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1 II (Basics of Probability Theory ad Radom Walks) (Preface),.,,,.,,,...,,.,.,,.,,. (Basics of Proability Theory). (Probability Spacees ad Radom Variables , (Expectatios, Meas) (Law of Large Numbers) (Radom Walks) 6 2. (Markov Chais) d (d-dimesioal Radom Walks) (Oe-dimesioal Ati-symmetric Radom Walks) ( 2.2(iii)) (Applicatos of Probability Thoery; Other Topics) 8 4. (Rui problem) (Prisoer s Dilemma)

2 (Basics of Proability Theory). (Probability Spacees ad Radom Variables,, (Ω, F, P ),, X,. (Ω, F, P ) (probability space) Ω ( ω Ω ) F ( 2 Ω ) Ω σ (σ-field); (2 Ω Ω ) (i) Ω F (ii) A F A c F (iii) A F ( =, 2,...) A F, A F (evet). P = P (dω) (Ω, F) (probability measure), i.e., ; P : F [0, ]. (i) P (Ω) = (ii) A F ( =, 2,...) P ( A ) = P (A ) (σ ). (Ω, F, P ),. (i) σ-., F σ- A, B, A F F, A B, A \ B, A B := (A \ B) (B \ A), A. lima lim sup A := A, lima lim if A := A F. N N N N (lim = if sup, lim = sup if.) = (ii) P ( ) = 0, A k F (k =, 2,..., ) P ( A k) = P (A k) ( ). (iii) A, B F; A B P (A) P (B) ( ). ( ) (iv) A F, A P A = lim P (A ). ( ) (v) A F, A P A = lim P (A ).. ( ) (vi) A F ( ) P A P (A ) ( ). (vii) (Borel-Catelli ) A F ( ), ( P (A ) < P ) lim sup A = 0, i.e., P ( ) lim if Ac =.

3 (Ω, F, P ) X = X(ω) : Ω R {X a} := {ω Ω; X(ω) a} F ( a R). (radom variable). X S = {a j } j R, {X = a j } F ( j ). X k (Ω, F, P ) (k =, 2,..., ). {X k } (idepedet) P (X a,, X a ) = P (X a ) P (X a ) ( a k R, k =,..., ). {X k } k N {X k } N. X k S = {a j } j, : P (X = b,, X = b ) = P (X = b ) P (X = b ) (b k S, k =,..., ). µ(a) = P (X A) X (distributio), F (x) = P (X x) X (distributio fuctio)..2, (Expectatios, Meas) X Z := Z {± }. X (expectatio) or (mea) EX = E[X] = XdP = X(ω)P (dω). () X 0 EX := P (X = ) + P (X = ). =0 (P (X = ) = 0 P (X = ) = 0. P (X = ) > 0 EX =.) (2) X X + := X 0, X := ( X) 0 ( X ± 0, X = X + X.) EX := EX + EX.,. EX = Z P (X = ), f : Z R, Ef(X) = f()p (X = ). (.) Z ;f()>0 ;f()<0 X, V (X) := E[(X EX) 2 ] = E[X 2 ] E[X] 2.. (Chebichev ) p. a > 0, P ( X a) E[ X p ] a p. [ ] P ( X a) = P ( X p a p ) p =. E X = P ( X = ) a P ( X = ) a a P ( X = ) = ap ( X a). Ω.2 X,..., X Z, E[Xk 2 ] < (k =,..., ). X,..., X, E[X j X k ] = E[X j ]E[X k ] (j k). 0 (E[X k ] = 0) ( ) 2 E X k = E[Xk]. 2 2

4 [ ] () j k P (X j = m, X k = ) = P (X j = m)p (X k = ) E[X j X k ] = m, mp (X j = m, X k = ) = m, mp (X j = m)p (X k = ) = E[X j ]E[X k ]. ( ) 2 (2) X k = X j X k () j k E[X j X k ] = E[X j ]E[X k ] = 0. X 2 k + j k.3 (Law of Large Numbers), /2.,., X =, X = 0. EX = /2 ( V (X ) = /2 )., X k,, /2..3 ( (Weak Law of Large Numbers)) X, X 2,... EX = m v := sup V (X ) < ϵ > 0, lim P ( ) X k m ϵ = 0, i.e., lim P ( ) X k m < ϵ =. [ ] {X } { X = X m} ( ). X k m = (X k m), X X m = 0, i.e., E[X ] = 0 V (X ) = E[X] 2, ( ) 2 E X k = E[Xk] 2 = V (X k ) sup V (X ) = v. ϵ > 0, ( P ) X k ϵ = P ( ) X k ϵ v ϵ 2 2 = v ϵ 2 0 E[( X k) 2 ] ϵ 2 2 ( ). X, X, ϵ > 0, P ( X X ϵ) 0 ( ), X X i pr., X X. P (X X) =, X X, P -a.s., X X. (a.s. almost surely ).2, i.e., X X, P -a.s. X X i pr.. 3

