Monday
R. Fokkink
Title: Submodular Search
Abstract: A submodular search game is defined motivated by search on a graph, but it takes away
the graph and replaces it by a cost function. This opens up the way to generalize existing games,
such as search on Eulerian networks as analysed by Gal or expanding search as analysed by Alpern
and Lidbetter. Submodular search games also relate to other optimization problems, to the benefit
of both.
This is joint work with Tom Lidbetter (LSE) and extends earlier joint work with Ken Kikuta (Hyogo)
and David Ramsey (Wroclaw).
E. Csóka
Title: Limits of some combinatorial problems
Abstract: We purify and generalize some techniques which were successful in the limit theory of
graphs and other discrete structures. We demonstrate how this technique can be used for solving
different combinatorial problems, by defining the limit problems of the Manickam--Miklós--Singhi
Conjecture, the Kikuta–Ruckle Conjecture and Alpern's Caching Game.
N. Zoroa
Title: New results on the study of the Weighted Concealment Game (N. Zoroa, A. Marín, M.J.
Fernández-Sáez)
Abstract: A spy gets information about a company through n different sources, S_1, S_2,…,S_n.
Throughout the day, he picks up information from each one of the sources, but the
information obtained from S_i (i = 1, 2, …, n) is reliable if it has been verified at least k_i
times. On the other hand, the security department of the enterprise performs daily
inspections, and if it detects a leak of information, it is activated a protection system that
invalidates the information that can be obtained during this day.
The spy must select the times of day that he will make the incursions bearing in mind,
first, that the importance of the obtained information depends of each source and,
secondly, that he cannot do more than s attempts a day.
Above situation can be modelled as a search game on the cartesian product of the sets
{1,2,...,n} and {1,2,...,m}, and has been partially solved in [1]. It can be considered a
Ruckle type game and is related to the accumulative games.
[1] N. Zoroa, P. Zoroa and M.J. Fernández-Sáez. Weighted search games. European Journal of
Operational Research 195 (2009) 394-411.
S. Gal
Title: Search - Pursue Games at Discrete Locations
Search - Pursue Games at Discrete Locations
Shmuel Gal
University of Haifa, Department of Statistics
[email protected]
Abstract
We present a family of games originating from the following situation. A
predator (searcher) looks for a prey (hider) in a search space consisting of n
locations. The hider chooses a location and the searcher inspects k di¤erent
locations, where k is a parameter of the game (the ‘giving-up time’for the
continuous version). If the predator visits a location i at which the prey hides,
then the game moves into a pursuit-evasion phase. In this phase capture is
not certain but occurs with probability pi :
We …rst analyze the one stage game. We show that for all k smaller
than an easily calculated threshold, it is optimal to hide with probability
proportional to 1=pi for each location i: If k exceeds the threshold, then the
optimal hiding strategy is always to stay at the location with the smallest pi :
We then extend this game to the repeated game Gk : During the k inspections among the di¤erent locations within a single patch, there can be
any of three events. First, if the searcher does not …nd the hider, then the
game ends with zero payo¤ for the searcher and a payo¤ of one to the hider.
Second, if the searcher …nds the hider and catches it, then the game ends
with a payo¤ of one to the searcher and zero to the hider. Finally, if the
searcher …nds the hider but does not catch it then the hider escapes to another patch and the process restarts. We show that in this game the optimal
hiding strategy is to always make all the locations equally "attractive" for
the searcher, no matter how large is k: This situation is quite di¤erent from
the one stage game in which solutions of this type occur only if k is below
the threshold.
We then consider discounted repeated games Gk; : We solve a family of
games depending on the discount factor ; 0
1: The two extreme cases
occur for = 0 in which we get the one stage game and for = 1 in which
we get the undiscounted repeated game. We obtain an explicit expression for
the discount factor above which the equally attractive solution is optimal.
1
In order to improve the realism of the repeated games model we introduce
two stochastic games. In the …rst game k we take into account the
probability ij of capture during ‡eeing from location i to a chosen location
j: At any stage, the stochastic game is at state i; i = 1; :::; n , if the hider
has been discovered but not captured at location i: In addition there are
two absorbing states corresponding to capture and to evasion. We present
the value equations and a value iteration algorithm that converges, with a
geometric rate, to the optimal solution of this game.
