Soumya Paul

Nash Equilibrium in Generalised Muller Games

We suggest that extending Muller games with preference ordering for players is a natural way to reason about unbounded duration games. In this context, we look at the standard solution concept of Nash equilibrium for non-zero sum games. We show that Nash equilibria always exists for such generalised Muller games on finite graphs and present a procedure to compute an equilibrium strategy profile. We also give a procedure to compute a subgame perfect equilibrium when it exists in such games.

A Probabilistic Analysis of the Efficiency of Automated Software Testing

We study the relative efficiencies of the random and systematic approaches to automated software testing. Using a simple but realistic set of assumptions, we propose a general model for software testing and define sampling strategies for random (R) and systematic (S0) testing, where each sampling is associated with a sampling cost: 1 and c units of time, respectively. The two most important goals of software testing are: (i) achieving in minimal time a given degree of confidence x in a programs correctness and (ii) discovering a maximal number of errors within a given time bound n̂. For both (i) and (ii), we show that there exists a bound on c beyond which R performs better than S0 on the average. Moreover for (i), this bound depends asymptotically only on x. We also show that the efficiency of R can be fitted to the exponential curve. Using these results we design a hybrid strategy H that starts with R and switches to S0 when S0 is expected to discover more errors per unit time. In our experiments we find that H performs similarly or better than the most efficient of both and that S0 may need to be significantly faster than our bounds suggest to retain efficiency over R.

Dynamic restriction of choices: a preliminary logical report

We study games in which the choices available to players are not fixed, and may change during the course of play. Specifically, we consider a model in which players may switch strategies, and a global (social) decision may remove some choices, based on the strategies being adopted by players. We propose a logical formalism in which such choices are specified, and a model of bounded memory strategies in which the eventual implications of such choices can be computed, and present preliminary results.

Imitation in Large Games

In games with a large number of players where players may have overlapping objectives, the analysis of stable outcomes typically depends on player types. A special case is when a large part of the player population consists of imitation types: that of players who imitate choice of other (optimizing) types. Game theorists typically study the evolution of such games in dynamical systems with imitation rules. In the setting of games of infinite duration on finite graphs wi th preference orderings on outcomes for player types, we explore the possibility of imitation as a vi able strategy. In our setup, the optimising players play bounded memory strategies and the imitators play according to specifications given by automata. We present algorithmic results on the eventual survival of types. 1 Summary Imitation is an important heuristic studied by game theorists in the analysis of large games, in both extensive form games with considerable structure, and repeated normal form games with large number of players. One reason for this is that notions of rationalit y underlying solution concepts are justified by players’ assumptions about how other players play, iterati vely. In such situations, players’ knowledge of the types of other players alters game dynamics. Skilled players can then be imitated by less skilled ones, and the former can then strategize about how the latter might play. In games with a large number of players, both strategies and outcomes are studied using distributions of player types. The dynamics of imitation, and strategizing of optimizers in the presence of imitators can give rise to interesting consequences. For instance, in the game of chess, if the player playing white somehow knows that her opponent will copy her move for move then the following simple sequence of moves allows her to checkmate her opponent 1 : 1.e3 e6 2.Qf3 Qf6 3.Qg3 Qg6 4.Nf3 Nf6 5.Kd1 Kd8 6.Be2 Be7 7.Re1 Re8

Message Exchange Games in Strategic Contexts

When two people engage in a conversation, knowingly or unknowingly, they are playing a game. Players of such games have diverse objectives, or winning conditions: an applicant trying to convince her potential employer of her eligibility over that of a competitor, a prosecutor trying to convict a defendant, a politician trying to convince an electorate in a political debate, and so on. We argue that infinitary games offer a natural model for many structural characteristics of such conversations. We call such games message exchange games, and we compare them to existing game theoretic frameworks used in linguistics—for example, signaling games—and show that message exchange games are needed to handle non-cooperative conversation. In this paper, we concentrate on conversational games where players’ interests are opposed. We provide a taxonomy of conversations based on their winning conditions, and we investigate some essential features of winning conditions like consistency and what we call rhetorical cooperativity. We show that these features make our games decomposition sensitive, a property we define formally in the paper. We show that this property has far-reaching implications for the existence of winning strategies and their complexity. There is a class of winning conditions (decomposition invariant winning conditions) for which message exchange games are equivalent to Banach- Mazur games, which have been extensively studied and enjoy nice topological results. But decomposition sensitive goals are much more the norm and much more interesting linguistically and philosophically.

