Transcription
Okay, so this is example one. Um, and um, the choice sets, I mean, the set of alternatives are coming from R plus square. All right. So basically, they are non-negative uh vectors of reals. So these are the alternatives; there are infinitely many of them. All right, it's not a finite set. So well, sometimes if the set is not finite, things uh may be complicated because it's a bit more abstract, but you shouldn't really worry about it if you know how to sort of formally make a logical deduction. All right. So this is our choice set; you call it X if you want to. So it's basically a vector, something like x y. All right.
Um, well, I'm going to define the following: So define the the binary relation on X as follows. A bundle—I'm going to call them bundles rather than alternatives—x y is at least as good as u v. U v is another bundle. All right, they both are in R square plus. So x y bundle is at least as good as the u v bundle if and only if, if and only if—so this is a definition, so therefore even if I said if, it means if—I don't live—if and only if uh x y vector is greater than or equal to u v. Okay.
Um, so here, that's it. I mean, that's the definition of this binary relation. This is a well-defined uh binary relation. And here, however, I want you to be careful about the notation. So this is um, I mean, this is at least as good as sign. All right. It looks like greater than or equal to, but it's not. And the reason is, I mean, different uh typing software have different symbols. If you use LaTeX, I don't know what it looks like in Word, for example, but you know there is this, I mean, this is not equal to sign. All right. So it's sort of uh uh, you know, a wavy thing. So this is greater than or equal to. All right. It's like, you know, the greater than or equal to that we use in real numbers, um, but this is at least as great. So be careful: at least as good as. Um, the reason why we use a different symbol is because we don't want to use greater than or equal to sign, all right, because the greater than or equal to sign is usually defined on reals, uh you know, real numbers or real vectors, but they're defined on reals. Uh, at least as good as, however, can be defined on anything. Right? My choice set, my alternative set, set of alternatives can be any set, like I don't know, uh cars available, cars available, partners, whatever it's like. So x y's do not really have to be numbers. So in that sense, and on top of that, I don't want to say greater than or equal to; I just want to say at least as good as. So for this purpose, I basically use a different symbol, and it's it's happens to be this one.
So here, um, however, it's sort of related, right? Greater than or equal to and at least as good as, because you know I say they're they're equivalent to. If you need, let's remember uh the, I mean, you have to—if you don't remember what it means by, you know, a vector is greater than or equal to another vector—so here is how I can define it maybe. So x y is greater than or equal to u v if and only if x is greater than or equal to u and y is greater than or equal to v. All right. So this is how we basically define greater than or equal to sign when it becomes a vectors of reals, as simple as this. Okay, but we don't we don't make any such definition here, right? I mean, I can't say x y is at least as good as u v if and only if x is at least as good as u and y is at least as—no, no, no. All right, be careful about this because this is not greater than u is or equal to. However, I say a vector is at least as good as another vector if it is actually sort of greater than or equal to the other one. All right.
So my first question would be: Is this binary relation a preference relation? I mean, is it one, reflexive? Why am I using blue? I don't know. One, is it reflexive? Two, is it uh complete, maybe the most important one? And three, is it transitive? Okay. So remember the definitions of these axioms or the axioms, and then of course, four—we can ask—so we can define a choice behavior by using this preference relation and then ask, does this choice behavior satisfy condition alpha, for example? I will come to this later. I don't know why it is five; it should be four, but let's first tackle these three questions: reflexivity, completeness, and transitivity. All right.
So, uh, is it reflexive? Well, what does reflexive binary relation mean? A binary relation—I assume you already know the definition, but let me just repeat it—a binary relation is reflexive if—actually that means if and only if—for any in this world, my binaural relation is R plus square, so I'm going to take a vector. All right, I'm not making a general general definition here. So for any vector x y, x y should be at least as good as x y itself. Right? So that's the definition of a binary relation being reflexive. That's it. Well, is it really the case? Well, this is sort of a direct proof. How do I know or how do I make sure that this is true? Well, remember it's if and only if this is true. So, um, let's see. So I have x y, some vector—by the way, I don't know what it is; that could be—all I know is that they're not negative. Right? So this is just a vector. What I know is that any vector—we know that—we know that any x y in R uh plus square, x y is at least—I mean, I'm sorry—greater than or equal to itself. Right? I mean, it is equal to itself, but you know what does equal to mean? Equal to means the first guy greater than or equal to the second, and the second is greater than or equal to the first, so therefore they're equal. So nevertheless, I don't need—so it's like if and only if and only q if and only if p. I'm sorry, p if and only if q is—remember p implies q and q implies p kind of. All right. So here there's no implication, but it's like one is greater than or equal to the second one, so I use just one side of it. So we know that, right? Every vector is greater than or equal to uh itself. So therefore, by definition of this preference relation, right, by definition of this preference relation, this vector is at least as good as itself, and this is true for any x y. So then that's it; we basically proved that this binary relation uh is reflexive. Okay, good.
