What keeps minimum wage rates in balance? I would posit that the minimum wage is determined more on a political than an economic basis.  The fact is that there are more wage earners than business owners and, generally speaking, the wage earners prefer an increase while the business owners prefer a minimum wage decrease (perhaps even to 0).  There are more wage earners to vote than business owners, and this makes it politically expedient, on the balance, to vote for a minimum wage increase.

First, is this true?

Second, if you find it to be true, why doesn't the minimum wage increase substantially more than it has over the years (focusing on the US)?  What is the counter balancing force that causes minimum wages to seek equilibrium at some something between "poverty" and "living wage" level (again, in the US)?  Or is even that assumption false?
How is the minimum wage determined?

Yes and no. It is a political choice, but with economic reasoning. In the end, that's the case for every economic rule, maybe with the exception of monetary rules, as we try to preserve independence for central bankers (but it isn't completely true for the US FEDs).

When setting the new minimum wage, you compare minimum wages with other states, and with your own (earlier) state, to get a feeling what's possible and what is not. In the end, the minimum wage is a trade-off between raising earnings for some parts of the population, and decreasing employment chance for others. 

How does the minimum wage affect employment?

The degree of trade-off (if you will, the transition rate between the two) is unclear. In the data, we tend to see no employment effect of minimum wages, but an increase in earnings (Dube et al (2010): (Ungated version). In fact, we see a large number of estimates around 0:



Many economists suspect (based on a strong prior that there should be an employment effect) that this is due to poor data (Matt Rognlie's comment), or long-term effects of minimum-wages (adjustment costs in technology) that we are not capable of measuring/estimating precisely (Sorkin (2015) (IDEAS)).

Why doesn't the minimum wage affect employment?

The counter-effect to some part is the political process. People are not stupid, they will accept raises in the minimum wage to some degree, but not if you go bananas. Also, there are natural counter-forces in the economy: 


minimum wages are nominal
minimum wages are lower bounds


To that end, if you increase the minimum wage, eventually inflation will wash off the real impact. Also, raises in the minimum wage tend to become substitutes for (non-forced) increases in the wage level.

Have a look at the following graph, which depicts the relative impact of minimum wages. The author plots the maximum of minimum wages at a time period, deflated by average hourly wage in the private sector among production and non-supervisory employees (no managers). It really looks as if any spike (raise in the minimum wages) gets deflated quite quickly, doesn't it?