we will not speak about maximum 我们不会谈论最大
strategies these make particular sense 这些策略特别有意义
in the context of zero-sum games but 在零和游戏的背景下，
actually are pretty look quite - to all 实际上看起来很漂亮-所有人
games what is the max min strategy it 游戏的最大最小策略是什么
simply puts a player's strategy that 简单地提出一个球员的策略，
maximizes their payoff assuming the 假设
other player is out to get them we will 另一个玩家要得到他们，我们将
we will concentrate primarily on the 我们将主要集中于
true player case here again because when 真正的玩家案例再次出现，因为
we get to zero-sum games they don't 我们去了零和游戏
really make only center the case of two 确实只使两个情况居中
players but keep in mind that one could 玩家，但请记住，
define this kind of more generally when 更一般地定义这种情况
we speak about maximum strategy so the 我们谈论最大策略，因此
maximum strategy is a strategy that 最大策略是
maximizes my worst case outcome and my 最大化我的最坏情况的结果，
maximum value or safety level is that 最大值或安全级别是
payoff that's guaranteed by the maximum 最高保证的回报
strategy and here it is defined formally 策略，在这里正式定义
the maximum strategy for player I is the 我的最大策略是
strategy Aswan that maximizes the 最大化阿斯旺战略
minimum that the other player remember 另一个玩家记住的最低限度
the - I is the player other than I would -我是我以外的球员
hold play 1 down - and the maximum value 按住播放1-并保持最大值
is defined solely to be the value of 仅定义为
that maximum strategy now why why would 现在最大的策略为什么会
we want to think about the maximum 我们想考虑最大
strategy one can think of it either as a 策略可以将其视为一种
simply a sort of a certain cautionary 只是某种警告
maybe the other people will make some 也许其他人会做一些
mistakes and not act in their own best 错误，而不是尽自己最大的努力
interest maybe I'm not sure exactly what 兴趣也许我不确定到底是什么
their payoffs are there a lot of 他们的收益很多
interpretations or you can simply be 解释或者你可以简单地
paranoid about about about them and 关于他们的偏执和
think that there are to get you and you 认为有得到你和你
know the you know the saying you know 知道你知道的话你知道
even 甚至
apparently they have enemies that's the 显然他们有敌人
max min strategy and just a confuse 最大最小策略，只是一个混乱
thing will also speak about the min max 事情也会谈到最小最大
strategy the min max strategy is 策略最小最大策略是
strategy against if you wish the other 反对对方是否愿意的策略
player in the two-player game is the 两人游戏中的玩家是
strategies that minimizes their payoff 使收益最小化的策略
on the assumption that they're trying to 假设他们正在尝试
maximize it and so here is the formal 最大化它，所以这是正式的
definition the min max strategy for 定义最小最大策略
player I is playing against the other 我正在与其他玩家对战的玩家
guy who pre-dawn by minus I is the 减去我之前黎明的家伙是
strategy that minimizes the maximum 最小化最大策略
payoff as tempted by the other guy of 另一个人的诱惑所带来的回报
the payoff to the other guy and the min 回报给另一个人和最小
max value is simply the value of that 最大值就是那个的值
mimic strategy the value to play one now 模仿策略现在玩一玩的价值
why why would player one want to want to 为什么玩家会想要
harm the other guy well you could it 伤害其他人就可以了
could just be able to get him that's a 也许能够让他得到
possibility or they could be playing a 可能性，或者他们可能正在玩
zero-sum game and in a zero-sum game 零和游戏以及零和游戏
hurting the other guy is tantamount to 伤害另一个人等于
improving your own your own path and so 改善自己的道路等
in the setting of zero-sum games max min 在零和游戏最大最小的设置中
and min max strategy makes a lot of 最小最大策略使很多
sense and in fact in a very famous 感，实际上是非常著名的
theorem due to genre diamond it proved 钻石流派定理证明
that in a zero-sum game by definition we 根据定义，在零和博弈中
consider only two player such games any 只考虑两个玩家这样的游戏
Nash equilibrium the player sees that 纳什均衡玩家看到
payoff that is equal to both his max min 收益等于他的最大最小值
