arXiv subjects

Steve Alpern

Publications and source records attributed to Steve Alpern.

16 recordsLinked to original sources

arXiv2026

Accelerating Network-Agent Dispersion: Territorial Behavior and Directionally Biased Lazy Random Walks

Territorial behavior can greatly accelerate decentralized agent dispersion on networks. This paper studies a network-agent dispersion problem in which m autonomous agents move in discrete time on a connected graph and seek a configuration in which no two agents occupy the same node. We focus on the dispersion case m = n, where successful configurations contain exactly one agent per node. In the baseline model, each agent follows a lazy random walk with a common laziness parameter p. This process defines a finite absorbing Markov chain, and the expected absorption time is used to measure dispersion efficiency. We introduce two local behavioral extensions: territorial behavior, in which an agent that is alone at a node claims that node and repels later arrivals, and directional bias, in which agents share a preferred direction of movement on paths and cycles. Exact calculations on three-agent path and cycle networks and Monte Carlo simulations on larger instances show that territorial behavior substantially reduces expected dispersion time, with larger relative reductions as network size increases. Directional bias alone has limited effect in most small-network cases, but when combined with territorial behavior it can produce large additional speedups. In particular, the simulations show reductions of 99.22% on L100 and 97.48% on C100 when all agents start from one node. These results show how simple local movement rules can strongly affect global dispersion time in decentralized networked multi-agent systems.

stat.AP↗

arXiv2026

Competitive Search with a Faulty Satnav (GPS): When Probability Matching is Rational

A divisible treasure is located at a node $H$ of a network. From a given start node a group of $n$ Searchers each seek to reach $H$ first, dividing the treasure equally with the other first arrivers. This type of search game is called competitive search, where the conflict is not between hider and searcher but between searchers. Examples are search for oil deposits or for a pilot downed over enemy territory. In our model, the Searchers have a common Satnav (GPS) which points to $H$ at each branch node with a known probability $p<1$ and each Searcher chooses the probability $q$ with which they follow the pointer. We consider a family of star graphs where the Searchers start at the center and $H$ lies at one of the $k$ leaf nodes. We show that for all parameter values $n,k,p,$ there is a unique trust probability $q$ which forms a symmetric equilibrium. The equilibrium $q$ is increasing in $p,$ decreasing in $n$ and increasing in $k$. Furthemore for fixed $k$ and $p$ we have $q$ equal to $p$ in the limit of $n$ tending to infinity. This last fact is a new example where what is known in behavioural science as probability matching is in fact rational.

cs.GT↗

arXiv2024

Search for an Immobile Hider on a Binary Tree with Unreliable Locational Information

Adversarial search of a network for an immobile Hider (or target) was introduced and solved for rooted trees by Gal (1979). In this zero-sum game, a Hider picks a point to hide on the tree and a Searcher picks a unit speed trajectory starting at the root. The payoff (to the Hider) is the search time. In Gal's model (and many subsequent investigations), the Searcher receives no additional information after the Hider chooses his location. In reality, the Searcher will often receive such locational information. For homeland security, mobile sensors on vehicles have been used to locate radioactive material stashed in an urban environment. In a military setting, mobile sensors can detect chemical signatures from land mines. In predator-prey search, the predator often has specially attuned senses (hearing for wolves, vision for eagles, smell for dogs, sonar for bats, pressure sensors for sharks) that may help it locate the prey. How can such noisy locational information be used by the Searcher to modify her route? We model such information as signals which indicate which of two branches of a binary tree should be searched first, where the signal has a known accuracy p<1. Our solution calculates which branch (at every branch node) is favored, meaning it should always be searched first when the signal is in that direction. When the signal is in the other direction, we calculate the probability the signal should be followed. Compared to the optimal Hider strategy in the classic search game of Gal, the Hider's optimal distribution for this model is more skewed towards leaf nodes that are further from the root.

