feat(web): add multiplayer lobby, game, and score screen routes

- lobby.$code.tsx: waiting room with live player list via lobby:state,
  copyable lobby code, host Start Game button (disabled until 2+ players),
  sends lobby:join on connect, lobby:leave on unmount
- game.$code.tsx: in-game view, sends game:ready on mount to get current
  question, handles game:question/answer_result/finished messages,
  reuses QuestionCard component, shows round results after each answer
- MultiplayerScoreScreen: final score screen sorted by score, highlights
  winner(s) with crown, handles ties via winnerIds array, Play Again
  navigates back to lobby, Leave goes to multiplayer landing
- GameRouteSearchSchema added to shared for typed lobbyId search param
  without requiring Zod in apps/web

apps/web/src/components/multiplayer/MultiplayerScoreScreen.tsx

@@ -0,0 +1,114 @@
+import { useNavigate } from "@tanstack/react-router";
+import type { LobbyPlayer } from "@lila/shared";
+
+type MultiplayerScoreScreenProps = {
+  players: LobbyPlayer[];
+  winnerIds: string[];
+  currentUserId: string;
+  lobbyCode: string;
+};
+
+export const MultiplayerScoreScreen = ({
+  players,
+  winnerIds,
+  currentUserId,
+  lobbyCode,
+}: MultiplayerScoreScreenProps) => {
+  const navigate = useNavigate();
+
+  const sortedPlayers = [...players].sort((a, b) => b.score - a.score);
+
+  const isWinner = winnerIds.includes(currentUserId);
+  const isTie = winnerIds.length > 1;
+
+  const winnerNames = winnerIds
+    .map((id) => players.find((p) => p.userId === id)?.user.name ?? id)
+    .join(" and ");
+
+  return (
+    <div className="min-h-screen bg-linear-to-b from-purple-100 to-pink-50 flex items-center justify-center p-6">
+      <div className="bg-white rounded-2xl shadow-lg p-8 w-full max-w-md flex flex-col gap-6">
+        {/* Result header */}
+        <div className="text-center flex flex-col gap-1">
+          <h1 className="text-2xl font-bold text-purple-800">
+            {isTie ? "It's a tie!" : isWinner ? "You win! 🎉" : "Game over"}
+          </h1>
+          <p className="text-sm text-gray-500">
+            {isTie ? `${winnerNames} tied` : `${winnerNames} wins!`}
+          </p>
+        </div>
+
+        <div className="border-t border-gray-200" />
+
+        {/* Score list */}
+        <div className="flex flex-col gap-2">
+          {sortedPlayers.map((player, index) => {
+            const isCurrentUser = player.userId === currentUserId;
+            const isPlayerWinner = winnerIds.includes(player.userId);
+            return (
+              <div
+                key={player.userId}
+                className={`flex items-center justify-between rounded-lg px-4 py-3 ${
+                  isCurrentUser
+                    ? "bg-purple-50 border border-purple-200"
+                    : "bg-gray-50"
+                }`}
+              >
+                <div className="flex items-center gap-3">
+                  <span className="text-sm font-medium text-gray-400 w-4">
+                    {index + 1}.
+                  </span>
+                  <span
+                    className={`text-sm font-medium ${
+                      isCurrentUser ? "text-purple-800" : "text-gray-700"
+                    }`}
+                  >
+                    {player.user.name}
+                    {isCurrentUser && (
+                      <span className="text-xs text-purple-400 ml-1">
+                        (you)
+                      </span>
+                    )}
+                  </span>
+                  {isPlayerWinner && (
+                    <span className="text-xs text-yellow-500 font-medium">
+                      👑
+                    </span>
+                  )}
+                </div>
+                <span className="text-sm font-bold text-gray-700">
+                  {player.score} pts
+                </span>
+              </div>
+            );
+          })}
+        </div>
+
+        <div className="border-t border-gray-200" />
+
+        {/* Actions */}
+        <div className="flex flex-col gap-3">
+          <button
+            className="rounded bg-purple-600 px-4 py-2 text-white hover:bg-purple-500"
+            onClick={() => {
+              void navigate({
+                to: "/multiplayer/lobby/$code",
+                params: { code: lobbyCode },
+              });
+            }}
+          >
+            Play Again
+          </button>
+          <button
+            className="rounded bg-gray-100 px-4 py-2 text-gray-700 hover:bg-gray-200"
+            onClick={() => {
+              void navigate({ to: "/multiplayer" });
+            }}
+          >
+            Leave
+          </button>
+        </div>
+      </div>
+    </div>
+  );
+};