In the second stochastic game k the number of inspections is reduced
by one after each inspection even if the hider eventually escapes to another
patch. At any stage, if the searcher has l (l k) remaining inspections to
make, then the stochastic game is at state l: This game always has a …nite
number of stages but is more complicated because the order of visiting the
locations is also important, in contrast to the previous versions in which
only the set of visited locations mattered. We present a recursive scheme for
computing the optimal solution of this game.
This is a joint work with Steve Alpern and Jerome Casas.
2
Tuesday
N. Nisse
Title: Spy-Game on graphs
Abstract: We define and study the following two-player game on a graph G. A set of k guards is
occupying some vertices of G while one spy is standing at some node. At each turn, first the spy
may move along at most s edges, where s is his speed. Then, each guard may move along one edge.
The spy and the guards may occupy same vertices. The spy has to escape the surveillance of the
guards, i.e., must reach a vertex at distance more than d (a predefined distance) from every guard.
Can the spy win against k guards? Similarly, what is the minimum distance d such that k guards may
ensure that at least one of them remains at distance at most d from the spy? This game generalizes
two well-studied games: Cops and robber games (when s = 1) and Eternal Dominating Set (when s is
unbounded). We consider the computational complexity of the problem, showing that it is NP-hard
and that it is PSPACE-hard in DAGs. Then, we establish tight tradeoff between the number of guards
and the required distance d when G is a path or a cycle. Our main result is that there exists A> 0 such
that n^{1+A} guards are required to win in any n*n grid.
Joint work with Nathann Cohen, Mathieu Hilaire, Nicolas A. Martins and Stéphane Pérennes
V.S. Subramanian
Title: HARE: Honey-Based Adversarial Reasoning Engine
Abstract: We develop a game-theoretic foundation for how adversaries will explore an enterprise
network and how they will attack it, based on the concept of a system vulnerability dependency
graph. Based on such a model of the adversary, we develop a mechanism by which the network can
be modified by the defender so as to induce deception by placing honey nodes and apparent
vulnerabilities into the network so as to minimize the expected impact of the adversary’s attacks
(according to multiple measures of impact). We also consider the case where the adversary learns
from blocked attacks using reinforcement learning. We run detailed experiments with real network
data (but with simulated attack data) and show that HARE (Honey-based Adversarial Reasoning
Engine) performs very well, even when the adversary’s behavior varies from what is optimal for him,
as well as under various other conditions. We also develop a method for the attacker to use
reinforcement learning when his activities are stopped by the defender. We propose two stopping
policies for the defender - Stop Upon Detection allows the attacker to learn about the defender’s
strategy and (according to our experiments) leads to significant damage in the long run, whereas
Stop After Delay allows the defender to introduce greater uncertainty into the attacker, leading to
better defendability in the long run. Joint work with Sushil Jajodia, Noseong Park, and Edoardo Serra.
D. Kirkpatrick
Title: Minimizing uncertainty in the relative location of moving points
Abstract: Search problems are conventionally understood in terms of minimizing uncertainty in the
location of one or more entities (perhaps moving in an adversarial fashion), using some form of
spatial queries. We consider, and encourage further consideration of, a natural modification in
which the goal is to reduce uncertainty, attributable to unmonitored motion, concerning the
relative, rather than absolute, location of a collection of moving entities.
For example, imagine a collection of point entities moving in one-, or higher, dimensional space each
with some (known, but possibly different) bound on their speed. If we know, by means of an
individual location query, the precise location of an individual entity at a particular time, then its
location, until the time that it is next queried, lies in a steadily-expanding region of uncertainty. We
consider the problem of minimizing measures of potential congestion—congestion of entity
configurations consistent with the current uncertainty of the collection—using individual queries
that are restricted to one query per unit of time.
We focus, in particular, on minimizing the ply of the uncertainty regions (defined as the maximum,
over all points p in the space, of the number of uncertainty regions that contain p), a natural
measure of worst-case co-location potential. This notion is studied in two settings, one where ply is
measured at some fixed time in the future, and the other where ply is measured continuously (i.e. at
all times). Competitive query strategies are described in terms of a notion of intrinsic ply (the
minimum ply achievable by any query strategy, even one that knows the trajectories of all entities).
