We can construct a tree of states, a/b meaning a games your team won out of b played: 
                    0/0
               1/1       0/1
          2/2       1/2       0/2
     3/3       2/3       1/3       0/3
4/4       3/4       2/4       1/4       0/4
     4/5       3/5       2/5       1/5
          4/6       3/6       2/6 
               4/7       3/7
At 4/4, 4/5, 4/6, and 4/7, we wish to be $1000 ahead; at 3/7, 2/6, 1/5, and 0/4, we wish to be $1000 behind; therefore, at 3/6, we must have $0, the average of $1000 (what we must have at 4/7) and $-1000 (what we must have at 3/7), and bet $1000, half the difference between $1000 and $-1000. If we move up the tree, filling in our desired worth at each state in this way, we get 
                     0 
              312.5     -312.5
          625        0       -625
     875       375      -375      -875
1000      750        0       -750      -1000
    1000       500      -500      -1000
         1000        0       -1000
              1000      -1000
Therefore, we should bet $312.50 on the first game.

David Moews (dmoews@fastmail.fm)

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