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Arkadi Predtetchinski

Publications and source records attributed to Arkadi Predtetchinski.

10 recordsLinked to original sources
arXiv2026

The Expected Number of Pairwise Stable Networks

This paper studies probabilistic properties of pairwise stability for a network model where individual utilities are random variables. We study the probability that a given network is pairwise stable and the expected number of pairwise stable networks. We provide a closed-form solution for the latter number. As the evaluation of the exact expression is computationally challenging for large populations, we provide tractable lower and upper bounds for this expression which allow us to pin down the asymptotic behavior of the expected number of pairwise stable networks up to a multiplicative constant. This asymptotic behavior is described by the number of networks $ 2^{n(n-1)/2} $ times $ (2/n+1)^{n} $, a sequence that tends to infinity fast. We normalize the number of pairwise stable networks by this sequence and show that the variance of the normalized number of pairwise stable networks converges to zero as $ n $ tends to infinity. We conclude that almost surely the number of pairwise stable networks tends to infinity, while the fraction of pairwise stable networks tends to $ 0 $ as $ n $ goes to infinity.

econ.TH↗
arXiv2026

Optimal strategies in Markov decision processes with finitely additive evaluations

We study infinite-horizon Markov decision processes (MDPs) where the decision maker evaluates each of her strategies by aggregating the infinite stream of expected stage-rewards. The crucial feature of our approach is that the aggregation is performed by means of a given diffuse charge (a diffuse finitely additive probability measure) on the set of stages. The results of Neyman [2023] imply that in this setting, in every MDP with finite state and action spaces, the decision maker has a pure optimal strategy as long as the diffuse charge satisfies the time value of money principle. His result raises the question of existence of an optimal strategy without additional assumptions on the aggregation charge. We answer this question in the negative with a counterexample. With a delicately constructed aggregation charge, the MDP has no optimal strategy at all, neither pure nor randomized.

math.OC↗
arXiv2024

Zero-one Laws for a Control Problem with Random Action Sets

In many control problems there is only limited information about the actions that will be available at future stages. We introduce a framework where the Controller chooses actions $a_{0}, a_{1}, \ldots$, one at a time. Her goal is to maximize the probability that the infinite sequence $(a_{0}, a_{1}, \ldots)$ is an element of a given subset $G$ of $\mathbb{N}^{\mathbb{N}}$. The set $G$, called the goal, is assumed to be a Borel tail set. The Controller's choices are restricted: having taken a sequence $h_{t} = (a_{0}, \ldots, a_{t-1})$ of actions prior to stage $t \in \mathbb{N}$, she must choose an action $a_{t}$ at stage $t$ from a non-empty, finite subset $A(h_{t})$ of $\mathbb{N}$. The set $A(h_{t})$ is chosen from a distribution $p_{t}$, independently over all $t \in \mathbb{N}$ and all $h_{t} \in \mathbb{N}^{t}$. We consider several information structures defined by how far ahead into the future the Controller knows what actions will be available. In the special case where all the action sets are singletons (and thus the Controller is a dummy), Kolmogorov's 0-1 law says that the probability for the goal to be reached is 0 or 1. We construct a number of counterexamples to show that in general the value of the control problem can be strictly between 0 and 1, and derive several sufficient conditions for the 0-1 ``law" to hold.

math.OC↗
arXiv2024

Approximations of expectations under infinite product measures

We are given a bounded Borel-measurable real-valued function on a product of countably many Polish spaces, and a product probability measure. We are interested in points in the product space that can be used to approximate the expected value of this function. We define two notions. A point is called a weak $\epsilon$-approximation, where $\epsilon \geq 0$, if the Dirac measure on this point, except in finitely many coordinates where another measure can be taken, gives an expected value that is $\epsilon$-close to the original expected value. A point is called a strong $\epsilon$-approximation if the same holds under the restriction that in those finitely many coordinates the measure is equal to the original one. We prove that both the set of weak 0-approximation points and the set of strong $\epsilon$-approximation points, for any $\epsilon>0$, have measure 1 under the original measure. Finally, we provide two applications: (i) in Game Theory on the minmax guarantee levels of the players in games with infinitely many players, and (ii) in Decision Theory on the set of feasible expected payoffs in infinite duration problems.

