That question is very difficult to answer, because there are many game species worldwide that we like to hunt, believing them to be native, but in reality they were introduced at some point, sometimes a very long time ago.
I don't have a favorite transplanted game species which is accepted as native. The game species I have hunted at home and worldwide was all native, but some of what is considered native here in Europe, is it not. The majority of the hunters believe this because the game is present in the areas for ages. However, the fallow deer is native from Asia and was perhaps introduced first by the Romans, it remains unclear whether the species existed in Europe prior to that, and the European mouflon is a domesticated sheep from the Near East and Central Asia, brought to Europe by Anatolian farmers thousands of years ago. Even the red deer, while native to Europe, did not establish itself unaided by human people in many parts of Western Europe. Some of the deer populations on the Scottish islands for example, were also introduced by humans from the continent, probably during the Stone Age.
