arXiv subjects

Bassel Tarbush

Publications and source records attributed to Bassel Tarbush.

9 recordsLinked to original sources

arXiv2026

The Computational Intractability of Not Worst Responding

Finding, counting, or determining the existence of Nash equilibria, where players must play optimally given each others' actions, are known to be computational intractable problems. We ask whether weakening optimality to the requirement that each player merely avoid worst responses -- arguably the weakest meaningful rationality criterion -- yields tractable solution concepts. We show that it does not: any solution concept with this minimal guarantee is ``as intractable'' as pure Nash equilibrium. In general games, determining the existence of no-worst-response action profiles is NP-complete, finding one is NP-hard, and counting them is #P-complete. In potential games, where existence is guaranteed, the search problem is PLS-complete. Computational intractability therefore stems not only from the requirement of optimality, but also from the requirement of a minimal rationality guarantee for each player. Moreover, relaxing the latter requirement gives rise to a tractability trade-off between the strength of individual rationality guarantees and the fraction of players satisfying them.

cs.GT↗

arXiv2026

Game connectivity and adaptive dynamics in many-action games

We study the typical structure of games in terms of their connectivity properties. A game is `connected' if it has a pure Nash equilibrium and there is a best-response path from every action profile which is not a pure Nash equilibrium to every pure Nash equilibrium; a game is generic if it has no indifferences. In previous work we showed that, among all $n$-player $k$-action generic games that admit a pure Nash equilibrium, the fraction that are connected tends to $1$ as $n$ gets sufficiently large relative to $k$. Here, we consider the large-$k$ regime, which behaves differently: we show that the connected fraction tends to $1-\zeta_n$ as $k$ gets large, where $\zeta_n>0$ is an explicit constant. Thus, a constant fraction of many-action games are \emph{not} connected. However, for $n\geq3$, $\zeta_n$ is small and tends to $0$ rapidly with $n$, so as $n$ increases all but a vanishingly small fraction of many-player-many-action games are connected. Since connectedness is conducive to equilibrium convergence, we find a simple adaptive dynamic that is guaranteed to converge to a pure Nash equilibrium in all but a vanishingly small fraction of generic games that have one. We rely on new probabilistic and combinatorial arguments to tackle the large-$k$ regime.

econ.TH↗

arXiv2025

Economic dynamics with differential fertility

We characterize the outcomes of a canonical deterministic model for the intergenerational transmission of capital that features differential fertility. A fertility function determines the relationship between parental capital and the number of children, and a transmission function determines the relationship between the capital of a parent and that of their children. Together these functions generate an evolving cross-sectional distribution of capital. We establish easy-to-verify conditions on the fertility and transmission functions that guarantee (a) that the dynamical system has a steady state distribution that is either atomless (exhibiting inequality) or degenerate (not exhibiting inequality), and (b) that the system converges to such states from essentially any initial distribution. Our characterization provides new insights into the link between differential fertility and long-run cross-sectional inequality, and it gives rise to novel comparative statics relating the two. We apply our results to several parametric examples and to a model of economic growth that features endogenous differential fertility.

econ.TH↗

arXiv2025

On the existence of pure epsilon-equilibrium

We show that for any $\epsilon>0$, as the number of agents gets large, the share of games that admit a pure $\epsilon$-equilibrium converges to 1. Our result holds even for pure $\epsilon$-equilibrium in which all agents, except for at most one, play a best response. In contrast, it is known that the share of games that admit a pure Nash equilibrium, that is, for $\epsilon=0$, is asymptotically $1-1/e\approx 0.63$. This suggests that very small deviations from perfect rationality, captured by positive values of $\epsilon$, suffice to ensure the general existence of stable outcomes. We also study the existence of pure $\epsilon$-equilibrium when the number of actions gets large. Our proofs rely on the probabilistic method and on the Chen-Stein method.

econ.TH↗

arXiv2024

Satisficing Equilibrium

In a satisficing equilibrium each agent $i$ plays one of her top $k_i$ actions in response to the actions of the other agents. Our concept unifies models of bounded rationality and yields predictions that differ from canonical solution concepts. We study its theoretical properties and show that it provides sharp predictions, exists in most games as well as in a broad new class of economic environments, admits standard epistemic and dynamic foundations, and is empirically falsifiable.

econ.TH↗

arXiv2023

Game Connectivity and Adaptive Dynamics

We analyse the typical structure of games in terms of the connectivity properties of their best-response graphs. Our central result shows that, among games that are `generic' (without indifferences) and that have a pure Nash equilibrium, all but a small fraction are \emph{connected}, meaning that every action profile that is not a pure Nash equilibrium can reach every pure Nash equilibrium via best-response paths. This has important implications for dynamics in games. In particular, we show that there are simple, uncoupled, adaptive dynamics for which period-by-period play converges almost surely to a pure Nash equilibrium in all but a small fraction of generic games that have one (which contrasts with the known fact that there is no such dynamic that leads almost surely to a pure Nash equilibrium in \emph{every} generic game that has one). We build on recent results in probabilistic combinatorics for our characterisation of game connectivity.

econ.TH↗

arXiv2021

Best-response dynamics, playing sequences, and convergence to equilibrium in random games

We analyze the performance of the best-response dynamic across all normal-form games using a random games approach. The playing sequence -- the order in which players update their actions -- is essentially irrelevant in determining whether the dynamic converges to a Nash equilibrium in certain classes of games (e.g. in potential games) but, when evaluated across all possible games, convergence to equilibrium depends on the playing sequence in an extreme way. Our main asymptotic result shows that the best-response dynamic converges to a pure Nash equilibrium in a vanishingly small fraction of all (large) games when players take turns according to a fixed cyclic order. By contrast, when the playing sequence is random, the dynamic converges to a pure Nash equilibrium if one exists in almost all (large) games.

econ.TH↗

arXiv2013

Agreeing on Decisions: An Analysis with Counterfactuals

Moses & Nachum ([7]) identify conceptual flaws in Bacharach's generalization ([3]) of Aumann's seminal "agreeing to disagree" result ([1]). Essentially, Bacharach's framework requires agents' decision functions to be defined over events that are informationally meaningless for the agents. In this paper, we argue that the analysis of the agreement theorem should be carried out in information structures that can accommodate for counterfactual states. We therefore develop a method for constructing such "counterfactual structures" (starting from partitional structures), and prove a new agreement theorem within such structures. Furthermore, we show that our approach also resolves the conceptual flaws in the sense that, within our framework, decision functions are always only defined over events that are informationally meaningful for the agents.

cs.GT↗

arXiv2012

Social network markets: the influence of network structure when consumers face decisions over many similar choices

In social network markets, the act of consumer choice in these industries is governed not just by the set of incentives described by conventional consumer demand theory, but by the choices of others in which an individual's payoff is an explicit function of the actions of others. We observe two key empirical features of outcomes in social networked markets. First, a highly right-skewed, non-Gaussian distribution of the number of times competing alternatives are selected at a point in time. Second, there is turnover in the rankings of popularity over time. We show here that such outcomes can arise either when there is no alternative which exhibits inherent superiority in its attributes, or when agents find it very difficult to discern any differences in quality amongst the alternatives which are available so that it is as if no superiority exists. These features appear to obtain, as a reasonable approximation, in many social network markets. We examine the impact of network structure on both the rank-size distribution of choices at a point in time, and on the life spans of the most popular choices. We show that a key influence on outcomes is the extent to which the network follows a hierarchical structure. It is the social network properties of the markets, the meso-level structure, which determine outcomes rather than the objective attributes of the products.

cs.SI↗