Equity is one of the foremost considerations in free allocation of power system carbon emission

while the impact of historical deficits is one of the key essentials in fair distribution. Based on Shapley values

a historical deficit sharing model capable of solving the allocation problem of historical deficits in the emission reduction year is established and axiomatized. In accordance with the principle that whoever causes the issue should be responsible for it

a historical emission deficit sharing model among the units is developed

which is capable of solving the reduction of unit's claims caused by historical deficits. On this basis

with the consistency constraint of the allocation method taken into account

an allocation model for initial emission allowances considering the historical emission deficit is established. Furthermore

the participants' gaming behaviors under the aggregation mechanism are studied

which proved the existence of game equilibrium. In the meantime

the properties of the historical emission deficits sharing model such as anonymity

efficiency and independence of irrelevant share are proved

and the fair ranking is discussed. In the end

it is proved by a case study that as the proposed method meets the basic requirements for fair allocation

it can be easily accepted by the participants.