1
00:00:00,089 --> 00:00:02,770
we will not speak about maximum
我们不会谈论最大

2
00:00:02,970 --> 00:00:05,919
strategies these make particular sense
这些策略特别有意义

3
00:00:06,120 --> 00:00:08,620
in the context of zero-sum games but
在零和游戏的背景下， 

4
00:00:08,820 --> 00:00:12,100
actually are pretty look quite - to all
实际上看起来很漂亮-所有人

5
00:00:12,300 --> 00:00:15,580
games what is the max min strategy it
游戏的最大最小策略是什么

6
00:00:15,779 --> 00:00:19,239
simply puts a player's strategy that
简单地提出一个球员的策略， 

7
00:00:19,439 --> 00:00:21,550
maximizes their payoff assuming the
假设

8
00:00:21,750 --> 00:00:26,970
other player is out to get them we will
另一个玩家要得到他们，我们将

9
00:00:27,170 --> 00:00:29,499
we will concentrate primarily on the
我们将主要集中于

10
00:00:29,699 --> 00:00:32,829
true player case here again because when
真正的玩家案例再次出现，因为

11
00:00:33,030 --> 00:00:34,570
we get to zero-sum games they don't
我们去了零和游戏

12
00:00:34,770 --> 00:00:36,608
really make only center the case of two
确实只使两个情况居中

13
00:00:36,808 --> 00:00:40,299
players but keep in mind that one could
玩家，但请记住， 

14
00:00:40,500 --> 00:00:42,729
define this kind of more generally when
更一般地定义这种情况

15
00:00:42,929 --> 00:00:46,768
we speak about maximum strategy so the
我们谈论最大策略，因此

16
00:00:46,969 --> 00:00:49,409
maximum strategy is a strategy that
最大策略是

17
00:00:49,609 --> 00:00:54,070
maximizes my worst case outcome and my
最大化我的最坏情况的结果， 

18
00:00:54,270 --> 00:00:58,119
maximum value or safety level is that
最大值或安全级别是

19
00:00:58,320 --> 00:01:00,338
payoff that's guaranteed by the maximum
最高保证的回报

20
00:01:00,539 --> 00:01:03,849
strategy and here it is defined formally
策略，在这里正式定义

21
00:01:04,049 --> 00:01:07,988
the maximum strategy for player I is the
我的最大策略是

22
00:01:08,188 --> 00:01:13,259
strategy Aswan that maximizes the
最大化阿斯旺战略

23
00:01:13,459 --> 00:01:16,450
minimum that the other player remember
另一个玩家记住的最低限度

24
00:01:16,650 --> 00:01:22,709
the - I is the player other than I would
 -我是我以外的球员

25
00:01:22,909 --> 00:01:28,778
hold play 1 down - and the maximum value
按住播放1-并保持最大值

26
00:01:28,978 --> 00:01:30,640
is defined solely to be the value of
仅定义为

27
00:01:30,840 --> 00:01:36,189
that maximum strategy now why why would
现在最大的策略为什么会

28
00:01:36,390 --> 00:01:37,390
we want to think about the maximum
我们想考虑最大

29
00:01:37,590 --> 00:01:43,168
strategy one can think of it either as a
策略可以将其视为一种

30
00:01:43,368 --> 00:01:46,019
simply a sort of a certain cautionary
只是某种警告

31
00:01:46,219 --> 00:01:49,808
maybe the other people will make some
也许其他人会做一些

32
00:01:50,009 --> 00:01:51,429
mistakes and not act in their own best
错误，而不是尽自己最大的努力

33
00:01:51,629 --> 00:01:54,969
interest maybe I'm not sure exactly what
兴趣也许我不确定到底是什么

