I think what Anderson does best is that he combines close control and strength like few other players. That's pretty simple and not too exciting to post but it's hugely important.
Even though his decision-making does need to improve significantly for him to realize his potential, you can see that he has great potential precisely because his tactical instincts are very good. Take the move he started off against Arsenal last week -- the one early in the match, which went Anderson-Berbatov-Ronaldo-Rooney and ended with Rooney skying it over the bar. It wasn't complicated -- just a dribble into space and a simple pass forward to Berbatov -- but it was a good example, I think, of the simple, fundamental football he is capable of playing -- the kind of football that is the most simple and therefore the most beautiful (to kind of paraphrase Cruyff). When he is making good passes, they rarely surprise but at the same time often impress me and give rise to admiration, if you understand the distinction I'm making.
When I look carefully at his particular talents, while I see clearly that he is quite a good footballer, he doesn't seem to stand out so far beyond other good footballers. What sets him apart and takes him to the next level is his competitiveness or whatever you want to call it -- he just has the air of a winner. Part of that is his energy and stamina -- he's physically capable of playing on all cylinders -- but the most important part is attitudinal.
In a game where offensive and defensive and playmaking and attacking roles are becoming more blurred, where interplay and interchange are becoming paramount, Anderson often looks to me like the perfect 21st-century midfielder.
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About comparing him to other players, he's not like today's Steven Gerrard, because outside of attitude what makes Gerrard special is that he's got a great shot. And there's no evidence right now that Anderson has that. I think Anderson is more like Essien than he is like any other midfielder currently playing. But I don't think he's really that much like Essien.
I understand why people compare him to Cesc, because they're about the same age and they play for top 4 teams in the same league, etc. But it's kind of a daft comparison. They're different players with different talents and to compare them as like players is to do both a disservice, like it would be to compare, say, Drogba and Del Piero. What they do share, I think, is that competitive quality that makes them both winners. I can't think of a much better central midfield (again, for the 21st century) than Anderson and Cesc together.