Comment by ffn | Hacker News Reader

Comment by ffn | Hacker News Reader

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>If one prefers A over B when comparing just A and B, then one should also prefer A over B when an additional option C is offered

That assumption seems fishy to me if in a real life application, in particular because we assume offering new choices can't fundamentally change the agent/voter's preferences isn't exactly true to life. I'll give a real life, but slightly historically oversimplified example:

Say I am a poor serf living in pre-revolution Russia about a century ago. I am usually offered the following two choices:

A. serve the bourgeois royal family and live off of their scraps

B. die like a dog

Then, one day, Lenin (or similar revolutionary) comes along and offers me:

C. fight for glorious USSR communism

In real life, I would think the offered choice of C would absolutely change my preferences for A and B.

Granted this doesn't invalidate the Arrow paradox in any way, I'm just saying the paradox isn't true to life because one of its core tenants doesn't quite hold.

> Granted this doesn't invalidate the Arrow paradox in any way, I'm just saying the paradox isn't true to life because one of its core tenants doesn't quite hold.

However, this plays rather fast and loose with the principles behind Arrow's theorem. First, you're violating the nondictatorial condition (you are the only person voting on your own choice), so it's kind of silly to apply the theorem here in the first place.

Secondly, you're saying that, as soon as Communism becomes an option (but only then), you'd rather die than serve the royal family.

I can see how this might be the case, but if so, you're fundamentally changing the meaning of choice B. It's no longer just the default state ("die of starvation"), but rather dying for a cause (Communism). It's not that the introduction of choice C suddenly means that you're now choosing B over A; it's that it also introduces choice D ("die as a martyr in the name of Communism", as contrasted with "die for no particular cause"). Your ranking is now C > D > A > B.

You have to understand that Arrow's theorem came about as the result of an effort to understand the way that firms make decisions, and the ways that voting structures of boards can influence corporations to make non-optimal decisions for the corporation. Your situation doesn't really apply to the definition of independence of irrelevant alternatives as implied by the context in which Arrow's theorem is applied (or the formal proof thereof).

>Secondly, you're saying that, as soon as Communism becomes an option (but only then), you'd rather die than serve the royal family.

Well, maybe. Supposing there's some information contained in option C, or the mere availability of option C, that reveals to you the futility of option A.

In other words, your preferences aren't necessarily transitive because the availability or unavailability of particular options are themselves a little information payload that could influence the decision.

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