More on cooperative games

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# More on cooperative games - PowerPoint PPT Presentation

More on cooperative games. Landowner-worker game, 2 workers possible revolution.

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### More on cooperative games

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
• 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. 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?
• 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 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.