6/7/12

is no voting system perfect

Arrow's impossibility theorem suggests that no voting system is perfect. I think the proof is misleading.

More formally, Arrow says that you can't have all three of the following (taken from Wikipedia):

• If every voter prefers alternative X over alternative Y, then the group prefers X over Y.
• There is no "dictator": no single voter possesses the power to always determine the group's preference.
• If every voter's preference between X and Y remains unchanged, then the group's preference between X and Y will also remain unchanged (even if voters' preferences between other pairs like X and Z, Y and Z, or Z and W change).

However, that last thing seems bad to me. Why would we want that? For example, let's say there are 10 options. Everyone seems to hate options X and Y, so they rank them last in their lists. However, 51% of people put X above Y, and 49% put Y above X. Let's say that our system interprets this is "X is better than Y". Seems reasonable. Both suck, but X is liked a little more than Y.

Now let's say that those 49% decide that they really like Y a lot, so they put Y first in their list, with X still last. The relationship between X and Y is still the same in everyone's list. However, it now seems reasonable for the system to rank Y above X, since 49% of people love Y and hate X, and 51% of people hate them both more-or-less equally.

If we remove that requirement, then we can have a voting system that meets the remaining two requirements. For instance, the system can rank options with points, where an option gets one point for being last in someone's list, two points for being second to last, and so on. There won't be a dictator, since nobody's ballot is valued differently from anyone else's. Also, if everyone puts X over Y in their list, then X will necessarily have more points than Y.

Of course, that doesn't mean this system is perfect. The problem in general with voting systems is that they seem to encourage collusion and strategy about how to vote, rather than everyone just ranking the options according to how they really feel. For instance, in the system I proposed, people might try to find out who the best competitor is against the option they want, and put that option last in their list, even if they don't actually value that option least.

With that said, there is a voting system that eliminates this sort of strategy and collusion, where everyone is incentivized to be honest. The trick is to pick a random voter, and go with their choice. This is not quite the same as a dictator, since no voter always determines the group's preference. This voting strategy even has some practical appeal, like letting a different family member decide what movie to watch every Friday. If you don't do this, then the one family member with odd movie taste never gets to watch a movie they like. Of course, maybe that's what you want. If a decision will affect an entire nation, maybe we never want to let some crazy person decide to raise taxes to 100%.

In any case, I feel like a good voting system may exist that involves some sort of currency that persists between multiple voting opportunities. That is to say, I think votes are a form of currency, but a rather silly form when everyone gets an equal amount of them for every decision, since it seems to cause "wasteful spending", where people spend all the votes they have however they can to get what they want now, because those votes become worthless after the decision is made, and they'll just get a new allotment of votes for the next decision. Note: this doesn't necessarily mean equating votes with money.