5 ( P (X X) = P { X X < } = P k k N N k N N k, lim P { X X } = P N k N N N ( = k, lim P X N X ) lim N k P N N (, /k ϵ > 0. ) { X X k } = 0 { X X k } = 0 { X X k } = 0.,.,.,,..3, i.e., X X i pr. { k }; X k X a.s.. (, (?) { k }; P ( X k X 2 ) k 2 k. Borel-Catelli P { X k X < 2 } k =..) N,.. k N.4 ( (Strog Law of Large Numbers)) X, X 2,... EX = m v := sup V (X ) < ( ) P lim X k = m =..2,,. ( ) P lim (X k EX k ) = 0 =. [ ] EX = 0, S = (X k /k), (i) Kolmogorov sup k S k S 0 ( ) i pr. (ii), {S } Cauchy,. (iii) Kroecker X k 0 P -a.s..,,. 4

6 [ sup E[X 4 ] < ] X X = X m m = 0, i.e., E[X ] = 0 ( ) 4 X k 0 E[Y 2 ] (E[Y 4 ]) /2 ( ; 0 V (X) = E[(X EX) 2 ] = E[X 2 ] (EX) 2.) ( ) 4 E X k = E[Xk] 4 + i j, i,j E[Xi 2 ]E[Xj 2 ] 2 sup E[Xk] 4 k Fubii ( ) 4 ( E ) 4 X k = 4 E X k 2 sup E[Xk] 4 < k P = ( = ( = ) ) ( X k < =,, P lim lim., δ > 0,. (X k EX k ) = 0 P -a.s. +δ = ) X k = 0 = δ 0 ( ).. R ( ) µ(dx) = g(x)dx g(x) = e (x m)2 2v 2πv, m, v (ormal dist.), (Gaussia dist.), N(m, v).. X, Y a R, P (X a) = P (Y a), X (d) = Y. (X = Y i the sese of distributio ).5 ( (Cetral Limit Theorem)) {X } (idepedet idetically distributed = i.i.d. ). EX = m, V (X ) = v (X k m) 0, v N(0, v), i.e., a < b,, v lim P ( a (X k m) b ) = b 2πv a e x2 2v dx. (X k m) 0, N(0, v),,, Fourier,. (. 5

7 2 (Radom Walks),,.,,. d, d. Z d ( j = (j,..., j d )) d (lattice). (X, P ) d (simple radom walk),, 2d,. Y = X X ( ) {X 0, Y, Y 2,...}, {Y }, P (Y = k) = /(2d) ( k = ), = 0 ( k = ). k = (k,..., k d ), k = k k2 d. {X 0, Y, Y 2,...}, k 0, k,..., k Z, P (X 0 = k 0, Y = k,..., Y = k ) = P (X 0 = k 0 )P (Y = k ) P (Y = k ). Z d {p k } k Z d (p k 0, p k = ), (X, P ), d. P (Y = k) = p k (, k Z d ). P j (X = k,..., X = k ) := P (X = k,..., X = k X 0 = j) P j (X, P j ) j d. 2. P (A B) := P (A B)/P (B) P (B) > 0. A, B F P (A B) = P (A). 2. (Markov Chais),,.. 2. S,,.,,,.,.,,,,.. S, (X, P ) = (X (ω), P (dω)) ( = 0,, 2,...) S (Stochastic Processes), 0, X, i.e., j S, {X h = j} F. (X, P ) (Markov Chai) : 6

