1. The document describes an algorithm for game theory optimization using Nash equilibriums. 2. It performs several iterations of evaluating different strategy combinations for players and selecting the combination that maximizes each player's payoff given other player's strategies. 3. The aim is to find a Nash equilibrium where no player can benefit by changing only their own strategy.

1. ήʔϜཧ࿦ BASIC ԋश1
3ਓήʔϜ

2. 1. φογϡ‫ߧۉ‬ͷఆٛ
2. 3ਓήʔϜͷφογϡ‫ߧۉ‬
3. ิ଍ɿ3ਓήʔϜͷઓུ‫ܗ‬ήʔϜ͔Β
ಛੑؔ਺‫ܗ‬ήʔϜ΁

3. φογϡ‫ߧۉ‬
ఆٛɿ࠷ద൓Ԡઓུ
ϓϨΠϠʔ Ҏ֎ͷઓུΛ ͱ͠‫ݻ‬ఆ͢Δ
Λຬͨ͢ͱ͖ ͸ ʹର͢Δ࠷ద൓Ԡઓུͱ͍͏
ఆٛɿφογϡ‫ߧۉ‬
ઓུͷ૊ ͕φογϡ‫͋Ͱߧۉ‬Δͱ͸
͢΂ͯͷϓϨΠϠʔʹରͯ͠
φογϡ‫ߧۉ‬͸
‫࠷ʹ͍ޓ‬ద൓Ԡઓུͱͳ͍ͬͯΔઓུͷ૊
i s−i
∈ S−i
fi
(s *i
, s−i
) ≥ fi
(si
, s−i
), ∀si
∈ Si
s *i
s−i
(s*1
, ⋯, s*n
) ∈ S
fi
(s*i
, s*−i
) ≥ fi
(si
, s*−i
) ∀si
∈ Si
AʘB C C
B
B
B

4. ҎԼͷήʔϜʹ͓͚Δ७ਮઓུφογϡ‫ߧۉ‬Λ‫ٻ‬ΊΑ
3ਓήʔϜͷφογϡ‫ߧۉ‬
͕७ઓུD
1ʘ2 C C
B
B
͕७ઓུD
1ʘ2 C C
B
B

5. ҎԼͷήʔϜʹ͓͚Δ७ਮઓུφογϡ‫ߧۉ‬Λ‫ٻ‬ΊΑ
ϓϨΠϠʔͷϓϨΠϠʔ
ͷ C
D
ʹର͢Δ࠷ద൓Ԡઓུ͸
B
3ਓήʔϜͷφογϡ‫ߧۉ‬
͕७ઓུD
1ʘ2 C C
B
B
͕७ઓུD
1ʘ2 C C
B
B

6. ҎԼͷήʔϜʹ͓͚Δ७ਮઓུφογϡ‫ߧۉ‬Λ‫ٻ‬ΊΑ
ϓϨΠϠʔͷϓϨΠϠʔ
ͷ C
D
ʹର͢Δ࠷ద൓Ԡઓུ͸
B·ͨ͸B
3ਓήʔϜͷφογϡ‫ߧۉ‬
͕७ઓུD
1ʘ2 C C
B
B
͕७ઓུD
1ʘ2 C C
B
B

7. ҎԼͷήʔϜʹ͓͚Δ७ਮઓུφογϡ‫ߧۉ‬Λ‫ٻ‬ΊΑ
ϓϨΠϠʔͷϓϨΠϠʔ
ͷ C
D
ʹର͢Δ࠷ద൓Ԡઓུ͸
B
3ਓήʔϜͷφογϡ‫ߧۉ‬
͕७ઓུD
1ʘ2 C C
B
B
͕७ઓུD
1ʘ2 C C
B
B

8. ҎԼͷήʔϜʹ͓͚Δ७ਮઓུφογϡ‫ߧۉ‬Λ‫ٻ‬ΊΑ
ϓϨΠϠʔͷϓϨΠϠʔ
ͷ C
D
ʹର͢Δ࠷ద൓Ԡઓུ͸
B·ͨ͸B
3ਓήʔϜͷφογϡ‫ߧۉ‬
͕७ઓུD
1ʘ2 C C
B
B
͕७ઓུD
1ʘ2 C C
B
B

9. ҎԼͷήʔϜʹ͓͚Δ७ਮઓུφογϡ‫ߧۉ‬Λ‫ٻ‬ΊΑ
ϓϨΠϠʔͷϓϨΠϠʔ
ͷ B
D
ʹର͢Δ࠷ద൓Ԡઓུ͸
C
3ਓήʔϜͷφογϡ‫ߧۉ‬
͕७ઓུD
1ʘ2 C C
B
B
͕७ઓུD
1ʘ2 C C
B
B

