The Core

Published by Mario Oettler on

Game Theory & Blockchain Cooperative Game Theory The Core

The core is a solution concept in cooperative game theory. The core comprises all distributions of values that are rational for the coalition parties. This means that accepting value distribution is beneficial for all participants in a coalition.

Numerical Example of the Core

Suppose three students are looking for an apartment. They can choose between the following options:

  1. Each student rents their own apartment
  2. Two students share one apartment, and the third one rents his own apartment
  3. All three students live in the same apartment.

Three single apartments would cost:

  • Student 1: 200 EUR
  • Student 2: 170 EUR
  • Student 3: 180 EUR

For option two, one single apartment and one shared apartment:

  • 330 EUR for the shared apartment
  • + rent for the single student.

For option three, three students sharing one apartment:

  • 450 EUR

The question is now, how much each student should contribute to the rent that the third option is chosen.

Step 1: Calculate cost savings for every coalition.

Each student stays in their own apartment.

v({1}) = v({2}) = v([3}) = 0

Nobody saves anything.

Two students share one appartement:

v({1,2}) =200+170-330 = 40

v({1,3}) = 200+180-330 = 50

v({2,3}) = 170+180-330 = 20

Three students share one appartement:

v({1,2,3}) = 200+170+180 – 450 = 100

Step 2: Avoid a blockade through a two-student coalition.

The saving s of the three-student coalition must be higher better than the two-student coalition for each two-student coalition.

  • s1 + s2 >= 40
  • s1+s3 >= 50
  • s2+s3 => 20

Step 3: Avoid a blockade through a single-student coalition.

The saving of a three-student coalition must be higher than zero.

Step 4: Check for Pareto optimality.

The solution must distribute the whole saving.

  • v({1,2,3}) = 100
  • s1+s2+s3 = 100

Hence,

  • 0<=s1<=80
  • 0<=s2<=50
  • 0<=s3<=60

Check if a distribution strategy lies inside the core.

Every student pays 150 EUR rent.

  • s1 = 50
  • s2 = 20
  • s3 = 30

They all fulfill the condition from step 4.

If the core is empty, no negotiation solution can satisfy every party.

The Core

The core comprises all distributions of values that are rational for the coalition parties. This means that accepting value distribution is beneficial for all participants in a coalition.

Option A

For example, three single apartments.

Option B

For example, one single apartment and one shared apartment.

Option C

For example, three students sharing one apartment.

Step 1: Calculate cost savings for every coalition.

Option A:
Option B:
Option C:

Step 2: Check Blockade through Option B.

Step 3: Check Blockade through Option A.

Step 4: Check for Pareto optimality

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