Apple - Third party application cannot write to disk

The logs are indeed getting created. I executed the app on a Mac running macOS High Sierra 10.13.4 (17E199). I have verified the log creation by precisely following the steps documented on the linked download web-page. Three important instructions to take note of from the linked web-page:

1. If logs are not being generated you are probably running a version of OS X that sandboxes untrusted applications. This prevents the simulator writing to disk. To resolve this, move the executable within the Applications folder.

2. This simulator can create log data in the donkey tub format. This is stored in the log dir at the root of the folder next to the executable. On the mac you will need to browse the package contents to see this folder. If this folder is missing, no data will be recorded.

3. Hit Auto Drive w Rec button.

Now, follow the steps as described below:

1. Double click the downloaded DonkeySimMac.zip file. It will extract DonkeySimMac/donkey_sim.app app bundle.

2. Move/copy donkey_sim.app to /Applications directory.

3. Run donkey_sim.app by right clicking (please don't double click) on the app icon. Upon first launch (only), you'll be presented with a dialog box stating

   "donkey_sim.app" is an application downloaded from the Internet. Are you sure you want to open it?
   

   Click Open to launch the app.

4. Select your preferred Graphics options and hit Play!.

5. Start scene of your choice by double clicking.

6. Hit Auto Drive w Rec button.

7. You'll be able to see a running counter against the label Log in bottom left corner of the app window. This verifies that the logs are getting generated.

8. Once you are done, click Stop → Exit → Quit.

9. Navigate to /Applications directory.

10. Right click on donkey_sim.app and select Show Package Contents.

11. Double click to open log directory.

12. Voila! Your logs are here.

Observation: Log generation only works when either Joystick/Keryboard w Rec or Auto Drive w Rec option is selected. (Yes, keyboard is mistyped as Keryboard!).

Did you try creating a new folder in the Applications folder (i.e. "Donkey Simulator") and then moving the application inside of that folder?

From reading the Mac install instructions, once you unzip the downloaded archive file you have a folder you then move to (the root level) /Applications folder. However, if your account is a standard user account then you will not have write access to the /Applications folder or any of its subfolders. You can open the Get Info window for the /Applications folder and add your user account and give it Read & Write access then do the same operation on the Donkey Simulator folder you placed in /Applications, but add the additional step of clicking the gear icon pop down and select "Apply to enclosed items..." so that your read & write access is applied to the logs subfolder as well. This should solve the issue.