8 (M) [ ], j 0, j,..., j, j, k S, P (X + = k X 0 = j 0,..., X = j, X = j) = P (X + = k X = j).. (M2) [ ], j, k S, P (X + = k X = j) = P (X = k X 0 = j).,. 0, j, k S, q () j,k = P (X = k X 0 = j), Q () = (q () j,k ) ( ) (-step trasitio probability (trasitio matrix)),, Q () Q = (q j,k ),, ( ). 2.. (i) q () j,k 0, k q() j,k = (j S), (ii), j 0, j,..., j S. P (X = j,..., X = j X 0 = j 0 ) = P (X = j X = j )P (X = j X 2 = j 2 ) P (X = j X 0 = j 0 ) = q j0,j q j,j = P (X m+ = j,..., X m+ = j X m = j 0 ) (m 0) (iii) Q (0) = I := (δ jk ) ( ), Q () = Q ( ). (Q ) jk = q j,j q j,j 2 q j,k j,...,j X 0 µ = {µ j }; µ j = P (X 0 = j) (iitial distributio),, j S, P (X 0 = j) = P P j, (X, P j ) j. ( P (X 0 = j) > 0, P j ( ) := P ( X 0 = j),.) 2.2 P j0 : ( ) P j0 z(x + = k X 0 = j 0,..., X = j, X = j) = P j0 (X + = k X = j) = P j0 (X 2 = k X = j) = q j,k 2.2 P j0 ( ) := P ( X 0 = j 0 ) P j0 (X 2 = k X = j) = q j,k. 2.3 µ = {µ j },. (i) P (X 0 = j 0, X = j,..., X = j ) = µ j0 q j0,j q j,j, (ii) P (X = k) = j S µ j q () j,k. 2.4 (i) {B k }, A, C, P (A B k) = P (A C) ( k ). P (A B k ) = P (A C). (ii) m,, j,..., j m, k 0, k,..., k S P (X + = j,..., X +m = j m X 0 = k 0, X = k,..., X = k ) = q k,j q j,j 2 q jm,j m ( 2.3 (i)),. P (X + = j,..., X +m = j m X 0 = k 0, X = k,..., X = k ) = P (X + = j,..., X +m = j m X = k ). 7

9 , j S (recurrece time): T j : T j = if{ ; X = j} (= if { } = ). j (recurret) j (trasiet) def P j (T j < ) =, def P j (T j < ) <. j, T j, j (positive-recurret) def E j [T j ] < ( P j (T j < ) = ), j (ull-recurret) def E j [T j ] =, P j (T j < ) =. E j [T j ] T j P j, : E j [T j ] = mp j (T j = m) + P j (T j = ). m= j (or,, ) (X ) (or,, ). 2.5 E j [T j ] < P j (T j < ) =. {X } Q = (q j,k ) π = {π j } π (statioary distributio) def π k = j π jq j,k (k S), π (reversible distributio) def π k q k,j = π j q j,k (j, k S) (i) π,, X π. (ii) π, {X } :, j 0,..., j S, P (X 0 = j 0,..., X = j ) = P (X 0 = j,..., X = j 0 ). {X } Q = (q j,k ) (irreducible) j, k,, q () j,k > 0.,,. (,,.), : 2.2 j, k S. (i) j : a) q () j,j =. =0 b) P j ({X } j ) =. 8

10 (ii) j : a) q () j,j <. (iii) =0 b) P j ({X } j ) = 0. {X },,,,,. (π j ) [ k, j π jq j,k = π k ], π j = /E j [T j ] ( ). (i), (ii) b), a), (iii). (iii). 2. (i) j S P j ({X } j ) =. (ii) j S P j ({X } j ) = 0..,,,., 0. m j T (m) j. P j (T (m) j T () j = T j, T (m) j = mi{ > T (m ) j ; X = j} (= if { } = ). < ) = P j (T j < ) m. s, t,, P j (T (m) j = s + t T (m ) j = s) = P j (T j = t). (, [ ]= P j (X s+t = j, X s+u j ( u t ) T (m ) j = s), {T (m ) j = s} {X,..., X s } ( 2.4), ( 2. (ii)) [ ]= P j (X s+t = j, X s+u j ( u t ) X s = j) =[ ].)., P j (T (m) j P j (T (m ) j = s, T (m) j P j (T (m) j < ) = P j (T (m ) j = = s + t) = P j (T (m ) j = s)p j (T j = t) s=m t= < T (m) j < ) P j (T (m ) j = P j (T (m ) j < )P j (T j < ) < ) = P j (T j < ) m. P j ({X } j ) = P j ( m P j (T j < ) =, 0. = s, T (m) j = s + t) {T (m) j < }) = lim m P j(t (m) j < ) = lim m P j(t j < ) m. 9