value and his min max value and and that 值和他的最小最大值，以及
means that so we will call it the value 意味着我们将其称为价值
of the game the value for player one is 在游戏中，玩家一的价值是
called the value of the game and that 称之为游戏的价值，
means that the the set of maximum 表示最大集
strategies are really the same a set of 一套策略真的一样
the min max strategies that is try to 尝试的最小最大策略
improve your worst-case situation is the 改善您最坏的情况是
same as 如同
to minimize the other guys best case 尽量减少其他人的最佳情况
situation and any maximum strategy 情况和任何最大策略
profile or minimax tragic profile 轮廓或极小悲剧轮廓
because they're the same constitute a 因为它们相同构成了一个
Nash equilibrium and furthermore those 纳什均衡，还有那些
are all the Nash equilibria that exists 是所有存在的纳什均衡
and so the payoffs it all Nash 所以回报全是纳什
equilibria is the same maybe the value 均衡是一样的，也许价值
of the game one way to get a concrete 游戏的一种获得具体方法的方法
feel for it is graphically and here's 感觉是图形化的，这是
the game of matching pennies this is a 匹配便士的游戏，这是一个
game where you each of us chooses heads 我们每个人都选择头的游戏
and tails on probability and if we if it 和概率的尾巴，如果我们
comes up either if we've both if if the 如果我们两者都出现，
result of our randomized ation I end up 我们随机化的结果我最终
choosing head and you are you end up 选择头，你就结束了
playing tail you win and confer and vice 打尾巴你赢了，赋予和副
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
but if we both chose ahead of us just 但是如果我们都选择了我们
tailed I win and so here are the payoffs 我赢了，所以这是回报
you see here the strategy spaces this is 你在这里看到的策略空间是
player 2 is kind of increasing their 玩家2有点增加他们的
probability of playing heads and this is 打头的概率，这是
player 1 and what this dimensioned 播放器1的大小
you have the value of the K of the path 您拥有路径K的值
to day one and the only Nash equilibria 到第一天，唯一的纳什均衡
is for both to randomized 5050 it's just 是为了将两者随机化为5050
right here it's kind of intriguing the 在这里，这很有意思
the the three-dimensional structure in 在三维结构
this way and you sort of see that it's 这样，您会发现它是
got to be an equilibrium in the sense 必须在某种意义上达到平衡
that player player 1 could be moving 玩家1可能正在移动
along this this curve but if as he does 沿着这条曲线，但如果他这样做
it his payoffs would only drop and so 它的收益只会下降，所以
he's trying to maximize the value would 他正在努力使价值最大化
do it and conversely player 2 can only 这样做，相反，玩家2只能
traverse around along this but if he 沿着这条路走，但如果他
does that the payoffs would only 这样做的收益只会
increase and he's trying to minimize the 增加，他正在尝试最小化
value so 如此重视
so you get a stable point which is which 所以你会得到一个稳定点
for obvious reasons is called a saddle 由于明显的原因被称为马鞍
point so although there are general 这样说虽然有一般
purpose procedure is for finding a Nash 目的程序是寻找纳什
equilibrium in particular in in two by 特别是在两个方面的均衡
two games we can use the max min 两个游戏，我们可以使用最大最小值
definition to to find it directly in 可以直接在其中找到的定义
zero-sum games and let's see how it 零和游戏，让我们看看它如何
happens in the game we'll call the 发生在游戏中，我们称之为
penalty kick game so in this game we 罚点球游戏，所以在这个游戏中我们
have a a penalty kicker and a goalie 有一个点球手和一个守门员
it's a zero-sum game and the goal of the 这是一个零和游戏，目标是
kicker is to score goal and the goal the 踢球者是进球，而进球是
goal of the goalie is to prevent it and 守门员的目标是预防和
let's assume that they each have two 假设它们每个都有两个
strategies kick to the left and kick to 策略向左踢并向
the right to a kicker and jump to the 有权获得踢球权并跳至
left and jump to the right for the 向左跳并向右跳
goalie the payoffs will be the each of 守门员的回报将是每个
those will determine a probability of 这些将决定
the kicker scoring a goal and we'll have 射门得分，我们将拥有
