The efficient extension operator is a mathematical tool used in cooperative game theory to extend a value defined on a restricted class of games to a larger class while preserving key properties such as efficiency. In the context of cooperative games with communication structures or restrictions, like those addressed by the Banzhaf and Myerson values, the efficient extension operator helps restore efficiency by ensuring that the total value allocated to all players equals the worth of the grand coalition. Specifically, for the Myerson value, which accounts for communication graphs among players, the efficient extension operator adjusts the value assignments so that the sum of individual payoffs matches the value of the entire networked coalition, thus restoring efficiency lost due to the imposed communication constraints. Similarly, for the Banzhaf value, which measures players' power indices, the efficient extension operator extends the value from simple games to more complex or restricted games, maintaining the efficiency property by appropriately redistributing the total worth. This operator systematically modifies the original value function to ensure that the sum of payoffs remains consistent with the total game value, thereby restoring efficiency in these cooperative game solutions.