34
00:01:55,170 --> 00:01:56,948
their payoffs are there a lot of
他们的收益很多

35
00:01:57,149 --> 00:01:58,918
interpretations or you can simply be
解释或者你可以简单地

36
00:01:59,118 --> 00:02:04,509
paranoid about about about them and
关于他们的偏执和

37
00:02:04,709 --> 00:02:06,939
think that there are to get you and you
认为有得到你和你

38
00:02:07,140 --> 00:02:08,380
know the you know the saying you know
知道你知道的话你知道

39
00:02:08,580 --> 00:02:09,509
even
甚至

40
00:02:09,709 --> 00:02:14,009
apparently they have enemies that's the
显然他们有敌人

41
00:02:14,209 --> 00:02:17,429
max min strategy and just a confuse
最大最小策略，只是一个混乱

42
00:02:17,628 --> 00:02:19,530
thing will also speak about the min max
事情也会谈到最小最大

43
00:02:19,729 --> 00:02:23,250
strategy the min max strategy is
策略最小最大策略是

44
00:02:23,449 --> 00:02:26,099
strategy against if you wish the other
反对对方是否愿意的策略

45
00:02:26,299 --> 00:02:28,920
player in the two-player game is the
两人游戏中的玩家是

46
00:02:29,120 --> 00:02:31,500
strategies that minimizes their payoff
使收益最小化的策略

47
00:02:31,699 --> 00:02:33,750
on the assumption that they're trying to
假设他们正在尝试

48
00:02:33,949 --> 00:02:36,659
maximize it and so here is the formal
最大化它，所以这是正式的

49
00:02:36,859 --> 00:02:39,209
definition the min max strategy for
定义最小最大策略

50
00:02:39,408 --> 00:02:40,920
player I is playing against the other
我正在与其他玩家对战的玩家

51
00:02:41,120 --> 00:02:43,890
guy who pre-dawn by minus I is the
减去我之前黎明的家伙是

52
00:02:44,090 --> 00:02:47,368
strategy that minimizes the maximum
最小化最大策略

53
00:02:47,568 --> 00:02:50,189
payoff as tempted by the other guy of
另一个人的诱惑所带来的回报

54
00:02:50,389 --> 00:02:54,360
the payoff to the other guy and the min
回报给另一个人和最小

55
00:02:54,560 --> 00:02:56,159
max value is simply the value of that
最大值就是那个的值

56
00:02:56,359 --> 00:03:01,319
mimic strategy the value to play one now
模仿策略现在玩一玩的价值

57
00:03:01,519 --> 00:03:08,159
why why would player one want to want to
为什么玩家会想要

58
00:03:08,359 --> 00:03:12,390
harm the other guy well you could it
伤害其他人就可以了

59
00:03:12,590 --> 00:03:15,270
could just be able to get him that's a
也许能够让他得到

60
00:03:15,469 --> 00:03:19,409
possibility or they could be playing a
可能性，或者他们可能正在玩

61
00:03:19,609 --> 00:03:23,118
zero-sum game and in a zero-sum game
零和游戏以及零和游戏

62
00:03:23,318 --> 00:03:26,599
hurting the other guy is tantamount to
伤害另一个人等于

63
00:03:26,799 --> 00:03:32,509
improving your own your own path and so
改善自己的道路等

64
00:03:32,709 --> 00:03:36,629
in the setting of zero-sum games max min
在零和游戏最大最小的设置中

65
00:03:36,829 --> 00:03:38,069
and min max strategy makes a lot of
最小最大策略使很多

66
00:03:38,269 --> 00:03:39,840
sense and in fact in a very famous
感，实际上是非常著名的

67
00:03:40,039 --> 00:03:44,159
theorem due to genre diamond it proved
钻石流派定理证明

68
00:03:44,359 --> 00:03:48,058
that in a zero-sum game by definition we
根据定义，在零和博弈中

69
00:03:48,258 --> 00:03:52,439
consider only two player such games any
只考虑两个玩家这样的游戏

