Consider the weighted voting system \( [q: 25,20,15,10,5] \). Find the Banzhaf power distribution of this weighted voting system when (a) \( q=47 \) (b) \( q=51 \) (c) \( q=59 \) (d) \( q=71 \). (a) Find the Banzhaf power distribution when \( q=47 \). \( \beta_{1}=\square, \beta_{2}=\square, \beta_{3}=\square, \beta_{4}=\square, \beta_{5}=\square \) (Type integers or simplified fractions.)

We are given the weighted voting game   [q : 25, 20, 15, 10, 5] with quota q = 47. The players’ weights are   Player 1 = 25   Player 2 = 20   Player 3 = 15   Player 4 = 10   Player 5 = 5 Our task is to compute the (normalized) Banzhaf power index for each player. In a weighted voting game a player is “critical” or “swing” in a winning coalition if when you remove that player the coalition’s total weight falls below the quota. The (raw) Banzhaf count for a player is the number of winning coalitions in which that player is critical. Once we have the counts, the normalized index is just the player’s count divided by the sum of all counts. We now find all winning coalitions (sets with total weight ≥ 47) and count for each player when his removal would turn a winning coalition into a losing one. ────────────────────────────── Step 1. List the Winning Coalitions and Identify Critical Players Note: There are 2⁵ = 32 possible coalitions. We will “group” them by size. A. Coalitions of size 1 and 2 – No single player, or pair of players, reaches 47 since the highest weight is 25 and even 25 + 20 = 45 < 47. B. Coalitions of size 3 We need to check all 3–player combinations. 1. Coalition {1, 2, 3}:   Total = 25 + 20 + 15 = 60 (winning).   • Remove 1: 20 + 15 = 35 (<47) → Player 1 is critical.   • Remove 2: 25 + 15 = 40 (<47) → Player 2 is critical.   • Remove 3: 25 + 20 = 45 (<47) → Player 3 is critical. 2. Coalition {1, 2, 4}:   Total = 25 + 20 + 10 = 55 (winning).   • Remove 1: 20 + 10 = 30 (<47) → Critical.   • Remove 2: 25 + 10 = 35 (<47) → Critical.   • Remove 4: 25 + 20 = 45 (<47) → Critical. 3. Coalition {1, 2, 5}:   Total = 25 + 20 + 5 = 50 (winning).   • Remove 1: 20 + 5 = 25 (<47) → Critical.   • Remove 2: 25 + 5 = 30 (<47) → Critical.   • Remove 5: 25 + 20 = 45 (<47) → Critical. 4. Coalition {1, 3, 4}:   Total = 25 + 15 + 10 = 50 (winning).   • Remove 1: 15 + 10 = 25 (<47) → Critical.   • Remove 3: 25 + 10 = 35 (<47) → Critical.   • Remove 4: 25 + 15 = 40 (<47) → Critical. (All other 3–player groups have totals less than 47.) So from size‑3 coalitions, the swings are:   Player 1 appears in {1,2,3}, {1,2,4}, {1,2,5}, {1,3,4}  → 4 swings   Player 2: in {1,2,3}, {1,2,4}, {1,2,5}  → 3 swings   Player 3: in {1,2,3}, {1,3,4}  → 2 swings   Player 4: in {1,2,4}, {1,3,4}  → 2 swings   Player 5: in {1,2,5}  → 1 swing C. Coalitions of size 4 There are 5 possibilities (since 5 choose 4 = 5). 1. Coalition {1, 2, 3, 4}:   Total = 25 + 20 + 15 + 10 = 70 (winning).   Check removals:   • Remove 1: 20 + 15 + 10 = 45 (<47) → Critical for 1.   • Remove 2: 25 + 15 + 10 = 50 (≥47) → Not critical.   • Remove 3: 25 + 20 + 10 = 55 (≥47) → Not critical.   • Remove 4: 25 + 20 + 15 = 60 (≥47) → Not critical. 2. Coalition {1, 2, 3, 5}:   Total = 25 + 20 + 15 + 5 = 65 (winning).   • Remove 1: 20 + 15 + 5 = 40 (<47) → Critical.   • Remove 2: 25 + 15 + 5 = 45 (<47) → Critical.   • Remove 3: 25 + 20 + 5 = 50 (≥47) → Not critical.   • Remove 5: 25 + 20 +