11 ,. j, k S, f (m) j,k := P j(t k = m) (m ) Q jk (s) := =0 q () j,k s ( s < ), F jk (s) := m= f (m) j,k s ( s ). {q () j,k } 0, {f (m) j,k } m (geeratig fuctios). F jk () = P j (T k < ). 2. j, k S, : q () j,k = m= f (m) j,k q m k,k ( ), Q jk (s) = δ jk + F jk (s)q kk (s) ( s < ). {T k = m} = {X m = k, X s k ( s m )} m= f (m) j,k q( m) k,k = = = P j (T k = m)p j (X = k X m = k) m= P j (T k = m)p j (X = k T k = m) m= P j (X = k, T k = m) m= = P j (X = k) = q () j,k. Q jk (s) = δ jk + = δ jk + = q () j,k s f (m) j,k q( m) k,k s = m= = δ jk + F jk (s)q kk (s). 2.2 j S =0 q () j,j =. Q jj (s)( F jj (s)) = ( s < ) F jj () = P j (T j < ) lim Q jj (s) = s =0 q () j,j s. : =0 q () j,j ( P j(t j < )) =. 0

12 2.8 j k j S =0 q () k,j < ( k S), k S; =0 q () k,j = j :. ( q() k,j = F kj() q() j,j.) 2.2 j j k [i.e., ; q () j,k > 0] P k(t j < ) =., i, j S P i (T j < ) = q i,j +. (, k S;k j q i,k P k (T j < ) P i (X = k, T j = ) = q i,k P k (T j = ) P i (T j < ) = P i (X = k, T j = ) = k S.) i = j j, k ; q j,k > 0, P k (T j < ) =., k 2 ; q k,k 2 > 0, i.e, q (2) j,k 2 > 0, P k2 (T j < ) =., q () j,k > 0 (k,..., k ); q j,k q k,k 2 q k2,k 3 q k,k > 0, : j, j k =0 q () k,j =. j, k S j k k j j k. 2.3 j, k S; j k, j,, k.,,,. l, m 0; q (l) j,k > 0, q(m) k,j > 0. j, =0 q (l+m+) j,j q (l) j,k q() k,k q(m) k,j ( 0) Q jj (s) s l+m q (l) j,k q(m) k,j Q kk(s). lim Q jj (s) = s =0 q () j,j < q () k,k <, k. j, k.

13 . 2. Q jj (s)( F jj (s)) = F jj (s) = Q jj (s)/q jj(s) 2. j. lim s Q jj (s) Q jj (s) 2 = F jj( ) = E j [T j ] <. Q kk (s) s l+m q (m) k,j q(l) j,k Q jj(s), Q jj(s) (l + m + )s l+m+ q (l+m+) =0 s l+m q (l) j,k q(m) k,j Q kk(s) Q kk (s) Q kk (s) 2 Q jj (s) s 3(l+m) (q (l) j,k )3 (q (m) k,j )3 Q jj (s) 2. j,j E k [T k ] = F kk( ) Q kk = lim (s) s Q kk (s) 2 < k. j, k. 2.0 k S E k [T k ] = F kk ( ) , 2.9 j, k S, q() j,k =. j, k S, q() j,k <., j, k S, q() j,k,. 2.2 d (d-dimesioal Radom Walks) (X, P ) d., {p k } k Z d Z d, {X 0, X X 0, X 2 X,...}, P (X X = k) = p k (, k Z d ). ( p k = /(2d).), d. Q = (q j,k ) q j,k = p k j [,,, ] 2. (X, P ) d. (i) X + X (X 0, X,..., X ), i.e., P (X + X = j, X 0 = k 0, X = k,..., X = k ) = P (X + X = j)p (X 0 = k 0, X = k,..., X = k ). k 0, k,..., k Z d, X + X X. (ii) P (X + = j X 0 = k 0, X = k,..., X = k ) = P (X + = j X = k ) = p j k {X }, q j,k = p k j.. 2

14 (iii). ( j k := j k + + j d k d j k, j = k.), Q = (q j,k ) = (p k j ),,,., : 2.3 d (i) d =, 2 (i.e., E j [T j ] = P j (T j < )), (ii) d ( ), q () 0,0. q (2+) 0,0 = 0, q (2) 0,0.. ( 2.2.) 2.4 d Q = (q j,k ) (i) d =, 2 q (2) 0,0 { / π (d = ) /(π) (d = 2) (ii) d = 3 C q (2) 0,0 C 3/2. 2.4,, : (d = 3 (3/π) 3 /4) q (2) 0,0 2 d d d/2 (π) d/2 ( ). a b ( ) def a /b ( ). 2.2 {a }, {b }, a b ( ) c, c 2 > 0; c b a c 2 b ( ).. [ (Stirlig s formula)]! 2π +/2 e ( ). 2.4 d =, : ( ) q (2) 2 0,0 = 2 2. d = 2 q (2) 0,0 = j,k 0;j+k= (2)! (j!k!) = ( 2 ) j=0 ( ) k 3

