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Athanasios Kehagias

Publications and source records attributed to Athanasios Kehagias.

13 recordsLinked to original sources
arXiv2025

Markov Chains of Evolutionary Games with a Small Number of Players

We construct and study the transition probability matrix of evolutionary games in which the number of players is finite (and relatively small) of such games. We use a simplified version of the population games studied by Sandholm. After laying out a general framework we concentrate on specific examples, involving the Iterated Prisoner's Dilemma, the Iterated Stag Hunt, and the Rock-Paper-Scissors game. Also we consider several revision protocols: Best Response, Pairwise Comparison, Pairwise Proportional Comparison etc. For each of these we explicitly construct the MC transition probability matrix and study its properties.

cs.GT↗
arXiv2023

On the Nash Equilibria of a Simple Discounted Duel

We formulate and study a two-player static duel game as a nonzero-sum discounted stochastic game. Players $P_{1},P_{2}$ are standing in place and, in each turn, one or both may shoot at the other player. If $P_{n}$ shoots at $P_{m}$ ($m\neq n$), either he hits and kills him (with probability $p_{n}$) or he misses him and $P_{m}$ is unaffected (with probability $1-p_{n}$). The process continues until at least one player dies; if nobody ever dies, the game lasts an infinite number of turns. Each player receives unit payoff for each turn in which he remains alive; no payoff is assigned to killing the opponent. We show that the the always-shooting strategy is a NE but, in addition, the game also possesses cooperative (i.e., non-shooting) Nash equilibria in both stationary and nonstationary strategies. A certain similarity to the repeated Prisoner's Dilemma is also noted and discussed.

cs.DM↗
arXiv2019

On positionality of trigger strategies Nash Equilibria in SCAR

We study the positionality of \emph{trigger strategies} Nash equilibria $\overlineσ$ for the $N$-player SCAR games $Γ_{N}(G|s_{0},γ,\varepsilon)$ (with $N\geq3$). Our study is exhaustive with respect to types of graphs $G$, initial states $s_{0}$ and values of $N,γ,\varepsilon$. We conclude that in the majority of cases, profiles $\overlineσ$ are nonpositional. Whenever $\overlineσ$ are positional a key role is played by paths and the $\varepsilon$, $γ$ values (especially whether $\varepsilon>0$ or not). A crucial concept in our analysis is the \emph{state cop number}, which is first introduced in the current paper.

math.CO↗
arXiv2018

A Note on the Nash Equilibria of Some Multi-Player Reachability / Safety Games

In this short note we study a class of multi-player, turn-based games with deterministic state transitions and reachability / safety objectives (this class contains as special cases "classic" two-player reachability and safety games as well as multi-player "stay--in-a-set" and "reach-a-set" games). Quantitative and qualitative versions of the objectives are presented and for both cases we prove the existence of a deterministic and memoryless Nash equilibrium; the proof is short and simple, using only Fink's classic result about the existence of Nash equilibria for multi-player discounted stochastic games.

cs.GT↗
arXiv2018

Generalized Cops and Robbers: A Multi-Player Pursuit Game on Graphs

We introduce and study the Generalized Cops and Robbers game (GCR), an N-player pursuit game in graphs. The two-player version is essentially equivalent to the classic Cops and Robbers (CR) game. The three-player version can be understood as two CR games played simultaneously on the same graph; a player can be at the same time both pursuer and evader. The same is true for four or more players. We formulate GCR as a discounted stochastic game of perfect information and prove that, for three or more players, it has at least two Nash Equilibria: one in positional deterministic strategies and another in non-positional ones. We also study the capturing properties of GCR Nash Equilibria in connection to the cop-number of a graph. Finally, we briefly discuss GCR as a member of a wider family of multi-player graph pursuit games with rather interesting properties.

cs.DM↗
arXiv2016

Selfish Cops and Passive Robber: Qualitative Games

Several variants of the cops and robbers (CR) game have been studied in the past. In this paper we examine a novel variant, which is played between two cops, each one independently trying to catch a "passive robber". We will call this the Selfish Cops and Passive Robber {SCPR} game. In short, SCPR is a stochastic two-player, zero-sum game where the opponents are the two cop players. We study sequential and concurrent versions of the SCPR game. For both cases we prove the existence of value and optimal strategies and present algorithms for the computation of these.

