National governments are some of the most powerful entities in the world today. But what are their missions (or what should they be)? What are they optimizing for? Democratic governments at least are supposed to represent the interests of the voting public, but this just means that the voters get to choose between one of several parties, each of which have their own mission.
The mission of parties on the right of the political spectrum seems to be maximization of GDP (or GDP per capita). On the left, things are a bit more complicated. The labour party's ideology is roughly social democracy, but what does that actually mean? Bouncing around on Wikipedia for a bit suggests that this ideology promotes social justice, of which the key tenet seems to be reducing inequality. So if we simplify the right down to "GDP maximization" then the corresponding simplification on the left is "equality maximization". Obviously these are gross caricatures of infinitely more nuanced policy sets, but bear with me.
Suppose we are considering two independent policy changes, A and B. Change A makes the poorest person in the country poorer by £1 and makes the richest person in the country richer by £2 (a net GDP increase of £1). Change B makes the poorest person in the country richer by £1 and makes the richest person in the country richer by £2 (a net GDP increase of £3). Both of these changes increase inequality. All else being equal, it seems that conservative policy would be to support both changes (since they both increase GDP) and labour policy would be to reject both (since they both increase inequality). I think that change A is bad and change B is good, and I think a lot of people feel the same way. I would prefer maximizing a metric which is increased by change B but decreased by change A.
What's an example of such a metric? Well, one example would be "how rich is the poorest person in the country?" It seems like quite a dumb metric at first - how can you judge the performance of a entire country on the outcome for a single person? But the more I've thought about it, the more sense it seems to make. The poorest are almost by definition those who need the most effort expended to help them. It doesn't take very much money to stop that person being the poorest, at which point you switch to the new poorest person. Then you need to raise up the poorest two people to the level of the initially third-poorest and so on.
Where do you get the money to help these poor people? Well, you can take it from the rich - the "poorest person" metric doesn't care about them one way or another to a first approximation (and rightly so - the rich don't need help from the government, they can help themselves). Now, the logical extrapolation of that is that we should take all the money and redistribute it evenly - give everybody N/M where N is the total amount of wealth and M is the number of people. That is a way to make a society that is extremely egalitarian but utterly impoverished. Without any inequality at all, there is practically no incentive for people to work hard to create wealth (if you did you'd only get an infinitesimal 1/M of it). Whenever this kind of thing has been tried in the past, that is the result.
So distributing all the money evenly does not maximize the wealth of the poorest - we can improve things for them even more by allowing some inequality and redistributing some of the wealth (but leaving enough to create incentives). So applying this metric does enforce some compromise between the capitalist and socialist sides.
As written above, the anti-poverty metric is very underspecified. Over what timescale should you maximize the wealth of the poorest? If you choose a very short timescale then that implies that you should just redistribute all the wealth evenly very soon (which is very bad in the longer term) and if you choose a very long timescale then that implies that you should aim to maximize GDP and then redistribute all the wealth evenly at some very far off point in the future (which doesn't actually do anything to help people that are hungry now). Perhaps the difference between left and right politics is that they really want the same things, but just disagree on the timescales that should be involved. As for me, I suspect the ideal timescale would be something between that of a typical political term and that of a typical human lifespan, but the exact length may depend on the specific policy under consideration.
There's more that governments do more than just the redistribution of wealth (or lack thereof). How can the other functions that we'd like governments to do be justified under this metric? Well, we can do so by taking proper care about how "wealth" and "poorest" are defined. There's more to wealth than just money. Somebody who is in need of medical treatment is (all else being equal) "poorer" than somebody who isn't, so governments should provide healthcare. Somebody who is a victim of crime (or who is living in fear of becoming one due to crime going unsolved) is also impoverished. Even a certain amount of military expenditure can be justified on the grounds that getting invaded by another country would be impoverishing. The more things we include in our definition of "wealth", the more government intervention is justified by the metric.