Individuals make socks, which are only valuable in pairs. A pair of socks sells for $10. Since a single sock is pointless, it cannot be sold for a positive price. (a) If there are two individuals, who both make 5 socks1 each, what describes the set of actions that are in the core? (b) If there are three individuals, who each make 5 socks, what describes the set of actions that are in the core? (c) For this part, assume that the socks do correspond to right or left feet and that only pairs of one right sock and one left sock can be sold for $10. If there are 4 individuals who each produce a left sock and 5 individuals who each produce a right sock, what is one "core" solution? (You don't need to describe them all, although they all share some things in common. For the answer you only need to give a single allocation that is in the core. It will help to label your players, maybe r1, r2, l1, l2, etc.) 

 

 (a) If there are two individuals, who both make 5 socks1 each, what describes the set of actions that are in the core?

Call the individuals P1 and P2. In this case, the individual participation constraints and the resource constraint tell the whole story: P1 ≥ 20, P2 ≥ 20, and P1 + P2 = 50. There are many combinations in the core - they could split evenly at $25 each, but also $21 to P1 and $29 to P2, $20.01 and $29.99, etc.

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(b) If there are three individuals, who each make 5 socks, what describes the set of actions that are in the core?

Here the participation constraints are Pi ≥ 20 (for any player by themselves), P1 + P2 ≥ 50, P1 + P3 ≥ 50, and P2 + P3 ≥ 50. The resource constraint is P1 + P2 + P3 = 70. It doesn't matter how you attack this, you'll run into the same issue. For example, solving the resource constraint for P1 gives P1 = 70 − P2 − P3. Plugging this into the first two player participation constraint gives 70 − P2 − P3 + P2 ≥ 50; that simplifies to P3 ≤ 20. The same is true of P1 and P2. But that causes a problem - that means that no two players ever have more than 40, which violates the two-player participation constraints. There are no outcome that simultaneously meets all the constraints, so there is no core solution to this game. Intuitively, this is similar to the three player majority game. Any set of two could split $50. The third player only creates $20 worth of extra value. But say you had P1 = P2 = 25 and P3 = 20. Then P3 and P1 could form a sub coalition where both are better off - ie, where P1 = 28 and P3 = 22. This ability to form a sub coalition with better outcomes for its members is true for any potential core solution

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(c) For this part, assume that the socks do correspond to right or left feet and that only pairs of one right sock and one left sock can be sold for $10. If there are 4 individuals who each produce a left sock and 5 individuals who each produce a right sock, what is one "core" solution? (You don't need to describe them all, although they all share some things in common. For the answer you only need to give a single allocation that is in the core. It will help to label your players, maybe r1, r2, l1, l2, etc.) 

You can, if you like, set up all the participation constraints. Each individual player must get more than 0, since that's what they get on their own. Any combination of two players with different types must get at least 10; combinations of three players must get at least 10 as long as there is at least one L and one R sock in the group, etc. What really matters here is the participation constraint for any group of 4 R and 4 L. This group could get $40 as a sub coalition, so they must get this much in a core solution. One version of this constraint is R1 + R2 + R3 + R4 + L1 + L2 + L3 + L4 ≥ 40, but then we need to repeat it with each combination of 4 right sock makers (ie, also R1 + R2 + R3 + R5 + L1 + L2 + L3 + L4 ≥ 40 and R1 + R2 + R4 + R5 + L1 + L2 + L3 + L4 ≥ 40 and R1+R3+R4+R5+L1+L2+L3+L4 ≥ 40 and R2+R3+R4+R5+L1+L2+L3+L4 ≥ 40). Our resource constraint is R1 + R2 + R3 + R4 + R5 + L1 + L2 + L3 + L4 = 40 (note the inclusion of all 5 right sock producers). If you solve for, say, R1 and substitute into the first of the resource constraints above, you'll get R5 ≤ 0. Keep doing this with different constraints and you'll get this for any R - Ri ≤ 0.

The intuition here is that the extra right sock would make someone "surplus" - if all five of the right sock people got a positive payoff, then any group of the 4 Left socks and 4 of the 5 Right socks could do better. But if one of the Right socks gets 0 and the other 4 get a positive payoff, the one who gets 0 would always be willing to undercut the 4 who get something, until they are all competed down to 0. Another way of thinking about it: the marginal value of the 5th right sock is 0. Everyone of the same type gets their marginal value in a core solution.