全球首次完成此三款棋牌遊戲的數學分析與策略證明 World's first mathematical analysis and strategic proof for these three games
| 語言 | 內容 |
|---|---|
| 🇹🇼 中文 | 本倉庫收錄三份獨立的策略分析研究報告，分別針對三款棋牌遊戲進行嚴格的數學建模、賽局理論分析和電腦模擬驗證，覆蓋完全資訊與不完全資訊兩大博弈範疇。 |
| 🇬🇧 English | This repository contains three independent strategic analysis reports on three different board/card games, featuring rigorous mathematical modeling, game-theoretic analysis, and computer simulations, covering both perfect-information and imperfect-information games. |
| 遊戲 / Game | 核心結論 / Core Finding | 研究方法 / Method |
|---|---|---|
| 雙步棋 / Shuang Bu Qi | 必然和棋，堵門策略為納許均衡 / Forced draw, blocking strategy is a Nash equilibrium | 圖論割集、100局蒙地卡羅 / Graph cut-set, 100 Monte Carlo games |
| 積分棋 / Score Chess | 先手優勢極大（95%勝率）/ First player has significant advantage (95% win rate) | 奇偶性理論、2000+局測試 / Parity theory, 2000+ game tests |
| 象棋麻將 / Xiangqi Mahjong | 已解決遊戲，outs策略為天花板 / Solved game, outs strategy is the ceiling | MCCFR、萬餘局驗證 / MCCFR, 10k+ game benchmarks |
| 中文 | English |
|---|---|
| 棋盤存在致命結構缺陷：節點9與10構成2節點割集，是黑方抵達所有目的地的唯一瓶頸 | The board has a fatal structural flaw: nodes 9 and 10 form a 2-node cut-set, the only bottleneck for black to reach all destinations |
| 任何一方佔住此兩節點即可永久封鎖對方，實現強制和棋 | Either player occupying these two nodes can permanently block the opponent, forcing a draw |
| 雙方同時堵門為納許均衡，偏離無益 | Mutual blocking is a Nash equilibrium; deviation yields no benefit |
| 項目 / Item | 結果 / Result |
|---|---|
| 模擬局數 / Games simulated | 100 局 / 100 games |
| 白方策略 / White strategy | 堵門 + 三層啟發式評分 / Blocking + 3-layer heuristic |
| 黑方策略 / Black strategy | 貪婪進攻 / Greedy offense |
| 結果 / Outcome | 100% 和棋 / 100% draws |
| 中文 | English |
|---|---|
| 先手優勢極大：第一步佔 (3,3) + 鏡像策略，勝率達 95% | First player has significant advantage: opening (3,3) + mirroring yields 95% win rate |
| 中心點具備雙重功能：破解鏡像 + 橋接鏈結構 | Center point (3,3) serves dual roles: breaking mirror + bridging chains |
| 後手所有已知反制策略均無效 | All known counter-strategies for second player fail |
| 注意：此為「強化實驗性弱解」，非嚴格數學必勝 | Note: This is a "strong empirical heuristic", not a mathematical forced win |
| 策略對局 / Matchup | 先手勝率 / First Player Win Rate |
|---|---|
| (3,3)+鏡像 vs 聰明隨機 / (3,3)+mirror vs smart random | 95.2% |
| 後手四種主動反制 / Four active counter-strategies | 全數下降 / All decreased |
| 中文 | English |
|---|---|
| 象棋麻將是一個「已解決的博弈」 | Xiangqi Mahjong is a solved game |
| 最優策略為 outs-smart_discard（1階outs近似） | Optimal strategy is outs-smart_discard (1st-order outs approximation) |
| 與理論最優的差距僅 0.5%，在統計雜訊範圍內 | Gap from theoretical optimum is only 0.5%, within statistical noise |
| 策略 / Strategy | vs 純隨機勝率 / vs Random Win Rate | 說明 / Note |
|---|---|---|
| 純隨機 / Random | 50%（基準 / baseline） | 完全無策略 / No strategy |
| smart_discard | 67% | 型態匹配 / Pattern matching |
| outs-smart_discard | 89% | 機率加權，目前最強 / Probability-weighted, strongest |
| 對手建模 / Opponent modeling | 89.5% | 效益僅 +0.5%，無實質意義 / Only +0.5%, negligible |
| 指標 / Metric | 數值 / Value |
|---|---|
| outs vs outs 莊家勝率 / Dealer win rate | 52.6% |
| 結構性先手優勢 / Structural first-player advantage | 5.2% |
| 對手建模效益 / Opponent modeling gain | +0.5%（統計雜訊 / statistical noise） |
| 2人版流局率（outs vs outs）/ Draw rate (2-player) | 0.5% |
├── 雙步棋強制和棋研究報告.docx ├── 積分棋策略分析報告_v3.docx ├── 象棋麻將AI研究報告_v14_完整證明版.docx └── 象棋麻將_數學證明附錄.docx
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中文： 本倉庫內容為學術研究目的，所有分析、結論與實驗設計均為原創。作者已盡合理努力確保分析與數據的正確性。惟研究結論可能隨未來更大規模驗證而修正，引用時請留意。
English: The contents of this repository are for academic research purposes only. All analyses, conclusions, and experimental designs are original. The author has made reasonable efforts to ensure the accuracy of the analysis and data. However, research conclusions may be subject to revision with future larger-scale validation. Please take this into consideration when citing.
歡迎透過 GitHub Issues 進行學術交流或提出問題。 For academic exchange or questions, please use GitHub Issues.
最後更新 / Last Updated：2026年5月