What should an employee do when their manager does not assign any tasks to him/her, although being aware of employee's idle state and dissatisfaction? Can the employee freely entertain himself, given the fact that he has completed all assignments?
If the employee wants to keep his/her job, I would suggest:
Looking round for tasks that need doing which you would like to do.
Suggesting to your boss that, as you've finished all your current work, you'd like to go ahead and do this other task, and asking if that's ok with them.
This is a boiled-down version of the 'assertive statement' approach to respectfully questioning authority from the discipline of 'crew resource management' (designed for aircraft cockpit crews but also extremely useful in other teamwork contexts).
The full version goes like this:
Opening or attention getter - Address the individual. "Hey Chief," or "Captain Smith," or "Bob," or however the name or title that will get the person's attention.
State your concern - Express your analysis of the situation in a direct manner while owning your emotions about it. "I'm concerned that we may not have enough fuel to fly around this storm system," or "I'm worried that the roof might collapse."
State the problem as you see it - "We're showing only 40 minutes of fuel left," or "This building has a lightweight steel truss roof, and we may have fire extension into the roof structure."
State a solution - "Let's divert to another airport and refuel," or "I think we should pull some tiles and take a look with the thermal imaging camera before we commit crews inside."
Obtain agreement (or buy-in) - "Does that sound good to you, Captain?"
I've put the most important bit in bold.
What kind of entertainment do you have in mind?
Telling the boss that you are ready, willing and able to help other employees who could use the help and helping these employees is a perfectly acceptable kind of entertainment.
Asking the boss for a list of lower priority or long delayed projects that you could work on and that could put the firm on a sounder footing. and working on them is a perfectly acceptable kind of entertainment.
Going through the list of to-do's, identifying skills you might need to acquire to do these tasks and acquiring these skills is a perfectly acceptable kind of entertainment.
Keep in mind that the general rule that applies for whatever else you choose to do during your downtime, inactivity results in unemployment.
Why are you asking "Can the employee freely entertain himself, given the fact that he has completed all assignments?" on this site? Why are you not asking YOUR boss that question? If I were your boss, what do you think I would/should do to you if you asked me that question?