Equity Gain

[Ellsworth T]

I know this isn't quite a probability question but an interesting one that can probably be answered by some of you math geniuses....

There is a question that is a prelude to my response that will go unwritten due its repitition througout my premise.

When I first read this question I asked myself if I, as an unbiased arbiter would prefer that one person receive 75% of a cake and one other receive 25%. Or should some of the cake be thrown away as a tithe to the gods so there could be a more equitable division of the cake. I will assume that both parties have an equal share in the production or discovery of said cake and the issue simply rests on the perceived advantages and/or disadvantages between differing division standards. One may argue that inefficiency is waste and that overall equity will be reduced, which is undesirable. However, I argue that there are intrinsic social instincts that have compelled behaviors that have been inefficient in isolation but that have produced perceived “equity” gain.

Now the math question that I have is, can one derive an algebraic equation that describes an increase in 'equity' amongst a reduction in equity, where 'equity' is the realization and satisfaction of a more proportionate division, or is this to semantical?

[BruceZ]

[ QUOTE ]
I know this isn't quite a probability question but an interesting one that can probably be answered by some of you math geniuses....

There is a question that is a prelude to my response that will go unwritten due its repitition througout my premise.

When I first read this question I asked myself if I, as an unbiased arbiter would prefer that one person receive 75% of a cake and one other receive 25%. Or should some of the cake be thrown away as a tithe to the gods so there could be a more equitable division of the cake. I will assume that both parties have an equal share in the production or discovery of said cake and the issue simply rests on the perceived advantages and/or disadvantages between differing division standards. One may argue that inefficiency is waste and that overall equity will be reduced, which is undesirable. However, I argue that there are intrinsic social instincts that have compelled behaviors that have been inefficient in isolation but that have produced perceived “equity” gain.

Now the math question that I have is, can one derive an algebraic equation that describes an increase in 'equity' amongst a reduction in equity, where 'equity' is the realization and satisfaction of a more proportionate division, or is this to semantical?

[/ QUOTE ]

The best way to divide a cake fairly between 2 people is to have one person cut the cake into 2 pieces, and have the other person choose a piece.

[Ellsworth T]

Maybe I should be a little more clear. You are a judge and must decide to choose between two different methods of dividing a cake.

Method 1- Person A gets 75% and Person B gets 25%.

Method 2- Person A gets 50% and Person B gets 40%. Now there is inefficiency loss of 10% because we decide to divvy the cake up this way. My hypothesis was then...

there are intrinsic social instincts that have compelled behaviors that have been inefficient in isolation but that have produced perceived “equity” gain. For example, we have graduated income tax for the numerous reasons that we do. Or more extreme, a soldier jumps on a grenade because if he doesn't 10 men die instead of one.

My question is now, can we derive an algebraic equation that describes these events.

[T50_Omaha8]

[ QUOTE ]
Method 1- Person A gets 75% and Person B gets 25%.

Method 2- Person A gets 50% and Person B gets 40%. Now there is inefficiency loss of 10% because we decide to divvy the cake up this way. My hypothesis was then...

[/ QUOTE ]
Efficiency loss is kind of subjective here. Wasting 10% of a cake is clearly inefficient, but if the increase in utility of an extra 15% of the cake for person A is larger than the decrease in utility wrought on person B for being deprived of 25% of a cake (which is quite possible due to the law of diminishing marginal utility), then the second situation is the more efficient one, sans any 'equity gain' effect.

I think this situation is more to the point:
Person A gets 60% and person B gets 40%
vs.
Person A gets 40% and person B gets 40%

Person B's traditional utility is hence unchanged while person A's clearly diminishes.

BUT let's assume that person B gets pissed from every % of cake person A gets more than he gets. Let's define a function g(x) to be the 'equity loss' in utility person B gets because person A gets unfairly more cake than he does (ie x represents the difference in their portions). Note that g(x) is by definition negative. Let's also define A as person A's share of the cake and B as person B's sare of the cake. Then we can define ua(x) and ub(x) to be person A, B's utility functions, respectively.

Our total welfare is then
ua(A) + ub(B) + g(A-B)

But if person A gets the same amount as person B, the welfare equation simplifies to
ua(B) + ub(B) + g(0) = ua(B) + ub(B)

Is this case better than if A > B? It depends.
ua(A) + ub(B) + g(A-B) is the unequal expression, while
ua(B) + ub(B) is the equal one, so if equal > unequal,
ua(B) + ub(B) > ua(A) + ub(B) + g(A-B)
ua(B) > ua(A) + g(A-B)
ua(B) - ua(A) > g(A-B), or
ua(A) - ua(B) < -g(A-B)

Which simply means the utility lost by person A from getting less must be outweighed by the increase in utility person B gets from person A getting less. This, of course, was more or less obvious from the start.

You need to make additional assumptions to get anything meaningful out of it, in other words.