Like the Macedonians and the Persians, the Carthaginians are strictly an Iron Age civilization since almost all of their bonuses apply during that age. As a whole, the Carthaginians somewhat struggle in the beginning of the game compared to other civilizations so they are definitely not recommended in short games especially in Random Map. If played in Deathmatches, the tide of the game changes since they rely on large amounts of resources to field an army.
The Carthaginians are suitable on land but are best used at water maps since their fast Transport speed and improved attack for Fire Galleys packs a powerful punch to civilizations with no bonuses on naval units. Packing a whole horde of Armored Elephants and Helepoli into a Heavy Transport escorted by countless Fire Galleys will make the opposing civilization gone for good. Since it takes effort to gather all these resources to supply these units with bonuses, the Carthaginians are best played in Deathmatches where large resource output are already provided in the beginning.
Since most of the Carthaginian army consists of War Elephants, Centurions and Helepoli, the total cost to train and build these units could overwhelm the player's economy if played in short games. A failed attack can be devastasting to the player since it takes many resources to field that much of an army. In addition, the Carthaginians lacks economic bonuses to keep up with high demand of resource production. Simply put, the Carthaginians will lose to those with a quick and powerful economy in short, early games.