game theoretic analysis of junk mail

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)

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Wei Dai!  Long time no read!  I hope to see more articles from
you.


 A million monkeys operating under the pseudonym 
 "Wei Dai <weidai@eskimo.com>" typed:
> 
> 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).


I've suggested it several times.  I'm curious who else has 
proposed it.


<snip on game model of problem>


Nice!  I look forward to part 2.  (And part 3, where you go 
ahead and account for the cost of signing up for digital 
payment systems and the cost of managing each 
deposit/payment...)



I think the major problems with the proposal are first making 
digital payment systems ubiquitous and second making the
"unsolicited mail good faith deposit" idea ubiquitous.



Hm.  If we could somehow make the _idea_ gain.. er.. 
"currency" then the problem of spammers would cause more 
people to adopt digital payment systems.  _That_ would be
making a blessing from a curse!


Regards,

Zooko Journeyman, a.k.a. Bryce

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