70
00:03:52,639 --> 00:03:55,200
Nash equilibrium the player sees that
纳什均衡玩家看到

71
00:03:55,400 --> 00:03:57,539
payoff that is equal to both his max min
收益等于他的最大最小值

72
00:03:57,739 --> 00:04:04,259
value and his min max value and and that
值和他的最小最大值，以及

73
00:04:04,459 --> 00:04:07,050
means that so we will call it the value
意味着我们将其称为价值

74
00:04:07,250 --> 00:04:09,330
of the game the value for player one is
在游戏中，玩家一的价值是

75
00:04:09,530 --> 00:04:11,368
called the value of the game and that
称之为游戏的价值， 

76
00:04:11,568 --> 00:04:13,709
means that the the set of maximum
表示最大集

77
00:04:13,908 --> 00:04:16,199
strategies are really the same a set of
一套策略真的一样

78
00:04:16,399 --> 00:04:18,329
the min max strategies that is try to
尝试的最小最大策略

79
00:04:18,528 --> 00:04:21,120
improve your worst-case situation is the
改善您最坏的情况是

80
00:04:21,319 --> 00:04:22,079
same as
如同

81
00:04:22,279 --> 00:04:24,389
to minimize the other guys best case
尽量减少其他人的最佳情况

82
00:04:24,589 --> 00:04:28,560
situation and any maximum strategy
情况和任何最大策略

83
00:04:28,759 --> 00:04:30,420
profile or minimax tragic profile
轮廓或极小悲剧轮廓

84
00:04:30,620 --> 00:04:33,120
because they're the same constitute a
因为它们相同构成了一个

85
00:04:33,319 --> 00:04:36,030
Nash equilibrium and furthermore those
纳什均衡，还有那些

86
00:04:36,230 --> 00:04:37,590
are all the Nash equilibria that exists
是所有存在的纳什均衡

87
00:04:37,790 --> 00:04:39,780
and so the payoffs it all Nash
所以回报全是纳什

88
00:04:39,980 --> 00:04:41,460
equilibria is the same maybe the value
均衡是一样的，也许价值

89
00:04:41,660 --> 00:04:46,980
of the game one way to get a concrete
游戏的一种获得具体方法的方法

90
00:04:47,180 --> 00:04:49,170
feel for it is graphically and here's
感觉是图形化的，这是

91
00:04:49,370 --> 00:04:51,509
the game of matching pennies this is a
匹配便士的游戏，这是一个

92
00:04:51,709 --> 00:04:56,160
game where you each of us chooses heads
我们每个人都选择头的游戏

93
00:04:56,360 --> 00:05:02,490
and tails on probability and if we if it
和概率的尾巴，如果我们

94
00:05:02,689 --> 00:05:06,930
comes up either if we've both if if the
如果我们两者都出现， 

95
00:05:07,129 --> 00:05:10,259
result of our randomized ation I end up
我们随机化的结果我最终

96
00:05:10,459 --> 00:05:11,850
choosing head and you are you end up
选择头，你就结束了

97
00:05:12,050 --> 00:05:15,180
playing tail you win and confer and vice
打尾巴你赢了，赋予和副

98
00:05:15,379 --> 00:05:16,860
versa as if I chose tail and you head
反之亦然，就好像我选择了尾巴，而你却是头

99
00:05:17,060 --> 00:05:19,650
but if we both chose ahead of us just
但是如果我们都选择了我们

100
00:05:19,850 --> 00:05:22,050
tailed I win and so here are the payoffs
我赢了，所以这是回报

101
00:05:22,250 --> 00:05:25,468
you see here the strategy spaces this is
你在这里看到的策略空间是

102
00:05:25,668 --> 00:05:29,400
player 2 is kind of increasing their
玩家2有点增加他们的

103
00:05:29,600 --> 00:05:31,250
probability of playing heads and this is
打头的概率，这是

104
00:05:31,449 --> 00:05:34,770
player 1 and what this dimensioned
播放器1的大小

105
00:05:34,970 --> 00:05:37,319
you have the value of the K of the path
您拥有路径K的值