15 , j=0 d = 3 ( ) 2 = k, 3 ( ) 2. q (2) 0,0 = j,k,m 0;j+k+m= (2)! (j!k!m!) q (2) (2)! 0,0 c 3 6 2!. c = max j,k,m 0;j+k+m= (j!k!m!). c,,. c c3 +3/2 3/2 e (c > 0 ). (), 3 (m!) 3 ( = 3m) c (m!) 2 ((m + )!) ( = 3m + ) (m!) ((m + )!) 2 ( = 3m + 2) (2),, c, c 2 > 0 c +/2 e! c 2 +/2 e (2), (), d = 3 ( ). d =, 2,. 2.5 d =, 2 Z d (i.e., E 0 [T 0 ] = ). 2.3 (i) α > α s Γ(α + ) ( s) α+ (s ). = (ii) α = s = log s. = α > log(/s) s (s ) : α = log x α s x dx = ( log s ) α Γ(α + ). 4

16 2.3 F 00(s) = Q 00(s)/Q 00 (s) 2, d =, q (2) 0,0 / π ( ), s Q 00 (s) = + Q 00(s) = = = s 2 q (2) 0,0 + 2s 2 q (2) 0,0 = = s 2 Γ(/2) π π ( s 2 ), /2 2s 2 2 Γ(3/2) π π ( s 2 ). 3/2 F 00(s) = Q 00(s) Q 00 (s) 2 2 πγ(3/2) Γ(/2) 2 s (s ). s 2 E 0 [T 0 ] = lim s F 00(s) =. d = 2 q (2) 0,0 /(π) ( ), s Q 00(s) E 0 [T 0 ] = lim s 2 Q 00 (s) π log s 2, [ 2 πs( s 2 ) 2 π( s 2 ) π( s 2 ) ( ) ] 2 log s 2 =. 2.3 (Oe-dimesioal Ati-symmetric Radom Walks) Z {X } p (0 < p < ), p. p /2, {X = X (p) }. d, d, ( 0 ). q j,j+ = q 0, = p, q j,j = q 0, = p, q () j,k = ( ) +j k p ( j+k)/2 ( p) (+j k)/2 ( + j k 2Z) 2 0 ( + j k 2Z + ) [ + l, m, l + m =. l m = k j.] 5

17 q (2) 0,0 = ( ) 2 (p( p)) (4p( p)) ( ) π 2.6. p /2 4p( p) < : 2.4 {X = X (p) } (0 < p <, p /2)., Y := X X EY = 2p, : ( ) X P lim = 2p =.,, ε > 0, N, N, (2p ε) < X < (2p +ε). 2.7 p > /2 j, u j (s) := F j0 (s) = m sm P j (T 0 = m) (0 < s < ) u (s) = psu 2 (s) + ( p)s u j (s) = psu j+ (s) + ( p)su j (s) (j 2) lim j u j(s) = 0,,.. ( ) j 4p( p)s F j0 (s) = 2 (0 < s < ) 2ps ( ) j p P j (T 0 < ) = (j ) p [ {X = j + }, {X = j }., P j (T 0 = m) = P j (T 0 = m X = j + )P j (X = j + ) + P j (T 0 = m X = j )P j (X = j ), P j (T 0 = m X = j ) j 2 P j (T 0 = m ), j = P (T 0 = m X = 0) = δ m., P j (T 0 = m) = { pp j+ (T 0 = m ) + ( p)p j (T 0 = m ) (j 2) pp 2 (T 0 = m ) + ( p)δ m (j = ) lim j P j (T 0 = m) = 0 ( m ). psx 2 x + ( p)s = 0 x = α, β (α < β), 0 < α < < β ( 2ps < 4p( p)s 2 ) u 2 αu = β(u α), u 2 βu = α(u β) u j+ αu j = β(u j αu j ), u j+ βu j = α(u j βu j ) (j 2), u j = (β j (u α) α j (u β))/(β α) j u = α. ] 2.8 u j := P j (T 0 < ) (j Z). p > /2 j, u j = P j (T 0 < ) =, j = 0 u 0 = pu + ( p), P 0 (T 0 < ) = u 0 = 2( p) <. 6

