You might be better kicking to the left or to the right depending on whether you're right handed or left handed.

你最好从左路还是右路射门,和你是左撇子还是右撇子有关

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Finally, let's do it in green, let's look at the payoffs, the expected payoffs if I shoot to the right.

最后,用绿线来表示吧,我们来看看我从右路射门的预期收益

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That's probably as important a decision as placing it, as deciding which direction to hit, and it turns out to matter.

踢球的力度与射门的方向,这两者是同样重要的

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The first lesson was don't shoot towards the middle of the goal and the second more general lesson was what?

结论一是不要从中路射门,第二个更普遍的结论是什么

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So right footed players find it easier to shoot to the left as facing, to shoot across the goal.

习惯用右脚射门的球员从左路进攻,会更容易把球射入球门

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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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There's no point at which the 6, at which it shooting in the middle gets a higher payoff.

在6那条线,即从中路射门的线上,任何一点都无法使你获得较高的收益

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Here, at least, we got somewhere, we got rid of the idea of shooting to the middle.

但现在,至少我们有点办法了,我们至少剔除了从中路射门的策略

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So you might think that maybe middle dominates left, but notice that middle has a higher payoff against left than shooting to the left.

你可能认为中路比左路好,那请你仔细看一下在门将扑向左路时,从中路射门的收益确实比左路高

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So the idea here is shooting to the left if you're right footed is the natural direction, so left here means the natural direction.

就是说如果你习惯用右脚射门,那么你从左路攻门就是自然方向,在这里左路表示自然方向

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And there is no belief you could possibly hold given these numbers in this game that could ever rationalize shooting the ball to the middle.

在给出的这些数据下,无论你认为门将有多大概率会扑向右路,从中路射门绝对不是一个合理的选择

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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 this line represents the expected payoff of shooting to the middle as a function of the probability that the goal keeper dives to the right.

这条线表示从中路射门时罚球者的收益,是门将扑向右路概率的一个函数

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It's the expected payoff to Player 1 if shooting to the left as it depends on the probability that the goal keeper dives to the right.

它表示参与人1,从左路射门的预期收益,与门将扑向右路的概率有关

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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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The lesson is, do not shoot to the middle. not shoot to the middle.

结论是无论如何千万不要从中路射门

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Conversely, if I shoot to the right and he or she dives to the right, then I score 40% of the time, so here's my payoff 4.

相反如果我从右路射门而门将扑向右路,我进球的概率是40%,收益是4

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So it starts with a possibility of shooting to the left.

我们从左路射门这种情况开始分析

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Is that right? So no: to say it another way, middle is not a best response to any belief I can hold about the goal keeper, to any belief.

我说的没错吧,或者换个说法,从中路射门无论在何种信念下,都不是一个最佳对策

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There are three ways, the goal--the attacker could kick the ball.

罚球队员可以选择三种不同射门路径

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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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So what we're going to do is we're going to look at some numbers that are approximately the probabilities of scoring when you kick the penalty kick in different directions.

接下来,我们来用数字表示,在不同角度踢点球射门,能够得分的概率

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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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And the idea here is that the number 4 represents 40% chance of scoring if you shoot the ball to the left of the goal and the goal keeper dives to the left.

在这里表示进球概率是40%,前提是你从左路射门,而门将也扑向左路

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And underlining what arises in this game, notice that in this game we're able to eliminate one of the strategies, in this case the strategy of shooting to the middle, even though nothing was dominated.

我们从这个博弈中可以得出,我们能够剔除其中的一个策略,在本案例中是从中路射门,尽管这里并没有劣势策略

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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 in reality, we need to take into account a little bit more, and in particular, we need to take into account the abilities of players to hit the ball accurately and/or hard.

所以分析实际情况时要深思熟虑,比如说,我们需要把罚球队员的,射门力度跟精准度考虑在内

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It turns out, if you look at those dotted lines, there's an area in the middle, the area between here and here, this little area here, you actually might be just fine shooting to the middle.

请大家看这条虚线,在这两点中间的范围内,就是这个小区间内,从中路射门也是个不错的选择

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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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Here it goes in the middle, and if you look carefully there, I didn't really make it clear enough, you can see, suddenly a strategy that looked crazy shooting to the middle, that suddenly started to seem okay.

这是中路射门的情况,如果你仔细看,我可能画的不是很清楚,你会发现曾经卖傻似的中路射门法,现在竟然一下子变成香饽饽了

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