GTO-5-07 - A Folk Theorem for Discounted Repeated Games.srt

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hi again folks so let's talk a little
大家好，再说一遍

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bit about a folk theorem now for a
现在谈谈一个民间定理

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discounted repeated game so we're in the
打折重复游戏，所以我们在

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case where there's a discount factor and
有折扣因素的情况下

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people care more about today than the
人们比今天更关心

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future or tomorrow and so forth and we
未来或明天等等，我们

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want to think about the expansion of the
想考虑的扩展

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logic that we just went through and some
我们刚刚经历的逻辑和一些

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examples but see whether that holds in a
例子，但看看是否在

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general setting of repeated games so
重复游戏的一般设置

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what's the folk theorem what's the
民间定理是什么

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extension to repeated games so take a
扩展到重复游戏，因此

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normal forma game and so there's
正常的形式游戏，所以有

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actually very many versions of folk
其实民间的版本很多

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theorems and we'll do a very particular
定理，我们将做一个非常特别的

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one which has I think the basic
我认为基本的

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intuition behind it and a fairly simple
背后的直觉和相当简单

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proof so the idea is is we're looking at
证明，所以我们的想法是

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some Nash equilibrium of the stage game
阶段游戏的纳什均衡

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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

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which is a Nash equilibrium of the stage
这是阶段的纳什均衡

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game and then also look for some
游戏，然后还寻找一些

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alternative strategy and here we have a
另类策略，在这里我们有一个

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couple of typos that should be in a
错别字应该在

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prime so look for some alternative
总理，所以寻找一些替代方案

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strategy a prime such that everybody
策略素数，使每个人

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gets a strictly higher payoff from a
从中获得更高的收益

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prime than a ok where a is the Nash
优胜于OK，那是纳什

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equilibrium then there exists some
平衡，然后存在一些

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discount factor below one such that if
低于1的折现系数，如果

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everybody has a discount factor above
每个人都有一个以上的折扣系数

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that level then there exists a sub-game
那个水平，然后有一个子游戏

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perfect equilibrium of the infinite
无限的完美平衡

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repetition of the game that has a prime
具有质数的游戏的重复

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played in every period on the
在每个时期都玩过

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equilibrium path so what this is telling
平衡路径，这说明了什么

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us is the logic that we went through in
我们是我们经历的逻辑

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those two examples of prisoners dilemmas
那两个囚徒困境的例子

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where we found the discount factory
我们找到折扣工厂的地方

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there half or at least 7 9 etc take any
有一半或至少7 9等采取任何

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game find a Nash equilibrium of that and
游戏找到那个的纳什均衡

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find something which is better than that
找到比这更好的东西

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which you'd like to sustain in an
您想维持在

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infinitely repeated game and you can do
无限重复的游戏，你可以做

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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

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examples in the general case where
一般情况下的例子

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there'll be a high enough discount
会有足够高的折扣

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factor that'll make that sustainable ok
使可持续发展的因素

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and and basically the proof of this
并基本上证明了这一点

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theorem is is very similar to what we
定理与我们非常相似

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went through in those examples so the
经历了那些例子，所以

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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

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as long as everybody plays it
只要大家玩

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if anybody deviates from that then we're
如果有人偏离了那我们

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gonna go to a grim trigger we're just
要去一个严峻的触发，我们只是

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gonna threaten to play the Nash
威胁要打纳什

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equilibrium a forever after which is
永远平衡之后

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giving us a lower payoff than a prime
给我们的回报比素数低

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and we just need to make sure that
我们只需要确保

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people care enough about the loss of the
人们足够在意损失

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future to offset the game the game from
未来来抵消游戏

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today so in terms of the the proof
今天，就证明而言

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checking that this is a sub-game
检查这是一个子游戏

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equilibrium for high enough discount
足够高的折扣的均衡

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factors what do we have to do well
因素我们要做好什么

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playing a forever if everyone anyone is
如果每个人都在玩一个永远

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ever deviated is part of a sub-game
曾经偏离的游戏是子游戏的一部分

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perfect continuation if we ever have a
如果我们有一个完美的延续

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deviation because it's Nash in every sub
偏差，因为它是每个子项中的纳什

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game so we need to check will anybody
游戏，所以我们需要检查任何人

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want to deviate from a prime if nobody
如果没有人想偏离素数

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has in the past and we can bound the
在过去，我们可以绑定

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game so an upper bound on the gain is
游戏，所以收益的上限是

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the maximum over all players and all
所有玩家和所有玩家的最大值

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possible deviations they could have of
他们可能会有的偏差

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the best of the gain and payoff that
最好的收益和回报

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they would get from that so that gives
他们会从中得到，所以给

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us a maximum possible gain the minimum
我们最大可能获得最小

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period per period loss so this is the
每个时期的损失，所以这是

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maximum they can gain from today well
他们今天最大的收获就是

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compared to the minimum they could lose
与他们可能损失的最低赔率相比

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from tomorrow
从明天起

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so the minimum they could lose is
因此他们可能损失的最低金额是

