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Let me use bullet points.
*First of all, he, like Ancelotti before him, realised Chelsea needed a creative CENTRAL midfielder. So he (supposedly, although I believe it was actually Emenalo) went after Modric and Pastore. This was good. However, when it became increasingly obvious that he was not going to get them he should have gone after other CREATIVE players rather than settling with Raul Meireles (absolutely anyone would have been ok) And when even that failed he could still have utilised Josh or Mikel (One of Chelsea's best passers for the last 5 seasons).
*He should not have attempted to change Chelsea's style of play so suddenly and so dramatically. Especially since he did not have the personnel for the change to be successful.
*The argument that Chelsea did not have money or were not willing to buy players is rubbish. Chelsea spent £74.5 million this season on players. Yes, £74.5 MILLION! Chelsea bought the likes Juan Mata (£23.5m), Romelu LUKAKU (£18m) and Meireles (£5m) over the summer, added with the captures of Gary Cahill (£7m) and Kevin De Bruyne (£7m) in January. Not to mention the signings of Romeu, Davila, Courtois, Islam Feruz, Bamford, Kenneth Omeruo, etc etc. Sadly, only Mata is good enough to start for a team with Champions League aspirations. While I love the fact that Chelsea is buying youth, we should have strengthened the first team with starters like our rivals at Man City, Man Utd, Arsenal, Tottenham, Liverpool and even Newcastle (Demba Ba). AVB was extremely naive/arrogant in believing that just Mata was enough. While Mata is clearly a fantastic player, he is not enough.
*AVB's handling of the Alex and Anelka situation was very poor. For someone that was touted as an excellent man manager he went about this the wrong way. The most foolish part was banning the players from the dressing room. Did he think the other players would be happy about his treatment of their colleagues? Colleagues that helped the team win a double not that long ago? I think not. Morale will inevitably have been damaged by this move.
*His insistence on playing Torres when Drogba was not around. He should have tried something different when it was obvious that Torres was not going to score. He could have played Sturridge, LUKAKU or even Feruz/Piazon/etc (and Anelka, if he hadn't got rid of him through incompetence...). He SHOULD HAVE TRIED something different. Anything. Heck, even if it meant playing Lampard/Mata as false nines or something.
*His funny decisions not to try more fringe players. When a player is not in form for some reason or the other (be it due to fitness or mental fragility etc) AVB SHOULD NOT HAVE continued using them. Arsenal have used over 30 players in the Premier League this season! While this is not necessarily a good thing, one only needs to look at the league table to see which team was wiser. Players like Bertrand, Kalou, Mikel, LUKAKU, Josh(in-a-basket), Ferreira and Anelka should have been given more of a chance especially when some of the people playing ahead of them were so poor Chelsea may as well not have been playing anyone in that position.
*Not getting the press on his side. By trying to alienate the English media, AVB created several issues for himself. I suppose he was trying to follow Mourinho's "siege mentality" us-against-the-world policy. However, that only works if the team is actually performing well. This resulted in the media constantly piling pressure on both AVB and the players. Stories of player dissatisfaction kept cropping up in the media. One thing Mourinho is good at is making sure he receives all the flak for his teams. AVB failed at protecting his players from the press. Therefore, players were deliberately misquoted, everything Chelsea did was spun negatively, David Luiz was constantly crucified, etc.
*Inability to change a game with substitution or tactics. Apart from the first Man Utd match and QPR (which we lost, BTW), I can't think of a single game that Villas Boas positively changed when Chelsea was losing. (I should probably check statistics but it's quite late and I really can't be assed at this point)
*Insert your own complaints here.
(This is my first fan post. Hurray!)