106
00:05:37,519 --> 00:05:41,420
to day one and the only Nash equilibria
到第一天，唯一的纳什均衡

107
00:05:41,620 --> 00:05:45,960
is for both to randomized 5050 it's just
是为了将两者随机化为5050 

108
00:05:46,160 --> 00:05:50,060
right here it's kind of intriguing the
在这里，这很有意思

109
00:05:50,259 --> 00:05:55,650
the the three-dimensional structure in
在三维结构

110
00:05:55,850 --> 00:06:01,560
this way and you sort of see that it's
这样，您会发现它是

111
00:06:01,759 --> 00:06:03,088
got to be an equilibrium in the sense
必须在某种意义上达到平衡

112
00:06:03,288 --> 00:06:08,278
that player player 1 could be moving
玩家1可能正在移动

113
00:06:08,478 --> 00:06:13,710
along this this curve but if as he does
沿着这条曲线，但如果他这样做

114
00:06:13,910 --> 00:06:18,300
it his payoffs would only drop and so
它的收益只会下降，所以

115
00:06:18,500 --> 00:06:20,009
he's trying to maximize the value would
他正在努力使价值最大化

116
00:06:20,209 --> 00:06:23,850
do it and conversely player 2 can only
这样做，相反，玩家2只能

117
00:06:24,050 --> 00:06:26,460
traverse around along this but if he
沿着这条路走，但如果他

118
00:06:26,660 --> 00:06:28,439
does that the payoffs would only
这样做的收益只会

119
00:06:28,639 --> 00:06:31,379
increase and he's trying to minimize the
增加，他正在尝试最小化

120
00:06:31,579 --> 00:06:32,920
value so
如此重视

121
00:06:33,120 --> 00:06:36,160
so you get a stable point which is which
所以你会得到一个稳定点

122
00:06:36,360 --> 00:06:38,590
for obvious reasons is called a saddle
由于明显的原因被称为马鞍

123
00:06:38,790 --> 00:06:42,340
point so although there are general
这样说虽然有一般

124
00:06:42,540 --> 00:06:44,500
purpose procedure is for finding a Nash
目的程序是寻找纳什

125
00:06:44,699 --> 00:06:48,490
equilibrium in particular in in two by
特别是在两个方面的均衡

126
00:06:48,689 --> 00:06:53,370
two games we can use the max min
两个游戏，我们可以使用最大最小值

127
00:06:53,569 --> 00:06:58,090
definition to to find it directly in
可以直接在其中找到的定义

128
00:06:58,290 --> 00:07:01,800
zero-sum games and let's see how it
零和游戏，让我们看看它如何

129
00:07:02,000 --> 00:07:05,340
happens in the game we'll call the
发生在游戏中，我们称之为

130
00:07:05,540 --> 00:07:08,379
penalty kick game so in this game we
罚点球游戏，所以在这个游戏中我们

131
00:07:08,579 --> 00:07:13,920
have a a penalty kicker and a goalie
有一个点球手和一个守门员

132
00:07:14,120 --> 00:07:17,939
it's a zero-sum game and the goal of the
这是一个零和游戏，目标是

133
00:07:18,139 --> 00:07:21,189
kicker is to score goal and the goal the
踢球者是进球，而进球是

134
00:07:21,389 --> 00:07:24,629
goal of the goalie is to prevent it and
守门员的目标是预防和

135
00:07:24,829 --> 00:07:26,710
let's assume that they each have two
假设它们每个都有两个

136
00:07:26,910 --> 00:07:29,410
strategies kick to the left and kick to
策略向左踢并向

137
00:07:29,610 --> 00:07:31,960
the right to a kicker and jump to the
有权获得踢球权并跳至