(Based on joint work with Daniel Busto, Will Evans, Maarten Löffler, and Frank Staals.)
Wednesday
L. Gasieniec
Title: Majority population protocols
Abstract: We start with the review of literature on majority population protocols. In the considered
problem a large group of simple asynchronous entities, each associated with one of the two basic
colours, is asked to decide which colour is attributed to the majority. Later we present some
extensions of such protocols to different models and tasks. We also show how to amend majority
protocols to report equality if neither of the original colours dominates the other in the population.
Finally we consider populations with elevated diversity (with larger number of colours) and provide
space efficient solutions to the absolute and relative majority problems.
R. Hohzaki
Title: Introduction to Search Allocation Games
Abstract: A search allocation game (SAG) is a search game, in which a searcher distributes search
efforts to detect a target and the target moves to evade the searcher. Search theory originated from
Koopman's book entitled "Search and Screening". He discussed an optimal distribution of search
efforts to detect targets such as submarines in an effective way. The SAG is a natural extension of his
problem to game-theoretical situations. We started our research on SAGs from the 1990's and have
proposed several models: SAGs with false contacts, with attributes of efforts, with energy supply
strategies, with multiple stages, and with a cooperative behavior between searchers. In this
presentation, we show the outlines of these models.
J. Howard
Title: A Short Solution to the Many-Person Silent Duel
Abstract: The classical zero-sum ‘silent duel’ game was formulated and solved by researchers at
RAND around 1948-1952. The story involved 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 probability of hitting the target decreases
with its distance. 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).
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 a short and simple solution (entirely
avoiding differential equations) to this game.”
D. Ramsey
Title: On a Large Population Mate Choice Game with Continuous Time
A large population mate choice game is considered where the number of males equals the number
of females. Each individual begins searching for a partner at the beginning of the season. Whenever
a partnership is formed, the pair leave the mating pool. Hence, the number of players still searching
and the distribution of their values change according to the profiles of strategies used. The rate of
finding prospective partners is non-decreasing in the proportion of individuals still searching. Various
dynamics satisfying this condition are considered.
The paper describes Nash equilibria which satisfy the optimality criterion, which states that a
searcher accepts a prospective partner if and only if his/her value is greater or equal to the expected
reward from future search.
R. Lindelauf
TBC
M. Renault
Title: Social foraging and non-persistent food patches.
Abstract: Many animals perform cooperative food finding; that is they forage for food in groups and
share the food patches that are found. One perceived benefit is that larger groups will find more
food. However, this must be balanced by the fact that there will be more group members with
whom the food is shared. In this context, the optimal group size can be seen as the group size that
minimizes the probability of starvation for its members, i.e. maximizes the probability that each
member receives more food than some threshold $R$.
Here, we consider a model in which the food patches are not persistent. That is, the group has a
limited time, $\tau$, to consume the patch once discovered. In the basic model considered here, we
show that the optimal group size is $\Theta(F/\rho)$, where $F$ is the amount of food in a patch
and $\rho = \min\{R,\tau\}$. We also consider models where $\tau$ depends on the group size and
where the amount of food in a patch is drawn from some distribution.
This is a joint work with Amos Korman and Yossi Yovel.
Thursday
C. Dürr
Title: The expanding search ratio of a graph
Abstract: We study the problem of searching for a hidden target in an environment that is modeled
by an edge-weighted graph. A sequence of edges is chosen starting from a given root vertex such
that each edge is adjacent to a previously chosen edge. This search paradigm, known as expanding
search was recently introduced for modeling problems such as coal mining in which the cost of reexploration is negligible. We define the search ratio of an expanding search as the maximum over all
vertices of the ratio of the time taken to reach the vertex and the shortest-path cost to it from the
root. Similar objectives have previously been studied in the context of conventional (pathwise)
search.
In this paper we address algorithmic and computational issues of minimizing the search ratio over all
expanding searches, for a variety of search environments, including general graphs, trees and starlike graphs. Our main results focus on the problem of finding the randomized expanding search with
minimum expected search ratio, which is equivalent to solving a zero-sum game between a Searcher
and a Hider. We solve these problems for certain classes of graphs, and obtain constant-factor
approximations for others.
