More on cooperative games

Landowner-worker game, 2 workers possible 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

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?

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.

There are 3 players. Person 1 has an object that is of no value to him 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.

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?

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.

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.

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.