Problem 193 Solution

Assume the matrix looks like this.

First, verify neither player is always better off with one or two strategies. If there is an obviously inferior pick for either player then the problem can be reduced to one such as problem 192. 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[(d-e)(h-i) - (e-f)(g-h)].
Player A should pick paper with a probability propotional to abs[(a-b)(h-i) - (b-c)(g-h)].
Player A should pick scissors with a probability propotional to abs[(a-b)(e-f) - (b-c)(d-e)].

Player B should pick rock with a probability propotional to abs[(b-e)(f-i) - (e-h)(c-f)].
Player B should pick paper with a probability propotional to abs[(a-d)(f-i) - (d-g)(c-f)].
Player B should pick scissors with a probability propotional to abs[(a-d)(e-h) - (d-g)(b-e)].

Here, again is the matrix for this problem.

Player A should pick rock with probability proportional to abs[(10*1) - (-2)*(-2)] = abs[10-4] = 6.
Player A should pick paper with probability proportional to abs[(6*1) - (-6)*(-2)] = abs[6-12] = 6.
Player A should pick scissors with probability proportional to abs[(6*-2) - (-6)*(10)] = abs[-12+60] = 48.

Player B should pick rock with probability proportional to abs[(2*-5) - (-8)*(6)] = abs[-10+48] = 38.
Player B should pick paper with probability proportional to abs[(-2*-5) - (4)*(6)] = abs[10-24] = 14.
Player B should pick scissors with probability proportional to abs[(-2*-8) - (4)*(2)] = abs[16-8] = 8.

Player A's average win will be (6/60)×(38/60)×6 + (6/60)×(14/60)×0 + (6/60)×(8/60)×6 + (6/60)×(38/60)×8 + (6/60)×(14/60)×-2 + (6/60)×(8/60)×0 + (48/60)×(38/60)×4 + (48/60)×(14/60)×6 + (48/60)×(8/60)×5 = 0.38 + 0+ 0.08+0.506666667 -0.046666667+ 0+2.026666667 +1.12 +0.533333333 = 4.6

The solution for the general case for an n × n matrix:

1. Call the original matrix m
2. For the row weights, make submatrix (r) of row differences, where r_i,j = m_i+1,j - m_i,j.
3. The weighting for row i is abs(mdeterm(r')), where r' is the submatrix s, omitting row i. Mdeterm is the Excel function for the determinant.
4. For the column weights, make submatrix (c) of row differences, where c_i,j = m_i,j+1 - m_i,j.
5. The weighting for column i is abs(mdeterm(c')), where c' is the submatrix c, omitting column i.

Consider the matrix:

1	7	0	3
0	0	3	5
1	2	4	1
6	0	2	0

The weighting for row 1 is the absolute value of the determinant of:

0	3	2
1	2	-3
-6	2	-2

=88

The weighting for row 2 is the absolute value of the determinant of:

6	-7	3
1	2	-3
-6	2	-2

=86

Here is the full solution:

1	7	0	3	88
0	0	3	5	86
1	2	4	1	78
6	0	2	0	101
89	62	119	83	

For help with this problem I recommend 'The Complete Strategyst' by J.D. Williams.