Marco Dall'Aglio

Strategic games on defense trees

In this paper we use defense trees, an extension of attack trees with countermeasures, to represent attack scenarios and game theory to detect the most promising actions attacker and defender. On one side the attacker wants to break the system (with as little efforts as possible), on the opposite side the defender want to protect it (sustaining the minimum cost). As utility function for the attacker and for the defender we consider economic indexes (like the Return on Investment (ROI) and the Return on Attack (ROA)). We show how our approach can be used to evaluate effectiveness and economic profitability of countermeasures as well as their deterrent effect on attackers, thus providing decision makers with a useful tool for performing better evaluation of IT security investments during the risk management process.

Finding maxmin allocations in cooperative and competitive fair division

We define a subgradient algorithm to compute the maxmin value of a completely divisible good in both competitive and cooperative strategic contexts. The algorithm relies on the construction of upper and lower bounds for the optimal value which are based on the convexity properties of the range of utility vectors associated to all possible divisions of the good. The upper bound always converges to the optimal value. Moreover, if two additional hypotheses hold: that the preferences of the players are mutually absolutely continuous, and that there always exists relative disagreement among the players, then also the lower bound converges, and the algorithm finds an approximately optimal allocation.

Maximin share and minimax envy in fair-division problems

For fair-division or cake-cutting problems with value functions which are normalized positive measures (i.e., the values are probability measures) maximin-share and minimax-envy inequalities are derived for both continuous and discrete measures. The tools used include classical and recent basic convexity results, as well as ad hoc constructions. Examples are given to show that the envyminimizing criterion is not Pareto optimal, even if the values are mutually absolutely continuous. In the discrete measure case, sufficient conditions are obtained to guarantee the existence of envy-free partitions.

How to allocate hard candies fairly

Abstract We consider the problem of allocating a finite number of indivisible items to two players with additive utilities. We design a procedure that looks for all the maximin allocations and makes repeated use of an extension of the Adjusted Winner, an effective procedure that deals with divisible items, to find new candidate solutions, and to suggest which items should be assigned to the players.

Interval Game Theoretic Division Rules

Interval bankruptcy problems arise in situations where an estate has to be liquidated among a fixed number of creditors and uncertainty about the amounts of the estate and the claims is modeled by intervals. We extend in the interval setting the classic results by Curiel, Maschler and Tijs [Bankruptcy games, Zeitschrift fur Operations Research, 31 (1987), A 143 { A 159] that characterize division rules which are solutions of the cooperative bankruptcy game.

When Lorenz met Lyapunov

We provide a simple characterization of the zonoid, lift zonoid, Lorenz zonoid of a random vector, as defined by Koshevoy and Mosler (J. Amer. Statist. Assoc. 91 (1996) 873; Bernoulli 4 (1998) 377). This characterization allows a direct use of Lyapunovs theorem on the range of nonatomic vector measures and therefore simplifies the proof of some results.

Bounds for α-optimal partitioning of a measurable space based on several efficient partitions

Abstract We provide a two-sided inequality for the α-optimal partition value of a measurable space according to a finite number of nonatomic finite measures. The result extends and often improves Legut [Inequalities for α-optimal partitioning of a measurable space, Proc. Amer. Math. Soc. 104 (1988)] since the bounds are obtained considering several partitions that maximize the weighted sum of the partition values with varying weights, instead of a single one. Furthermore, we show conditions that make these bounds sharper.

Ranking And Unranking Permutations WithApplications

Permutation theory has many applications in several fields of science and technology and it also has a charm in itself. Mathematica is particularly suitable for writing combinatorial algorithms because it provides many easy-to-use tools for handling lists. Several combinatorial built-in functions which involve permutations and combinations are available as standard add-on Mathematica packages. They are grouped under the name of DiscreteMath and are described by their author Steven Skiena in his book [7]. In this paper we focus on ranking and unranking procedures and we examine and implement alternative algorithms to the RankPermutations and NthPermutations already contained in the above mentioned add-on packages. Moreover we provide some applications of these ranking procedures in different topics such as probability, statistics and elementary calculus. Let /„={(), 1, ..., n-\ } be the set of the nonnegative integers smaller than n and let A be a rc-set. We denote the set of all permutations of A by the n!-set: