https://www.nicovideo.jp/watch/sm36115578 https://youtu.be/nqtNKK4uI_8

2. S K

3. S K

4. S K

5. S K
S

6. S K

7. S K

8. S K
K

9. S K

10. S K

11. S K
K S
S

12.
✤
✤
S K

13. A B
((1/2,1/2), (1/2,1/2))

14. EA(p, q)=pq(-1)+p(1-q)+(1-p)q+(1-p)(1-q)(-1)
=-1+2p+2q-4pq=2(1-2p)q+2p-1
minEA(p, q)
q
A B
EA(p, q)
p<1/2 q=0, EA(p, 0)= 2p-1
p=1/2 q , EA(1/2, q)= 0
p>1/2 q=1, EA(p, 1)= -2p+1
max minEA(p, q)
p q

15. EA(p, q)=2(2q-1)p-2q+1
maxEA(p, q)
p
A B
EA(p, q)
q<1/2 p=0, EA(0, q)= 1-2q
q=1/2 p , EA(p, 1/2)= 0
q>1/2 p=1, EA(1, q)= 2q-1
min maxEA(p, q)
q p

16.
A B

17. S K

18. S K
!ES(p, q) = 0 * pq + 50 * (1 − p)q + (−50) * p(1 − q) + 25 * (1 − p)(1 − q)
p
1 − p
q 1 − q
! = 25(1 + p)q − 50p + 25
! K !
→S !
→S !
25(1 + p) > 0 q = 0
−50p + 25
p = 0
S !
25
(p,1 − p) = (0,1)

19. S K
!ES(p, q) = 0 * pq + 50 * (1 − p)q + (−50) * p(1 − q) + 25 * (1 − p)(1 − q)
p
1 − p
q 1 − q
! = 25(q − 2)p + 25q + 25
! S !
→K !
→K !
25(q − 2) < 0 p = 0
25q + 25
q = 0
K !
25
(q,1 − q) = (0,1)

20. S K

21. S K
p
1 − p
q 1 − q !ES(p, q) = (−30) * pq + 50 * (1 − p)q + (−20) * p(1 − q) + (−60) * (1 − p)(1 − q)
! = (−120p + 110)q + 40p − 60
11
12 10
p
−
70
3
p = 11/12

22. S K
p
1 − p
q 1 − q !ES(p, q) = (−30) * pq + 50 * (1 − p)q + (−20) * p(1 − q) + (−60) * (1 − p)(1 − q)
! = (−120q + 40)p + 110q − 60
1
3
10
q
−
70
3
q = 1/3

23. S K
p
1 − p
q 1 − q !ES(p, q) = (−30) * pq + 50 * (1 − p)q + (−20) * p(1 − q) + (−60) * (1 − p)(1 − q)
!EK(p, q) = 30 * pq + (−50) * (1 − p)q + 20 * p(1 − q) + 60 * (1 − p)(1 − q)
1
3
11
12
! = (−120q + 40)p + 110q − 60
! = (120p − 110)q − 40p + 60
((11/12,1/12), (1/3,2/3))
−
70
3

24. See you next time