Tue, 13 Apr, 2021

An Axiom of Fairness

By Sudarshan Guruacharya

Consider this scenario: Two people are given 100 rupees. Now we ask the following question: How will these two people divide this 100 rupees among each other, given that we stipulate a condition that none of them will get anything if negotiation between them fails? That is, we take back the Rs. 100 away from them if they fail to reach an agreement. Common sense dictates that these two people divide the sum according to 50-50 rule. In other words, each person gets half of the 100 rupees, which is 50 rupees. This solution is attractive because it fits our intuitions of fairness. Thus we can even claim that the proposed solution is a "fair" solution. However, it is worthwhile to consider why this particular 50-50 division should be considered as a fair division. One possible answer to this question is to say that each person is in some way "equal" to each other. Hence, they should get equal amount during the division of the money. But this begs a further question: in what way are they "equal" to each other? To answer this question let us call these two people Alice and BoB. We can make this idea of being "equal" more precise by considering symmetric situations. Let us first consider Alice alone. If Alice stays alone, she gets nothing; so her self worth is zero. Now if Bob comes along to join her, then the value of their partnership becomes 100 rupees. That is, they get a chance to retain the 100 rupees. Thus we say that Bob's marginal (or incremental) contribution to this partnership is 100 rupees. Then again, we can switch the order of consideration. In the second situation, We can first consider Bob alone, due to which his self worth is zero. If Alice comes and partners with him, then their group value becomes 100 rupees. Thus, in this second situation, we say that Alice's marginal contribution to the partnership is 100 rupees. But we have no reason to prefer any one of these two order of consideration -- whether Alice comes first and is joined later by Bob, or Bob comes first and is later joined by Alice. Since both cases are equally likely, if we average the marginal contributions of Alice and Bob in these two cases, we find that both Alice and Bob makes an average marginal contribution of 50 rupees. This is their worth or value that they each individually on average contributes to the partnership. Thus, it is fair to divide the 100 rupees using 50-50 rule.