138
00:07:32,160 --> 00:07:33,639
left and jump to the right for the
向左跳并向右跳

139
00:07:33,839 --> 00:07:38,110
goalie the payoffs will be the each of
守门员的回报将是每个

140
00:07:38,310 --> 00:07:41,040
those will determine a probability of
这些将决定

141
00:07:41,240 --> 00:07:45,338
the kicker scoring a goal and we'll have
射门得分，我们将拥有

142
00:07:45,538 --> 00:07:48,100
that probability being the payoff the
那是回报的可能性

143
00:07:48,300 --> 00:07:49,949
value of the game that is a pair of two
是两个的游戏价值

144
00:07:50,149 --> 00:07:54,968
pair 1 and therefore minus the after we
对1，因此减去之后

145
00:07:55,168 --> 00:07:58,509
player 2 namely the goalie and so here
玩家2即守门员，所以在这里

146
00:07:58,709 --> 00:08:01,360
they are so for example if the kicker
例如踢球者

147
00:08:01,560 --> 00:08:03,400
kicks left and the goalie guesses
向左踢，守门员猜测

148
00:08:03,600 --> 00:08:06,069
correctly and jump left also then the
正确地向左跳然后

149
00:08:06,269 --> 00:08:08,770
goalie has not too bad a chance of
守门员的机会还不错

150
00:08:08,970 --> 00:08:13,319
suffering the shot namely probably 0.4
遭受打击大概是0.4 

151
00:08:13,519 --> 00:08:16,838
if he jumps to the wrong side his
如果他跳到错误的一侧，他的

152
00:08:17,038 --> 00:08:20,259
probability of stopping it is much lower
停止它的可能性要低得多

153
00:08:20,459 --> 00:08:25,360
it's point 2 similarly if the kicker
如果踢球者也类似地指向第2点

154
00:08:25,560 --> 00:08:28,860
decides to kick to the right if the
决定向右踢

155
00:08:29,060 --> 00:08:31,750
goalie guess is wrong
守门员猜错了

156
00:08:31,949 --> 00:08:33,939
his probability is low even lower than
他的概率甚至比

157
00:08:34,139 --> 00:08:36,218
if you get strong in the other case and
如果您在其他情况下变强，并且

158
00:08:36,418 --> 00:08:38,049
if you guys right is probability is
如果你们是对的，则概率是

159
00:08:38,250 --> 00:08:40,328
higher although not quite a size that
较高，尽管尺寸不大

160
00:08:40,528 --> 00:08:44,439
had to get right in the left case C's is
在C的左情况下必须正确

161
00:08:44,639 --> 00:08:45,609
bitter
苦

162
00:08:45,809 --> 00:08:49,240
at stopping shots when the kicker kicks
踢脚时停止投篮

163
00:08:49,440 --> 00:08:53,879
to the left so how does the clicker
向左，那么点击器如何

164
00:08:54,080 --> 00:08:56,019
maximize these minimum that's what we're
最大化这些最小值，这就是我们的目标

165
00:08:56,220 --> 00:09:00,578
after so here is the expression right we
之后，这里是正确的表达

166
00:09:00,778 --> 00:09:04,620
want the maximum value of the following
想要以下的最大值

167
00:09:04,820 --> 00:09:08,319
so they each have some mixed strategy of
所以他们每个人都有一些混合策略

168
00:09:08,519 --> 00:09:10,719
playing left and right with some
和一些左右演奏

169
00:09:10,919 --> 00:09:13,479
probability so s1 L is a probability
 s1 L是一个概率

170
00:09:13,679 --> 00:09:16,509
that the kicker kicks to the left and s
踢脚向左踢，并且

171
00:09:16,710 --> 00:09:19,740
2 L is the probability that the goalie
 2 L是守门员的概率

172
00:09:19,940 --> 00:09:22,659
jumps to the left and as we saw in that
跳到左边，就像我们看到的那样

173
00:09:22,860 --> 00:09:27,609
case the value is 0.6 and similarly the
情况下该值为0.6，类似地