cs.DM↗

arXiv2022

Continuous Patrolling Games

We study a patrolling game played on a network $Q$, considered as a metric space. The Attacker chooses a point of $Q$ (not necessarily a node) to attack during a chosen time interval of fixed duration. The Patroller chooses a unit speed path on $Q$ and intercepts the attack (and wins) if she visits the attacked point during the attack time interval. This zero-sum game models the problem of protecting roads or pipelines from an adversarial attack. The payoff to the maximizing Patroller is the probability that the attack is intercepted. Our results include the following: (i) a solution to the game for any network $Q$, as long as the time required to carry out the attack is sufficiently short, (ii) a solution to the game for all tree networks that satisfy a certain condition on their extremities, and (iii) a solution to the game for any attack duration for stars with one long arc and the remaining arcs equal in length. We present a conjecture on the solution of the game for arbitrary trees and establish it in certain cases.

cs.DM↗

arXiv2021

The Faulty GPS Problem: Shortest Time Paths in Networks with Unreliable Directions

This paper optimizes motion planning when there is a known risk that the road choice suggested by a Satnav (GPS) is not on a shortest path. At every branch node of a network Q, a Satnav (GPS) points to the arc leading to the destination, or home node, H - but only with a high known probability p. Always trusting the Satnav's suggestion may lead to an infinite cycle. If one wishes to reach H in least expected time, with what probability q=q(Q,p) should one trust the pointer (if not, one chooses randomly among the other arcs)? We call this the Faulty Satnav (GPS) Problem. We also consider versions where the trust probability q can depend on the degree of the current node and a `treasure hunt' where two searchers try to reach H first. The agent searching for H need not be a car, that is just a familiar example -- it could equally be a UAV receiving unreliable GPS information. This problem has its origin not in driver frustration but in the work of Fonio et al (2017) on ant navigation, where the pointers correspond to pheromone markers pointing to the nest. Neither the driver or ant will know the exact process by which a choice (arc) is suggested, which puts the problem into the domain of how much to trust an option suggested by AI.

cs.AI↗

arXiv2021

Social Distancing, Gathering, Search Games: Mobile Agents on Simple Networks

During epidemics, the population is asked to Socially Distance, with pairs of individuals keeping two meters apart. We model this as a new optimization problem by considering a team of agents placed on the nodes of a network. Their common aim is to achieve pairwise graph distances of at least D, a state we call socially distanced. (If D=1, they want to be at distinct nodes; if D=2 they want to be non-adjacent.) We allow only a simple type of motion called a Lazy Random Walk: with probability p (called the laziness parameter), they remain at their current node next period; with complementary probability 1-p , they move to a random adjacent node. The team seeks the common value of p which achieves social distance in the least expected time, which is the absorption time of a Markov chain. We observe that the same Markov chain, with different goals (absorbing states), models the gathering, or multi-rendezvous problem (all agents at the same node). Allowing distinct laziness for two types of agents (searchers and hider), extends the existing literature on predator-prey search games to multiple searchers. We consider only special networks: line, cycle and grid. Keywords: epidemic, random walk, dispersion, rendezvous search

math.OC↗

arXiv2021

Optimizing Voting Order on Sequential Juries: A Median Voter Theorem and Beyond

We consider an odd-sized "jury", which votes sequentially between two states of Nature (say A and B, or Innocent and Guilty) with the majority opinion determining the verdict. Jurors have private information in the form of a signal in [-1,+1], with higher signals indicating A more likely. Each juror has an ability in [0,1], which is proportional to the probability of A given a positive signal, an analog of Condorcet's p for binary signals. We assume that jurors vote honestly for the alternative they view more likely, given their signal and prior voting, because they are experts who want to enhance their reputation (after their vote and actual state of Nature is revealed). For a fixed set of jury abilities, the reliability of the verdict depends on the voting order. For a jury of size three, the optimal ordering is always as follows: middle ability first, then highest ability, then lowest. For sufficiently heterogeneous juries, sequential voting is more reliable than simultaneous voting and is in fact optimal (allowing for non-honest voting). When average ability is fixed, verdict reliability is increasing in heterogeneity. For medium-sized juries, we find through simulation that the median ability juror should still vote first and the remaining ones should have increasing and then decreasing abilities.

