Sony isn't interested in directly offering backward compatibility so PlayStation 4 gamers can play their PlayStation 3 titles on the new generation console. However, Sony wants gamers to check out the PlayStation Now streaming subscription service that provides access to hundreds of PS3 games.
There are some PS4 gamers and media members that think Sony should investigate rolling out a backward compatibility feature for the PS4 - just to make gamers happy. Sony has worked hard to create a good gaming ecosystem, trying to keep sometimes fickle fans happy.
If it doesn't offer backward compatibility, Sony runs the risk of leaving gamers disappointed, though with other methods to play PS3 titles - for free or paid rental - I think the Japanese electronics company has done enough already.
Regardless of what PlayStation gamers want, it seems unlikely that Sony will try to follow in Microsoft's footsteps anytime soon.