Uh, by the way, in a question like this—is it reflexive?—well, you either—if you think it is reflexive or complete or transitive—you either provide the proof of it, or if you think it is not reflexive or complete or transitive, you give a counter example. Does anybody know what counter example mean? Anybody or anybody heard of it? Okay. Are you there? By the way, I don't hear anybody. Okay, yeah, I just wanted to make sure that you guys are there. All right. So uh, counter example is is not uh is not a proof method. Right? I'm just showing that an argument is false by sort of representing one case where the argument fails to hold. That's it. All right. So it's a counter-example means I am giving an example for the counter argument or the negation of the argument. So if you if if—so if if you want to show that something is complete, but in fact you believe it's not complete, so what you do? Well, the negation of completeness is—well, it's not complete—so therefore I am showing one case, one example where this completeness idea fails to hold. So that example is is called counter example because here I proved reflexivity; I I I mean, I can't find any counter example. Right? What does that mean? It would mean you need to find a vector x y, a vector of reals where it's not greater than or equal to itself, but trust me, you can't find such a real vector. All right.
So the next is then completeness. Is this binary relation complete? What do you think? Anybody? Do you have any hunch? Okay, you guys are quiet today. Fine. Uh, a binary relation is complete if for any vectors in my uh set of alternatives—either x y—oops, I'm sorry—for any two, right? It should be for any x y u v in R square plus such that either x y at least as good as u v or u v at least as good as x y. Uh, by the way, what does that mean? That means either this is true or this is true. Right? This is p or q statement. So either this is true or this is true. I just—we we usually don't put blah blah is true or blah blah is true because this is read as uh either this or this means either this is true or this is true. So, um, well, obviously it's a matter of experience, right? If you have never uh sort of thought about this before, you probably don't know whether you should start searching for a country example or searching for a proof. All right. How should I start? Well, it's a matter of choice. I would recommend you to first try to find a counter example. I mean, try to show that for some examples, for some x y and u v, this doesn't hold. Try to come up with an example. So for any argument, try to disprove them first. All right. So try to come up with counter examples. The reason is, well, when you work on, you know, finding counter examples, actually you're going to learn a lot. You'll learn a lot how things work in this environment, and so if you can't find—after some search, you know, few attempts—if you can't find a counter example, that actually may mean that the statement is true, and so you may start proving the argument. All right. And when you sort of try to prove it, you may stuck at some point, and you know the proof, the the logical deductions you had previously may actually lead to another uh sort of example that you did not consider before, and so that might be your counter example. All right. So you basically go back and forth between proving the argument, trying to find a country example. So there's no, you know, one correct way. However, because our time is limited, what I'm going to do here is we'll start with finding a a counter example. So remember the the definition of the binary relation back in of of your mind is like it actually means the vector is greater than or equal to. So is this—the vectors being greater than or equal to another one—is it a complete relation? Right? I mean, you may not know it, but for example, can you compare a vector 1 2 and 2 1? Which one is bigger, greater than or equal to? Can I say 1 2 is greater than or equal to 2 1? Is it true? Or can I say or 2 1 is greater than or equal to 1 2? Is this true? Well, remember the definition of a vector being greater than or equal to another one; it means the first component should be greater than or equal to—well, it fails because one is less than two. So this is not true. All right. So this is false clearly. All right. What about this? Two is greater than or equal to one? Yes, good. One is greater than—no, it is not—so therefore this vector is not greater than or equal to 1 2. So you know what? This one is also false. So you know what? I found an example for x y 1 two and u v is two one, which are an element of R square uh plus. Right? So they are in the element of uh I'm sorry, in my set of alternatives, and neither this nor this is true. So therefore, this binary relation—I'm sorry, by the way, remember if this is false because by the definition if and only if—so this must be false as well. Right? And again, if—because this is false because the definition is an if and only if statement—this must be false as well. So therefore, I found an example where uh you know, both these uh both of these are false. So that means I actually found the counter example, meaning well one, this relationship, this binary relation is not complete. Right? So it's uh it's not complete, and you know what? Here's my example. Well, is this the only example? Well, no. I mean, you can find millions of examples. Instead of one two, make it two five, five two. Right? It doesn't have to be sort of the mirror image of the same thing. It could be, for example, 1 3 and then, for example, 2 uh 5. So again—oh, I'm sorry—2 5 is greater than this guy, so it should be something like—so one is bigger than the other, the other is smaller. All right. So two is bigger than one here, so therefore this number should be less than three, for example, uh I don't know, zero. All right. So whenever you you find such an example, there are millions of them; you cannot really compare them uh according to this sign greater than or equal to. So that means equivalently, I cannot compare them according to this binary relation, and hence this binary relation is not complete.