Coalitions: General Count and Three-Player Example

Note: The transcript appears partial and contains some garbled text. The notes below reconstructs a standard three-player coalition problem and explains the general count of coalitions, followed by the typical majority-winning coalitions for a 3-player example. Where the transcript is unclear, I state the assumption explicitly.

Key Concepts

• Coalition: any nonempty subset of players that form an alliance to achieve a goal (e.g., pass a measure in a vote).
• Winning coalition: a coalition that meets or exceeds the quota required to win a vote.
• Total number of coalitions (nonempty): for N players, $2^N-1$.
• In a three-player example, common quota is a majority (2 or more of 3).
• Notation: players often denoted as $\mathcal{P} = {P<em>1, P</em>2, P<em>3, \dots}$; coalitions written as ${P</em>i, P_j, \dots}$.

Counting coalitions (general)

• Number of all nonempty coalitions among N players:
  • $\text{Total coalitions} = 2^N - 1$.
  • Reason: every player can be either in or out of a coalition, giving $2^N$ subsets, minus the empty set.
• For a fixed N (e.g., N = 3):
  • Number of coalitions of size 1: $\binom{3}{1} = 3$
  • Number of coalitions of size 2: $\binom{3}{2} = 3$
  • Number of coalitions of size 3: $\binom{3}{3} = 1$
  • Total: $3 + 3 + 1 = 7 = 2^3 - 1$.
• Step (c) interpretation from transcript (assumed): the general count of coalitions is $2^N - 1$.

Three-Player example (P1, P2, P3)

• Assumed setup based on common classroom problems:
  • Players: $\mathcal{P} = {P<em>1, P</em>2, P_3}$.
  • Quota: majority rule for N = 3, so you need at least 2 votes to win: $q = 2$.
• All nonempty coalitions for this set:
  • Size 1 coalitions: ${P<em>1}, {P</em>2}, {P_3}$
  • Size 2 coalitions: ${P<em>1, P</em>2}, {P<em>1, P</em>3}, {P<em>2, P</em>3}$
  • Size 3 coalition: ${P<em>1, P</em>2, P_3}$
• Step (a) likely involved listing these coalitions; the transcript fragment suggests listing some coalitions like ${P<em>1, P</em>2}, {P<em>1, P</em>3}, {P<em>1, P</em>2, P_3}$ (incomplete in the excerpt).

Step (d): Circle all winning coalitions in step (a)

• Under the majority quota (q = 2), the winning coalitions are those with at least 2 members:
  • ${P<em>1, P</em>2}$
  • ${P<em>1, P</em>3}$
  • ${P<em>2, P</em>3}$
  • ${P<em>1, P</em>2, P_3}$
• How to identify them from step (a): circle the coalitions with size 2 or 3.

Step (e): Write all winning coalitions and underline each coalition

• All winning coalitions listed explicitly (underlined):
  • {P1, P2}
  • {P1, P3}
  • {P2, P3}
  • {P1, P2, P3}
• Note on formatting: underline is shown here with HTML tags to reflect the instruction to underline each winning coalition.

Formulas and numerical references (LaTeX)

• General count of coalitions: $2^N - 1$
• For N = 3:
  • Total nonempty coalitions: $2^3 - 1 = 7$
  • By size: $\binom{3}{1} = 3, \; \binom{3}{2} = 3, \; \binom{3}{3} = 1$
• Winning coalitions under majority ($q = 2$):
  • Count: $\binom{3}{2} + \binom{3}{3} = 3 + 1 = 4$
  • Winning coalitions: ${P<em>1, P</em>2}, {P<em>1, P</em>3}, {P<em>2, P</em>3}, {P<em>1, P</em>2, P_3}$

Connections to foundational principles and real-world relevance

• Coalition formation is a fundamental concept in voting theory and social choice.
• Power distribution in simple majority games can be analyzed using indices like Shapley–Shubik or Banzhaf, which quantify each player’s influence within all winning coalitions.
• Practical relevance: parliamentary coalitions, governance, alliance-building in committees, and any decision-making body with a quota.
• If the quota were different (e.g., unanimity q = N), the set of winning coalitions would change (e.g., only the grand coalition if q = N).

Notes on the transcript and ambiguities

• The excerpt contains garbled text (e.g., "8-1=", "2N-1" missing exponents, and incomplete coalition lists). The analysis above follows a standard 3-player majority example consistent with the visible parts:
  • Step (c) yields $2^N - 1$.
  • For N = 3, there are 7 nonempty coalitions; under a 2-vote majority, 4 are winning.
• If the original step (a) enumerated coalitions differently or used a different quota, the specific winning coalitions would differ accordingly. The reconstruction here uses the common majority-quota interpretation for a 3-player case.