174
00:09:27,809 --> 00:09:30,370
value 8.8 ISINs if they if we end up in
如果它们以8.8表示，则价值8.8 ISIN 

175
00:09:30,570 --> 00:09:34,719
this situation point 9 and point 7 in
这种情况下的第9点和第7点

176
00:09:34,919 --> 00:09:39,009
these situations okay so this is the
这些情况好吧，这就是

177
00:09:39,210 --> 00:09:42,819
expression that we somehow need to
我们需要某种表达的表情

178
00:09:43,019 --> 00:09:51,159
compute so what is the minimum that the
计算，那么最小的是

179
00:09:51,360 --> 00:09:52,829
kicker should keep in mind
踢球者应牢记

180
00:09:53,029 --> 00:09:57,399
so the kicker says I'm going to pick I'm
所以踢手说我要选我

181
00:09:57,600 --> 00:10:01,629
gonna play my strategy s 1 whatever it
无论如何都会发挥我的策略

182
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is when I play it player 2 is gonna play
当我播放时，播放器2将播放

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s 2 so as to minimize my payoff so let
 s 2以使我的收益最小化

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00:10:12,089 --> 00:10:16,240
me write down this entire expression and
我写下整个表情

185
00:10:16,440 --> 00:10:19,740
I'm just this is simply copying it over
我只是将其复制过来而已

186
00:10:19,940 --> 00:10:26,740
replacing its expression such as s 1r by
将其表达式替换为s 1r 

187
00:10:26,940 --> 00:10:31,839
1 minus s 1 l that's all it's doing and
 1减去1公升

188
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the same thing for s 2 R and so we have
对于s 2 R同样的事情，所以我们有

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00:10:39,320 --> 00:10:42,339
this is our expression and player 1s
这是我们的表情和玩家1 

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00:10:42,539 --> 00:10:45,609
ain't enough water player to do if I
如果我没有足够的水玩家来做

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00:10:45,809 --> 00:10:50,559
were to play my strategy s 1 and this is
玩我的策略s 1，这是

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00:10:50,759 --> 00:10:52,779
simply rearranging the terms nothing
简单地重新排列术语

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00:10:52,980 --> 00:10:56,740
else going on here as a function of s 2
否则这里是s 2的函数

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L because
 L因为

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00:10:58,679 --> 00:11:01,599
player one says I'm gonna put I'm gonna
玩家一说我要放我去

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00:11:01,799 --> 00:11:04,419
pick s one what would s to be best
选择一个最好的

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response namely its minimum and so this
响应就是它的最小值，所以这

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00:11:08,519 --> 00:11:10,870
is arranging it as a function of s two
正在将其安排为s两个的函数

199
00:11:11,070 --> 00:11:14,259
strategy and now all it remained is to
策略，现在剩下的就是

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look for the minimum of the strategy and
寻找最小的策略， 

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00:11:17,458 --> 00:11:21,099
the minimum is taken by luck taking the
最小的是运气

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00:11:21,299 --> 00:11:22,779
first derivative with respect to s two
关于s 2的一阶导数

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00:11:22,980 --> 00:11:26,589
holding s 1 is a constant and so we have
持有s 1是一个常数，所以我们有

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00:11:26,789 --> 00:11:30,179
this expression and then we solve for
这个表达式，然后我们解决

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for s 1 and we get that s 1 of L is is
对于s 1，我们得到L的s 1是

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00:11:33,600 --> 00:11:38,439
1/2 and therefore s 1 of R is a half as
 1/2，因此R的s 1为

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00:11:38,639 --> 00:11:43,240
well and so by this maximum calculation
好吧，这样最大的计算

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00:11:43,440 --> 00:11:47,349
we see that player the kicker figures
我们看到那个球员的数字

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out that in equilibrium
在平衡中

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00:11:49,080 --> 00:11:51,609
they better randomize 1/2 1/2 between
他们最好将1/2 1/2 

