Step 1: Run the game as Administrator

- Right-click the game in your Steam Library.
- Go to Properties, then to the Local Files tab.
- Click Browse Local Files.
- Locate the game executable (the application file).
- Right-click it and go to Properties.
- Click the Compatibility tab.
- Check the "Run this program as an administrator" box.
- Click Apply. Having done this, restart Steam and try launching the game again.

Step 2: Run Steam as Administrator

- Navigate your Steam installation file (By default, it is C:\Program Files (x86)\Steam).
- Right-click the Steam executable.
- Follow the above steps to set it to Run as Administrator.
- Restart Steam.

Step 3: Verify Integrity of Game Cache

- Now that Steam is running as Admin, it should have no permission issues.
- Go to your Steam LIBRARY, right-click Dying Light, select Properties, open the Local Files tab and select VERIFY INTEGRITY OF GAME CACHE

If the above steps do not help, create a new Windows User account and see if you can install the game properly, please.