econ.TH↗

arXiv2020

The Uniformed Patroller Game

Patrolling Games were introduced by Alpern, Morton and Papadaki (2011) to model the adversarial problem where a mobile Patroller can thwart an attack at some location only by visiting it during the attack period, which has a prescribed integer duration m. Here, we modify the problem by allowing the Attacker to go to his planned attack location early and observe the presence or the absence there of the Patroller (who wears a uniform). To avoid being too predictable, the Patroller may sometimes remain at her base when she could have been visiting a possible attack location. The Attacker can then choose to delay attacking for some number of periods d after the Patroller leaves his planned attack location. As shown here, this extra information for the Attacker can reduce thwarted attacks by as much as a factor of four in specific models. Our main finding, is that the attack should begin in the second period the Patroller is away (d = 2) and that the Patroller should never attack the same location in consecutive periods.

cs.CR↗

arXiv2020

Search and Delivery Man Problems: When Are Depth-First Paths Optimal?

Let h be a probability measure on the nodes and arcs of a network Q, viewed either as the location of a hidden object to be found or as the continuous distribution of customers receiving packages. We wish to find a trajectory starting from a specified root, or depot O that minimizes the expected search or delivery time. We call such a trajectory optimal. When Q is a tree, we ask for which h there is an optimal trajectory that is depth-first, and we find sufficient conditions and in some cases necessary and sufficient conditions on h. A consequence of our analysis is a determination of the optimal depot location in the Delivery Man Problem, correcting an error in the literature. We concentrate mainly on the search problem, with the Delivery Man Problem arising as a special case.

math.OC↗

arXiv2019

A Game Model of Search and Pursuit

Shmuel Gal and Jerome Casas have recently introduced a game theoretic model that combines search and pursuit by a predator for a prey animal. The prey (hider) can hide in a finite number of locations. The predator (searcher) can inspect any k of these locations. If the prey is not in any of these, the prey wins. If the prey is found at an inspected location, a pursuit begins which is successful for the predator with a known capture probability which depends on the location. We modify the problem so that each location takes a certain time to inspect and the predator has total inspection time k. We also consider a repeated game model where the capture probabilities only become known to the players over time, as each successful escape from a location lowers its perceived value capture probability.

cs.GT↗

arXiv2019

A Functional Equation of Tail-balance for Continuous Signals in the Condorcet Jury Theorem

Consider an odd-sized jury, which determines a majority verdict between two equiprobable states of Nature. If each juror independently receives a binary signal identifying the correct state with identical probability $p$, then the probability of a correct verdict tends to one as the jury size tends to infinity (Condorcet, 1785). Recently, the first two authors developed a model where jurors sequentially receive signals from an interval according to a distribution, which depends on the state of Nature and on the juror's "ability", and vote sequentially. This paper shows that to mimic Condorcet's binary signal, such a distribution must satisfy a functional equation related to tail-balance, that is, to the ratio $α(t)$ of the probability that a mean-zero random variable satisfies $X >t$ given that $|X|>t$. In particular, we show that under natural symmetry assumptions the tail-balances $α(t)$ uniquely determine the distribution.

math.PR↗

arXiv2019

The Uniformed Patroller Game

In the recently introduced network patrolling game, an Attacker carries out an attack on a node of her choice, for a given number m of consecutive periods. The parameter m indicates the difficulty of the attack at a given node. To thwart such an attack, the Patroller adopts a walk on the network, hoping to be at the attacked node during one of the m periods. If this occurs, the attack is interrupted and the Patroller wins; otherwise the Attacker wins. In the original setting, the Attacker has no knowledge of the Patroller's location at any time. Here, to model the important alternative where the Patroller can be identified when he is at the Attacker's node (e.g. the Patroller wears a uniform), we allow the Attacker to initiate her attack after waiting for a chosen number d of consecutive periods in which the Patroller has been away. We solve this version of the game for various networks: star, line, circle and a mixture. We restrict the Patroller to Markovian strategies, which cover the whole network.