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00:11:51,809 --> 00:11:53,698
left and right
左和右

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00:11:53,899 --> 00:11:59,349
what does what is the goalie kind of
什么是守门员

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00:11:59,549 --> 00:12:02,258
figure out well he was trying to
弄清楚他正在尝试

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minimize the kickers maximum right
最小化踢脚者最大右脚

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00:12:05,278 --> 00:12:06,399
that's one was a look of it he's doing
那是他正在做的一件事

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00:12:06,600 --> 00:12:10,089
the min/max strategy and so here is the
最小/最大策略，所以这是

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00:12:10,289 --> 00:12:12,549
minimax strategy for player 2 just
玩家2的minimax策略

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00:12:12,750 --> 00:12:15,429
writing it down and as before we'll
写下来，像以前一样

219
00:12:15,629 --> 00:12:22,959
simply will simply rewrite it's one of
只是将简单地重写它是其中之一

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00:12:23,159 --> 00:12:25,328
our heart as 1 minus s 1 of L and so on
我们的心是L的1减去1 

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for s 2 of our and we'll rearrange the
对于我们的s 2，我们将重新排列

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term this time as a function of waist 1
这次根据腰围1 

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00:12:33,570 --> 00:12:38,078
s 1 of L because player 2 is saying
 L之1，因为玩家2在说

224
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player player one player two says player
球员球员一球员二说球员

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00:12:40,799 --> 00:12:45,339
1 is whatever I choose namely s 2 player
 1是我选择的2位玩家

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00:12:45,539 --> 00:12:47,109
1 will want to bet respond namely to
 1将要投注回应

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00:12:47,309 --> 00:12:49,419
maximize so if you write it down as a
最大化，所以如果您将其记为

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00:12:49,620 --> 00:12:52,959
function of s 1 choice and now figuring
 s 1选择的功能，现在确定

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out what the maximum would be well
找出最大的好

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here's the maximum and when you solve
这是最大值，当您解决时

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00:12:58,589 --> 00:13:02,169
for L 2 now you get that the
对于L 2现在你得到

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00:13:02,370 --> 00:13:05,919
randomization in equilibrium for for s 2
 s 2的平衡随机化

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is 4 if player 2 is 1/4 and 3/4
如果玩家2是1/4和3/4，则为4 

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00:13:10,519 --> 00:13:13,839
so this illustrates how we can use the
所以这说明了我们如何使用

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00:13:14,039 --> 00:13:18,599
maxima theorem to actually compute the
极大定理实际计算

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00:13:18,799 --> 00:13:22,149
the Nash equilibrium is zero-sum games
纳什均衡是零和博弈

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00:13:22,350 --> 00:13:25,120
at least in two by two games in general
一般至少两乘两局

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we can use the min/max theorem to
我们可以使用最小/最大定理

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00:13:29,309 --> 00:13:34,479
compute the equilibria of zero-sum game
计算零和博弈的均衡

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00:13:34,679 --> 00:13:36,519
and we do it
而我们做到了

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00:13:36,720 --> 00:13:41,969
by simply laying out a linear program
通过简单地布置一个线性程序

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00:13:42,169 --> 00:13:46,229
that captures the game and here it is so
捕获了游戏，这里就是

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00:13:46,429 --> 00:13:50,679
u 1 star is going to be the value of the
 u 1星将成为

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00:13:50,879 --> 00:13:54,819
game that is that payoff to play 1 in
就是玩1的回报的游戏

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equilibrium and so we're gonna specify
平衡，所以我们要指定

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00:13:59,340 --> 00:14:01,359
from players to point of view we could
从玩家的角度来看，我们可以

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00:14:01,559 --> 00:14:02,669
have done it the other way around also
反过来也做到了

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00:14:02,870 --> 00:14:06,539
so what player 2 is saying is simply
所以玩家2所说的只是

