Motivated by the sequence form formulation of Koller et al. (GEB'96), this paper defines {\em... more Motivated by the sequence form formulation of Koller et al. (GEB'96), this paper defines {\em bilinear games}, and proposes efficient algorithms for its rank based subclasses. Bilinear games are two-player non-cooperative single-shot games with compact polytopal strategy sets and two payoff matrices (A,B) such that when (x,y) is the played strategy profile, the payoffs of the players are xAy and
We analyze the complexity of computing pure strategy Nash equilibria (PSNE) in sym- metric games ... more We analyze the complexity of computing pure strategy Nash equilibria (PSNE) in sym- metric games with a fixed number of actions. We restrict ourselves to "compact" representations, meaning that the number of players can be exponential in the representation size. We show that in the general case, where utility functions are represented as arbitrary circuits, the problem of decid- ing
Proceedings of the 11th ACM conference on Electronic commerce - EC '10, 2010
We analyze the complexity of computing pure strategy Nash equilibria (PSNE) in symmetric games wi... more We analyze the complexity of computing pure strategy Nash equilibria (PSNE) in symmetric games with a fixed number of actions. We restrict ourselves to "compact" representations, meaning that the number of players can be exponential in the representation size. We show that in the general case, where utility functions are represented as arbitrary circuits, the problem of deciding the existence
This paper proposes the Multi-Operation Patrol Scheduling System (MOPSS), a new system to generat... more This paper proposes the Multi-Operation Patrol Scheduling System (MOPSS), a new system to generate patrols for transit system. MOPSS is based on five contributions. First, MOPSS is the first system to use three fundamentally different adversary models for the threats of fare evasion, terrorism and crime, generating three significantly different types of patrol schedule. Second, to handle uncertain interruptions in the execution of patrol schedules, MOPSS uses Markov decision processes (MDPs) in its scheduling. Third, MOPSS is the first system to account for joint activities between multiple resources, by employing the well known SMART security game model that tackles coordination between defender’s resources. Fourth, we are also the first to deploy a new Opportunistic Security Game model, where the adversary, a criminal, makes opportunistic decisions on when and where to commit crimes. Our fifth, and most important, contribution is the evaluation of MOPSS via real-world deployments, providing data from security games in the field.
Nash equilibria of two-player games are much easier to compute in practice than those of n-player... more Nash equilibria of two-player games are much easier to compute in practice than those of n-player games, even though the two problems have the same asymptotic complexity. We used a recent constructive reduction to solve general games using a two-player algorithm. However, the reduction increases the game size too much to be practically usable. An open problem is to find
ABSTRACT Bayesian normal form’s tabular representation of type profile distribution P can be stra... more ABSTRACT Bayesian normal form’s tabular representation of type profile distribution P can be straightforwardly represented as a BN, e.g. by creating a random variable representing θ as the only parent of the ∑ random variables θ1,..., θn. To represent utility functions, we create an action graph G with i |Θi||Ai | action nodes; in other words, all type-action sets A ′ i,θi are disjoint. Each action ai ∈ Ai of the Bayesian normal form corresponds to |Θi | action nodes in the BAGG, one for each type instantiation θi. For each player i and each type θi ∈ Θi, each action node α ∈ A ′ has incoming i,θi edges from all action nodes from type-action sets A ′ j,θj for all j ̸ = i, θj ∈ Θj, i.e. all action nodes corresponding to ai ∈ Ai, the utility function of the other players. For each action node α ∈ A ′ i,θi uα is defined as follows: given configuration c (α) we can infer the action profile a ′ −i ∈ A ′ −i of the BAGG, which then tells us the corresponding a−i and θ−i of the Bayesian normal form, which gives us the utility ui(a, θ). The number of utility values stored in this BAGG is the same as the Bayesian
Motivated by the sequence form formulation of Koller et al. (GEB'96), this paper defines {\em... more Motivated by the sequence form formulation of Koller et al. (GEB'96), this paper defines {\em bilinear games}, and proposes efficient algorithms for its rank based subclasses. Bilinear games are two-player non-cooperative single-shot games with compact polytopal strategy sets and two payoff matrices (A,B) such that when (x,y) is the played strategy profile, the payoffs of the players are xAy and
We analyze the complexity of computing pure strategy Nash equilibria (PSNE) in sym- metric games ... more We analyze the complexity of computing pure strategy Nash equilibria (PSNE) in sym- metric games with a fixed number of actions. We restrict ourselves to "compact" representations, meaning that the number of players can be exponential in the representation size. We show that in the general case, where utility functions are represented as arbitrary circuits, the problem of decid- ing
Proceedings of the 11th ACM conference on Electronic commerce - EC '10, 2010
We analyze the complexity of computing pure strategy Nash equilibria (PSNE) in symmetric games wi... more We analyze the complexity of computing pure strategy Nash equilibria (PSNE) in symmetric games with a fixed number of actions. We restrict ourselves to "compact" representations, meaning that the number of players can be exponential in the representation size. We show that in the general case, where utility functions are represented as arbitrary circuits, the problem of deciding the existence
This paper proposes the Multi-Operation Patrol Scheduling System (MOPSS), a new system to generat... more This paper proposes the Multi-Operation Patrol Scheduling System (MOPSS), a new system to generate patrols for transit system. MOPSS is based on five contributions. First, MOPSS is the first system to use three fundamentally different adversary models for the threats of fare evasion, terrorism and crime, generating three significantly different types of patrol schedule. Second, to handle uncertain interruptions in the execution of patrol schedules, MOPSS uses Markov decision processes (MDPs) in its scheduling. Third, MOPSS is the first system to account for joint activities between multiple resources, by employing the well known SMART security game model that tackles coordination between defender’s resources. Fourth, we are also the first to deploy a new Opportunistic Security Game model, where the adversary, a criminal, makes opportunistic decisions on when and where to commit crimes. Our fifth, and most important, contribution is the evaluation of MOPSS via real-world deployments, providing data from security games in the field.
Nash equilibria of two-player games are much easier to compute in practice than those of n-player... more Nash equilibria of two-player games are much easier to compute in practice than those of n-player games, even though the two problems have the same asymptotic complexity. We used a recent constructive reduction to solve general games using a two-player algorithm. However, the reduction increases the game size too much to be practically usable. An open problem is to find
ABSTRACT Bayesian normal form’s tabular representation of type profile distribution P can be stra... more ABSTRACT Bayesian normal form’s tabular representation of type profile distribution P can be straightforwardly represented as a BN, e.g. by creating a random variable representing θ as the only parent of the ∑ random variables θ1,..., θn. To represent utility functions, we create an action graph G with i |Θi||Ai | action nodes; in other words, all type-action sets A ′ i,θi are disjoint. Each action ai ∈ Ai of the Bayesian normal form corresponds to |Θi | action nodes in the BAGG, one for each type instantiation θi. For each player i and each type θi ∈ Θi, each action node α ∈ A ′ has incoming i,θi edges from all action nodes from type-action sets A ′ j,θj for all j ̸ = i, θj ∈ Θj, i.e. all action nodes corresponding to ai ∈ Ai, the utility function of the other players. For each action node α ∈ A ′ i,θi uα is defined as follows: given configuration c (α) we can infer the action profile a ′ −i ∈ A ′ −i of the BAGG, which then tells us the corresponding a−i and θ−i of the Bayesian normal form, which gives us the utility ui(a, θ). The number of utility values stored in this BAGG is the same as the Bayesian
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