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I'm working on understanding a textbook on Mechanism Design and I came across a part that I didn't quite understand, which is underlined in RED:

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What does the underlined part refer to? I know it's an interval of some sort, but I'm clueless beyond that.

Joebevo
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1 Answers1

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Interval Notation is a method of specifying a chunk of the real numbers using the endpoint values and brackets of varying types to say what the nature of that end point is. In this case, $[\theta_i,\overline {\theta_i} \cdot \overline {m_i}]$ means all the numbers between $\theta_i$ and $\overline {\theta_i} \cdot \overline {m_i}$, including both endpoints.

Dan Uznanski
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  • I know that, but what is the dot doing there between theta(i) and m(i)? And why is there a bar over theta? – Joebevo Aug 10 '18 at 02:01
  • The overline difference I don't know; there might be context in the rest of the passage. I suspect the dot might be misplaced, the difference between \overline {\theta_i} \cdot \overline {m_i} and \overline {\theta_{i\cdot}} \overline {m_i}, which would be a common typo in automatic typesetters where you don't use tex commands directly. – Dan Uznanski Aug 10 '18 at 02:11
  • What I got from another place in the book: $\overline {\theta_i}$ is the highest valuation of agent $i$, while $\underline {\theta_i}$ is the same agent's lowest valuation. So the lower bound makes sense. I'm still trying to figure out what the dot and the $\overline{m_i}$ are doing there. – Joebevo Aug 10 '18 at 02:27
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    Ahhh. Got it. This a major typo. The square brackets should be closed after $\overline {\theta_i}$ and there should be a period after that, as it ends the sentence. $\overline {m_i}$ begins a new sentence! This is awful..how am I going to get through the book??! – Joebevo Aug 10 '18 at 02:49
  • That's just bonkers. – Dan Uznanski Aug 10 '18 at 03:15
  • You should add your own answer saying that and accept it, by the way, so we have a more permanent record of what's happened. And you'll get some rep! – Dan Uznanski Aug 10 '18 at 03:25
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    Ordinarily, @Joebevo, I counsel tolerance in cases like these: after all, texts are created and produced by fallible mortals, not divinities. But I agree with your frustration. If this is representative of the accuracy and reliability of the book, you are in for a rough ride. Good luck. And don’t forget to answer your own question, for just the reasons that Dan says. – Lubin Aug 10 '18 at 03:34