249
00:14:06,740 --> 00:14:13,870
says for each of the actions of player 1
说出玩家1的每个动作

250
00:14:14,070 --> 00:14:16,959
each action that player 1 might consider
玩家1可能考虑的每个动作

251
00:14:17,159 --> 00:14:23,049
I want to find a mixed strategy s 2 so
我想找到一个混合策略2 

252
00:14:23,250 --> 00:14:25,990
here's my mixed strategy s 2 it will
这是我的混合策略2 

253
00:14:26,190 --> 00:14:30,509
look at all my pure strategies K and
看看我所有的纯策略K和

254
00:14:30,710 --> 00:14:34,269
make sure that the probability that's a
确保这是一个概率

255
00:14:34,470 --> 00:14:35,799
probability distribution over those some
这些概率的分布

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00:14:36,000 --> 00:14:38,609
they sum to one and they're not negative
他们加一而不是负数

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00:14:38,809 --> 00:14:41,379
so what I'd like to do is that the best
所以我想做的就是最好的

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00:14:41,580 --> 00:14:46,299
response in my strategy by player 1 for
我的策略由玩家1回应

259
00:14:46,500 --> 00:14:48,189
any of these actions will never exceed
这些动作中的任何一个都不会超过

260
00:14:48,389 --> 00:14:50,289
this value of the game because I'm
游戏的价值，因为我

261
00:14:50,490 --> 00:14:53,069
trying to minimize so I'm going to find
试图最小化，所以我要找到

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00:14:53,269 --> 00:14:58,199
the lowest you that has a property that
拥有最低财产的最低

263
00:14:58,399 --> 00:15:00,669
player 1 doesn't have a profitable
玩家1没有获利

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00:15:00,870 --> 00:15:08,679
deviation by any of his of his these
他的任何一个偏离这些

265
00:15:08,879 --> 00:15:12,539
pure strategies so when I look at the
纯粹的策略，所以当我看着

266
00:15:12,740 --> 00:15:22,149
payoff for player 2 when I play a 2k and
我玩2k时玩家2的收益

267
00:15:22,350 --> 00:15:23,899
he plays a
他扮演一个

268
00:15:24,100 --> 00:15:25,849
J that would that Jaden I'm considering
我会考虑让Jaden考虑的J 

269
00:15:26,049 --> 00:15:28,248
right now and I multiplied the
现在，我乘以

270
00:15:28,448 --> 00:15:30,618
probability of in my mixed strategy
在我的混合策略中的概率

271
00:15:30,818 --> 00:15:35,089
playing a 2k I don't want that play one
玩2k我不想玩

272
00:15:35,289 --> 00:15:37,549
that other player player one to have a
另一个玩家拥有一个

273
00:15:37,750 --> 00:15:40,099
profitable deviation so it's got to be
有利可图的偏差，所以它必须是

274
00:15:40,299 --> 00:15:42,618
that his expected payoff will be no
他的预期收益不会是

275
00:15:42,818 --> 00:15:47,649
greater than the value you want star so
大于您想要的星号

276
00:15:47,850 --> 00:15:50,419
clearly this is a correct formulation of
显然，这是正确的表述

277
00:15:50,620 --> 00:15:52,938
the game and it is a linear program as
游戏，它是一个线性程序

278
00:15:53,139 --> 00:15:56,919
we know their programs are efficiently
我们知道他们的计划是有效的

279
00:15:57,120 --> 00:15:59,118
solvable
可解

280
00:15:59,318 --> 00:16:05,479
in theory by a interior methods that is
从理论上讲是通过内部方法

281
00:16:05,679 --> 00:16:08,828
provably polynomial in practice by
在实践中证明多项式

282
00:16:09,028 --> 00:16:13,339
procedures that are worst-case
最坏情况的程序

283
00:16:13,539 --> 00:16:18,539
exponential but in practice work well
指数的，但实际上效果很好