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instead of getting a prime they're not
而不是得到一个素数，他们不是

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going to go to a so that's that and and
去一个就是这样，并且

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take that the minimum across different
把不同的最小值

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players for this and one question is
玩家对此，一个问题是

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sort of why this the question here is
有点为什么这是这里的问题

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you know is really the minimum relative
你知道真的是最小的亲戚

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to the Nash equilibrium or could they
到纳什均衡还是

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gain so think about this a little bit
获得，所以请对此一点点思考

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why wouldn't they want to change from
他们为什么不想要从

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the Nash equilibrium in the future right
未来权利中的纳什均衡

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so the idea there is there they're not
所以那里没有他们的想法

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going to be able to to help themselves
能够自助

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by trying to change away from the
通过尝试改变

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punishment because that is a Nash
惩罚，因为那是纳什

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equilibrium so they're already getting
平衡，所以他们已经在

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the best possible payoff if the other
最好的回报，如果其他

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people follow through with the
人们遵循

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punishment so we've got the maximum
惩罚，所以我们得到了最大的

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possible game the minimum possible loss
可能的游戏最小可能的损失

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so if I deviate and given what other
所以如果我偏离并给了其他

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players are doing the maximum possible
玩家正在尽最大努力

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net gain overall is our gain the M today
总体净收益是我们今天的收益M 

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but I could lose up to M tomorrow
但明天我可能输给M 

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in the future I'll lose at least them in
将来我至少会失去他们

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the future and this should be an AI then
未来，这应该是一个人工智能

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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

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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

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has to be non-negative sorry has to be
必须是非负的，对不起

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negative in order for players not to
消极的，以使球员不

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want to deviate so what do we need we
想偏离，所以我们需要什么

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need that M is less than or equal to
需要M小于或等于

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this so M over m is less than or equal
因此，M大于m小于或等于

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to beta I over 1 minus beta I and that
超过1的beta I减去beta I， 

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gives us a lower bound on beta I it has
给了我们它的beta的下限

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to be at least M over m plus m so that's
至少M超过m加m，所以

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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

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that we went with fairly loose bounds
我们的界限相当宽松

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here but if everybody has a high enough
在这里，但如果每个人都足够高

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discount factor then you can sustain
折扣系数，那么您可以维持

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cooperation so this is just a
合作，所以这只是一个

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straightforward generalization of the
的直接概括

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examples we looked through before and
我们之前看过的例子

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it's showing us that we can sustain
告诉我们我们可以维持

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cooperation in an infinitely repeated
合作无限重复

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setting this is provided people have
设置这个是提供给人们

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enough patience for the future now
现在对未来有足够的耐心

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there's many bells and whistles on this
有很多风吹草动

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one thing to think about you could you
想一想你能

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can sustain fairly complicated play if
可以维持相当复杂的比赛，如果

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you if you'd like so let's take a look
如果您愿意的话，让我们看一下

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at the game we looked at before so the
在我们之前看过的游戏中， 

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prisoner's dilemma but now we've got
囚徒的困境，但现在我们有了

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this very high payoff from deviating one
从偏离中获得很高的回报

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thing you can notice is the total of the
您可以注意到的是

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payoffs here the players together get
玩家们在这里得到的回报

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ten if they cooperate they're only
十，如果他们合作，他们只是

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getting six in total so here actually
一共六个

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playing this makes one of the players
玩这个使球员之一

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really well off so if they played this
真的很好，所以如果他们玩这个

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in perpetuity they get three three
他们永远得到三三

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suppose they trying to do following they
假设他们试图跟随他们

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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

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so in in periods one three five and so
所以在一期三五等

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forth will play cooperate by the first
第四将发挥合作的第一

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player defect for the other so the
另一个玩家的缺陷

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second player is going to get tens in
第二个玩家将获得

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those periods but then we'll play will
那些时期，但是我们将演奏

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reverse it and the even periods
扭转它和偶数期

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right so now roughly on average players
现在大概是普通玩家

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are getting five each instead of three
每人得到五个而不是三个

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each right so what we'll do is what
每一项权利，所以我们要做的是

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we'll alternate and as long as we're
我们会交替，只要我们

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you're continuing to abide by these
你继续遵守这些

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rules where we nicely do this then in in
规定我们在哪里做的很好

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the future as long as everybody does
只要每个人都做的未来

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this we'll continue to do it
这我们会继续做

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if anybody deviates from this then we'll
如果有人偏离这一点，那么我们将

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just go to defect defect right and you
只是去缺陷缺陷吧，你

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00:07:06,000 --> 00:07:07,660
can check and see what kinds of discount
可以检查看看有哪些折扣

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factors you need and you know are there
你需要的因素，你知道在那里

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different discount factors you need for
您需要的不同折扣系数

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the first player that the player is
该玩家是的第一个玩家

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getting the CD in the first period
在第一期获得CD 

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versus the second player and so forth
与第二个玩家等

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you can go through that and actually
你可以经历一下，实际上

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this kind of thing is is something that
这种事情是

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people worry about in regulatory
人们担心监管

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settings so for instance you know
设置，例如，您知道

