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THIS INFO IS PRE-1.3 AND AS SUCH IS NOT ACCURATE FOR THE CURRENT VERSION!
Paradox Mages are powerful spellcasters who manipulate the cosmic forces of the universe to bewilder, disable and annihilate their opponents. While their spells may sometimes be unstable or dangerous to use, they are capable of both awesome destruction and debilitating effects, from stopping foes in their tracks to erasing them from reality.
Unlike Archmages, whose skills lean to a strategy of shielding up and raining all manner of elemental death on opponents, the playstyle of Paradox Mages is more standoffish and centred around disabling enemies before going in for the kill. You have a limited number of damaging spells, most of which have long cooldowns or high Paradox costs (or both!) making varying your attacks a necessity. Additionally, Paradox Mages lack the Archmage's powerful shielding skills and numerous escape options (though your main one, Dimensional Step, is quite good). So, what can you do?
The Paradox Mage's specialty is controlling the battlefield. Your skills can confuse whole groups of enemies, slow or stun them for extended periods, teleport surrounding enemies away, push them away or pull them towards you (or toward another point entirely), pin them to the ground, remove them from time entirely for several turns, dramatically lower their resistances as you attempt to wipe them from existence, and even put their skills on cooldown. You can quickly refresh your own skills, speed yourself up, raise your resistances and saves, split the timeline to scout ahead, heal and remove physical ailments, and even create a clone of yourself to fight alongside you.
Archmages look almost boring in comparison.
Essentially, playing as a Paradox Mage involves tactical use of disables and hammering enemies with attack spells, using your support skills to stay in the fight and keep the pressure on, all whilst keeping a good eye on your Paradox to make sure you don't blow yourself up. Speaking of which...
Paradox and You
Paradox Mages use the Paradox resource, which represents how much damage has been caused to spacetime. Unlike Mana, which is drained by spells and recharges slowly or with skills, Paradox starts out at zero and is raised by using Chronomancy spells. As Paradox increases, the damage and effectiveness of your skills rises with it - but so does the chance that your spells may fail to work, backfire onto yourself, or cause one of several random anomalies, all of which can be devastating in the middle of a pitched battle. Needless to say, keeping your Paradox at a manageable level is a crucial part of playing the class. Your Willpower stat will allow you to raise your Paradox higher with a lower failure chance.
A few skills can lower Paradox, the main two being Spacetime Tuning (which can take many turns, precluding its use in a fight) and Static History (which is less effective but instant, and also prevents spell failure for a short time after use).
You're best off keeping your Paradox as low as possible. Sustained effects will create a minimum level, but this will still be quite low (you only have three of them, and two together still won't raise your Paradox high enough for any spell failures even with low Willpower). The higher damage and effectiveness is not worth an attack blowing up in your face, or your Dimensional Step failing at a critical moment, which will probably mean your death. Never take an eye off your levels, use Static History as often as you can, and you will cut down the inevitable screwups a great deal. It is also helpful to put spacetime tuning on autocast (right click on talent.)