Computing tournament solutions using relation algebra and RelView

We describe a simple computing technique for the tournament choice problem. It rests upon relational modeling and uses the BDD-based computer system RelView for the evaluation of the relation-algebraic expressions that specify the solutions and for the visualization of the computed results. The Copeland set can immediately be identified using RelView’s labeling feature. Relation-algebraic specifications of the Condorcet non-losers, the Schwartz set, the top cycle, the uncovered set, the minimal covering set, the Banks set, and the tournament equilibrium set are delivered. We present an example of a tournament on a small set of alternatives, for which the above choice sets are computed and visualized via RelView. The technique described in this paper is very flexible and especially appropriate for prototyping and experimentation, and as such very instructive for educational purposes. It can easily be applied to other problems of social choice and game theory.     

Applying relational algebra and RelView to coalition formation

We present an application of relational algebra to coalition formation. This leads to specifications, which can be executed with the help of the RelView tool after a simple translation into the tool’s programming language. As an example we consider a simplification of the situation in Poland after the 2001 elections.     

On the role of chance moves and information in two-person games

The value of information has been the subject of many studies in a strategic context. The central question in these studies is how valuable the information hidden in the chance moves of a game is for one or more of the players. Generally speaking, only the extra possibilities that are beneficial for the players have been considered so far. In this note we study the value of information for a special class of two-person games. For these games we also investigate how “badly” the players can do, both with and without knowing the result of the chance move. In this way one can determine to what extent the players are restricted in their possibilities by the fact that some information is hidden in the chance moves of the games. This allows for a comparison of the influence of the chance move to the control that the players have over the game result.     

A general model for voting systems with multiple alternatives

In this paper, the extension of simple games to the vector case is proposed. Games with multiple qualitative criteria and multi-criteria simple games are introduced as a natural tool for modelling voting systems and related social-choice situations. After formally defining these games, the special class of monotonic multi-criteria simple games is characterized. We show that these games enable the formulation and analysis of several collective decision models proposed in the literature. Furthermore, our model can be applied to group-decision problems which cannot be analyzed in the existing frameworks.     

Applying relation algebra and RelView to measures in a social network

We present an application of relation algebra to measure agents’ ‘strength’ in a social network with influence between agents. In particular, we deal with power, success, and influence of an agent as measured by the generalized Hoede–Bakker index and its modifications, and by the influence indices. We also apply relation algebra to determine followers of a coalition and the kernel of an influence function. This leads to specifications, which can be executed with the help of the BDD-based tool RelView after a simple translation into the tool’s programming language. As an example we consider the present Dutch Parliament.     

Matroidal games

The theory ofmatriods consists of generalization of basic notions oflinear algebra likedependence, basis andspan. In this paper we point out that every non-trivial matroid represents a simple game though the converse need not be true. The class of simple games which possess the matroidal structure is designated asmatroidal games. In matroidal games, we have a generalization of the concept of complete exchangeability of players observed in purely size dependent games. Invoking the well developed theory of matroids, we study the combinatorial structure of matroidal games.     

The Shapley–Shubik index for multi-criteria simple games

In this paper we address multi-criteria simple games which constitute an extension of the basic framework of voting systems and related social-choice situations. For these games, we propose the extended Shapley–Shubik index as the natural generalization of the Shapley–Shubik index in conventional simple games, and establish an axiomatic characterization of this power index.     

Infinite Player Noncooperative Games with Vector Payoffs Under Relative Pseudomonotonicity

In this paper we consider the Nash equilibrium problem for infinite player games with vector payoffs in a topological vector space setting. By employing new concepts of relative (pseudo)monotonicity, we establish several existence results of solutions for usual and normalized vector equilibria. The results strengthen existence results for vector equilibrium problems, which were based on classical pseudomonotonicity concepts. They also extend previous results for vector variational inequalities and finite player games under relative (pseudo)monotonicity.     

Adaptive testing for specification coverage and refinement in CPS models

Design of correct cyber–physical systems (CPS) is of uttermost importance for safety-critical applications. This crucial yet extremely challenging property is often addressed in practice by simulation-based methods. The simulation activity can be made more systematic and rigorous by using formal specifications to express requirements and guide the testing of the system.In this paper, we develop a procedure for generating tests from formal specifications given in Signal Temporal Logic (STL), a declarative language used to express CPS requirements. The proposed test generation method is adaptive with the aim at achieving specification coverage. We devise to this goal cooperative reachability games, which we enhance with numerical optimization to facilitate exercising various parts of specifications. The resulting approach is effective in finding specification violations, but also in increasing confidence (via coverage) that the specification is satisfied. In the latter case, we also propose a method for automatically refining the specification into its part that is actually implemented, thus gaining additional insight into the system-under-test.     

Simple games and weighted games: A theoretical and computational viewpoint

It is a well-known result in the theory of simple games that a game is weighted if and only if it is trade robust. In this paper we propose a variant of trade robustness, that we call invariant-trade robustness, which is enough to determine whether a simple game is weighted. To test whether a simple game is invariant-trade robust we do not need to consider all winning coalitions; a reduced subset of minimal winning coalitions is enough.We make a comparison between the two methods (trade robustness and invariant-trade robustness) to check whether a simple game is weighted. We also provide by means of algorithms a full classification using both methods, for simple games with less than 8 voters according to the maximum level of (invariant-)trade robustness they achieve.     

On Dedekind’s problem for complete simple games

We state an integer linear programming formulation for the unique characterization of complete simple games, i.e. a special subclass of monotone Boolean functions. In order to apply the parametric Barvinok algorithm to obtain enumeration formulas for these discrete objects we provide a tailored decomposition of the integer programming formulation into a finite list of suitably chosen sub-cases. As for the original enumeration problem of Dedekind on Boolean functions we have to introduce some parameters to be able to derive exact formulas for small parameters. Recently, Freixas et al. have proven an enumeration formula for complete simple games with two types of voters. We will provide a shorter proof and a new enumeration formula for complete simple games with two minimal winning vectors.     

