We study teams of agents that play against Nature towards achieving a common objective. The agents are assumed to have imperfect information due to partial observability, and have no communication during the play of the game. We propose a natural notion of higher-order knowledge of agents. Based on this notion, we define a class of knowledgebased strategies, and consider the problem of synthesis of strategies of this class. We introduce a multi-agent extension, MKBSC, of the well-known knowledge-based subset construction applied to such games. Its iterative applications turn out to compute higherorder knowledge of the agents. We show how the MKBSC can be used for the design of knowledge-based strategy profiles, and investigate the transfer of existence of such strategies between the original game and in the iterated applications of the MKBSC, under some natural assumptions. We also relate and compare the "intensional" view on knowledge-based strategies based on explicit knowledge representation and update, with the "extensional" view on finite memory strategies based on finite transducers and show that, in a certain sense, these are equivalent.