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imagine that you have a situation where
想象你有一种情况

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you've got companies bidding for
你有公司竞标

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government contracts and they're
政府合同，他们是

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repeating you know they're doing this
再说一遍，你知道他们正在这样做

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repeatedly over time and one way they
随着时间的推移反复出现，一种方法是

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could do it is to say okay look we can
可以说是可以的，我们可以

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compete against each other and bid very
互相竞争并非常出价

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high every day or have to bid you know
每天都很高，或者必须出价

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to give them a the government of low
给他们一个低下的政府

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cost every day if there's a procurement
如果有采购，每天都会花费

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contract but what they could do
合同，但他们能做什么

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alternatively is say okay look I'll let
或者说好吧，我会让

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you in the contract today you let me win
你今天的合同，你让我赢

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it tomorrow and we'll just alternate and
明天，我们将轮流和

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as long as we keep cooperating we won't
只要我们保持合作，我们就不会

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compete with each other we'll enjoy high
相互竞争，我们将享受高

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payoffs but if that ever breaks down
回报，但如果失败了

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then we're gonna go back to competition
然后我们要回到竞争

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so there's situations where regulators
因此，在某些情况下，监管机构

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worry and in fact there's some various
担心，实际上有一些

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cases that have some evidence that
有证据表明

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companies will tend to do this to try
公司将倾向于这样做尝试

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and game the system and an increase
和游戏系统，并增加

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payoff so so you can see that the kind
回报，所以你可以看到那种

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of logic and what has to be true in
逻辑和什么必须是真实的

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order for that to happen
为了做到这一点

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okay so repeated games we've had a
好吧，如此反复的比赛，我们有一个

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fairly detailed look at these things
比较详细地看这些东西

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players can condition their future play
玩家可以限制他们未来的比赛

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on past actions that allows them to
根据他们过去的行动

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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

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in a static game it produces new
在静态游戏中，它会产生新的

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equilibria in the game folk theorems
博弈定理中的均衡

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partly referring to the fact that these
部分是指这些

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were
是

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known for a long time in kind of
长期以来以

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folklore and game theory before they
民俗学和博弈论

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were actually written down
实际上被写下来

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there's many equilibria in these things
这些东西之间有很多平衡

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and they're based on on key ingredients
它们基于关键成分

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having observation of what other players
观察其他玩家

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do and being able to react to that and
能够并且能够对此做出反应

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having sufficient value in the future
将来有足够的价值

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either by limit of the means which is an
要么通过手段限制

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extreme value or high enough discount
极高价值或足够高的折扣

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factors so that players really care
使玩家真正关心的因素

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about the future now repeated games have
关于未来现在重复的游戏有

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actually been a fairly active area of
实际上是一个相当活跃的领域

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research recently there's a lot of other
最近的研究还有很多其他的

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interesting questions on these what
关于这些有趣的问题是什么

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happens if you don't always see what
如果您不总是看到什么就会发生

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other players do you only see sometimes
其他玩家您有时只会看到

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there's some noises and things what
有一些噪音和其他东西

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happens if there's uncertain payoffs
如果不确定的回报会发生

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over time there are payoffs are varying
随着时间的流逝，收益是不同的

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there's a whole series of issues there
那里有一系列的问题

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there's also issues about things like
还有一些类似的问题

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renegotiation so you know the logic here
重新谈判，所以你知道这里的逻辑

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has been okay if if we anybody ever
如果我们有人曾经一直没事

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deviates then we go to a bad equilibrium
偏离然后我们达到一个糟糕的平衡

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forever after so suppose that happens
如此假设之后永远

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somebody deviates and then we you know
有人偏离，然后我们知道

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after about a few periods we say well
大约过了一段时间后，我们说好了

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this is kind of silly why are we why are
这有点傻，为什么我们为什么

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we hurting ourselves let's go back to
我们伤害自己让我们回到

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the original agreement let's forget
原始协议让我们忘记

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about things by Ron's be bygones so we
关于罗恩（Ron）过去的事情，所以我们

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can do better by just starting all over
可以从头开始做得更好

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right okay well that's wonderful the
好吧，那太好了

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difficulty is that if if we now believe
困难在于如果我们现在相信

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that if we deviate eventually we're
如果我们最终偏离了我们

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going to be forgiven then that changes
将被原谅，然后改变

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the whole nature of the game and changes
游戏的整体性质和变化

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the the the incentives at the beginning
一开始的激励措施

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and so incorporating that kind of logic
所以结合了这种逻辑

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is is quite complicated and another area
是相当复杂的另一个领域

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of research and in these so repeated
研究，并在这些如此重复

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games are very interesting they have
游戏非常有趣

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lots of applications there's some
很多应用程序中有一些

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interesting logic which comes out of
有趣的逻辑

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them sometimes you can sustain
他们有时你可以维持

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cooperation or better payoffs and you
合作或更好的回报，您

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can in the static setting sometimes you
可以在静态设置中有时

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can't and we've seen some of the
不能，我们已经看到了一些

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features that affect that
影响那个的功能