More on cooperative games

1. More on cooperative games

2. Landowner-worker game, 2 workerspossible revolution • Let x1,x2,x3 be an allocation of the output f(k) from k people working on landowner’s land. Two workers could revolt, kill landowner, and take land. Output after revolution is less than two workers with no revolt. • With no revolution, f(1)=1, f(2)=3, f(3)=4.With a revolution, output with 2 workers is 1.5. • What’s in the core? All work, no revolution. • Then x2+x3≥1.5, otherwise {23} would gain by revolt. So x1≤2.5 • Also x1+x3 ≥3 —otherwise {13} could do better by themselves. Therefore x2≤1. Why? • Similarly, x2≤1. • Then it must be that x2≥.5 and x3≥.5 • Sample core allocations: • x2=x3=1, x1=1 • x2=1, x3=.5, x1=2.5

3. One owner two possible buyers • Owner (person 1) has an object that is worthless to him, worth $1 to either of two possible buyers (persons 2 and 3). Persons 2 and 3 each start out with more than $1. Trade is possible. • Two outcomes are in the core. Person 1 sells object to 2 for $1. Person 1 sells object to 3 for $1. • Why is nothing else in the core?

4. Previous example except that • Person 2 values object at 1. Person 3 values it at $v<1. What is in core? Person 2 gets the object and pays person 1 a price p that is between v and 1.

5. There are 3 players.Person 1 has an object that is of no value to him. It is worth $10 to person 2, and $6 to person 3. Which of these outcomes in in the core? • Person 1 sells to Person 2 at $5. • Person 1 sells to Person 3 at $6. • Person 1 sells to Person 2 at $7. • Person 1 sells to Person 2 at $11. • None of these.

6. House Allocation Problem • N-people, each owns a house. Each has preferences over other houses. • Coalitions can allocate houses owned by their members. • What is the core? • How do you find the core?

7. Top trading cycle • Everybody points at his favorite house. • Those who point at their own house are assigned their own house and removed from consideration. • Find cycles. Each person in a cycle can get his favorite house. Make these assignments and eliminate cycle members from consideration. • Iterate until everybody is placed.

8. Top trading cycle and core • Top trading cycle is in the core. • Strong core—No coalition can take an action that some of its members prefer to the core allocation and all are at least as well off as in the core. • Top trading cycle outcome is only allocation in the strong core of the house allocation problem.

9. Matching games • Roommate Problem: 4 students Al, Bob, Chuck, Don. Two two-person rooms. A core assignment is one such that no two persons can do better by rooming together than with their assigned partners. Preferences Al--Bob, Chuck, Don Bob--Chuck, Al, Don Chuck—Al, Bob, Don Show that the core is empty.