Your description of incomplete information is off. What you give as the definition of incomplete information is one type of imperfect information, where nature is added as a player.
A game has incomplete information when one player has more information than another about payoffs. Since Harsanyi, incomplete information has been seen as a special case of imperfect information with nature randomly assigning types to each player according to a commonly known distribution and payoffs given types being commonly known.
Your description of incomplete information is off. What you give as the definition of incomplete information is one type of imperfect information, where nature is added as a player.
A game has incomplete information when one player has more information than another about payoffs. Since Harsanyi, incomplete information has been seen as a special case of imperfect information with nature randomly assigning types to each player according to a commonly known distribution and payoffs given types being commonly known.
Thanks, you’re right and I didn’t understand what the Harsanyi transformation was. Edited the post again.