The junk mail problem (also known as spam) is well known to just about everyone who receives e-mail. There has also been many solutions proposed. Noticeably, the idea of having e-mail senders include ecash payments with their mail has come up several times (I believe as the result of independent discovery). In order to evaluate the effectiveness of this proposal, I will construct a game theoretic model of the interaction between the sender and the recipient of an e-mail, and compare the solutions with and without the ecash payment option. The Model Players: A - Sender, B - Recipient A: Send mail? / \ no / \ yes / \ (0,0) B: Read mail? / \ no / \ yes / \ (0,0) B: Accept offer? / \ no / \ yes / \ (0,-c) (s,r-c) Assumptions: - sending the e-mail is costless to the sender - c (the cost of reading a piece of e-mail) is known to both the sender and the recipient. - s and r (the profit of the proposed deal to sender and receiver, respectively, both assumed to be non-negative) are distributed according to the probability density function f(s,r). The sender knows s and r before sending the e-mail, but the receiver does not learn s and r until he reads the e-mail. Solution of the game: To solve this game, we apply the method of backward induction. In the last stage of the game, B decides whether to accept A's offer. Clearly he always accepts since r-c >= -c. Therefore, in the next to last stage, B knows that the expected payoff if he reads the mail is the expected value of r-c, E(r)-c, so he will read if E(r)-c > 0. Finally, we come to A's decision. If A knows that B will not read, then he is indifferent between sending and not sending. However, if we assume that there is a small probability that B will read and accept irrationally, then we can conclude that A always sends the mail. Conclusions To summerize, if E(r) > c, B always reads the mail and accepts the offer, otherwise B never reads. A always sends regardless of the value of the parameters. Now we can see the outcome is not socially optimal. For example if E(r) < c, both A and B would be better off if A only sends when r>c. The above model is not very realistic. The most unrealistic assumption is that the sender knows the exactly value of his offer to the recipient. However I believe the model captures the essence of the junk mail problem. Next time I will analyze the proposed solution of adding the option of a pre-payment. For those who want to try it themselves, I give the game tree here: 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,r-c)