10. ҎԼͷήʔϜʹ͓͚Δ७ਮઓུφογϡ‫ߧۉ‬Λ‫ٻ‬ΊΑ
ϓϨΠϠʔͷϓϨΠϠʔ
ͷ B
D
ʹର͢Δ࠷ద൓Ԡઓུ͸
C
3ਓήʔϜͷφογϡ‫ߧۉ‬
͕७ઓུD
1ʘ2 C C
B
B
͕७ઓུD
1ʘ2 C C
B
B

11. ҎԼͷήʔϜʹ͓͚Δ७ਮઓུφογϡ‫ߧۉ‬Λ‫ٻ‬ΊΑ
ϓϨΠϠʔͷϓϨΠϠʔ
ͷ B
D
ʹର͢Δ࠷ద൓Ԡઓུ͸
C·ͨ͸C
3ਓήʔϜͷφογϡ‫ߧۉ‬
͕७ઓུD
1ʘ2 C C
B
B
͕७ઓུD
1ʘ2 C C
B
B

12. ҎԼͷήʔϜʹ͓͚Δ७ਮઓུφογϡ‫ߧۉ‬Λ‫ٻ‬ΊΑ
ϓϨΠϠʔͷϓϨΠϠʔ
ͷ B
D
ʹର͢Δ࠷ద൓Ԡઓུ͸
C
3ਓήʔϜͷφογϡ‫ߧۉ‬
͕७ઓུD
1ʘ2 C C
B
B
͕७ઓུD
1ʘ2 C C
B
B

13. ҎԼͷήʔϜʹ͓͚Δ७ਮઓུφογϡ‫ߧۉ‬Λ‫ٻ‬ΊΑ
ϓϨΠϠʔͷϓϨΠϠʔ
ͷ B
C
ʹର͢Δ࠷ద൓Ԡઓུ͸
D
3ਓήʔϜͷφογϡ‫ߧۉ‬
͕७ઓུD
1ʘ2 C C
B
B
͕७ઓུD
1ʘ2 C C
B
B

14. ҎԼͷήʔϜʹ͓͚Δ७ਮઓུφογϡ‫ߧۉ‬Λ‫ٻ‬ΊΑ
ϓϨΠϠʔͷϓϨΠϠʔ
ͷ B
C
ʹର͢Δ࠷ద൓Ԡઓུ͸
D
3ਓήʔϜͷφογϡ‫ߧۉ‬
͕७ઓུD
1ʘ2 C C
B
B
͕७ઓུD
1ʘ2 C C
B
B

15. ҎԼͷήʔϜʹ͓͚Δ७ਮઓུφογϡ‫ߧۉ‬Λ‫ٻ‬ΊΑ
ϓϨΠϠʔͷϓϨΠϠʔ
ͷ B
C
ʹର͢Δ࠷ద൓Ԡઓུ͸
D
3ਓήʔϜͷφογϡ‫ߧۉ‬
͕७ઓུD
1ʘ2 C C
B
B
͕७ઓུD
1ʘ2 C C
B
B

16. ҎԼͷήʔϜʹ͓͚Δ७ਮઓུφογϡ‫ߧۉ‬Λ‫ٻ‬ΊΑ
ϓϨΠϠʔͷϓϨΠϠʔ
ͷ B
C
ʹର͢Δ࠷ద൓Ԡઓུ͸
D
3ਓήʔϜͷφογϡ‫ߧۉ‬
͕७ઓུD
1ʘ2 C C
B
B
͕७ઓུD
1ʘ2 C C
B
B

17. ҎԼͷήʔϜʹ͓͚Δ७ਮઓུφογϡ‫ߧۉ‬Λ‫ٻ‬ΊΑ
Ҏ্ΑΓ
७ਮઓུφογϡ‫ߧۉ‬͸
B
C
D
͓Αͼ B
C
D
3ਓήʔϜͷφογϡ‫ߧۉ‬
͕७ઓུD
1ʘ2 C C
B
B
͕७ઓུD
1ʘ2 C C
B
B

18. ಛੑؔ਺‫ܗ‬ήʔϜΛߟ͍͑ͨ
ఏ‫·ؚʹ಺ܞ‬ΕΔརಘͷ࿨Λಛੑؔ਺ͷ஋ͱ͢Δ
ͨͩ͠
ఏ‫֎ܞ‬ͷϓϨΠϠʔ͕࠷ѱͳߦಈΛऔΔͱߟ͑ɺ
ͦͷத͔Βఏ‫࠷ͯͬͱʹܞ‬΋େ͖͘ͳΔ஋Λಛੑؔ਺஋ͱ͢ΔʢϚοΫεϛχʣ
ิ଍ɿ3ਓήʔϜͷઓུ‫ܗ‬ήʔϜ͔Βಛੑؔ਺‫ܗ‬ήʔϜ΁
͕७ઓུD
1ʘ2 C C
B
B
͕७ઓུD
1ʘ2 C C
B
B

