11 6
1 1 2 (game theory) 1 2.1........................................ 1 2.2...................................... 3 3 2 3 3.1........................................ 3 3.2................................... 9 4 14
1 ( ).,. TV, ( ).,,.,. 2 (game theory), 1944, John von Neumann, Oskar Morgenstern, ( ).,,,,,,,,,. 2.1, (game theory).,,. (player),.,,,,., 2, 3, n. (ratonal),.. ( ), ( ). (coalton),,. (strategy),, (outcome). 1
(preference order),,. (payoff), (utlty).. (rule),,,. (game wth complete nformaton),,., (common knowlegde). ex),, etc. (ntellgent),. (soluton),,.,., (game n strategc form), (game n extensve form), (game n coaltonal form).,.... 2
2.2 2.1.,,.. 3 2. 3.1. (1) 2. (2) 0. (3). (4) 1, 1. (5),. 1 2 P 1, P 2. 3.1 P 1 P 2 Π 1 = { = 1, 2,..., m} Π 2 = { = 1, 2,..., n}, P 1, P 2. P 1, P 2, f 1 (, ), f 2 (, )., ( 0, ), f 1 (, ) + f 2 (, ) = 0 ( ) 3
., a = f 1 (, ) = f 2 (, ), a P 1 P 2., a., A = (a ) = a 11 a 1 a 1n... a 1 a a n... a m1 a m a mn, (payoff matrx)., (matrx game) m n., A = (a ) P 1 (P 2 )., P 1. P 2.., P 1, P 2.,. 3.1.1 3.1.. P 1 P 2 7 1 2 5 0 4 5 1 3,, P 1 P 2, P 1., P 2, P 1 = 1,, mn (7, 1, 2) = 2 ( = 3) P 2., P 1 = 2, = 3, 4
, mn ( 5, 0, 4) = 5 ( = 2) mn (5, 1, 2) = 1 ( = 3) P 2.,, = 3 P 1. ( 2, 5, 1) = 1, P 1 = 3, 1., P 2. P 1, P 2 = 1, (7, 5, 5) = 7 ( = 1) P 1, P 2., P 2 = 2, = 3,, ( 1, 0, 1) = 1 ( = 2) ( 2, 4, 3) = 4 ( = 3) P 1.,, = 2. mn (7, 1, 4) = 1 P 1, P 2 1. 1( P 2 1),. A = (a ), P 1, mn (a 1, a 2,..., a n ) = mn a, P 1 (securty level)., P 1, v 1 = (mn = mn a 1, mn a 2,..., mn a m ) a 5
v 1. v 1 (mn value)., P 1 (mn strategy)., P 2,, (a 1, a 2,..., a m ) = a, P 2., P 2, v 2 = mn ( = mn a 1, a a 2,..., a n ) v 2. v 2 (mn value), P 2 (mn strategy)., v 1, v 2,. (mn prncple), (mn prncple), 2., 2. 3.1.2, A = (a ), mn a = mn a, (strctly determned).,,, (, )., (, ) (equlbrum pont), v(a) = a (value). (, ) v(a),. 3.1.3 A = (a ), mn a mn a 6
, v 1 = v 2.,. A = (a ),,, a 0 a 0 0 a 0, ( 0, 0 ) (saddle pont), a 0 0 (saddle pont value). 3.2., 1.,.. A = (a ), = 1,..., m, ( 0, 0 ),, = 1,..., n, A a 0 a 0 0 a 0, a 0 = a 0 0 = mn a 0.,, mn a a 0, mn a 0 mn a,.,, mn a a 0 0 mn mn a mn a = a 0 0 = mn a mn a a.,, ( 0, 0 ), a 0 0.. mn a = mn a, 1 (, )., mn a = mn a 7