18 3.,. 3. ( 2.2 (iii)) 3.. π = (π j ) π j = /E j [T j ] > 0,. i, j S, 2. Q ij (s) = δ ij + F ij (s)q jj (s) i j s lim( s)q jj (s) = lim s s F jj (s) = F jj ( ) = E j [T j ]. lim( s)q ij (s) = F ij() s E j [T j ]., 2.2 F ij () = P i (T j < ) = i, j S, lim( s)q ij (s) = s E j [T j ] (=: π j ). E j [T j ] < 0 < π j. j S ( s)q ij (s) = Fatou j S π j, k S, j S π j q j,k lim if s ( s)q ij (s)q j,k j S lim( s) s =0 s q (+) i,k = lim s ( s)s (Q ik (s) δ ik ) = π k, k π j q j,k = π k (k S) j S. π j ( s)q jk (s) = ( s) j S =0 s j S π j q () j,k = π k. (3) s Lebesgue j π j π k = π k j π j =. π = (π j ). π = (π j ), (3) s π k = F jk () π j E k [T k ] + π k E k [T k ] E k [T k ] j k 7

19 . k; π k > 0 E k [T k ] <,. k S E k [T k ] <, 2.2 F jk () = P j (T k < ) = ( j, k S),., π k = /E k [T k ] > 0 (k S). π = (π j ). 3. Fubii s)q ij (s) =. j S( 3.2 Fatou Lebsgue. [], ( ), (Applicatos of Probability Thoery; Other Topics),,,. 4. (Rui problem) A, B 2, a, b,., A, B,. A 0 < p <., q = p., 0,. A P A.,, P P = 0,..,. 4. () p = /2. P A = b a + b, P B = a a + b. (2) p /2. r := q/p = /p ( ). P A = ra r a+b r a+b, P B = ra r a+b. (3), P = 0,,,,. a A u a.,. u a = pu a+ + qu a (a ), u 0 =, u a+b = 0. (4) 4.. 8

20 , 2. 0,,, a + b, B, 0.,,. Y, A,, A Y =. Y =, i.e., P (Y = ) = p, P (Y = ) = q. X 0 = a, X = X 0 + Y k ( ), A. T 0 = if{ 0; X = 0}, T a+b = if{ 0; X = a + b}, u a = P a (T 0 < T a+b ) = P (T a < T a+b X 0 = a)., Y = ± X = a ±, P a (T 0 < T a+b ) = P a (T 0 < T a+b Y = )P a (Y = ) + P a (T 0 < T a+b Y = )P a (Y = ), (4), a,, a + b, = P a+ (T 0 < T a+b )P (Y = ) + P a (T 0 < T a+b )P (Y = ). u u 0 = u a+ u a = (q/p)(u a u a ) = r a (u u 0). { r k (u u 0) (r = ) (u u 0) = r (u r u 0 ) (r ), = a + b,, u 0 =, u a+b = 0, u u 0 = /(a + b) if r =, = ( r)/( r a+b ) if r., u a = a/(a + b) = b/(a + b) if r =, = ( r a )/( r a+b ) = (r a r a+b )/( r a+b ) if r.., N, B b, v a = E a [N] A a, A. v = + pv + + qv ( a a + b ), v 0 = v a+b = 0., v a = ab (r = ), {(a + b) } ra p q r a+b a (r ). r =, (v a+ v a ) (v a v a ) + 2 = 0, v a = v a v a (a ), v a+ v a + 2 = 0. a =, v + v + 2 = 0., v + = v 2 = v 2. (v 0 = 0.) v + = + v k = ( + )(v )., v = (v ( ))., 0 = v a+b = (a + b)(v (a + b )) v = a + b., v a = a(v (a )) = ab. 9

21 r. p v a+ + = q v a, pv + = qv, v = /(q p)., p( v a+ v) = q( v a v). v a+ v = (q/p)( v a v) = r a ( v v)., v a v = r a (v v). (v 0 = 0.) v a = a v a = av + a r k (v v) = av + ra (v v). r, 0 = v a+b = (a + b)v + ra+b r (v v), v v = (a + b)v r r a+b v a = av (a + b)v ( ra r = v a (a + b) ) ra. a+b r a+b r =,, p = q = /2, a =, b = 00, A P A = 00/0, 00, /2. 4.2,, 3,,.,, 2,.,,,.,,, /3,, 2, /2., 2,,,.,,. /3,, 2/3., /2, 3, /3,,,, 2,, 2/3.,, ( ).,,,,,..,. (,,,.) 4.3 (Prisoer s Dilemma),, 2, (i), 2 2 (ii),, 0 20

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