# Understanding the Choice Rule in MWG

I am reading the Microeconomics Theory book by MWG, and I am having a tough time interpreting what things mean to a real life example, so any help would be appreciated.

For example, it gave this. Suppose X = {x, y, z} and B, the budget set, = {{x, y}, {x, y, z}}. It says we can define a choice rule so that C({x, y}) = {x} and C({x, y, z}) = {x, y}.

I am having trouble understanding what C({x, y, z}) = {x, y} means. Does that mean that when faced with the choice of x, y, and z, the person is indifferent between x and y? It doesn't mean he chooses both right? It just means he will choose either or so it's random which one of the two he chooses?

Furthermore, how can we even have a situation where if the consumer is choosing between x and y, he'll always choose x, but then when he is faced between choosing x, y, and z, he will choose either x or y? Can someone give me a real life example of how this would be possible? That usually helps me understand better.

Thank you so much! I appreciate any responses.

I am having trouble understanding what $$C(\{x, y, z\}) = \{x, y\}$$ means.

MWG already explain this on p.10:

When [$$C(B)$$ contains more than one element], the elements of $$C(B)$$ are the alternatives in $$B$$ that the decision maker might choose; that is, they are her acceptable alternatives in $$B$$. In this case, the set $$C(B)$$ can be thought of as containing those alternatives that we would actually see chosen if the decision maker were repeatedly to face the problem of choosing an alternative from set $$B$$.

[H]ow can we even have a situation where if the consumer is choosing between $$x$$ and $$y$$, he'll always choose $$x$$, but then when he is faced between choosing $$x$$, $$y$$, and $$z$$, he will choose either $$x$$ or $$y$$? Can someone give me a real life example of how this would be possible?

Consider a voting situation as follows. Individuals $$x$$, $$y$$, and $$z$$ are potential candidates for the position of a department chair. A five-member committee is to select the chair by voting via simple majority rule. Let's label the committee members $$1$$ to $$5$$ and suppose that they each have a private ranking of the three candidates as follows ($$\succ_i$$ is member $$i$$'s ranking): \begin{align} x\succ_1 y\succ_1 z \\ y\succ_2 x\succ_2 z \\ z\succ_3 x\succ_3 y \\ y\succ_4 z\succ_4 x \\ x\succ_5 z\succ_5 y \end{align} Suppose that each member will vote for his/her top-ranked candidate. Then when only two candidates $$x$$ and $$y$$ are under consideration, $$x$$ will get three votes (from members $$1$$, $$3$$ and $$5$$) and thus always be chosen. But when all three candidates are being considered, $$x$$ and $$y$$ will each get two votes ($$x$$ from $$1$$ and $$5$$, $$y$$ from $$2$$ and $$4$$) and thus both be "acceptable" candidates.

• I suppose I could have made the example based on the current moderator election at Econ.SE, which also happens to have three candidates. But I had a hard time deciding who's $z$ :) Oct 25 '20 at 5:02
• The very reasonable voting example may miss the point that people are supposed to be independent of irrelevant alternatives, that is to say, OP's confusion is fair. Individual agents should NOT suddenly become interested in y after the addition of z. Another example could be cars in colors RB vs RBG, where the buyer would always pick R in RB but the presence of the nearby G car in choice RBG really makes the blue "pop". It should't matter- but it sometimes does. Oct 26 '20 at 13:21
• @RegressForward: I'm not sure I understand what you mean by the comment "people are supposed to be independent of irrelevant alternatives". But in my example, I would say that it's not that voting members "suddenly become interested in $y$ after the addition of $z$", but rather that some of them become less interested in $x$ because the newly available $z$ draws vote(s) away from $x$. Oct 26 '20 at 14:52
• Right, but the question by OP is about individual agents (so voting problems are out). OP seems to be reading MWG's discussion of the irrelevance of independent alternatives, which for individual agents is typically assumed to hold. Oct 26 '20 at 16:16
• Thank you for explaining this!
– Alex
Oct 27 '20 at 22:51