引言
飞行棋是经典的棋盘游戏，玩家通过掷骰子决定棋子移动步数，目标是让自己的四枚棋子全部绕棋盘一圈并到达终点。看似依赖运气的游戏背后，隐藏着丰富的概率决策问题。本文将使用Java建模飞行棋的马尔可夫决策过程（MDP），并通过动态规划计算每一步的最优行动策略。
算法原理
飞行棋规则建模
标准飞行棋棋盘包含52个格子（环形），四枚棋子从基地出发，需要掷出6点才能起飞。棋子按照顺时针方向前进，经过自己颜色的跳跃格时可飞跃前进，到达对手棋子位置时可将其撞回基地。
马尔可夫决策过程
飞行棋可以建模为MDP，包含以下要素：
- 状态空间 ( S )：所有棋子的位置组合
- 行动空间 ( A )：当有多枚可移动棋子时选择移动哪一枚
- 转移概率 ( P(s’|s,a) )：由骰子点数决定
- 奖励函数 ( R(s,a,s’) )：到达终点、撞击对手、被撞击等的收益
动态规划求解
对于有限时界的MDP，可以使用逆向归纳法（Backward Induction）求解最优策略：
[ V_t(s) = \max_a \sum_{s’} P(s’|s,a) [R(s,a,s’) + V_{t+1}(s’)] ]
其中 ( V_t(s) ) 表示在时刻 ( t ) 处于状态 ( s ) 的最优期望收益。
Java实现
import java.util.*;
/**
* 飞行棋游戏状态
*/
class LudoState {
// 每位玩家4枚棋子的位置，-1表示在基地，0-51表示棋盘上，52表示到达终点
private final int[][] positions;
private final int currentPlayer;
public LudoState(int[][] positions, int currentPlayer) {
this.positions = new int[4][4];
for (int p = 0; p < 4; p++) {
System.arraycopy(positions[p], 0, this.positions[p], 0, 4);
}
this.currentPlayer = currentPlayer;
}
public int[][] getPositions() { return positions; }
public int getCurrentPlayer() { return currentPlayer; }
/**
* 判断某枚棋子是否在基地
*/
public boolean isInBase(int player, int piece) {
return positions[player][piece] == -1;
}
/**
* 判断某枚棋子是否已到达终点
*/
public boolean isFinished(int player, int piece) {
return positions[player][piece] >= 52;
}
/**
* 获取棋子在棋盘上的实际位置（考虑玩家起始偏移）
*/
public int getBoardPosition(int player, int piece) {
int pos = positions[player][piece];
if (pos < 0 || pos >= 52) return -1;
return (pos + player * 13) % 52;
}
@Override
public boolean equals(Object o) {
if (this == o) return true;
if (!(o instanceof LudoState)) return false;
LudoState other = (LudoState) o;
return currentPlayer == other.currentPlayer &&
Arrays.deepEquals(positions, other.positions);
}
@Override
public int hashCode() {
return 31 * Arrays.deepHashCode(positions) + currentPlayer;
}
}
/**
* 马尔可夫决策过程求解器
*/
public class LudoMDPSolver {
private static final int NUM_PLAYERS = 4;
private static final int PIECES_PER_PLAYER = 4;
private static final int BOARD_SIZE = 52;
private static final int DICE_SIDES = 6;
// 奖励函数参数
private static final double REWARD_FINISH = 100.0; // 棋子到达终点
private static final double REWARD_KICK = 20.0; // 撞击对手棋子
private static final double REWARD_JUMP = 10.0; // 经过跳跃格
private static final double REWARD_PROGRESS = 1.0; // 每前进一步的奖励
private static final double PENALTY_KICKED = -15.0; // 被撞击回基地
private static final double PENALTY_RISK = -2.0; // 停留在易被撞击位置
// 跳跃格位置（相对于每位玩家起点）
private static final int[] JUMP_SPOTS = {0, 13, 26, 39};
// 动态规划表：state -> expected value
private Map<LudoState, Double> valueTable;
private Map<LudoState, Integer> policyTable; // 最优行动
public LudoMDPSolver() {
this.valueTable = new HashMap<>();
this.policyTable = new HashMap<>();
}
/**
* 计算执行某个行动后的期望价值
* @param state 当前状态
