Search

This paper treats of discrete-time Markov decision processes with recursive utility defined using a non-linear aggregator of Epstein-Zin type with the Kreps-Porteus certainty equivalent operator. According to the classification introduced by Marinacci and Montrucchio, the aggregators that we consider are of Thompson type. We focus on the existence and uniqueness of a solution to the Bellman equation. If the action spaces of an agent are singletons, the problem concerns the existence and uniqueness of a recursive utility in the sense of Koopmans. The models of this type were recently studied by Ren and Stachurski who applied Du's fixed point theorem for increasing and convex or concave operators acting on an ordered Banach space. We show that, in two cases considered by Ren and Stachurski, the existence and uniqueness of solution to the Bellman equation follows from the Banach contraction mapping principle. We allow the per-period utilities in the models to be unbounded. Therefore, we work with the weighted supremum norm. Since we apply the Banach fixed point theorem for contraction mappings acting on a standard complete metric space, we need not assume any boundary conditions, which are present when the Thompson metric or Du's theorem are used. Moreover, our results give better bounds for the geometric convergence of the value iteration algorithm leading to the solution of the Bellman equation than those obtained by Du's fixed point theorem. Some illustrative examples are given. This paper also contains also new results on the existence and uniqueness of solutions to the Bellman equation in two cases of parameters in the Epsten-Zin aggregator, where the application of Du's theorem is not possible. Our results are based on a relatively simple reformulation of the problem exploiting the structure of Epstein-Zin preferences.

When we go for a walk with friends, we can observe an interesting effect: From step lengths to arm movements - our movements unconsciously align; they synchronize. Prior research found that this synchronization is a crucial aspect of human relations that strengthens social cohesion and trust. Generalizing from these findings in synchronization theory, we propose a dynamical approach that can be applied in the design of non-humanoid robots to increase trust. We contribute the results of a controlled experiment with 51 participants exploring our concept in a between-subjects design. For this, we built a prototype of a simple non-humanoid robot that can bend to follow human movements and vary the movement synchronization patterns. We found that synchronized movements lead to significantly higher ratings in an established questionnaire on trust between people and automation but did not influence the willingness to spend money in a trust game., Comment: To appear in Proceedings of the 2023 CHI Conference on Human Factors in Computing Systems (CHI 23), April 23-28, 2023, Hamburg, Germany. ACM, New York, NY, USA, 14 pages

The principle of similarity, or homophily, is often used to explain patterns observed in complex networks such as transitivity and the abundance of triangles (3-cycles). However, many phenomena from division of labor to protein-protein interactions (PPI) are driven by complementarity (differences and synergy). Here we show that the principle of complementarity is linked to the abundance of quadrangles (4-cycles) and dense bipartite-like subgraphs. We link both principles to their characteristic motifs and introduce two families of coefficients of: (1) structural similarity, which generalize local clustering and closure coefficients and capture the full spectrum of similarity-driven structures; (2) structural complementarity, defined analogously but based on quadrangles instead of triangles. Using multiple social and biological networks, we demonstrate that the coefficients capture structural properties related to meaningful domain-specific phenomena. We show that they allow distinguishing between different kinds of social relations as well as measuring an increasing structural diversity of PPI networks across the tree of life. Our results indicate that some types of relations are better explained by complementarity than homophily, and may be useful for improving existing link prediction methods. We also introduce a Python package implementing efficient algorithms for calculating the proposed coefficients., Comment: Authors' copy of the published version

In this paper, we consider constrained discounted stochastic games with a countably generated state space and norm continuous transition probability having a density function. We prove existence of approximate stationary equilibria and stationary weak correlated equilibria. Our results imply the existence of stationary Nash equilibrium in $ARAT$ stochastic games.

In this paper, we consider a large class of constrained non-cooperative stochastic Markov games with countable state spaces and discounted cost criteria. In one-player case, i.e., constrained discounted Markov decision models, it is possible to formulate a static optimisation problem whose solution determines a stationary optimal strategy (alias control or policy) in the dynamical infinite horizon model. This solution lies in the compact convex set of all occupation measures induced by strategies, defined on the set of state-action pairs. In case of n-person discounted games the occupation measures are induced by strategies of all players. Therefore, it is difficult to generalise the approach for constrained discounted Markov decision processes directly. It is not clear how to define the domain for the best-response correspondence whose fixed point induces a stationary equilibrium in the Markov game. This domain should be the Cartesian product of compact convex sets in locally convex topological vector spaces. One of our main results shows how to overcome this difficulty and define a constrained non-cooperative static game whose Nash equilibrium induces by a stationary Nash equilibrium in the Markov game. This is done for games with bounded cost functions and positive initial state distribution. An extension to a class of Markov games with unbounded costs and arbitrary initial state distribution relies on approximation of the unbounded game by bounded ones with positive initial state distributions. In the unbounded case, we assume the uniform integrability of the discounted costs with respect to all probability measures induced by strategies of the players, defined on the space of plays (histories) of the game. Our assumptions are weaker than those applied in earlier works on discounted dynamic programming or stochastic games using so-called weighted norm approaches.

In this paper, we study a Markov decision process with a non-linear discount function and with a Borel state space. We define a recursive discounted utility, which resembles non-additive utility functions considered in a number of models in economics. Non-additivity here follows from non-linearity of the discount function. Our study is complementary to the work of Ja\'skiewicz, Matkowski and Nowak (Math. Oper. Res. 38 (2013), 108-121), where also non-linear discounting is used in the stochastic setting, but the expectation of utilities aggregated on the space of all histories of the process is applied leading to a non-stationary dynamic programming model. Our aim is to prove that in the recursive discounted utility case the Bellman equation has a solution and there exists an optimal stationary policy for the problem in the infinite time horizon. Our approach includes two cases: $(a)$ when the one-stage utility is bounded on both sides by a weight function multiplied by some positive and negative constants, and $(b)$ when the one-stage utility is unbounded from below.

It is hard to overstate the importance that the concept of symmetry has had in every field of physics, a fact alluded to by the Nobel Prize winner P.W. Anderson, who once wrote that physics is the study of symmetry. Whereas the idea of symmetry is widely used in science in general, very few (if not almost no) applications has found its way into the field of finance. Still, the phenomenon appears relevant in terms of for example the symmetry of strategies that can happen in the decision making to buy or sell financial shares. Game theory is therefore one obvious avenue where to look for symmetry, but as will be shown, also technical analysis and long term economic growth could be phenomena which show the hallmark of a symmetry, Comment: 27 pages, 9 figures