hi again folks so let's talk a little 大家好，再说一遍
bit about a folk theorem now for a 现在谈谈一个民间定理
discounted repeated game so we're in the 打折重复游戏，所以我们在
case where there's a discount factor and 有折扣因素的情况下
people care more about today than the 人们比今天更关心
future or tomorrow and so forth and we 未来或明天等等，我们
want to think about the expansion of the 想考虑的扩展
logic that we just went through and some 我们刚刚经历的逻辑和一些
examples but see whether that holds in a 例子，但看看是否在
general setting of repeated games so 重复游戏的一般设置
what's the folk theorem what's the 民间定理是什么
extension to repeated games so take a 扩展到重复游戏，因此
normal forma game and so there's 正常的形式游戏，所以有
actually very many versions of folk 其实民间的版本很多
theorems and we'll do a very particular 定理，我们将做一个非常特别的
one which has I think the basic 我认为基本的
intuition behind it and a fairly simple 背后的直觉和相当简单
proof so the idea is is we're looking at 证明，所以我们的想法是
some Nash equilibrium of the stage game 阶段游戏的纳什均衡
so take a stage game find a profile 所以参加一个舞台游戏找到一个个人资料
which is a Nash equilibrium of the stage 这是阶段的纳什均衡
game and then also look for some 游戏，然后还寻找一些
alternative strategy and here we have a 另类策略，在这里我们有一个
couple of typos that should be in a 错别字应该在
prime so look for some alternative 总理，所以寻找一些替代方案
strategy a prime such that everybody 策略素数，使每个人
gets a strictly higher payoff from a 从中获得更高的收益
prime than a ok where a is the Nash 优胜于OK，那是纳什
equilibrium then there exists some 平衡，然后存在一些
discount factor below one such that if 低于1的折现系数，如果
everybody has a discount factor above 每个人都有一个以上的折扣系数
that level then there exists a sub-game 那个水平，然后有一个子游戏
perfect equilibrium of the infinite 无限的完美平衡
repetition of the game that has a prime 具有质数的游戏的重复
played in every period on the 在每个时期都玩过
equilibrium path so what this is telling 平衡路径，这说明了什么
us is the logic that we went through in 我们是我们经历的逻辑
those two examples of prisoners dilemmas 那两个囚徒困境的例子
where we found the discount factory 我们找到折扣工厂的地方
there half or at least 7 9 etc take any 有一半或至少7 9等采取任何
game find a Nash equilibrium of that and 游戏找到那个的纳什均衡
find something which is better than that 找到比这更好的东西
which you'd like to sustain in an 您想维持在
infinitely repeated game and you can do 无限重复的游戏，你可以做
the same logic that we did in those 与我们在那些逻辑中所做的相同的逻辑
examples in the general case where 一般情况下的例子
there'll be a high enough discount 会有足够高的折扣
factor that'll make that sustainable ok 使可持续发展的因素
and and basically the proof of this 并基本上证明了这一点
theorem is is very similar to what we 定理与我们非常相似
went through in those examples so the 经历了那些例子，所以
idea is is you know we'll play a prime 想法是你知道我们会发挥主要作用
as long as everybody plays it 只要大家玩
if anybody deviates from that then we're 如果有人偏离了那我们
gonna go to a grim trigger we're just 要去一个严峻的触发，我们只是
gonna threaten to play the Nash 威胁要打纳什
equilibrium a forever after which is 永远平衡之后
giving us a lower payoff than a prime 给我们的回报比素数低
and we just need to make sure that 我们只需要确保
people care enough about the loss of the 人们足够在意损失
future to offset the game the game from 未来来抵消游戏
today so in terms of the the proof 今天，就证明而言
checking that this is a sub-game 检查这是一个子游戏
equilibrium for high enough discount 足够高的折扣的均衡
factors what do we have to do well 因素我们要做好什么
playing a forever if everyone anyone is 如果每个人都在玩一个永远
ever deviated is part of a sub-game 曾经偏离的游戏是子游戏的一部分
perfect continuation if we ever have a 如果我们有一个完美的延续
deviation because it's Nash in every sub 偏差，因为它是每个子项中的纳什
game so we need to check will anybody 游戏，所以我们需要检查任何人
want to deviate from a prime if nobody 如果没有人想偏离素数
has in the past and we can bound the 在过去，我们可以绑定
game so an upper bound on the gain is 游戏，所以收益的上限是
the maximum over all players and all 所有玩家和所有玩家的最大值
possible deviations they could have of 他们可能会有的偏差
the best of the gain and payoff that 最好的收益和回报
they would get from that so that gives 他们会从中得到，所以给
us a maximum possible gain the minimum 我们最大可能获得最小
period per period loss so this is the 每个时期的损失，所以这是
maximum they can gain from today well 他们今天最大的收获就是
compared to the minimum they could lose 与他们可能损失的最低赔率相比
from tomorrow 从明天起
so the minimum they could lose is 因此他们可能损失的最低金额是
instead of getting a prime they're not 而不是得到一个素数，他们不是
going to go to a so that's that and and 去一个就是这样，并且
take that the minimum across different 把不同的最小值
