Utilitarian vs. Egalitarian Allocation in Game Theory in AI

Question

Let's say I have two players: Player A and Player B and they have preferences over what resources (let's just be general and use the term 'resource'). Their preferences could be:

   {p}     {q}     {p,q}     {}        
A  10      15      20        0  
B  5       5       10        1

This says that the two players can have one resource, both or none. The greater the number the more the player wants it.

I believe the 'utilitarian' view would be to maximise the allocation overall so this would be the following two allocations:

A: {p,q} and B: {}

because it adds to 21 even though B is not very happy [happiness 1 :-( ].

My question is what would the egalitarian (see wiki: 1) allocations be (if there are any)? I'm not sure how this would be properly calculated from the above table?

Thanks :).

Answer 1

In a strictly egalitarian solution everyone receives the same value. That is not possible in the example.

However, the egalitarian social welfare solution is defined as that which maximizes the minimun value received by any agent (see page.79 of my multiagent systems textbook)

In this case, there is a tie between two solutions:

                      A     B 
A: {p}   B: {q}      10     5
A: {q}   B: {p}      15     5

You can further refine egalitarian social welfare solution concept as you see fit.

Answer 2

Assuming: 1. That the same resource cannot be given to both players. 2. That resources are identified as either p or q. 3. That all resources must be allocated to one player or the other. 4. That "egalitarian allocations" mean that both players get the same score.

Then it is not possible, since none of the scores are the same in the A and B rows, except for a combination that would supply p to both players.

Assuming instead: 4. That "egalitarian allocations" mean that the difference between the scores of the two players is minimized.

Then the possibilities are:

                      A     B   Difference
A: {p,q} B: {}       20     1           19
A: {p}   B: {q}      10     5            5
A: {q}   B: {p}      15     5           10
A: {}    B: {p, q}    0    10           10

Giving a minimum difference of 5 for A:{p} B:{q}

Assuming instead: 3. That all resources may be allocated to one player or the other, or given to neither.

Then the possibilities are:

                      A     B   Difference
A: {p,q} B: {}       20     1           19
A: {p}   B: {q}      10     5            5
A: {p}   B: {}       10     1            9
A: {q}   B: {p}      15     5           10
A: {q}   B: {}       15     1           14
A: {}    B: {p, q}    0    10           10
A: {}    B: {p}       0     5            5
A: {}    B: {q}       0     5            5
A: {}    B: {}        0     1            1

Giving a minimum difference of 1 for A:{} B:{}