math.PR↗
arXiv2022

Regularity of the minmax value and equilibria in multiplayer Blackwell games

A real-valued function $\varphi$ that is defined over all Borel sets of a topological space is \emph{regular} if for every Borel set $W$, $\varphi(W)$ is the supremum of $\varphi(C)$, over all closed sets $C$ that are contained in $W$, and the infimum of $\varphi(O)$, over all open sets $O$ that contain $W$. We study Blackwell games with finitely many players. We show that when each player has a countable set of actions and the objective of a certain player is represented by a Borel winning set, that player's minmax value is regular. We then use the regularity of the minmax value to establish the existence of $\varepsilon$-equilibria in two distinct classes of Blackwell games. One is the class of $n$-player Blackwell games where each player has a finite action space and an analytic winning set, and the sum of the minmax values over the players exceeds $n-1$. The other class is that of Blackwell games with bounded upper semi-analytic payoff functions, history-independent finite action spaces, and history-independent minmax values. For the latter class, we obtain a characterization of the set of equilibrium payoffs.

math.OC↗
arXiv2021

Random perfect information games

The paper proposes a natural measure space of zero-sum perfect information games with upper semicontinuous payoffs. Each game is specified by the game tree, and by the assignment of the active player and of the capacity to each node of the tree. The payoff in a game is defined as the infimum of the capacity over the nodes that have been visited during the play. The active player, the number of children, and the capacity are drawn from a given joint distribution independently across the nodes. We characterize the cumulative distribution function of the value $v$ using the fixed points of the so-called value generating function. The characterization leads to a necessary and sufficient condition for the event $v \geq k$ to occur with positive probability. We also study probabilistic properties of the set of Player I's $k$-optimal strategies and the corresponding plays.

cs.GT↗
arXiv2020

Games characterizing limsup functions and Baire class 1 functions

We consider a real-valued function $f$ defined on the set of infinite branches $X$ of a countably branching pruned tree $T$. The function $f$ is said to be a \textit{limsup function} if there is a function $u \colon T \to \mathbb{R}$ such that $f(x) = \limsup_{t \to \infty} u(x_{0},\dots,x_{t})$ for each $x \in X$. We study a game characterization of limsup functions, as well as a novel game characterization of functions of Baire class 1.

math.GN↗
arXiv2019

Reachability and safety objectives in Markov decision processes on long but finite horizons

We consider discrete-time Markov decision processes in which the decision maker is interested in long but finite horizons. First we consider reachability objective: the decision maker's goal is to reach a specific target state with the highest possible probability. Formally, strategy $\sigma$ overtakes another strategy $\sigma'$, if the probability of reaching the target state within horizon $t$ is larger under $\sigma$ than under $\sigma'$, for all sufficiently large $t\in\NN$. We prove that there exists a pure stationary strategy that is not overtaken by any pure strategy nor by any stationary strategy, under some condition on the transition structure and respectively under genericity. A strategy that is not overtaken by any other strategy, called an overtaking optimal strategy, does not always exist. We provide sufficient conditions for its existence. Next we consider safety objective: the decision maker's goal is to avoid a specific state with the highest possible probability. We argue that the results proven for reachability objective extend to this model. We finally discuss extensions of our results to two-player zero-sum perfect information games.

math.OC↗
arXiv2019

The doubling metric and doubling measures

We introduce the so--called doubling metric on the collection of non--empty bounded open subsets of a metric space. Given a subset $U$ of a metric space $X$, the predecessor $U_{*}$ of $U$ is defined by doubling the radii of all open balls contained inside $U$, and taking their union. If $U$ is open, the predecessor of $U$ is an open set containing $U$. The directed doubling distance between $U$ and another subset $V$ is the number of times that the predecessor operation needs to be applied to $U$ to obtain a set that contains $V$. Finally, the doubling distance between $U$ and $V$ is the maximum of the directed distance between $U$ and $V$ and the directed distance between $V$ and $U$.

math.GN↗