Master Equation for Discrete-Time Stackelberg Mean Field Games with single leader

In this paper, we consider a discrete-time Stackelberg mean field game with a leader and an infinite number of followers. The leader and the followers each observe types privately that evolve as conditionally independent controlled Markov processes. The leader commits to a dynamic policy and the followers best respond to that policy and each other. Knowing that the followers would play a mean field game based on her policy, the leader chooses a policy that maximizes her reward. We refer to the resulting outcome as a Stackelberg mean field equilibrium (SMFE). In this paper, we provide a master equation of this game that allows one to compute all SMFE. Based on our framework, we consider two numerical examples. First, we consider an epidemic model where the followers get infected based on the mean field population. The leader chooses subsidies for a vaccine to maximize social welfare and minimize vaccination costs. In the second example, we consider a technology adoption game where the followers decide to adopt a technology or a product and the leader decides the cost of one product that maximizes his returns, which are proportional to the people adopting that technologyComment: 25 pages. arXiv admin note: text overlap with arXiv:2005.0199

A Stackelberg viral marketing design for two competing players

A Stackelberg duopoly model in which two firms compete to maximize their market share is considered. The firms offer a service/product to customers that are spread over several geographical regions (e.g., countries, provinces, or states). Each region has its own characteristics (spreading and recovery rates) of each service propagation. We consider that the spreading rate can be controlled by each firm and is subject to some investment that the firm does in each region. One of the main objectives of this work is to characterize the advertising budget allocation strategy for each firm across regions to maximize its market share when competing. To achieve this goal we propose a Stackelberg game model that is relatively simple while capturing the main effects of the competition for market share. {By characterizing the strong/weak Stackelberg equilibria of the game, we provide the associated budget allocation strategy.} In this setting, it is established under which conditions the solution of the game is the so-called ``winner takes all". Numerical results expand upon our theoretical findings and we provide the equilibrium characterization for an example.Comment: This paper appears in: IEEE Control Systems Letter

Solving Structured Hierarchical Games Using Differential Backward Induction

Many real-world systems possess a hierarchical structure where a strategic plan is forwarded and implemented in a top-down manner. Examples include business activities in large companies or policy making for reducing the spread during pandemics. We introduce a novel class of games that we call structured hierarchical games (SHGs) to capture these strategic interactions. In an SHG, each player is represented as a vertex in a multi-layer decision tree and controls a real-valued action vector reacting to orders from its predecessors and influencing its descendants' behaviors strategically based on its own subjective utility. SHGs generalize extensive form games as well as Stackelberg games. For general SHGs with (possibly) nonconvex payoffs and high-dimensional action spaces, we propose a new solution concept which we call local subgame perfect equilibrium. By exploiting the hierarchical structure and strategic dependencies in payoffs, we derive a back propagation-style gradient-based algorithm which we call Differential Backward Induction to compute an equilibrium. We theoretically characterize the convergence properties of DBI and empirically demonstrate a large overlap between the stable points reached by DBI and equilibrium solutions. Finally, we demonstrate the effectiveness of our algorithm in finding \emph{globally} stable solutions and its scalability for a recently introduced class of SHGs for pandemic policy making

Public Goods Games in Disease Evolution and Spread

Cooperation arises in nature at every scale, from within cells to entire ecosystems. In the framework of evolutionary game theory, public goods games (PGGs) are used to analyse scenarios where individuals can cooperate or defect, and can predict when and how these behaviours emerge. However, too few examples motivate the transferal of knowledge from one application of PGGs to another. Here, we focus on PGGs arising in disease modelling of cancer evolution and the spread of infectious diseases. We use these two systems as case studies for the development of the theory and applications of PGGs, which we succinctly review and compare. We also posit that applications of evolutionary game theory to decision-making in cancer, such as interactions between a clinician and a tumour, can learn from the PGGs studied in epidemiology, where cooperative behaviours such as quarantine and vaccination compliance have been more thoroughly investigated. Furthermore, instances of cellular-level cooperation observed in cancers point to a corresponding area of potential interest for modellers of other diseases, be they viral, bacterial or otherwise. We aim to demonstrate the breadth of applicability of PGGs in disease modelling while providing a starting point for those interested in quantifying cooperation arising in healthcare.Comment: 12 pages, 2 figures, 3 table

State-controlled epidemic in a game against a novel pathogen

The pandemic reminded us that the pathogen evolution still has a serious effect on human societies. States, however, can prepare themselves for the emergence of a novel pathogen with unknown characteristics by analysing potential scenarios. Game theory offers such an appropriate tool. In our game-theoretical framework, the state is playing against a pathogen by introducing non-pharmaceutical interventions to fulfil its socio-political goals, such as guaranteeing hospital care to all needed patients, keeping the country functioning, while the applied social restrictions should be as soft as possible. With the inclusion of activity and economic sector dependent transmission rate, optimal control of lockdowns and health care capacity management is calculated. We identify the presence and length of a pre-symptomatic infectious stage of the disease to have the greatest effect on the probability to cause a pandemic. Here we show that contrary to intuition, the state should not strive for the great expansion of its health care capacities even if its goal is to provide care for all requiring it and minimize the cost of lockdowns

