and you shall know a webcomic by the questions he (frequently) answers

Amazing Super Powers is still funny when sober? Signs point to yes.

on envelopes

The two-envelopes problem, via xkcd. I'm not goint to re-hash it here, because it makes me dizzy, and the wikipedia page is fine. It's a 'paradox' of probability/game theory - like Monty Hall problem but without goats, cars, or a convincing solution.

There doesn't seem to be any dispute that if you can take a 50-50 bet that will either double or halve your money, then you should - your average profit is 25%. On the other hand, looking in your envelope doesn't give you any useful information, and switching without looking can't possibly earn you money in the long run. So what's going on?

Clearly the odds bet isn't 50-50! And the problem is that in order to fill the envelopes, we have to randomly pick an amount of money $X to put in the small envelope. But what does random mean? We can't have a uniform distribution over an infinite range - it doesn't make sense. You're trying to pick a number at random between 1 and infinity, whatever you pick will be in the bottom 1%.

Once we look in the envelope and see $Y, we know X = Y or X = Y/2. It's only a 50-50 bet if those possibilities are equally likely, and there aren't any probability distributions that have P(Y) = P(Y/2) for all Y.

So the game, as stated, can never be played. Any real randomizer for the game is going to give the player worse than 50-50 odds in some cases, and they won't make profit by switching in the long run.

that other election

So New Zealand just had our election. My take on the results: disappointing, not surprising, and not the end of the world.

The process was pretty uneventful compared to certain recent elections. Most of the difference can, I think, be attributed to a less polarised political culture; but part is the proportional representation system that prevents some votes being worth much more than others according to which part of the country you live in. The importance of a few 'swing' states (Ohio, Florida, et al) makes it tempting to try to tip the scales, and the rush of attention, money, and pressure to them provides the opportunity. If every vote is worth the same, election fever is spread out and doesn't boil over. (You like that metaphor? I mixed it myself).

mmp is broken

But we don't quite use proportional representation. New Zealand's system gives everyone a vote for a party, and a vote for a local MP. The party votes decide the proportional makeup of the parties in parliament. The electorate votes decide who your 'local MP' is. This is a carry-over from our old system where a parliament was made up of all the local MPs.

The upshot is that if the Banana party wins half the votes, they get 60 seats (out of 120) to fill with bananas. If Alice, Bob, and Carol from the Banana party all won local seats, then they have to be included. The other 57 seats are filled from a party list.

But there's a situation called 'overhang' that happens when a party wins too many electoral seats. Suppose the Banana party wins 65 electorates - more seats than their proportional fair share. What happens is 5 extra seats are created, so parliament has 125 instead of 120. All the electorate MPs are seated, and of course they don't get any list MPs. The proportionality has been distorted - the party with 50% of the vote got 52% of the seats.

This is pretty much impossible, there are only 70 electorate seats and to win that many without getting more than 50% of the vote isn't going to happen. But it can happen with smaller parties, in particular those with regional appeal. In our last election in 2005, we had an overhang for the first time. The Maori party won 4 electorate seats, but only had 3 seats worth of party vote.

and getting worse

This time around they got 5 electorate seats with basically the same party vote, creating an overhang of two seats. So far, so slightly odd. Now I'm going to do something very odd and assume both politicians and voters are rational.

Suppose you're a Maori party supporter. You know they tend to win more electorate seats than their party share. If you give them your party vote, you're not going to increase the number of seats they get, you're just going to decrease the size of their overhang by one. This increases their power a little (5/121 seats beats 5/122) but not much. Suppose they have a naturally allied party that doesn't have an overhang. You're better off giving that party your party vote, to better increase the size of your bloc and therefore the chances Maori make it into a coalition government.

Now suppose you're the Maori party. You realise this vote-splitting is better for you too, so you put out a TV ad directing supporters to vote that way. The Maori party vote drops to almost zero, they have the same number of seats, and more powerful allies.

If you're the allied party, you realise that running local candidates helps your profile a bit, but doesn't actually increase your share of parliamentary power. The Maori party winning electoral seats does help you, though, because it increases your voting bloc. So you stop running electoral candidates against them, and possibly encourage your popular local candidates to switch parties.

So given two allied parties, once one achieves an overhang, rational behavior leads to one becoming completely electorate-based, and one completely party-based. We have two single-seat parties in parliament at the moment, although both got enough party votes to 'cover' their electoral seat and avoid overhang. I think at least Jim Anderton may explicitly aim for an overhang next time, effectively running as an independent.

major parties?

Could a major party split in half to achieve the same thing? There are some practical problems:

• You can't give list seats to people if they lose their electorate, if they're not in the same party. You'd probably just want to split off the party's 'safe' seats.
• People will see it as cheating, and react negatively. A minor party has less choice, and more dedicated supporters, so can get away with gaming the system. A major party is likely to suffer blowback. However once one party succeeds with it, the others need to follow suit to be competitive.
• You'd lose some support just due to the confusion - we're assuming rational and informed voters, but not everyone's going to split their vote in the desired way. For example, they might vote for the 'electorate' party.

If a major party did this the rules would almost certainly be changed quickly. The temptation is great, though - splitting your supporters' electorate votes to an overhang party increases its power by around 57%, and not taking advantage of this leaves votes on the table.

solutions?

Wikipedia discusses forms of MMP without overhang seats. The only one that sounds unreasonable is a system used in some German states, where extra party seats are allocated to restore proportionality. Supposing a party won 0.1% of the vote and 1 electorate seat (which would no longer be strategically clever, but can easily happen). To restore proportionality you need to inflate parliament to 1000 members!

My personal opinion is that sacrificing proportionality to provide strong voices to particular groups is misguided. If parliament can protect against injustice to minorities in general, then it can protect against injustice to certain parts of the country. (And if a parliament doesn't protect minorities from mob rule, it needs to be redesigned). If the need for advocacy is there and local governments aren't strong enough, perhaps the electorate vote could be changed to non-voting advocates to parliament.