Analyze Optimal Funding Allocations Across Election Districts

## Prompt Two players allocate nonnegative integer funding units across labeled election districts. In each district, the player with more funding wins that district; equal funding is a tie. The overall winner is the player who wins more districts than the opponent. ### Constraints & Assumptions - Every player's allocation must sum to that player's full budget. - For random allocation questions, all weak compositions of the stated budget across labeled districts are equally likely. - For “possible to win,” you may choose the alignment that is most favorable to your allocation; for “cannot win,” no alignment may produce more district wins than losses. - An evenly distributed opponent allocation differs by at most one unit across districts. ### Clarifying Questions to Ask - Are districts symmetric, or do any carry different values? Treat them as symmetric here. - Does an overall tie count as success? It does not. - Does “minimum” ask for a possible favorable alignment or a guaranteed win against every alignment? ### Part 1 — Three Districts and Equal Budgets With three districts and four units for each player, count the opponent's possible allocations. If the opponent chooses uniformly among them, compare your allocations `(2,1,1)` and `(4,0,0)`, and explain why rotations of `(2,1,1)` have equal performance. #### What This Part Should Cover - Stars-and-bars count for labeled weak compositions. - Symmetry of rotated allocations against a uniform symmetric opponent. - Why concentrating all four units can win at most one district. ### Part 2 — Minimum Budget for a Possible Win There are eight districts. The opponent has nine units and distributes them as evenly as possible. Find the minimum budget with which you can possibly win under a favorable alignment, and give an allocation and matchup that proves feasibility. #### What This Part Should Cover - The opponent multiset `(2,1,1,1,1,1,1,1)` up to rotation. - A nine-unit allocation that produces strictly more wins than losses. - A lower-bound argument showing eight units cannot do so. ### Part 3 — Opponent Budget That Makes a Win Impossible You have nine units across eight districts. The opponent again distributes as evenly as possible. Find the smallest opponent budget that makes it impossible for you to win even under the most favorable alignment. #### What This Part Should Cover - Comparison of opponent budgets eleven and twelve. - A favorable nine-unit allocation that still beats the eleven-unit distribution. - Why the twelve-unit distribution forces at least as many losses as wins. ```hint Count wins and losses, not only majorities Tied districts contribute to neither side. A winning allocation may deliberately concede some districts and tie others instead of winning five outright. ``` ### What a Strong Answer Covers - Exact combinatorial counts and explicitly matched allocations. - A feasibility construction plus a separate lower-bound or impossibility proof. - Careful distinction between labeled rotations, random symmetry, and favorable alignment. - Checks that all allocation vectors use the complete stated budget. ### Follow-up Questions 1. How would you compute the best response for arbitrary district count and budgets? 2. How does the answer change if a tie in the overall election is considered success? 3. What if districts have unequal electoral values rather than one vote each?