players for this and one question is 玩家对此，一个问题是
sort of why this the question here is 有点为什么这是这里的问题
you know is really the minimum relative 你知道真的是最小的亲戚
to the Nash equilibrium or could they 到纳什均衡还是
gain so think about this a little bit 获得，所以请对此一点点思考
why wouldn't they want to change from 他们为什么不想要从
the Nash equilibrium in the future right 未来权利中的纳什均衡
so the idea there is there they're not 所以那里没有他们的想法
going to be able to to help themselves 能够自助
by trying to change away from the 通过尝试改变
punishment because that is a Nash 惩罚，因为那是纳什
equilibrium so they're already getting 平衡，所以他们已经在
the best possible payoff if the other 最好的回报，如果其他
people follow through with the 人们遵循
punishment so we've got the maximum 惩罚，所以我们得到了最大的
possible game the minimum possible loss 可能的游戏最小可能的损失
so if I deviate and given what other 所以如果我偏离并给了其他
players are doing the maximum possible 玩家正在尽最大努力
net gain overall is our gain the M today 总体净收益是我们今天的收益M
but I could lose up to M tomorrow 但明天我可能输给M
in the future I'll lose at least them in 将来我至少会失去他们
the future and this should be an AI then 未来，这应该是一个人工智能
we've got beta I over 1 minus beta I and 我们的Beta I超过1减去Beta I，并且
so if you go ahead and you know set this 因此，如果您继续进行，并且知道设置
has to be non-negative sorry has to be 必须是非负的，对不起
negative in order for players not to 消极的，以使球员不
want to deviate so what do we need we 想偏离，所以我们需要什么
need that M is less than or equal to 需要M小于或等于
this so M over m is less than or equal 因此，M大于m小于或等于
to beta I over 1 minus beta I and that 超过1的beta I减去beta I，
gives us a lower bound on beta I it has 给了我们它的beta的下限
to be at least M over m plus m so that's 至少M超过m加m，所以
a not a tight lower bound in the sense 从某种意义上讲不是一个严格的下限
that we went with fairly loose bounds 我们的界限相当宽松
here but if everybody has a high enough 在这里，但如果每个人都足够高
discount factor then you can sustain 折扣系数，那么您可以维持
cooperation so this is just a 合作，所以这只是一个
straightforward generalization of the 的直接概括
examples we looked through before and 我们之前看过的例子
it's showing us that we can sustain 告诉我们我们可以维持
cooperation in an infinitely repeated 合作无限重复
setting this is provided people have 设置这个是提供给人们
enough patience for the future now 现在对未来有足够的耐心
there's many bells and whistles on this 有很多风吹草动
one thing to think about you could you 想一想你能
can sustain fairly complicated play if 可以维持相当复杂的比赛，如果
you if you'd like so let's take a look 如果您愿意的话，让我们看一下
at the game we looked at before so the 在我们之前看过的游戏中，
prisoner's dilemma but now we've got 囚徒的困境，但现在我们有了
this very high payoff from deviating one 从偏离中获得很高的回报
thing you can notice is the total of the 您可以注意到的是
payoffs here the players together get 玩家们在这里得到的回报
ten if they cooperate they're only 十，如果他们合作，他们只是
getting six in total so here actually 一共六个
playing this makes one of the players 玩这个使球员之一
really well off so if they played this 真的很好，所以如果他们玩这个
in perpetuity they get three three 他们永远得到三三
suppose they trying to do following they 假设他们试图跟随他们
say in odd periods we'll play a CD right 说在奇怪的时期我们会播放CD权利
so in in periods one three five and so 所以在一期三五等
forth will play cooperate by the first 第四将发挥合作的第一
player defect for the other so the 另一个玩家的缺陷
second player is going to get tens in 第二个玩家将获得
those periods but then we'll play will 那些时期，但是我们将演奏
reverse it and the even periods 扭转它和偶数期
right so now roughly on average players 现在大概是普通玩家
are getting five each instead of three 每人得到五个而不是三个
each right so what we'll do is what 每一项权利，所以我们要做的是
we'll alternate and as long as we're 我们会交替，只要我们
you're continuing to abide by these 你继续遵守这些
rules where we nicely do this then in in 规定我们在哪里做的很好
the future as long as everybody does 只要每个人都做的未来
this we'll continue to do it 这我们会继续做
if anybody deviates from this then we'll 如果有人偏离这一点，那么我们将
just go to defect defect right and you 只是去缺陷缺陷吧，你
can check and see what kinds of discount 可以检查看看有哪些折扣
factors you need and you know are there 你需要的因素，你知道在那里
different discount factors you need for 您需要的不同折扣系数
the first player that the player is 该玩家是的第一个玩家
getting the CD in the first period 在第一期获得CD
versus the second player and so forth 与第二个玩家等
you can go through that and actually 你可以经历一下，实际上
this kind of thing is is something that 这种事情是
people worry about in regulatory 人们担心监管