* @param action 选择的棋子索引（0-3）
* @param diceRoll 骰子点数
* @return 即时奖励
*/
private double computeImmediateReward(LudoState state, int action, int diceRoll) {
int player = state.getCurrentPlayer();
int[] playerPos = state.getPositions()[player];
int piecePos = playerPos[action];
double reward = 0;
// 处理起飞：只有掷出6且棋子在基地才能起飞
if (piecePos == -1) {
if (diceRoll == 6) {
// 起飞到起点
reward += REWARD_PROGRESS;
} else {
return 0; // 无法移动
}
}
// 计算新位置
int newPos = piecePos == -1 ? 0 : piecePos + diceRoll;
// 到达终点奖励
if (newPos >= 52) {
reward += REWARD_FINISH;
return reward;
}
// 每前进一步的进度奖励
reward += diceRoll * REWARD_PROGRESS * 0.1;
// 检查是否经过跳跃格
int boardPos = (newPos + player * 13) % BOARD_SIZE;
for (int jumpSpot : JUMP_SPOTS) {
int actualJump = (jumpSpot + player * 13) % BOARD_SIZE;
if (boardPos == actualJump) {
reward += REWARD_JUMP;
break;
}
}
// 检查是否可以撞击对手
for (int opponent = 0; opponent < NUM_PLAYERS; opponent++) {
if (opponent == player) continue;
int[] oppPos = state.getPositions()[opponent];
for (int p = 0; p < PIECES_PER_PLAYER; p++) {
if (oppPos[p] >= 0 && oppPos[p] < 52) {
int oppBoardPos = (oppPos[p] + opponent * 13) % BOARD_SIZE;
if (boardPos == oppBoardPos) {
reward += REWARD_KICK;
}
}
}
}
// 评估新位置的风险（是否容易被对手撞击）
int riskCount = 0;
for (int opponent = 0; opponent < NUM_PLAYERS; opponent++) {
if (opponent == player) continue;
// 简化的风险评估：统计对手能一步到达此位置的骰子点数数量
for (int roll = 1; roll <= 6; roll++) {
int[] oppPos = state.getPositions()[opponent];
for (int p = 0; p < PIECES_PER_PLAYER; p++) {
if (oppPos[p] >= 0 && oppPos[p] < 52) {
int potentialPos = oppPos[p] + roll;
if (potentialPos < 52) {
int potBoardPos = (potentialPos + opponent * 13) % BOARD_SIZE;
if (potBoardPos == boardPos) {
riskCount++;
}
}
}
}
}
}
reward += riskCount * PENALTY_RISK;
return reward;
}
/**
* 执行行动并生成新状态
*/
private LudoState executeAction(LudoState state, int action, int diceRoll) {
int[][] newPositions = new int[NUM_PLAYERS][PIECES_PER_PLAYER];
for (int p = 0; p < NUM_PLAYERS; p++) {
System.arraycopy(state.getPositions()[p], 0, newPositions[p], 0, PIECES_PER_PLAYER);
}
int player = state.getCurrentPlayer();
int piecePos = newPositions[player][action];
// 处理起飞
if (piecePos == -1 && diceRoll == 6) {
newPositions[player][action] = 0;
} else if (piecePos >= 0) {
int newPos = piecePos + diceRoll;
// 检查并处理撞击
if (newPos < 52) {
int newBoardPos = (newPos + player * 13) % BOARD_SIZE;
for (int opponent = 0; opponent < NUM_PLAYERS; opponent++) {
if (opponent == player) continue;
for (int p = 0; p < PIECES_PER_PLAYER; p++) {
if (newPositions[opponent][p] >= 0 && newPositions[opponent][p] < 52) {
int oppBoardPos = (newPositions[opponent][p] + opponent * 13) % BOARD_SIZE;
if (newBoardPos == oppBoardPos) {
newPositions[opponent][p] = -1; // 撞回基地