that probability being the payoff the 那是回报的可能性
value of the game that is a pair of two 是两个的游戏价值
pair 1 and therefore minus the after we 对1，因此减去之后
player 2 namely the goalie and so here 玩家2即守门员，所以在这里
they are so for example if the kicker 例如踢球者
kicks left and the goalie guesses 向左踢，守门员猜测
correctly and jump left also then the 正确地向左跳然后
goalie has not too bad a chance of 守门员的机会还不错
suffering the shot namely probably 0.4 遭受打击大概是0.4
if he jumps to the wrong side his 如果他跳到错误的一侧，他的
probability of stopping it is much lower 停止它的可能性要低得多
it's point 2 similarly if the kicker 如果踢球者也类似地指向第2点
decides to kick to the right if the 决定向右踢
goalie guess is wrong 守门员猜错了
his probability is low even lower than 他的概率甚至比
if you get strong in the other case and 如果您在其他情况下变强，并且
if you guys right is probability is 如果你们是对的，则概率是
higher although not quite a size that 较高，尽管尺寸不大
had to get right in the left case C's is 在C的左情况下必须正确
bitter 苦
at stopping shots when the kicker kicks 踢脚时停止投篮
to the left so how does the clicker 向左，那么点击器如何
maximize these minimum that's what we're 最大化这些最小值，这就是我们的目标
after so here is the expression right we 之后，这里是正确的表达
want the maximum value of the following 想要以下的最大值
so they each have some mixed strategy of 所以他们每个人都有一些混合策略
playing left and right with some 和一些左右演奏
probability so s1 L is a probability s1 L是一个概率
that the kicker kicks to the left and s 踢脚向左踢，并且
2 L is the probability that the goalie 2 L是守门员的概率
jumps to the left and as we saw in that 跳到左边，就像我们看到的那样
case the value is 0.6 and similarly the 情况下该值为0.6，类似地
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
this situation point 9 and point 7 in 这种情况下的第9点和第7点
these situations okay so this is the 这些情况好吧，这就是
expression that we somehow need to 我们需要某种表达的表情
compute so what is the minimum that the 计算，那么最小的是
kicker should keep in mind 踢球者应牢记
so the kicker says I'm going to pick I'm 所以踢手说我要选我
gonna play my strategy s 1 whatever it 无论如何都会发挥我的策略
is when I play it player 2 is gonna play 当我播放时，播放器2将播放
s 2 so as to minimize my payoff so let s 2以使我的收益最小化
me write down this entire expression and 我写下整个表情
I'm just this is simply copying it over 我只是将其复制过来而已
replacing its expression such as s 1r by 将其表达式替换为s 1r
1 minus s 1 l that's all it's doing and 1减去1公升
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
this is our expression and player 1s 这是我们的表情和玩家1
ain't enough water player to do if I 如果我没有足够的水玩家来做
were to play my strategy s 1 and this is 玩我的策略s 1，这是
simply rearranging the terms nothing 简单地重新排列术语
else going on here as a function of s 2 否则这里是s 2的函数
L because L因为
player one says I'm gonna put I'm gonna 玩家一说我要放我去
pick s one what would s to be best 选择一个最好的
response namely its minimum and so this 响应就是它的最小值，所以这
is arranging it as a function of s two 正在将其安排为s两个的函数
strategy and now all it remained is to 策略，现在剩下的就是
look for the minimum of the strategy and 寻找最小的策略，
the minimum is taken by luck taking the 最小的是运气
first derivative with respect to s two 关于s 2的一阶导数
holding s 1 is a constant and so we have 持有s 1是一个常数，所以我们有
this expression and then we solve for 这个表达式，然后我们解决
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
1/2 and therefore s 1 of R is a half as 1/2，因此R的s 1为
well and so by this maximum calculation 好吧，这样最大的计算
we see that player the kicker figures 我们看到那个球员的数字
out that in equilibrium 在平衡中
they better randomize 1/2 1/2 between 他们最好将1/2 1/2
left and right 左和右
what does what is the goalie kind of 什么是守门员