19. ಛੑؔ਺‫ܗ‬ήʔϜΛߟ͍͑ͨ
v({1,2,3}) = 6
v({1,2}) = max (min(2,1),
ิ଍ɿ3ਓήʔϜͷઓུ‫ܗ‬ήʔϜ͔Βಛੑؔ਺‫ܗ‬ήʔϜ΁
͕७ઓུD
1ʘ2 C C
B
B
͕७ઓུD
1ʘ2 C C
B
B

20. ಛੑؔ਺‫ܗ‬ήʔϜΛߟ͍͑ͨ
v({1,2,3}) = 6
v({1,2}) = max (min(2,1), min(5,0)
ิ଍ɿ3ਓήʔϜͷઓུ‫ܗ‬ήʔϜ͔Βಛੑؔ਺‫ܗ‬ήʔϜ΁
͕७ઓུD
1ʘ2 C C
B
B
͕७ઓུD
1ʘ2 C C
B
B

21. ಛੑؔ਺‫ܗ‬ήʔϜΛߟ͍͑ͨ
v({1,2,3}) = 6
v({1,2}) = max (min(2,1), min(5,0), min(2,0)
ิ଍ɿ3ਓήʔϜͷઓུ‫ܗ‬ήʔϜ͔Βಛੑؔ਺‫ܗ‬ήʔϜ΁
͕७ઓུD
1ʘ2 C C
B
B
͕७ઓུD
1ʘ2 C C
B
B

22. ಛੑؔ਺‫ܗ‬ήʔϜΛߟ͍͑ͨ
v({1,2,3}) = 6
v({1,2}) = max (min(2,1), min(5,0), min(2,0), min(4,1)) = max(1,0,0,1) = 1
ิ଍ɿ3ਓήʔϜͷઓུ‫ܗ‬ήʔϜ͔Βಛੑؔ਺‫ܗ‬ήʔϜ΁
͕७ઓུD
1ʘ2 C C
B
B
͕७ઓུD
1ʘ2 C C
B
B

23. ಛੑؔ਺‫ܗ‬ήʔϜΛߟ͍͑ͨ
v({1,2,3}) = 6
v({1,2}) = max (min(2,1), min(5,0), min(2,0), min(4,1)) = max(1,0,0,1) = 1
v({1,3}) = max (min(2,2), min(2,3), min(3,3), min(4,1)) = max(2,2,3,0) = 3
ิ଍ɿ3ਓήʔϜͷઓུ‫ܗ‬ήʔϜ͔Βಛੑؔ਺‫ܗ‬ήʔϜ΁
͕७ઓུD
1ʘ2 C C
B
B
͕७ઓུD
1ʘ2 C C
B
B

24. ಛੑؔ਺‫ܗ‬ήʔϜΛߟ͍͑ͨ
v({1,2,3}) = 6
v({1,2}) = max (min(2,1), min(5,0), min(2,0), min(4,1)) = max(1,0,0,1) = 1
v({1,3}) = max (min(2,2), min(2,3), min(3,3), min(4,1)) = max(2,2,3,0) = 3
v({2,3}) = max (min(2,2), min(0,3), min(2,3), min(3,1)) = max(2,0,2,1) = 2
ิ଍ɿ3ਓήʔϜͷઓུ‫ܗ‬ήʔϜ͔Βಛੑؔ਺‫ܗ‬ήʔϜ΁
͕७ઓུD
1ʘ2 C C
B
B
͕७ઓུD
1ʘ2 C C
B
B

25. ಛੑؔ਺‫ܗ‬ήʔϜΛߟ͍͑ͨ
v({1,2,3}) = 6
v({1,2}) = max (min(2,1), min(5,0), min(2,0), min(4,1)) = max(1,0,0,1) = 1
v({1,3}) = max (min(2,2), min(2,3), min(3,3), min(4,1)) = max(2,2,3,0) = 3
v({2,3}) = max (min(2,2), min(0,3), min(2,3), min(3,1)) = max(2,0,2,1) = 2
v({1}) = v({2}) = v({3}) = 0
ิ଍ɿ3ਓήʔϜͷઓུ‫ܗ‬ήʔϜ͔Βಛੑؔ਺‫ܗ‬ήʔϜ΁
͕७ઓུD
1ʘ2 C C
B
B
͕७ઓུD
1ʘ2 C C
B
B

26.
͜ͷಛੑؔ਺Λ༻͍ͯ
ίΞ΍ਔ
γϟʔϓϨΠ஋Λ‫ٻ‬ΊͯΈΑ͏
v({1,2,3}) = 6
v({1,2}) = 1 v({1,3}) = 3 v({2,3}) = 2
v({1}) = v({2}) = v({3}) = 0
ิ଍ɿ3ਓήʔϜͷઓུ‫ܗ‬ήʔϜ͔Βಛੑؔ਺‫ܗ‬ήʔϜ΁
͕७ઓུD
1ʘ2 C C
B
B
͕७ઓུD
1ʘ2 C C
B
B

27. ήʔϜཧ࿦ BASIC ԋश1
3ਓήʔϜ
࣍ճɿ ԋश2