Did Judas betray Jesus willfully? Luke 22:3 (NIV) Then Satan entered Judas, called Iscariot, one of the Twelve.


Does this verse imply that 


Judas had no more control over his own actions as he was possessed by the devil.    
Judas yielded himself to the temptation of 30 pieces of silver offered by the priests for betraying Jesus.


The first option would mean that Judas was possessed and hence had no control over his own will whereas the second option would mean that the betrayal was a self-willed one.
Just because it was already prophesied, does not mean that Judas did not make a conscious choice. A man can most definitely choose to be used by Satan or by the Holy Spirit. Judas, by his actions, was already choosing, as he was choosing to steal from the purse.(John 12:6) This showed that even though he was with Jesus, the Holy Spirit was not working within him. He made a series of bad decisions and the devil took the opportunity.

God is blamed for many things, but just because He allows it to happen, does not mean He causes it to happen. Many times we put ourselves in this position. Isaiah 59:1-2 says:

Surely the arm of the Lord is not too short to save,
    nor his ear too dull to hear.
But your iniquities have separated
    you from your God;
your sins have hidden his face from you,
    so that he will not hear.


Judas put himself in this position. In a position where God allowed Satan use Judas.