Except in many instances when government rigs the game with crony capitalism, income is mostly a result of one's productivity and the value that people place on that productivity. Far more important than income inequality is productivity inequality. That suggests that if there's anything to be done about income inequality, we should focus on how to give people greater capacity to serve their fellow man, namely raise their productivity.
To accomplish that goal, let's look at a few things that we shouldn't do. Becoming a taxicab owner-operator lies within the grasp of many, but in New York City, one must be able to get a license (medallion), which costs $700,000. There are hundreds of examples of government restrictions that reduce opportunity. What about the grossly fraudulent education received by so many minority youngsters? And then we handicap them further with laws that mandate that businesses pay them wages that exceed their productivity, which denies them on-the-job training.
Think back to my poker example. If one is concerned about the game's result, which is more just, taking some of Tom's winnings and redistributing them to Dick and Harry or teaching Dick and Harry how to play better? If left to politicians, they'd prefer redistribution. That way, they could get their hands on some of Tom's winnings. That's far more rewarding to them than raising Dick's and Harry's productivity.