Assume the matrix looks like this.
| Problem 192 | ||
| Player A | Player B | |
| Rock | Paper | |
| Rock | a | b |
| Paper | c | d |
First, verify neither player is always better off with one strategy. If a>c and b>d then player A will always choose rock. Likewise, if a>b and c>d, the player B will always choose paper. Assuming there is no obvious pick for either player then it becomes a random strategy.
Player A should pick rock with a probability propotional to abs(c-d), and paper with probability proportional to abs(a-b).
Player B should pick rock with a probability propotional to abs(b-d), and paper with probability proportional to abs(a-c).
Here, again is the matrix for this problem.
| Problem 192 | ||
| Player A | Player B | |
| Rock | Paper | |
| Rock | 7 | 3 |
| Paper | 2 | 11 |
Player A should pick rock with probability abs(2-11)/(abs(2-11)+abs(7-3)) = 9/(9+4) = 9/13.
Player A should pick paper with probability abs(7-3)/(abs(2-11)+abs(7-3)) = 9/(9+4) = 4/13.
Player B should pick rock with probability abs(3-11)/(abs(3-11)+abs(7-2)) = 8/(8+5) = 8/13.
Player B should pick paper with probability abs(7-2)/(abs(3-11)+abs(7-2)) = 5/(8+5) = 5/13.
Player A's average win will be (9/13)×(8/13)×7 + (9/13)×(5/13)×3 + (4/13)×(8/13)×2 + (4/13)×(5/13)×11 = 2.982248521 + 0.798816568 + 0.378698225 + 1.301775148 = 5.461538462
For help with this problem I recommend 'The Complete Strategyst' by J.D. Williams.
Michael Shackleford, A.S.A.