Rig = a ratio between hours of pay credit vs time spent at work.
Duty Rig = the ratio of hours paid to hours on duty.
Example: Minimum of 1 hour pay credit for each 2 hours on duty. 1:2
Trip Rig = the ratio of hours paid to hours away from base in a pairing.
Example: Minimum of 1 hour pay for each 3.5 hours away from base on a given assignment. 1:3.5
Minimum Day = ratio of hours paid for each calendar day at work irrespective of actual flying.
Example: Min guarateee = 5.0 hours pay credit for each day on any duty.
Note: Rigs can be calculated in a wide variety of ways, e.g., pay the greater of on a daily basis; look back over an entire bid period to achieve an "average" rig effect (not the best for pilots), etc. It depends on how your contract is written.
Example: A) With a "look back" minimum day of 4.0, one might fly 2 hours on one day, 4 hours on another day, 8 hours on a third day. At the end of the month, each day on duty must pay at least 4.0 hours on the average. B) The same 4.0 minimum day calculated on a "look forward" basis, would credit four hours for each day, plus credit actual hours on days with more than 4.0.
In "A" (look back) over a 30 day bid period with 20 days on duty (10 days of 4 hours flying each and 10 days of 3 hours flying each) The min day would pay a max credit of 80 hours for the month.
The same schedule with a "look forward" would pay 10 3-hour days riged to = 40 hours, plus 10 4-hour days totaling 40 hours. 40 + 40 = monthly pay of 80 hours credit for pay. Any day during that period that the pilot flew more than 4 hours actual would result in higher pay for the month.
The duty rig guarantees, for each duty period, that the pay credit will never be less (1:2) than 1 hours pay for each two hours on duty. If the actual ratio turns out to be 1.5:2, the pilot is paid the actual which is higher than the "rig".
Usually the "trip rig" never applies unless the ratio of pay credits to hours away from base is greater than both the "duty rig' and the "min day", individiually.
Example- You are scheduled for a 3-day trip (58 hours TAFB). 1st day you are on duty for 12 hours and you fly 7 hours actual. Actual is higher than the duty rig and you are credited with 7 hours. 2nd day you are on duty for 12 hours but you only fly 2 hours actual. Duty rig pays 6 hours 3rd day you are on duty for 4 hours but you only fly 1 hour actual home. Duty rig is higher and pays 2.0. Your pay for the trip is as follows.
Without the rigs you get paid 7.0 + 2.0 + 1.0 = 10 hours actual.
With the Duty Rig only you get: Day 1 = 7.0 hours (actual), Day 2 = 6.0 hours (rig), Day 3 = 2.0 (rig) . Total for the trip = 7 + 6 + 2 = 15 hours credit due to the duty rig.
Min Day - Day 1 = 7.0 (actual), Day 2 = 4.0. Day 3 = 4.0. Total for trip 15.0
Total TAFB for the 3 days was Day 1/24 + Day 2/24 + Day 3/10 = 58 hours. The trip rig pays 1 hour for each 3.5 hours. 58/3.5 = 16.57 hours credit for the trip.
In this example you are paid by the trip rig, which is the greater of the other 3. This trip is "unproductive" as you spend most of your time sitting in the hotel doing nothing on days 2 & 3. On day 2 and 3 the company should have scheduled you for no less than 5 hours actual flying each. Had they done so, none of the rigs would apply.
As others have stated, a combination of all 3 is usually required to produce "productive" scheduling. If you can only achieve 1 of the 3 rigs contractually, go for the min day & make sure its a look forward and not less than 4.0 hours.
The company retains the flexibility to schedule as it chooses but pays a financial penalty if it does not schedule efficiently. The pilot is guaranteed a minimum compensation for each "schedule" produced by the company.
The concept is pretty simple. If I am available to the company for work, it is the company's responsibility to use me effectively. Whether they do or not, I am guaranteed a minimum compensation for each time period of my availability.
Rigs level the palying field and make "flying" compatible with other types of work. Eight hours on the job as and IT will result in 8 hours pay. It doesn't matter if you actually fix computers for 8 hours or only for 4 hours each day. The company decides what work you will do.
Since pilots wages are based on "flight hours" only, without a Rig, the pilot is paid nothing at all even though he is "at work" and available to fly. That's not reasonable. Rigs iron out the difference.
Companies resist rigs because they would like you to be available 24/7 but to only pay you when you are actually in the air. They should not be permitted to exploit employees in that fashion.
If the company chooses to keep you hanging around the airport with nothing to do, or sitting in a hotel a thousand miles from home with nothing to do, that should not be your problem. It is their problem. Rigs encourage them to solve it.
I hope I got the math right.