When Gradle says "program type already present"

Today I helped someone whose Android project was failing to build with a "program type already present" error about the dependencies. I found a Stack Overflow answer that seems to say that the problem is caused by two dependencies needing different versions of the same sub-dependency. To figure out which dependencies are conflicting, you can look at the output of running gradlew app:dependencies (e.g. in the Android Studio terminal). It might be possible to exclude the sub-dependency from the dependency requesting the older version using the Gradle exclude module directive in that SO answer.