Final answer:
In a weighted voting system, to ensure all three players have veto power, the smallest value of q is 8. For the condition where the second player has veto power but the third does not, the smallest value of q is 3.
Step-by-step explanation:
The question involves finding the smallest value of q in a weighted voting system that will ensure certain veto power dynamics among players. It's a part of the mathematics discipline, particularly topics related to voting theory or game theory.
Part (a) asks for the smallest value of q for which all three players have veto power. In a weighted voting system like [q: 12, 6, 2], the numbers represent the weight of each player's vote. Veto power means that the decision cannot pass without the player's approval. For all players to have veto power, q must be less than the sum of the weights of the players minus the largest weight plus one. Therefore, q must be less than 6 + 2 + 1 = 9. The smallest integral value that satisfies this condition is q = 8.
Part (b) asks for the value of q when the second player (P2) has veto power but the third player (P3) does not. To achieve this, q must be more than the weight of P3 but less than or equal to the combined weights of P1 and P2. Thus, q must be greater than 2 but less than or equal to 12 + 6 = 18. The smallest integral value of q that satisfies this condition is q = 3.