L. Mouatadid
Title: Graph Searches on Structured Families of Graphs
Abstract: Graph searching, a mechanism to traverse a graph visiting one vertex at a time in a specific
manner, is a powerful tool used to extract structure from families of graphs. In this talk, we study
how graph searching is used to produce vertex orderings of different graph families. These orderings
expose structure that we exploit to develop efficient linear and near-linear time algorithms for some
NP-hard problems (independent set, colouring, hamiltonicity for instance). The talk will survey the
applications of two specific graph searches: lexicographic breadth first search (LBFS) and
lexicographic depth first search (LDFS), on some perfect and non-perfect classes of graphs. In
particular we describe a Hamilton path certifying algorithm on cocomparability graphs using LDFS.
S. Alpern
Title: Searching a Network Using Combinatorial Paths
Abstract. Let Q be a network with given arc lengths and a distinguished starting point O. A Hider
chooses any point H in Q while the Searcher picks a unit speed covering path S(t) with S(0)=O. The
payoff is the capture time T=T(H,S)=min (t: S(t)=H), with Hider the maximizer. Such games have
been solved for only for two classes of networks: (i) Weakly Eulerian networks, where removing
bridges leaves Eulerian networks, were solved by Shmuel Gal in 2000; (ii) Networks consisting of two
nodes connected by an odd number of unit length arcs were solved by Liljana Pavolivic a few years
earlier. Since 2000 no new networks have been solved. To break this logjam, we restrict the Searcher
to combinatorial paths consisting of sequences of arcs, the usual paths of graph theory. This enables
us to solve the search game for some additional networks. Inter alia, we find a network where
optimal search involves traversing an arc three times, contradicting a conjecture of Gal.
M. Baykal-Gursoy
Title: Robust Search/Protect Games
Abstract: We consider the protection of an N-node network against an intelligent attacker as a nonzero sum non-cooperative game with uncertain performance measures. We develop a distributionfree model of incomplete-information games, both with and without private information, in which
the players use a robust optimization approach to contend with payoff uncertainty. Depending on
the objective of the adversary and existence of private information, we present three models for this
game. We then prove existence and uniqueness of the Nash equilibrium for the first two models and
characterize the shape of the Nash equilibrium for the third model. Finally we apply the proposed
approach on real data to determine the optimal defensive resource allocation.
P. Leone
Title: Algorithmic Solutions to Scent trails Rendezvous
Abstract: In the talk we discuss extensions of the classical problem of rendezvous on the line and we
propose algorithms to compute approximations to the optimal solutions of the problems. The first
extension consists in providing letters to the players that can be dropped off at any time. The game
stops when the players rendezvous or find the letters. The second extension provides marks to the
players that can be dropped off on the way as well. The game stops only when the players
rendezvous. Marks, or absence of marks, are used to indicate the presence/absence of the
players. We describe how we can find solutions to these problems that are arbitrarily close to the
optimal ones.
Friday
K. Papadaki
Title: Patrolling continuous networks
Abstract: We define a zero sum game between a patroller who wants to protect a continuous
network and an attacker who wants to disrupt the network. The patroller picks a walk on a network
that she periodically repeats and the attacker picks a point on the network and a time to attack. The
attack is successful if the attacker spends time r at the attack point uninterrupted by the patroller,
otherwise the attack is intercepted. We present the value and optimal mixed strategies for various
types of networks.
S. Katsikas
Title: The Uniformed Patroller Problem (Steve Alpern, Stamatios Katsikas)
Abstract: In the recently introduced network patrolling game, an Attacker carries out an attack on a
node of his choice for a given number m of consecutive time periods, where m denotes the
difficulty of the attack. To thwart the attack, a Patroller adopts a walk on the network,
hoping to be at the attacked node during one of the periods of the attack. If this occurs,
the attack is intercepted, and the Patroller wins the game. In this paper we assume that
the Attacker goes to his chosen node early, and observes in each period whether or not the
Patroller is there; hence we introduce the adjective uniformed (alternatively we could say
that the Patroller is noisy in contrast to silent, to denote this new information structure).
The Attacker can choose to wait until the Patroller is away from the attack node for a chosen
number of periods before starting the attack. We solve this game for star, line, and cycle
networks, for various randomised strategies allowed for the Patroller.