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So I shoot to the left and the goalie dives to the left, my payoff is what? It's 4.
我从左路射门而门将也扑向左路时,我的收益是4
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So on the horizontal axis is my belief, and my belief is essentially the probability that the goalie dives to the right.
横轴表示的是我的信念,我的信念表示,我认为门将扑向右路的概率
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So we're allowing you to, for example, to hold the belief that it's equally likely that the goalie dives left or dives right.
所以,打个比方,这里也可以包括,门将扑向左右两路概率相同的情况
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So this is the guy who is shooting, he's called the shooter and this is the goalie.
这边代表罚球队员,称他为射手吧,这边代表门将
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Certainly the goalie staying put is an issue, as I said we'll deal with that in the problem set, but there's another issue here.
门将守在中路的确是个问题,我们会在习题集中研究它,但除此之外还有一个问题
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At the end of the day, the goalie doesn't know the decision of the shooter and the shooter doesn't know the decision of the goalie.
直到比赛结束之前,门将都不会知道你会从哪路罚点球,罚球队员也不知道门将会扑向哪路
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If I shoot to the left and the goalie dived to the right, then I score 90% of the time, so my payoff is .9.
如果我从左路射门而门将扑向右路,那我有90%的概率进球,即我的收益就是9
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So if I shoot to the right and the goalie dives to the left, then I score with probability point 9 or my payoff is 9.
如果我从右路射门而门将扑向左路,我进球的概率是90%,即收益是9
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Player II was a goalie, he could dive to the left or the right.
门将可以选择扑向左侧或者右侧
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It has as lower payoff if the goalie dives to the right.
如果门将扑向右边中路反而不如左路了
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And here's my green line representing my expected payoff as the shooter, from shooting to the right, as a function of the probability that the goalie dives to the right.
这条绿线表示作为一个罚球球员,我从右路射门的收益,是门将扑向右路概率的一个函数
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The goalie can dive to the left or the right.
门将可以选择扑向左路或者右路
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In principle the goalie could stay in the middle.
原则上说门将也可以守在中路不动
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Good, so one thing that's clearly missing here is I'm ignoring that in fact right footed players find it easier to shoot to their left, which is actually the goalie's right.
很好,我们忽略了这一点,我们忽视了实际上惯用右脚的球员,从左路射门会更容易些,罚球球员的左路就是门将的右路
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That's if the goalie dives to the left.
这是门将扑向左路时的情况
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The goalie might stay in the middle.
门将也可能守在中路
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Notice here, just to underline something which came up at the end last time, that doesn't just mean beliefs of the form, the goalie's going to dive left or the goalie's going to dive right.
我再提一下,上一讲最后的结论,大家需要引起重视,我并不是说信念是这种形式的,即门将会扑向左路或者门将会扑向右路
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And if the goalie dives to the right, you're best off shooting to the left, second best off shooting to the middle, and worst off by shooting to the right; and that's kind of common sense.
如果门将扑向右边,你最好从左路射门,其次从中路射门,从右路射门是最差的选择,这些都是常识了吧
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So, not surprisingly in this game, it turns out that if the goalie dived to the left you're best off shooting to the right, second best off shooting to the middle, and worst off shooting to the left.
因此毫无争议地说,如果门将扑向左路,你最好选择从右路射门,其次是从中路射门,最差的选择是从左路射门
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If I shoot to the left and there's no probability of the goalie diving to the right, which means that they dive to the left, then my payoff is 4, meaning I score 40% of the time.
如果我从左路射门,而门将不会去扑向右路,也就是说他会扑向左路,此时我的收益是4,也就是说我进球的概率是40%
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So the goalie is going to shoot to the right with the probability less than a ?, sorry he's going to dive to the right with the probability less than a ?, you should shoot to the right.
如果门将从右路射门的概率,小于?的话,抱歉,是门将扑向右路的概率小于?,你应该从右路射门
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What I suggest we do is we do what we did last time and we start to draw a picture to figure out what my expected payoff is, depending on what I believe the goalie is going to do.
我建议用上次的方法,作图来表示在我对门将的策略,存在某种信念下,我的预期收益
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So the thing that I hope jumps out at you from this picture is no great guesses about figuring out this is a ?, so if the probability that the goalie's going to jump to the right is less than a ?, then the best you can do is represented by this green line, which is shoot to the right.
我希望大家都能从图中看出来,得出这一点的横坐标是?不难,如果门将扑向右路的概率,小于?的时候,此时你的最佳策略用这段绿线表示,即你应该从右路射门
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Conversely, if you think the goalie's going to shoot to the right with probability more than a ?, then the best you can do is represented by the pink line, and that's shooting to the left, or if you think the goalie's going to dive to the right with the probability more than a ?, the best you can do, your best response is to shoot to the left.
相反如果你认为门将扑向右侧的概率,大于?的情况下,此时你最佳策略用粉线表示,即从左路射门,也就是说如果你觉得门将扑向右路的,概率大于?的话,你最好选择从左路射门
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