Is it fair to split 50/50 of the realtor's fee when the two roommates pay the rent proportionately?

My roommate and I agreed when we searched for a new apartment that (1) it would be fine if the new place may require a realtor's fee, and (2) the rent would be paid proportionately based on the space to be occupied. We found a charming apartment with 1/2 month rent as a realtor's fee. We agreed the proportion of the rent, however, as for the realtor's fee, we didn't clarified our assumptions. Now when we tried to settle our moving cost, it became clear that our assumptions were different. My roommate thinks that the realtor's fee should be split based on the proportion we pay our rent. My assumption is that the realtor's fee is to be split equally by the two of us because this is the fee for the realtor's labor to "find" the place for us equally and it should not be related the proportion of the rent payment between the two. Your answers are welcome!!