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B: matrix(
[ 0 , 1/4, 3/4 ],
[ 1/10, 0 , 1/4 ],
[ 9/10, 3/4, 0 ]
);
F: matrix(
[ 0, -1, 1 ],
[ 1, 0, -1 ],
[ -1, 1, 0 ]
);
G: matrix(
[ 0, -1, 1 ],
[ 1, 0, -1 ],
[ -1, 1, 0 ]
);
U: matrix([1], [1], [1]);
p: (1/3) * U;
ratsimp(expand((B * G) . p));
ratsimp(expand(transpose(U) . (B * G) . p));
/* find the eigenvector */
/* find the eigenvector */
B0: B - identfor(B);
r: expand(B0[1] / B0[1,1]);
B0[1]: r;
B0[2]: B0[2] - B0[2,1] * r;
B0[3]: B0[3] - B0[3,1] * r;
B0: expand(B0);
r: B0[2] / B0[2,2];
B0[2]: r;
B0[3]: B0[3] - B0[3,2] * r;
B0: ratsimp(expand(B0));
B0[1]: B0[1] - B0[1,2] * B0[2];
B0: ratsimp(expand(B0));
l: 78 + 12 * epsilon + 80 * epsilon^2;
D: ratsimp(expand(l*B0));
n: ratsimp(expand(l -D[1,3] -D[2,3]));
p: (1/n) * matrix(
[ -B0[1,3]*l ],
[ -B0[2,3]*l ],
[ l ]
);
p: ratsimp(expand(p));
/* compute the expected gain */
G*B;
ratsimp(expand(transpose(U). (G*B) . p));
/* modified game */
Btilde: B - epsilon * F;
ratsimp(expand((Btilde * G) . p));
gain: ratsimp(expand(transpose(U) . (Btilde * G) . p));
/* find the eigenvector */
B0: Btilde - identfor(Btilde);
r: expand(B0[1] / B0[1,1]);
B0[1]: r;
B0[2]: B0[2] - B0[2,1] * r;
B0[3]: B0[3] - B0[3,1] * r;
B0: expand(B0);
r: B0[2] / B0[2,2];
B0[2]: r;
B0[3]: B0[3] - B0[3,2] * r;
B0: ratsimp(expand(B0));
B0[1]: B0[1] - B0[1,2] * B0[2];
B0: ratsimp(expand(B0));
l: 78 + 12 * epsilon + 80 * epsilon^2;
D: ratsimp(expand(l*B0));
n: ratsimp(expand(l -D[1,3] -D[2,3]));
pepsilon: (1/n) * matrix(
[ -B0[1,3]*l ],
[ -B0[2,3]*l ],
[ l ]
);
pepsilon: ratsimp(expand(pepsilon));
/* taylor series expansion of the eigenvector */
taylor(pepsilon, epsilon, 0, 3);
/* expectation */
G*Btilde;
gainepsilon: ratsimp(expand(transpose(U). (G*Btilde) . pepsilon));
taylor(gainepsilon, epsilon, 0, 5);
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