It has been proven that autonomy leads to greater productivity. If you want engagement from employees, autonomy is your friend.
The problem is that managers can't simply export methods of granting autonomy from one firm to another and expect them to work. Autonomy is perceived differently within individual organizations. Company culture has to support autonomy and show the results of how it benefits the entire team, otherwise it is viewed as disorganized chaos. Daniel Pink outlines a great example of how culture supports autonomy at an Australian software company Atlassian below:
Once a quarter, the managers at Atlassian tell the employees that they have 24 hours to work on whatever they'd like, however they'd like and with whoever they want to. All that is asked of them is that the results of their work be presented to the entire company at the end of the 24 hours. That one day of autonomy usually leads to hundreds of new product ideas that never emerged before and hundreds of bug fixes for their existing software.
All this is accomplished in one day, and without any monetary incentives. The idea here is that if you give people the freedom to work on what they want, amazing things will emerge. However, the culture in the company has to support autonomy. In other words, the value of the autonomy needs to be clear to the entire team. In the case of Atlassian the employees could see the outcome of their efforts and their benefit to the company at the end of 24 hours when they presented their work.
Give autonomy to your employees, support it, and watch amazing stuff emerge.