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Civ V Greece Guide: Greek Civilization Basics
The Greek civilization, with its leader, Alexander, comes with two unique units instead of a unique building and has a leader trait that synergizes extremely well with city-state focused play. Alexander's trait, Hellenic League, decreases the rate at which influence with city-states drops and increases the rate at which it recovers. This, combined with the patronage policy tree, can be very nearly gamebreaking, allowing Alexander to have twice as many city-state allies as would otherwise be possible.
In addition to this, Alexander comes with two classical era unique units: the Hoplite, and the Companion Cavalry. These units are more powerful versions of the Spearman and the Horseman, with significant bonuses that make early-game dominance a practical guarantee. Ordinary civilization strategy often involves an early game rush of these units, making Greece even more powerful. Because of Alexander's synergy with the new city-state system and his powerful early-game presence, this is often a good civilization for beginners to whet their teeth on.
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Alexander's Unique Units
The Hoplite replaces the Spearman, which is available fairly early on from Bronze Working, and has two more strength than its regular counterpart. 9 strength makes the Spearman just slightly worse than the Pikeman, which is a medieval era units and much farther down the tech path. The Hoplite is excellent for facing just about every classical era unit and very powerful when attacking cities, essential to Greek civilization strategy.
The Companion Cavalry replaces the Horseman, which is available early on as well from Horseback Riding and is one of the most powerful combat units, even into the medieval era. The Companion Cavalry has two more strength, one more movement, and increased Great General production. Two extra strength puts it at double that of Spearman, negating their bonus versus mounted units. This, combined with the Great General that will likely result from their use, allows Greece to have an almost undefeatable classical-era push if timed correctly.
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Civ V Greece Guide - Hellenic League, Diplomatic Victory and You
Hellenic League is deceptively powerful and possibly the best leader trait in the entire game for cultural, diplomatic and tech victories (there are better warmonger traits, but Hellenic League is excellent even for that). For early game civilization strategy, you want to focus on making friends with as many Maritime city-states as possible, to fuel growth of your cities and allow your citizens to work production and gold tiles, or become specialists. With the right city-states and Hellenic League, Alexander can found successful cities on the polar ice cap or in the middle of the desert. While other civilizations can follow this strategy, Alexander can do it the best. If you're going for a cultural victory, keep your empire small and befriend as many cultural city-states as you can afford, as well as militaristic ones to help you defend while your cities build culture buildings.
While Alexander works well with all victory types, he excels at diplomatic victory. City-states that are your allies will vote for you in the UN elections, so beeline for the United Nations and save as much gold as possible. When the vote is coming up (if you don't already have enough city-state allies), splurge and buy as many alliances as you need to win (with the Patronage tree, this should be apallingly easy). Whether you have a sprawling empire or a small, focused one, this should be achievable well beofe your AI opponents are even in competition with you, if you play intelligently.