Probabilistic consideration on chances of invasion between states -- a symmetry


acrobat form here

(please use monospaced font because 3 lines are used for each expression.)
Japanese Constitution outlaws any armament, though no small armed forces and weapons already exist. There are constantly debates to change the Constitution to legitimate this army and even to send it abroad.

Hawks attack the defenders of the Constitution by putting a question "What can you do without army when our country is invaded by another countries?" Though it is a natural question and has to be answered anyhow, it should be reminded that the following question must be put equally. "What can we do when our army invades another countries?" These possibilities are mathematically equal, if one assumes no a priori difference in the nature of countries --agressive or peaceful.

The following is not a realistic discussion but only a simple mathematical one. But I believe this is sufficient to remind these two questions should be posed equally.

One might think that the possibility of being invaded is greater, because the number of potential invader to his/her country is many but his/her country which may invade is unique. If you don't think so, you need not read hereafter.

Suppose we have n+1 states whose probabilities of invasion to other states are all equal.

Let p be this probability of invasion by a state to any other states in a certain period of time. p is also an expected value of the frequency of invasion by a state. Then the expected frequency of invasion by a state to one of other states is p/n.

Let us calculate the expected frequency of invasion to the state by _all_ other states.

The probability P(k) of being invaded simultaneously by particular k states is 
            (please use monospaced font because 
             3 lines are used for each expression.)

          p  k     p  n-k
 P(k) = ( - )  (1- - )
          n        n

and its number of cases is  C .
                           n k

Then the total expectation of frequency of invasion by all of other states is 

      n                  n          p  k    p  n-k
<f> = ·  k   C   P(k) =  ·  k  C  ( - ) (1- - )
      k=1   n  k         k=1   n k  n       n

      n   n-k                r  p  k  p  r
    = ·   ·  k  C     C  (-1) ( - ) ( - )
      k=1 r=0   n k n-k r       n     n

      n   n-k         n!            r  p  k+r
    = ·   ·   ----------------- (-1) ( - )
      k=1 r=0  (k-1)!(n-k-r)!r!        n

Changing the summation indices k and r into s=k+r and r, this expression becomes

      n   s-1         n!            r  p  s
<f> = ·   ·   ----------------- (-1) ( - )
      s=1 r=0  (s-r-1)!(n-s)!r!        n

      n     n!     p  s  s-1      1          r  
    = ·   ------ ( - )   ·  ----------- (-1) 
      s=1 (n-s)!   n     r=0 (s-r-1)!r!        

        n!     p  1
    = ------ ( - ) 
      (n-1)!   n   

       n     n!     p  s  s-1      1          r  
     + ·   ------ ( - )   ·  ----------- (-1) 
       s=2 (n-s)!   n     r=0 (s-r-1)!r!        


The last summation of the 2nd term is zero because 
it is simply an expansion of 

           s-1
    (1 - 1)

divided by (s-1)!.
So that,  is simply equal to p.

This result is anticipated because "defensive war" and "aggression" are only different names for a same event, the war.