Problem 83 Solution

If you confess there is probability p of two confessions and (1-p) probability that only you implicate your partner. If you don't confess there is probability p that neither of you implicate each other and a (1-p) probability that only you are implicated. Let s donate one unit of suffering. One year in jail yourself would cause you 2 units of suffering, and one year of your partner serving would cause you 1 unit of suffering.

The table below shows the total suffering given the four possibilities:


                      Your partner
                 -----------------------
                 confess     not confess
  
Y  confess:     10*2+10=30    0+20=20
o
u  not confess: 2*20+0=40     2*3+3=9

So given the probability p of your partner confessing if you confess the expected, or average, amount of suffering would be 30p + 20(1-p)=10p+20. If you don't confess the expected amount of suffering would be 40p + 9(1-p)=31p+9. To find the indifference point equate the two expressions:

10p+20 = 31p+9
11 = 21p
p=11/21 =~ 52.4%.

Thus if you think your partner's probability of confessing is greater than 11/21 you should confess, otherwise you should keep your mouth shut.

What is a much more interesting question, with no answer that fits everybody, is what would you do if you assumed your partner had the same morals and priorities as you, and that he assumes the same thing of you.

Michael Shackleford, A.S.A.