Infinite stage and finite stage games are considered in a tree-like graph in which a certain simultaneous game corresponds to each vertex. A definition of a strong Nash transferable equilibrium is given. In the case of infinite stage games, a regularization procedure is introduced which enables a strong transferable equilibrium to be constructed. A strong transferable equilibrium is found in explicit form for the specific case o the n-person, repeated, infinite stage "Prisoner's dilemma" game. A new class of Nash equilibria, based on the use of penalty strategies, is defined in the case of finite stage games. Explicit analytical formulae are obtained for the number of stages required for the penalty. It is shown that the payoffs in a given equilibrium exceed the payoffs in the classical absolute equilibrium. (C) 2004 Elsevier Ltd. All rights reserved.