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
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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)*E(r-c|r>=d).
However, in equilibrium it is not possible that P(r 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
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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.
