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Prestige is the standing of your ruling character in the world and it determines your winning rank. Prestige is accumulated by your characters during their lifetime through titles, events, ambitions and war. Upon death of your character, prestige is added to your score. Prestige is very important, and here is why:
Your diplomatic requests will be answered more favorably if you have more prestige.
You are an attractive husband (wife) to most wanna-be your spouses if you have more prestige.
Losing a war or making certain decisions costs prestige, so the more the better. It's like a currency.
Each 100 of prestige equals +1 opinion towards you by all of your vassals. (+20 max) This is very needed!
1000 Prestige allows you to fulfill the Exalted among Men ambition, granting a +1 permanent bonus to your diplomacy stat.
Claiming someone's title costs prestige, so this is very important for your expansion wars. Prestige is added to your dynasty score.
How to get more prestige
Marrying to someone or marrying your daughter to someone who has higher title than you earns prestige.
Winning a war, be it suppressing your independent vassal or full scale conflict, earns you lots of prestige. Equally, winning battles also gains you prestige, and lots of major battles can add up to a prestige gain (or loss!) equal to some wars!
Each title you hold, be it count, duke etc, gets you monthly prestige increase. The higher the title the better. (0.02 per barony; 0.10 per county; 0.20 per duchy; 0.80 per kingdom 1.60 per empire) Creating a title or Usurping a title also costs money and rewards prestige.
Some events and decisions, or certain ambitions will reward you prestige for your actions. Going on a Grand Hunt grants prestige.
Character who is on a council, earns some prestige each month. Honorary titles earn (or cost in the case of Court Jester) a very small amount of prestige.
Getting a trait "Proud" gets you 0.5 monthly prestige increase, "Scarred" 0.10 monthly prestige increase