figure out well he was trying to 弄清楚他正在尝试
minimize the kickers maximum right 最小化踢脚者最大右脚
that's one was a look of it he's doing 那是他正在做的一件事
the min/max strategy and so here is the 最小/最大策略，所以这是
minimax strategy for player 2 just 玩家2的minimax策略
writing it down and as before we'll 写下来，像以前一样
simply will simply rewrite it's one of 只是将简单地重写它是其中之一
our heart as 1 minus s 1 of L and so on 我们的心是L的1减去1
for s 2 of our and we'll rearrange the 对于我们的s 2，我们将重新排列
term this time as a function of waist 1 这次根据腰围1
s 1 of L because player 2 is saying L之1，因为玩家2在说
player player one player two says player 球员球员一球员二说球员
1 is whatever I choose namely s 2 player 1是我选择的2位玩家
1 will want to bet respond namely to 1将要投注回应
maximize so if you write it down as a 最大化，所以如果您将其记为
function of s 1 choice and now figuring s 1选择的功能，现在确定
out what the maximum would be well 找出最大的好
here's the maximum and when you solve 这是最大值，当您解决时
for L 2 now you get that the 对于L 2现在你得到
randomization in equilibrium for for s 2 s 2的平衡随机化
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
so this illustrates how we can use the 所以这说明了我们如何使用
maxima theorem to actually compute the 极大定理实际计算
the Nash equilibrium is zero-sum games 纳什均衡是零和博弈
at least in two by two games in general 一般至少两乘两局
we can use the min/max theorem to 我们可以使用最小/最大定理
compute the equilibria of zero-sum game 计算零和博弈的均衡
and we do it 而我们做到了
by simply laying out a linear program 通过简单地布置一个线性程序
that captures the game and here it is so 捕获了游戏，这里就是
u 1 star is going to be the value of the u 1星将成为
game that is that payoff to play 1 in 就是玩1的回报的游戏
equilibrium and so we're gonna specify 平衡，所以我们要指定
from players to point of view we could 从玩家的角度来看，我们可以
have done it the other way around also 反过来也做到了
so what player 2 is saying is simply 所以玩家2所说的只是
says for each of the actions of player 1 说出玩家1的每个动作
each action that player 1 might consider 玩家1可能考虑的每个动作
I want to find a mixed strategy s 2 so 我想找到一个混合策略2
here's my mixed strategy s 2 it will 这是我的混合策略2
look at all my pure strategies K and 看看我所有的纯策略K和
make sure that the probability that's a 确保这是一个概率
probability distribution over those some 这些概率的分布
they sum to one and they're not negative 他们加一而不是负数
so what I'd like to do is that the best 所以我想做的就是最好的
response in my strategy by player 1 for 我的策略由玩家1回应
any of these actions will never exceed 这些动作中的任何一个都不会超过
this value of the game because I'm 游戏的价值，因为我
trying to minimize so I'm going to find 试图最小化，所以我要找到
the lowest you that has a property that 拥有最低财产的最低
player 1 doesn't have a profitable 玩家1没有获利
deviation by any of his of his these 他的任何一个偏离这些
pure strategies so when I look at the 纯粹的策略，所以当我看着
payoff for player 2 when I play a 2k and 我玩2k时玩家2的收益
he plays a 他扮演一个
J that would that Jaden I'm considering 我会考虑让Jaden考虑的J
right now and I multiplied the 现在，我乘以
probability of in my mixed strategy 在我的混合策略中的概率
playing a 2k I don't want that play one 玩2k我不想玩
that other player player one to have a 另一个玩家拥有一个
profitable deviation so it's got to be 有利可图的偏差，所以它必须是
that his expected payoff will be no 他的预期收益不会是
greater than the value you want star so 大于您想要的星号
clearly this is a correct formulation of 显然，这是正确的表述
the game and it is a linear program as 游戏，它是一个线性程序
we know their programs are efficiently 我们知道他们的计划是有效的
solvable 可解
in theory by a interior methods that is 从理论上讲是通过内部方法
provably polynomial in practice by 在实践中证明多项式
procedures that are worst-case 最坏情况的程序
exponential but in practice work well 指数的，但实际上效果很好