cs.DM↗
arXiv2015

Simultaneously Moving Cops and Robbers

In this paper we study the concurrent cops and robber (CCCR) game. CCCR follows the same rules as the classical, turn-based game, except for the fact that the players move simultaneously. The cops' goal is to capture the robber and the concurrent cop number of a graph is defined the minimum number of cops which guarantees capture. For the variant in which it it required to capture the robber in the shortest possible time, we let time to capture be the payoff function of CCCR; the (game theoretic) value of CCCR is the optimal capture time and (cop and robber) time optimal strategies are the ones which achieve the value. In this paper we prove the following. (1) For every graph G, the concurrent cop number is equal to the "classical" cop number. (2) For every graph G, CCCR has a value, the cops have an optimal strategy and, for every epsilon>0, the robber has an epsilon-optimal strategy.

cs.DM↗
arXiv2014

Cops and Robbers, Game Theory and Zermelo's Early Results

We provide a game theoretic framework for the game of cops and robbers (CR). Within this framework we study certain assumptions which underlie the concepts of optimal strategies and capture time. We also point out a connection of these concepts to early work by Zermelo and D. Konig. Finally, we discuss the relationship between CR and related pursuit games to reachability games.

cs.DM↗
arXiv2014

The Role of Visibility in Pursuit / Evasion Games

The cops-and-robber (CR) game has been used in mobile robotics as a discretized model (played on a graph G) of pursuit/evasion problems. The "classic" CR version is a perfect information game: the cops' (pursuer's) location is always known to the robber (evader) and vice versa. Many variants of the classic game can be defined: the robber can be invisible and also the robber can be either adversarial (tries to avoid capture) or drunk (performs a random walk). Furthermore, the cops and robber can reside in either nodes or edges of G. Several of these variants are relevant as models or robotic pursuit / evasion. In this paper, we first define carefully several of the variants mentioned above and related quantities such as the cop number and the capture time. Then we introduce and study the cost of visibility (COV), a quantitative measure of the increase in difficulty (from the cops' point of view) when the robber is invisible. In addition to our theoretical results, we present algorithms which can be used to compute capture times and COV of graphs which are analytically intractable. Finally, we present the results of applying these algorithms to the numerical computation of COV.

cs.DM↗
arXiv2013

Bad Communities with High Modularity

In this paper we discuss some problematic aspects of Newman's modularity function QN. Given a graph G, the modularity of G can be written as QN = Qf -Q0, where Qf is the intracluster edge fraction of G and Q0 is the expected intracluster edge fraction of the null model, i.e., a randomly connected graph with same expected degree distribution as G. It follows that the maximization of QN must accomodate two factors pulling in opposite directions: Qf favors a small number of clusters and Q0 favors many balanced (i.e., with approximately equal degrees) clusters. In certain cases the Q0 term can cause overestimation of the true cluster number; this is the opposite of the well-known under estimation effect caused by the "resolution limit" of modularity. We illustrate the overestimation effect by constructing families of graphs with a "natural" community structure which, however, does not maximize modularity. In fact, we prove that we can always find a graph G with a "natural clustering" V of G and another, balanced clustering U of G such that (i) the pair (G; U) has higher modularity than (G; V) and (ii) V and U are arbitrarily different.

cs.SI↗
arXiv2012

Cops and Invisible Robbers: the Cost of Drunkenness

We examine a version of the Cops and Robber (CR) game in which the robber is invisible, i.e., the cops do not know his location until they capture him. Apparently this game (CiR) has received little attention in the CR literature. We examine two variants: in the first the robber is adversarial (he actively tries to avoid capture); in the second he is drunk (he performs a random walk). Our goal in this paper is to study the invisible Cost of Drunkenness (iCOD), which is defined as the ratio ct_i(G)/dct_i(G), with ct_i(G) and dct_i(G) being the expected capture times in the adversarial and drunk CiR variants, respectively. We show that these capture times are well defined, using game theory for the adversarial case and partially observable Markov decision processes (POMDP) for the drunk case. We give exact asymptotic values of iCOD for several special graph families such as $d$-regular trees, give some bounds for grids, and provide general upper and lower bounds for general classes of graphs. We also give an infinite family of graphs showing that iCOD can be arbitrarily close to any value in [2,infinty). Finally, we briefly examine one more CiR variant, in which the robber is invisible and "infinitely fast"; we argue that this variant is significantly different from the Graph Search game, despite several similarities between the two games.

cs.DM↗
arXiv2011

Some remarks on cops and drunk robbers

The cops and robbers game has been extensively studied under the assumption of optimal play by both the cops and the robbers. In this paper we study the problem in which cops are chasing a drunk robber (that is, a robber who performs a random walk) on a graph. Our main goal is to characterize the "cost of drunkenness." Specifically, we study the ratio of expected capture times for the optimal version and the drunk robber one. We also examine the algorithmic side of the problem; that is, how to compute near-optimal search schedules for the cops. Finally, we present a preliminary investigation of the invisible robber game and point out differences between this game and graph search.

cs.DM↗