fix(know-your-world): correctly identify local player for cursor sharing

Use player.isLocal flag to find the player that belongs to the current
viewer, instead of incorrectly using the first player in the game.

This fixes network cursors not showing because the localPlayerId was
wrong for non-host players.

🤖 Generated with [Claude Code](https://claude.com/claude-code)

Co-Authored-By: Claude <noreply@anthropic.com>

apps/web/src/arcade-games/know-your-world/components/PlayingPhase.tsx

@@ -1,5 +1,6 @@
 'use client'
 
+import { useMemo } from 'react'
 import { css } from '@styled/css'
 import { Panel, PanelGroup, PanelResizeHandle } from 'react-resizable-panels'
 import { useKnowYourWorld } from '../Provider'
@@ -12,11 +13,20 @@ import { useGameMode } from '@/lib/arcade/game-sdk'
 export function PlayingPhase() {
   const { state, clickRegion, giveUp, otherPlayerCursors, sendCursorUpdate } = useKnowYourWorld()
   const { data: viewerId } = useViewerId()
-  const { activePlayers } = useGameMode()
+  const { activePlayers, players } = useGameMode()
 
-  // Find the local player ID (first player that belongs to this viewer)
-  // In most cases, each user controls one player
-  const localPlayerId = Array.from(activePlayers)[0] || ''
+  // Find the local player ID (the player that belongs to this viewer)
+  // Look for a player marked as isLocal, or fall back to first active player
+  const localPlayerId = useMemo(() => {
+    for (const playerId of activePlayers) {
+      const player = players.get(playerId)
+      if (player?.isLocal) {
+        return playerId
+      }
+    }
+    // Fallback: return first active player (shouldn't happen in normal flow)
+    return Array.from(activePlayers)[0] || ''
+  }, [activePlayers, players])
 
   const mapData = getFilteredMapDataSync(
     state.selectedMap,