}
}
}
}
}
newPositions[player][action] = Math.min(newPos, 52);
}
// 下一个玩家（简化为轮流，掷出6可继续）
int nextPlayer = (diceRoll == 6) ? player : (player + 1) % NUM_PLAYERS;
return new LudoState(newPositions, nextPlayer);
}
/**
* 判断游戏是否结束（当前玩家的所有棋子都到达终点）
*/
private boolean isGameOver(LudoState state) {
int player = state.getCurrentPlayer();
for (int p = 0; p < PIECES_PER_PLAYER; p++) {
if (state.getPositions()[player][p] < 52) {
return false;
}
}
return true;
}
/**
* 获取当前状态下所有合法行动
*/
private List<Integer> getLegalActions(LudoState state) {
List<Integer> actions = new ArrayList<>();
int player = state.getCurrentPlayer();
int[] pos = state.getPositions()[player];
for (int i = 0; i < PIECES_PER_PLAYER; i++) {
// 基地中的棋子需要掷出6才能起飞
if (pos[i] == -1) {
actions.add(i); // 将在评估时根据骰子点数判断是否可行
} else if (pos[i] < 52) {
actions.add(i);
}
}
return actions;
}
/**
* 使用动态规划求解单步最优策略
* 对于完整游戏，由于状态空间巨大，这里使用蒙特卡洛采样近似
*/
public int solveOptimalAction(LudoState state) {
List<Integer> legalActions = getLegalActions(state);
if (legalActions.isEmpty()) return -1;
double bestValue = Double.NEGATIVE_INFINITY;
int bestAction = legalActions.get(0);
for (int action : legalActions) {
double expectedValue = 0;
// 对骰子的6种结果求期望
for (int dice = 1; dice <= DICE_SIDES; dice++) {
double reward = computeImmediateReward(state, action, dice);
// 如果能执行该行动，累加后续状态的期望价值
LudoState nextState = executeAction(state, action, dice);
if (nextState != null && !isGameOver(nextState)) {
// 使用启发式评估后续状态
double heuristic = evaluateState(nextState, state.getCurrentPlayer());
expectedValue += (reward + heuristic) / DICE_SIDES;
} else {
expectedValue += reward / DICE_SIDES;
}
}
if (expectedValue > bestValue) {
bestValue = expectedValue;
bestAction = action;
}
}
return bestAction;
}
/**
* 启发式状态评估函数
* 评估状态对指定玩家的有利程度
*/
private double evaluateState(LudoState state, int player) {
double score = 0;
int[] pos = state.getPositions()[player];
for (int p = 0; p < PIECES_PER_PLAYER; p++) {
if (pos[p] >= 52) {
score += REWARD_FINISH * 0.5; // 已完成的棋子
} else if (pos[p] >= 0) {
// 进度奖励：越靠前越好
score += pos[p] * REWARD_PROGRESS;
// 接近终点额外奖励
if (pos[p] > 45) score += (pos[p] - 45) * 2;
}
}
// 对手进度的惩罚
for (int opp = 0; opp < NUM_PLAYERS; opp++) {
if (opp == player) continue;
int[] oppPos = state.getPositions()[opp];
for (int p = 0; p < PIECES_PER_PLAYER; p++) {
if (oppPos[p] >= 52) {
score -= REWARD_FINISH * 0.3;
}
}
}
return score;
}
/**
* 蒙特卡洛模拟评估某个行动的长期价值
* @param state 当前状态
* @param action 要评估的行动
* @param simulations 模拟次数
*/
public double monteCarloEvaluate(LudoState state, int action, int simulations) {
double totalReward = 0;
Random rand = new Random();
for (int sim = 0; sim < simulations; sim++) {
LudoState simState = state;
double simReward = 0;
double discount = 1.0;