Strategic Interactions in Antiviral Drug Use During an Influenza Pandemic

Background: The evolution of antiviral drug resistance during influenza pandemics has created widespread concern. Use of antiviral drugs is a main contributor to the evolution of drug-resistant strains. Moreover, there are recent examples of influenza viruses acquiring drug resistance seemingly without incurring a fitness penalty that reduces their transmission rate. This creates the possibility of strategic (game theoretical) interaction between jurisdictions making decisions about use of antiviral drug stockpiles. Methods: We developed and analyzed a 2-player 2-strategy game theoretical model. Each ‘player’ (an authority in a health jurisdiction) can choose to treat with antiviral drugs at a low rate or a high rate. High treatment rates are more likely to cause emergence of a drug-resistant strain, and once a drug-resistant strain has evolved, it can spread between the two jurisdictions. We determine the Nash equilibria of the game. Results: We show that there is a coordination game between the jurisdictions, where both players choosing a low treatment rate, or both choosing a high treatment rate, are the only stable outcomes. The socially optimal outcome occurs if both players cooperate by choosing a low treatment rate, thereby avoiding generating drug-resistant mutants. However, such cooperation may fail to materialize if the jurisdictions are closely connected through travel; if the drug-resistant mutant is tolerated (not seen as undesirable); or if the antiviral drug has partial efficacy against transmission of the drug-resistant strain. Conclusions: Inter-jurisdictional cooperation could be essential during a severe influenza pandemic, but we know little about how jurisdictions will interact in a scenario where highly pathogenic, drug-resistant mutant strains are able to transmit as effectively as non-resistant strains. Therefore, strategic multi-population interactions during influenza pandemics should be further studied.Canadian Institutes of Health Researc

Public Goods Games in Disease Evolution and Spread

Cooperation arises in nature at every scale, from within cells to entire ecosystems. In the framework of evolutionary game theory, public goods games (PGGs) are used to analyse scenarios where individuals can cooperate or defect, and can predict when and how these behaviours emerge. However, too few examples motivate the transferal of knowledge from one application of PGGs to another. Here, we focus on PGGs arising in disease modelling of cancer evolution and the spread of infectious diseases. We use these two systems as case studies for the development of the theory and applications of PGGs, which we succinctly review and compare. We also posit that applications of evolutionary game theory to decision-making in cancer, such as interactions between a clinician and a tumour, can learn from the PGGs studied in epidemiology, where cooperative behaviours such as quarantine and vaccination compliance have been more thoroughly investigated. Furthermore, instances of cellular-level cooperation observed in cancers point to a corresponding area of potential interest for modellers of other diseases, be they viral, bacterial or otherwise. We aim to demonstrate the breadth of applicability of PGGs in disease modelling while providing a starting point for those interested in quantifying cooperation arising in healthcare

How specificity and presentation of data affect our rational decision-making ability, oriented to a pharmaceutical perspective.

This dissertation aims to show the influence of factors on our perception and consequent evaluation of data, respectively our assessment of situations. Furthermore, it deals with the question to what extent rationally abstracted processes are common in the medical-pharmaceutical field. Overall, this dissertation indicates that limited evidence of abstracted approaches in the medical-pharmaceutical context can be found. Furthermore it is shown that drug evaluations, in particular the risk evaluation( even in itself) , are subject to strong subjective factors that distort the results.Ziel dieser Arbeit ist es, den Einfluss von Faktoren auf unsere Wahrnehmung und die daraus resultierende Bewertung von Daten und Situationen aufzuzeigen. Ergänzend, inwieweit rationale abstrahierte Prozesse im medizinisch-pharmazeutischen Bereich üblich sind. Insgesamt zeigt die Dissertation, dass abstrahierende Ansätze im medizinisch-pharmazeutischen Kontext nur in begrenztem Umfang zu finden sind. Außerdem wird gezeigt, dass Arzneimittelbewertungen, insbesondere die Risikobewertung (auch an sich), starken subjektiven Faktoren unterliegen, die die Ergebnisse verzerren

Optimizing Resource Allocation for COVID-19 Vaccination Planning within Long-Term Care Facilities: A Refined Multilevel Linear Programming Approach

In this paper, we introduce an algorithm designed to solve a Multilevel MOnoObjective Linear Programming Problem (ML(MO)OLPP). Our approach is a refined adaptation of Sinha and Sinha's linear programming method, incorporating the development of an "interval reduction map" that precisely refines decision variable intervals based on the influence of the preceding level's decision maker. Each construction stage is meticulously examined. The effectiveness of the algorithm is validated through a detailed numerical example, illustrating its practical applicability in resource management challenges. With a specific focus on vaccination planning within long-term care facilities and its relevance to the COVID-19 pandemic, our study addresses the optimization of resource allocation, placing a strong emphasis on the equitable distribution of COVID-19 vaccines