., mn a = a.,, mn a = a a a,, a a.,, a a a mn a,, a a, a a a, (, ) A. 3.3. 2,,., ( 0, 0 ) (, ), ( 0, ), (, 0 ).,. 3.1.4 3.2. P 1 P 2 5 2 1 3 0 4 2 1 2 8
. P 1, = 1 = 3, = 1 P 2 = 3.,, P 1 = 3. P 2. = 1 = 2, = 2 P 1 = 1 P 2., P 2 = 1.,. P 2 [ ] 2 1 P 1 0 4 (domnate). A = (a ), 2 k, a > a k, k.,, a a k,,, a = a k, k (weakly domnate). 2 l. 3.2 3.2.1.,,.,,.,.,, 1.., p, q, E(p, q). p, q (mxed strategy).,, (pure strategy). P 1 ( ) 1/4. (80% + 0% + 10% + 30%)/4 = 30% 9
1 80% 0% 10% 30% P 2 ( ). (0% + 30%)/2 = 15% 1, P 1, P 2,, P 1 P 2,.,,. 3.2.2 1 Π 1 = { = 1, 2,..., m} 2 Π 2 = { = 1, 2,..., n}, a 11 a 1n A = (a ) =.. a m1 a mn, 1 p = (p 1, p 2,..., p m ), p 1 0, p 2 0,..., p m 0, p 1 + p 2 +... + p m = 1, 2 q = (q 1, q 2,..., q n ), q 1 0, q 2 0,..., q n 0, q 1 + q 2 +... + q n = 1.,, q T q,. E(p, q) = paq T = a 11 p 1 q 1 + a 12 p 1 q 2 +... + a mn p m q n P 1,, mn q E(p, q) 10
, p, v 1 = mn E(p, q) p q, 1., P 2, q E(p, q) p, q, v 2 = mn q E(p, q) p, 2.,. p mn q E(p, q) mn q E(p, q) p 3.4., 1 2 S 1, S 2,. p S 1 mn q S 2 paq T = mn q S 2 p S 1 paq T, (p, q ).,,.., v = v(a) = p Aq T.,, {(p, q ), v}. 3.5. (p, q ), (p, q ) E(p, q) = paq T., p S 1, q S 2, E(p, p ) E(p, q ) E(p, q). 11
3.6., (p, q ), (p 0, p 0 ), (p, q 0 ), (p 0, q ).. 3. 5, paq T p Aq T p Aq T (1) paq 0T p 0 Aq 0T p 0 Aq T (2). (1), p = p 0, q = q 0, p 0 Aq T p Aq T p Aq 0T (2), p = p, q = q, p Aq 0T p 0 Aq 0T p 0 Aq T, p Aq T = p 0 Aq T,.,.,, p Aq 0T = p Aq T = p 0 Aq 0T, paq 0T p Aq 0T p Aq T paq T p 0 Aq T p 0 Aq T., (p, q 0 ), (p 0, q ). 3.7. v(a), (p, q ), = 1, 2,..., m, = 1, 2,..., n, E(, q ) v(a) E(p, ). 12
. {(p, q ), v(a)},, 3. 5, v(a) = E(p, q ) E(p, q ) v(a) E(p, q) p, q., S 1 p, m m E(, q )p v(a)p = v(a) =1 =1, E(p, q ) v(a)., S 2 q, 2, p = p, q = q, v(a) E(p, q) E(p, q ) v(a), v(a) E(p, q ),, v(a) = E(p, q ). E(p, q ) E(p, q ) E(p, q), (p, q ), v(a). 3.8. {(p, q ), v},. E(, q ) = mn E(p, ) = v. 3. 7 = 1,..., n, v E(p, ), 13
v mn E(p, )., v < mn E(p, ),, v < E(p a st, ), n n v q < E(p, )q, =1 v, =1 v < E(p, q ) v = mn E(p, ), v = E(, q ),. 3.1.,. mn a v(a) mn a 4,,.,.,,,.,,,,.,. 14
[1],,, 2003. [2],,, 2011. [3],,, 2008. 15