the sender to include an ecash deposit with
his email.  (Note that there is a slight change of notation from the game
tree given last time.) 

      A: Send mail?
        /     \
    no /       \ yes
      /         \
    (0,0)    A: Decide deposit d
                |
                |
                |
             B: Read mail?
               /   \
           no /     \ yes
             /       \
         (-d,d)   B: Accept offer?
                    /  \
                no /    \ yes
                  /      \
               (-d,d-c)  (s,r-c)

Solution

We again apply the method of backward induction.  In the last stage B
accepts if r >= d.  Therefore in the next to last stage, B knows that if
he reads, his expected payoff is P(r<d)*(d-c)  + P(r>=d)*E(r-c|r>=d). 
However, in equilibrium it is not possible that P(r<d) > 0 since A is
always better off by offering a deposit of 0 instead of any deposit
greater than r. Therefore B reads if E(r|r>=d)-c >= d.  Now we come to A's
deposit decision.  A knowns that if he offers any d such that r >= d and
E(r|r>=d)-c >= d, B will read and accept.  A is indifferent between any
such d, so he might as well offer the smallest such d if it exists.  If it
doesn't exist, A offers d=0.  Finally A again always sends regardless of
the parameters, since A can get a payoff of at least 0 by sending, and may
do better if there is a small probability of B making a mistake.

Conclusions

We saw that if there exists a d such that r >= d and E(r|r>=d)-c >= d, A
offers the least such d, and B reads and accepts.  Otherwise A offers d=0,
and B does not read.  Interestingly, if E(r) > c, d=0 satisfies r >= d and
E(r|r>=d)-c >= d, so we reach the same outcome as before.  However, if
E(r) < c, the outcome of the new game represents a Pareto-improvement
since for realistic distributions of (s,r)  it seems likely that for all
sufficiently large r there exist d such that r >= d and E(r|r>=d)-c >= d,
and for these values of r both the sender and the receiver do better than
they did in the previous model.  Let's call the smallest such r t. 
Unfortunately the outcome is still not Pareto-optimal if t > c. 

This conclusion opens the question of whether a better solution exists.
One possibility is the following (called the pre-payment solution).

      A: Send mail?
        /     \
    no /       \ yes
      /         \
    (0,0)    A: Decide pre-payment p
                |
                |
                |
             B: Read mail?
               /   \
           no /     \ yes
             /       \
         (-p,p)   B: Accept offer?
                    /  \
                no /    \ yes
                  /      \
               (-p,p-c)  (s-p,r+p-c)

If there is enough interest, I'll follow up with a comparison between the
pre-payment solution and the deposit solution.