In a comment, you said:
Okay, then what about the living wage, the sum of the expenses that are absolutely necessary to meet one's basic needs? This can't be gamed, unless someone starts producing very cheap basic need products, which would be a win.
Why can't it be gamed? Consider a simple tax on groceries, rent, utilities, and clothing paid by the buyer. It cranks up the living wage, so the official makes more money for making people worse off. That's the opposite of what you want. And in general, you want the living wage to be low, so people have more excess over the living wage. This would encourage politicians to make the living wage high.
The average and median wage can be gamed by increasing the minimum wage so sharply as to ban many jobs. Sure, most everyone is unemployed, but the few people who still have jobs have a lot of money.
Per capita Gross Domestic Product (GDP) is calculated by the government, who then get more money when they calculate a high value.
In general, averages mean that giving more money to rich people is as effective as giving more money to poor people.
Medians reward giving money to middle class people at the expense of either the poor or the rich. And of course, as noted in a comment, medians incent politicians to encourage the poor to emigrate.
Others have already explained why tying it to the minimum wage could be counter-productive: minimum wages in general distort the economy. They are bad tools for their goal. Further, a high minimum wage may prevent people from getting entry level jobs. No entry level job means no promotion for experience. The best way to keep wages up is not to ban bad jobs but to encourage good jobs. If everyone can get a good, high-paying job, there would be no reason to take low-paying jobs unless they are actually better in some way.
The best way to keep government wages down is to vote for people who will pay themselves reasonably. If you don't have that, then no system will prevent them from paying themselves. They'll just change the system.