// 执行指定行动
int dice = rand.nextInt(DICE_SIDES) + 1;
simReward += computeImmediateReward(simState, action, dice) * discount;
simState = executeAction(simState, action, dice);
// 后续随机模拟
for (int step = 0; step < 20 && !isGameOver(simState); step++) {
discount *= 0.95;
List<Integer> legal = getLegalActions(simState);
if (legal.isEmpty()) break;
int randomAction = legal.get(rand.nextInt(legal.size()));
dice = rand.nextInt(DICE_SIDES) + 1;
simReward += computeImmediateReward(simState, randomAction, dice) * discount;
simState = executeAction(simState, randomAction, dice);
}
totalReward += simReward;
}
return totalReward / simulations;
}
/**
* 主程序：演示MDP决策
*/
public static void main(String[] args) {
LudoMDPSolver solver = new LudoMDPSolver();
// 初始化游戏状态：所有棋子都在基地
int[][] initialPositions = new int[4][4];
for (int p = 0; p < 4; p++) {
Arrays.fill(initialPositions[p], -1);
}
// 设置一个中间状态用于测试
initialPositions[0][0] = 10; // 玩家0的第1枚棋子在位置10
initialPositions[0][1] = -1; // 玩家0的第2枚棋子在基地
initialPositions[0][2] = 25; // 玩家0的第3枚棋子在位置25
initialPositions[0][3] = 48; // 玩家0的第4枚棋子接近终点
// 对手棋子位置
initialPositions[1][0] = 15;
initialPositions[2][0] = 23;
initialPositions[3][0] = 36;
LudoState state = new LudoState(initialPositions, 0);
System.out.println("=== 飞行棋MDP最优策略求解 ===");
System.out.println("当前玩家: 0");
System.out.println("玩家0棋子位置: " + Arrays.toString(initialPositions[0]));
// 求解最优行动
int optimalAction = solver.solveOptimalAction(state);
System.out.println("\nDP决策推荐移动棋子: " + optimalAction);
// 蒙特卡洛验证
System.out.println("\n=== 蒙特卡洛模拟验证 ===");
for (int action = 0; action < 4; action++) {
if (initialPositions[0][action] < 52) {
double mcValue = solver.monteCarloEvaluate(state, action, 1000);
System.out.printf("移动棋子 %d 的蒙特卡洛期望收益: %.2f%n", action, mcValue);
}
}
System.out.println("\n=== 各骰子点数下的即时奖励分析 ===");
for (int dice = 1; dice <= 6; dice++) {
System.out.printf("\n骰子点数 = %d:%n", dice);
for (int action = 0; action < 4; action++) {
if (initialPositions[0][action] < 52) {
double reward = solver.computeImmediateReward(state, action, dice);
System.out.printf(" 移动棋子 %d: 奖励 = %.2f%n", action, reward);
}
}
}
}
}
策略分析
关键决策场景
-
起飞决策：当基地中有棋子且掷出6时，应选择起飞还是移动已在棋盘上的棋子？MDP分析表明，早期阶段优先起飞以获取更多移动选择；后期阶段则优先推进靠前的棋子。
-
撞击权衡：移动棋子撞击对手可能获得高即时奖励，但如果新位置暴露在高风险区域，长期期望可能不如保守推进。
-
终点冲刺：当棋子接近终点时，即使骰子点数可能超出终点，也应优先移动该棋子，因为完成棋子的边际价值最高。
复杂度分析
| 指标 | 复杂度 | 说明 |
|---|---|---|
| 状态空间 | ( O(54^{16}) ) | 每枚棋子54种可能位置，16枚棋子 |
| 单步决策 | ( O(A \cdot D \cdot P) ) | ( A ) 为合法行动数，( D=6 ) 为骰子面数，( P ) 为玩家数 |
| 蒙特卡洛模拟 | ( O(S \cdot H \cdot A) ) | ( S ) 模拟次数，( H ) 模拟深度 |
总结
本文将飞行棋建模为马尔可夫决策过程，利用动态规划的期望最大化原理求解每步最优行动。通过定义包含进度奖励、撞击收益和风险惩罚的复合奖励函数，算法能够在起飞、推进、撞击等决策中做出平衡的选择。蒙特卡洛模拟进一步验证了单步DP策略的长期有效性。
对于完整游戏的最优策略，由于状态空间极其庞大，实际应用中通常结合蒙特卡洛树搜索（MCTS）与策略网络来近似求解。本文的DP框架为此类高级方法提供了坚实的理论基础。
