The Complexity of Nash Equilibria in Simple Stochastic Multiplayer Games

We analyse the computational complexity of finding Nash equilibria in simple stochastic multiplayer games. We show that restricting the search space to equilibria whose payoffs fall into a certain interval may lead to undecidability. In particular, we prove that the following problem is undecidable: Given a game G, does there exist a pure-strategy Nash equilibrium of G where player 0 wins with probability 1. Moreover, this problem remains undecidable if it is restricted to strategies with (unbounded) finite memory. However, if mixed strategies are allowed, decidability remains an open problem. One way to obtain a provably decidable variant of the problem is restricting the strategies to be positional or stationary. For the complexity of these two problems, we obtain a common lower bound of NP and upper bounds of NP and PSPACE respectively.Comment: 23 pages; revised versio

Decision Problems for Nash Equilibria in Stochastic Games

We analyse the computational complexity of finding Nash equilibria in stochastic multiplayer games with $\omega$-regular objectives. While the existence of an equilibrium whose payoff falls into a certain interval may be undecidable, we single out several decidable restrictions of the problem. First, restricting the search space to stationary, or pure stationary, equilibria results in problems that are typically contained in PSPACE and NP, respectively. Second, we show that the existence of an equilibrium with a binary payoff (i.e. an equilibrium where each player either wins or loses with probability 1) is decidable. We also establish that the existence of a Nash equilibrium with a certain binary payoff entails the existence of an equilibrium with the same payoff in pure, finite-state strategies.Comment: 22 pages, revised versio

Percentile Queries in Multi-Dimensional Markov Decision Processes

Markov decision processes (MDPs) with multi-dimensional weights are useful to analyze systems with multiple objectives that may be conflicting and require the analysis of trade-offs. We study the complexity of percentile queries in such MDPs and give algorithms to synthesize strategies that enforce such constraints. Given a multi-dimensional weighted MDP and a quantitative payoff function $f$, thresholds $v_i$ (one per dimension), and probability thresholds $\alpha_i$, we show how to compute a single strategy to enforce that for all dimensions $i$, the probability of outcomes $\rho$ satisfying $f_i(\rho) \geq v_i$ is at least $\alpha_i$. We consider classical quantitative payoffs from the literature (sup, inf, lim sup, lim inf, mean-payoff, truncated sum, discounted sum). Our work extends to the quantitative case the multi-objective model checking problem studied by Etessami et al. in unweighted MDPs.Comment: Extended version of CAV 2015 pape

Non-Zero Sum Games for Reactive Synthesis

In this invited contribution, we summarize new solution concepts useful for the synthesis of reactive systems that we have introduced in several recent publications. These solution concepts are developed in the context of non-zero sum games played on graphs. They are part of the contributions obtained in the inVEST project funded by the European Research Council.Comment: LATA'16 invited pape

The Complexity of Nash Equilibria in Stochastic Multiplayer Games

We analyse the computational complexity of finding Nash equilibria in turn-based stochastic multiplayer games with omega-regular objectives. We show that restricting the search space to equilibria whose payoffs fall into a certain interval may lead to undecidability. In particular, we prove that the following problem is undecidable: Given a game G, does there exist a Nash equilibrium of G where Player 0 wins with probability 1? Moreover, this problem remains undecidable when restricted to pure strategies or (pure) strategies with finite memory. One way to obtain a decidable variant of the problem is to restrict the strategies to be positional or stationary. For the complexity of these two problems, we obtain a common lower bound of NP and upper bounds of NP and PSPACE respectively. Finally, we single out a special case of the general problem that, in many cases, admits an efficient solution. In particular, we prove that deciding the existence of an equilibrium in which each player either wins or loses with probability 1 can be done in polynomial time for games where the objective of each player is given by a parity condition with a bounded number of priorities

Training infection control and hospital hygiene professionals in Europe, 2010 : agreed core competencies among 33 European countries

The harmonisation of training programmes for infection control and hospital hygiene (IC/HH) professionals in Europe is a requirement of the Council recommendation on patient safety. The European Centre for Disease Prevention and Control commissioned the ‘Training Infection Control in Europe’ project to develop a consensus on core competencies for IC/HH professionals in the European Union (EU). Core competencies were drafted on the basis of the Improving Patient Safety in Europe (IPSE) project’s core curriculum (CC), evaluated by questionnaire and approved by National Representatives (NRs) for IC/HH training. NRs also re-assessed the status of IC/HH training in European countries in 2010 in comparison with the situation before the IPSE CC in 2006. The IPSE CC had been used to develop or update 28 of 51 IC/HH courses. Only 10 of 33 countries offered training and qualification for IC/ HH doctors and nurses. The proposed core competencies are structured in four areas and 16 professional tasks at junior and senior level. They form a reference for standardisation of IC/HH professional competencies and support recognition of training initiatives.peer-reviewe

How many infection control staff do we need in hospitals?

During a one-day workshop experienced infection control practitioners (ICPs) and medical microbiologists debated how much time was needed for the delivery of infection control activities in a model hospital. They agreed a standard of one full-time equivalent (FTE) ICP per 178 hospital beds and one FTE medical microbiologist per 806 hospital beds. This is 40% and 24% more than the usual standard, respectively. Now that official numbers of hospital beds have become an inadequate parameter for work delivered by hospitals, a new standard is proposed, with the number of admissions as the denominator. This is one FTE ICP per 5000 admissions and one medical microbiologist or epidemiologist per 25000 admission