The Rossoneri were held to a 2-2 draw at the Stadio Picchi, having first taken the lead and then gone behind.
"I compliment Livorno because they put in a very good performance," Allegri told Sky Sport Italia.
"We showed a lack of determination and it is difficult to win a match like that. We conceded the second goal from a throw-in and that is not good enough."
For breaking stories and all the great banter like us on Facebook: facebook.com/tribalfootball