A logic of sights

We study labeled transition systems where at each state an agent is aware of and hence reasons about only a part of the entire system (called the ‘sight’). We develop a logic for such systems: the ‘logic of sights’. We explore its model theory, give an axiomatization and prove its completeness. We show that the logic is a fragment of the loosely guarded fragment of first-order logic. We show that the satisfiability problem of the logic is PSPACE-complete and the combined complexity of its model-checking problem is in PTIME. Finally we discuss its relation to other logics as well as extensions.

A Decomposition-based Approach towards the Control of Boolean Networks

We study the problem of computing a minimal subset of nodes of a given asynchronous Boolean network that need to be controlled to drive its dynamics from an initial steady state (or attractor) to a target steady state. Due to the phenomenon of state-space explosion, a simple global approach that performs computations on the entire network, may not scale well for large networks. We believe that efficient algorithms for such networks must exploit the structure of the networks together with their dynamics. Taking such an approach, we derive a decomposition-based solution to the minimal control problem which can be significantly faster than the existing approaches on large networks. We apply our solution to both real-life biological networks and randomly generated networks, demonstrating promising results.

Approximate Probabilistic Verification of Hybrid Systems

Hybrid systems whose mode dynamics are governed by non-linear ordinary differential equations (ODEs) are often a natural model for biological processes. However such models are difficult to analyze. To address this, we develop a probabilistic analysis method by approximating the mode transitions as stochastic events. We assume that the probability of making a mode transition is proportional to the measure of the set of pairs of time points and value states at which the mode transition is enabled. To ensure a sound mathematical basis, we impose a natural continuity property on the non-linear ODEs. We also assume that the states of the system are observed at discrete time points but that the mode transitions may take place at any time between two successive discrete time points. This leads to a discrete time Markov chain as a probabilistic approximation of the hybrid system. We then show that for BLTL (bounded linear time temporal logic) specifications the hybrid system meets a specification iff its Markov chain approximation meets the same specification with probability 1. Based on this, we formulate a sequential hypothesis testing procedure for verifying–approximately–that the Markov chain meets a BLTL specification with high probability. Our case studies on cardiac cell dynamics and the circadian rhythm indicate that our scheme can be applied in a number of realistic settings.

Neighbourhood structure in large games

We study repeated normal form games where the number of players is large and suggest that it is useful to consider a neighbourhood structure on the players. The structure is given by a graph G whose nodes are players and edges denote visibility. The neighbourhoods are maximal cliques in G. The game proceeds in rounds where in each round the players of every clique X of G play a strategic form game among each other. A player at a node v strategises based on what she can observe, i.e., the strategies and the outcomes in the previous round of the players at vertices adjacent to v. Based on this, the player may switch strategies in the same neighbourhood, or migrate to another neighbourhood. Player types, giving the rationale for such switching, are specified in a simple modal logic. We show that given the initial neighbourhood graph and the types of the players in the logic, we can effectively decide if the game eventually stabilises. We then prove a characterisation result for these games for arbitrary types using potentials. We then offer some applications to the special case of weighted co-ordination games where we can compute bounds on how long it takes to stabilise.

Dynamic restriction of choices: synthesis of societal rules

We study a game model to highlight the mutual recursiveness of individual rationality and societal rationality. These are games that change intrinsically based on the actions / strategies played by the players. There is an implicit player - the society, who makes actions available to players and incurs certain costs in doing so. If and when it feels that an action a is being played by a small number of players and/or it becomes too expensive for it to maintain the action a, it removes a from the set of available actions. This results in a change in the game and the players strategise afresh taking this change into account. We study the question: which actions of the players should the society restrict and how should it restrict them so that the social cost is minimised in the eventuality? We address two variations of the question: when the players are maximisers, can society choose an order of their moves so that social cost is minimised, and which actions may be restricted when players play according to given strategy specifications.