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In Elite Dangerous players can change their Reputation with the three major factions (Federation, Empire, Alliance) to either Allied, Friendly, Cordial, Neutral, Unfriendly, or Hostile. All players start out as Neutral with all three major factions and by doing missions or other influence increasing activities can improve their reputation with that faction. When a player performs activities that would decrease a faction's influence (e.g. pirating or failing missions) the player's reputation with that faction will decrease. When a player has a negative (current value unknown) reputation with a faction they will be attacked on sight by NPCs that represent that faction.
Local Factions and Reputation Edit
Who's in control of a system has a major bearing on how your actions affect your reputation with them and the other minor factions. There are usually several minor factions vying for control in each system, but outside the immediate local space around their starports and outposts, only one can claim to be "in control" of a system. This is displayed in the system view when you've no stellar body selected.
Normally it's the faction with the highest influence in the system, except in special situations like civil war.
This is important for criminal law, because each faction only cares about crimes committed in its own jurisdiction. It doesn't matter if you are an infamous pirate according to say, the "Bangwa Freedom Party", if you are in the jurisdiction of the "Bagwa Independents" and you've committed no crimes against them, you are an innocent pilot.
In addition, most ships are aligned with a minor faction (you can see which one by performing a basic scan).
So if a ship is wanted in the current jurisdiction, but is aligned with a different minor faction for which it has no criminal record, then destroying it will cause you to lose reputation with the faction it is aligned with; in their opinion, you've just destroyed an innocent citizen of theirs.
Gaining and Losing Reputation Edit
Your reputation goes up whenever you do good things for the faction. Completing missions, join combat zones on their side when they're at war. If they control a station, then doing anything profitable at that station gives you a reputation boost: selling cargo for a profit, handing in bounty vouchers, even handing in exploration data, will all give you a rep boost.
Negative rep is doing the opposite. Failing or abandoning their missions, selling trade goods at a loss, blowing up their ships or (if they own a station) Authority ships, that sort of thing.
Reputation Decay Edit
Reputation decay only applies to the Major factions, Alliance, Federation & Empire.
When your status with a major faction is above 75% Friendly, it will slowly decay down to 75%, by a percentage point every couple of days. Reputation decay happens whether you're logged into the game or not. To stay Allied you have to keep doing good deeds that gain you reputation for the major faction.
There's also anti-decay for negative reputation. When your reputation with a super power is below neutral but under 25% it will gradually climbs up to 25%. To clear negative reputation with a major faction you must do the following: (1) If your status with a major faction is at Hostile, you must wait for the Reputation decay mechanic to bring it back up to unfriendly (2) Once your status with a major faction is a at Unfriendly, you need to complete missions and do other positive reputation actions until you are back to Neutral Status.
There is no reputation decay for minor factions. 
- Reputation is different than Rank