 We therefore obtain the model \(M[\Pri_a(p)]\) as pictured above. We note that the product update-induced mapping   \[ (M,w) \mapsto (M[A],(w,e)) \]   from the initial situation \((M,w)\) to the resultant situation \((M[A],(w,e))\) has the following effect: we go from an initial situation \((M,w)\) in which neither agent knows whether p is true to a resultant situation \((M[A],(w,e))\) in which a knows p is true but b mistakenly believes everyone’s knowledge is unchanged. This is of course just what we want of the private announcement of p to agent a. 