math.OC↗

arXiv2017

A Short Solution to the Many-Player Silent Duel with Arbitrary Consolation Prize

The classical constant-sum 'silent duel' game had two antagonistic marksmen walking towards each other. A more friendly formulation has two equally skilled marksmen approaching targets at which they may silently fire at distances of their own choice. The winner, who gets a unit prize, is the marksman who hits his target at the greatest distance; if both miss, they share the prize (each gets a 'consolation prize' of one half). In another formulation, if they both miss they each get zero. More generally we can consider more than two marksmen and an arbitrary consolation prize. This non-constant sum game may be interpreted as a research tournament where the entrant who successfully solves the hardest problem wins the prize. We give the first complete solution to the many-player problem with arbitrary consolation prize: moreover (by taking particular values for the consolation prize), our theorem incorporates various special results in the literature, and our proof is simpler than any of these.

cs.GT↗

arXiv2017

Periodic Patrols on the Line and Other Networks

We consider a patrolling game on a graph recently introduced by Alpern et al. (2011) where the Patroller wins if he is at the attacked node while the attack is taking place. This paper studies the periodic patrolling game in the case that the attack duration is two periods. We show that if the Patroller's period is even, the game can be solved on any graph by finding the fractional covering number and fractional independence number of the graph. We also give a complete solution to the periodic patrolling game on line graphs of arbitrary size, extending the work of Papadaki et al. (2016) to the periodic domain. This models the patrolling problem on a border or channel, which is related to a classical problem of operational research going back to Morse and Kimball (1951). A periodic patrol is required to start and end at the same location, for example the place where the Patroller leaves his car to begin a foot patrol.

math.OC↗

arXiv2016

Search-and-Rescue Rendezvous

We consider a new type of asymmetric rendezvous search problem in which Agent II needs to give Agent I a `gift' which can be in the form of information or material. The gift can either be transfered upon meeting, as in traditional rendezvous, or it can be dropped o? by II at a location he passes, in the hope it will be found by I. The gift might be a water bottle for a traveller lost in the desert; a supply cache for Lieutenant Scott in the Antarctic; or important information (left as a gift). The common aim of the two agents is to minimize the time taken for I to either meet II or find the gift. We find optimal agent paths and droppo? times when the search region is a line, the initial distance between the players is known and one or both of the players can leave gifts. When there are no gifts this is the classical asymmetric rendezvous problem solved by Alpern and Gal in 1995 [10]. We exhibit strategies solving these various problems and use a `rendezvous algorithm' to establish their optimality.

cs.GT↗

arXiv2016

Constant Factor Approximate Solutions for Expanding Search on General Networks

We study the classical problem introduced by R. Isaacs and S. Gal of minimizing the time to find a hidden point $H$ on a network $Q$ moving from a known starting point. Rather than adopting the traditional continuous unit speed path paradigm, we use the ``expanding search'' paradigm recently introduced by the authors. Here the regions $S\left( t\right) $ that have been searched by time $t$ are increasing from the starting point and have total length $t$. Roughly speaking the search follows a sequence of arcs $a_{i}$ such that each one starts at some point of an earlier one. This type of search is often carried out by real life search teams in the hunt for missing persons, escaped convicts, terrorists or lost airplanes. The paper which introduced this type of search solved the adversarial problem (where $H$ is hidden to take a long time to find) for the cases where $Q$ is a tree or is 2-arc-connected. This paper solves the game on some additional families of networks. However the main contribution is to give strategy classes which can be used on any network and have expected search times which are within a factor close to 1 of the value of the game (minimax search time). We identify cases where our strategies are in fact optimal.

math.OC↗