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In this note, we prove the existence of a Markov perfect equilibrium in a non-stationary version of a paternalistic bequest game. The method we advocate is general and allows to study models with unbounded state space and unbounded utility functions. We cover both, the stochastic and deterministic cases. We provide a characterization of the set of all Markov perfect equilibria by means of a set-valued recursive equation involving the best response operator. In the stationary case, we show that there exists a set of strategies that is invariant under the best response mapping.

The existence of Markov perfect equilibria for a deterministic bequest game with bounded from below utility function was established independently by Bernheim and Ray [2] and Leininger [8] . The aim of this paper is to provide a simpler proof that, in contrast to [2] , [8] , also works for models with unbounded, e.g., isoelastic utility functions. Our approach employs the concept of strict single crossing property, introduced by Milgrom and Shannon in [10] , and includes supermodular functions as special cases.

We examine the average minimax control problem on a general state space with an unbounded cost function. The controller is uncertain about his model in the sense that he deals with a range of alternative models as possibly true. The aim of this paper is to establish the optimality inequality and to provide a minimax strategy for the controller. A new feature in our approach is an application of a generalised Tauberian relation that, in turn, allows us to impose relatively weak assumptions.

In this paper we study a Markov decision model with quasi-hyperbolic discounting and transition probability function depending on an unknown parameter. Assuming that the set of parameters is finite, the sets of states and actions are Borel and the transition probabilities satisfy some additivity conditions and are atomless, we prove the existence of a non-randomised robust Markov perfect equilibrium.

In this paper, we study discounted Markov decision processes on an uncountable state space. We allow a utility (reward) function to be unbounded both from above and below. A new feature in our approach is an easily verifiable rate of growth condition introduced for a positive part of the utility function. This assumption, in turn, enables us to prove the convergence of a value iteration algorithm to a solution to the Bellman equation. Moreover, by virtue of the optimality equation we show the existence of an optimal stationary policy.

We consider Markov control processes with Borel state space and Feller transition probabilities, satisfying some generalized geometric ergodicity conditions. We provide a new theorem on the existence of a solution to the average cost optimality equation.