settings so for instance you know 设置，例如，您知道
imagine that you have a situation where 想象你有一种情况
you've got companies bidding for 你有公司竞标
government contracts and they're 政府合同，他们是
repeating you know they're doing this 再说一遍，你知道他们正在这样做
repeatedly over time and one way they 随着时间的推移反复出现，一种方法是
could do it is to say okay look we can 可以说是可以的，我们可以
compete against each other and bid very 互相竞争并非常出价
high every day or have to bid you know 每天都很高，或者必须出价
to give them a the government of low 给他们一个低下的政府
cost every day if there's a procurement 如果有采购，每天都会花费
contract but what they could do 合同，但他们能做什么
alternatively is say okay look I'll let 或者说好吧，我会让
you in the contract today you let me win 你今天的合同，你让我赢
it tomorrow and we'll just alternate and 明天，我们将轮流和
as long as we keep cooperating we won't 只要我们保持合作，我们就不会
compete with each other we'll enjoy high 相互竞争，我们将享受高
payoffs but if that ever breaks down 回报，但如果失败了
then we're gonna go back to competition 然后我们要回到竞争
so there's situations where regulators 因此，在某些情况下，监管机构
worry and in fact there's some various 担心，实际上有一些
cases that have some evidence that 有证据表明
companies will tend to do this to try 公司将倾向于这样做尝试
and game the system and an increase 和游戏系统，并增加
payoff so so you can see that the kind 回报，所以你可以看到那种
of logic and what has to be true in 逻辑和什么必须是真实的
order for that to happen 为了做到这一点
okay so repeated games we've had a 好吧，如此反复的比赛，我们有一个
fairly detailed look at these things 比较详细地看这些东西
players can condition their future play 玩家可以限制他们未来的比赛
on past actions that allows them to 根据他们过去的行动
react to things in ways that they can't 以他们无法做到的方式对事物做出反应
in a static game it produces new 在静态游戏中，它会产生新的
equilibria in the game folk theorems 博弈定理中的均衡
partly referring to the fact that these 部分是指这些
were 是
known for a long time in kind of 长期以来以
folklore and game theory before they 民俗学和博弈论
were actually written down 实际上被写下来
there's many equilibria in these things 这些东西之间有很多平衡
and they're based on on key ingredients 它们基于关键成分
having observation of what other players 观察其他玩家
do and being able to react to that and 能够并且能够对此做出反应
having sufficient value in the future 将来有足够的价值
either by limit of the means which is an 要么通过手段限制
extreme value or high enough discount 极高价值或足够高的折扣
factors so that players really care 使玩家真正关心的因素
about the future now repeated games have 关于未来现在重复的游戏有
actually been a fairly active area of 实际上是一个相当活跃的领域
research recently there's a lot of other 最近的研究还有很多其他的
interesting questions on these what 关于这些有趣的问题是什么
happens if you don't always see what 如果您不总是看到什么就会发生
other players do you only see sometimes 其他玩家您有时只会看到
there's some noises and things what 有一些噪音和其他东西
happens if there's uncertain payoffs 如果不确定的回报会发生
over time there are payoffs are varying 随着时间的流逝，收益是不同的
there's a whole series of issues there 那里有一系列的问题
there's also issues about things like 还有一些类似的问题
renegotiation so you know the logic here 重新谈判，所以你知道这里的逻辑
has been okay if if we anybody ever 如果我们有人曾经一直没事
deviates then we go to a bad equilibrium 偏离然后我们达到一个糟糕的平衡
forever after so suppose that happens 如此假设之后永远
somebody deviates and then we you know 有人偏离，然后我们知道
after about a few periods we say well 大约过了一段时间后，我们说好了
this is kind of silly why are we why are 这有点傻，为什么我们为什么
we hurting ourselves let's go back to 我们伤害自己让我们回到
the original agreement let's forget 原始协议让我们忘记
about things by Ron's be bygones so we 关于罗恩（Ron）过去的事情，所以我们
can do better by just starting all over 可以从头开始做得更好
right okay well that's wonderful the 好吧，那太好了
difficulty is that if if we now believe 困难在于如果我们现在相信
that if we deviate eventually we're 如果我们最终偏离了我们
going to be forgiven then that changes 将被原谅，然后改变
the whole nature of the game and changes 游戏的整体性质和变化
the the the incentives at the beginning 一开始的激励措施
and so incorporating that kind of logic 所以结合了这种逻辑
is is quite complicated and another area 是相当复杂的另一个领域
of research and in these so repeated 研究，并在这些如此重复
games are very interesting they have 游戏非常有趣
lots of applications there's some 很多应用程序中有一些
interesting logic which comes out of 有趣的逻辑
them sometimes you can sustain 他们有时你可以维持
cooperation or better payoffs and you 合作或更好的回报，您
can in the static setting sometimes you 可以在静态设置中有时
can't and we've seen some of the 不能，我们已经看到了一些
features that affect that 影响那个的功能