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.