16 research outputs found
Randomized Pursuit-Evasion with Local Visibility
We study the following pursuit-evasion game: One or more hunters are seeking to capture an evading rabbit on a graph. At each round, the rabbit tries to gather information about the location of the hunters but it can see them only if they are located on adjacent nodes. We show that two hunters su#ce for catching rabbits with such local visibility with high probability. We distinguish between reactive rabbits who move only when a hunter is visible and general rabbits who can employ more sophisticated strategies. We present polynomial time algorithms that decide whether a graph G is hunter-win, that is, if a single hunter can capture a rabbit of either kind on G
Variations on Cops and Robbers
We consider several variants of the classical Cops and Robbers game. We treat
the version where the robber can move R > 1 edges at a time, establishing a
general upper bound of N / \alpha ^{(1-o(1))\sqrt{log_\alpha N}}, where \alpha
= 1 + 1/R, thus generalizing the best known upper bound for the classical case
R = 1 due to Lu and Peng. We also show that in this case, the cop number of an
N-vertex graph can be as large as N^{1 - 1/(R-2)} for finite R, but linear in N
if R is infinite. For R = 1, we study the directed graph version of the
problem, and show that the cop number of any strongly connected digraph on N
vertices is at most O(N(log log N)^2/log N). Our approach is based on
expansion.Comment: 18 page
Adaptive Resource Allocation in Jamming Teams Using Game Theory
In this work, we study the problem of power allocation and adaptive
modulation in teams of decision makers. We consider the special case of two
teams with each team consisting of two mobile agents. Agents belonging to the
same team communicate over wireless ad hoc networks, and they try to split
their available power between the tasks of communication and jamming the nodes
of the other team. The agents have constraints on their total energy and
instantaneous power usage. The cost function adopted is the difference between
the rates of erroneously transmitted bits of each team. We model the adaptive
modulation problem as a zero-sum matrix game which in turn gives rise to a a
continuous kernel game to handle power control. Based on the communications
model, we present sufficient conditions on the physical parameters of the
agents for the existence of a pure strategy saddle-point equilibrium (PSSPE).Comment: 6 pages, 2 figures, submitted to RAWNET/WNC3 201
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 -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
Optimal Mixed Strategies to the Zero-sum Linear Differential Game
This paper exploits the weak approximation method to study a zero-sum linear
differential game under mixed strategies. The stochastic nature of mixed
strategies poses challenges in evaluating the game value and deriving the
optimal strategies. To overcome these challenges, we first define the mixed
strategy based on time discretization given the control period . Then,
we design a stochastic differential equation (SDE) to approximate the
discretized game dynamic with a small approximation error of scale
in the weak sense. Moreover, we prove that the game
payoff is also approximated in the same order of accuracy. Next, we solve the
optimal mixed strategies and game values for the linear quadratic differential
games. The effect of the control period is explicitly analyzed when the payoff
is a terminal cost. Our results provide the first implementable form of the
optimal mixed strategies for a zero-sum linear differential game. Finally, we
provide numerical examples to illustrate and elaborate on our results