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

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Landowner worker game 2 workers possible revolution
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

One owner two possible buyers
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
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.

More on cooperative games

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
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
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 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
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.