Reformulation of Public Help Index θ Using Null Player Free Winning Coalitions

Area of Science:

• Game Theory
• Social Choice Theory
• Mathematical Economics

Background:

• The Public Help Index θ (PHI θ) was previously defined using winning coalitions.
• Understanding power distribution in simple games is crucial for various analytical models.

Purpose of the Study:

• To reformulate the PHI θ index using null player free winning coalitions.
• To provide a transparent method for analyzing power distribution among null and non-null players.
• To explore potential computational cost reductions in PHI θ index algorithms.

Main Methods:

• Reformulation of the PHI θ index based on null player free winning coalitions.
• Analysis of simple games defined by these coalitions.
• Comparison with existing indices like the Holler index and the g^ index.

Main Results:

• A new, transparent representation of the PHI θ index is established.
• The reformulation clearly distinguishes power assigned to null versus non-null players.
• Potential for reduced space complexity in computing the PHI θ index is identified.

Conclusions:

• The new PHI θ representation enhances understanding of power dynamics in simple games.
• This approach offers computational advantages, particularly when null players are present.
• Further investigation into relationships with other power indices is warranted.