Directed and weighted majority games

In this paper we deal with several classes of simple games; the first class is the one of ordered simple games (i.e. they admit of a complete desirability relation). The second class consists of all zero-sum games in the first one.First of all we introduce a natural partial order on both classes respectively and prove that this order relation admits a rank function. Also the first class turns out to be a rank symmetric lattice. These order relations induce fast algorithms to generate both classes of ordered games.Next we focus on the class of weighted majority games withn persons, which can be mapped onto the class of weighted majority zero-sum games withn+1 persons.To this end, we use in addition methods of linear programming, styling them for the special structure of ordered games. Thus, finally, we obtain algorithms, by combiningLP-methods and the partial order relation structure. These fast algorithms serve to test any ordered game for the weighted majority property. They provide a (frequently minimal) representation in case the answer to the test is affirmative.     

The Shapley–Shubik index for simple games with multiple alternatives

When analyzing mathematically decision mechanisms ruled by voting it is sometimes convenient to include abstention as a possible alternative for the voters. In classical simple games, abstention, if considered, is formally equivalent to voting against the proposal. Simple games with alternatives are useful to study voting systems where abstention does not favour any of the options. In this work, we axiomatically characterize the Shapley–Shubik index for simple games with alternatives and apply it to an example taken from real life. This work has been partially supported by Grant MTM 2006–06064 of the Education and Science Spanish Ministry and the European Regional Development Fund, and Grant SGR 2005–00651 of the Catalonia Government.     

Regular simple games

Using Kelley's intersection number (and a variant of it) we define two classes of simple games, the regular and the strongly regular games. We show that the strongly regular games are those in which the set of winning coalitions and the set of losing coalitions can be strictly separated by a finitely additive probability measure. This, in particular, provides a combinatorial characterization for the class of finite weighted majority games within the finite simple games. We also prove that regular games have some nice properties and show that the finite regular games are exactly those simple games which are uniquely determined by their counting vector. This, in particular, generalizes a result of Chow and Lapidot.     

Hybrid defensive resource allocations in the face of partially strategic attackers in a sequential defender–attacker game

Many models have been developed to study homeland security games between governments (defender) and terrorists (attacker, adversary, enemy), with the limiting assumption of the terrorists being rational or strategic. In this paper, we develop a novel hybrid model in which a centralized government allocates defensive resources among multiple potential targets to minimize total expected loss, in the face of a terrorist being either strategic or non-strategic. The attack probabilities of a strategic terrorist are endogenously determined in the model, while the attack probabilities of a non-strategic terrorist are exogenously provided. We study the robustness of defensive resource allocations by comparing the government’s total expected losses when: (a) the government knows the probability that the terrorist is strategic; (b) the government falsely believes that the terrorist is fully strategic, when the terrorist could be non-strategic; and (c) the government falsely believes that the terrorist is fully non-strategic, when the terrorist could be strategic. Besides providing six theorems to highlight the general results, we find that game models are generally preferred to non-game model even when the probability of a non-strategic terrorist is significantly greater than 50%. We conclude that defensive resource allocations based on game-theoretic models would not incur too much additional expected loss and thus more preferred, as compared to non-game-theoretic models.     

Impartial achievement and avoidance games for generating finite groups

We study two impartial games introduced by Anderson and Harary and further developed by Barnes. Both games are played by two players who alternately select previously unselected elements of a finite group. The first player who builds a generating set from the jointly selected elements wins the first game. The first player who cannot select an element without building a generating set loses the second game. After the development of some general results, we determine the nim-numbers of these games for abelian and dihedral groups. We also present some conjectures based on computer calculations. Our main computational and theoretical tool is the structure diagram of a game, which is a type of identification digraph of the game digraph that is compatible with the nim-numbers of the positions. Structure diagrams also provide simple yet intuitive visualizations of these games that capture the complexity of the positions.     

On the use of binary decision diagrams for solving problems on simple games

Simple games are yes/no cooperative games which arise in many practical applications. Recently, we have used reduced ordered binary decision diagrams and quasi-reduced ordered binary decision diagrams (abbreviated as Robdds and Qobdds, respectively) for the representation of simple games and for the computation of some power indices. In the present paper, we continue this work. We show how further important computational problems on simple games can be solved using Qobdds, viz. the identification of some key players, the computation of the desirability relation on individuals, the test whether a simple game is proper and strong, respectively, and the computation of Qobdd-representations for the sets of all minimal winning coalitions, all shift-minimal winning coalitions and all blocking coalitions, respectively. Applications of these solutions include the computation of recent power indices based on shift-minimal winning coalitions and the test for linear separability of a directed simple game.     

Traveling salesman games

In this paper we discuss the problem of how to divide the total cost of a round trip along several institutes among the institutes visited. We introduce two types of cooperative games—fixed-route traveling salesman games and traveling salesman games—as a tool to attack this problem. Under very mild conditions we prove that fixed-route traveling salesman games have non-empty cores if the fixed route is a solution of the classical traveling salesman problem. Core elements provide us with fair cost allocations. A traveling salesman game may have an empty core, even if the cost matrix satisfies the triangle inequality. In this paper we introduce a class of matrices defining TS-games with non-empty cores.     

A set of axioms for a value for partition function games

Myerson (1977) derived an efficient value for games in partition function form. In this paper, we present a set of axioms which characterize a different efficient value for such games. This latter value assigns value 0 to dummies and assigns nonnegative values to players in monotone simple games.