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Evolutionary game theory mathematically conceptualizes and analyzes biological interactions where one’s fitness not only depends on one’s own traits, but also on the traits of others. Typically, the individuals are not overtly rational and do not select, but rather inherit their traits. Cancer can be framed as such an evolutionary game, as it is composed of cells of heterogeneous types undergoing frequency-dependent selection. In this article, we first summarize existing works where evolutionary game theory has been employed in modeling cancer and improving its treatment. Some of these game-theoretic models suggest how one could anticipate and steer cancer’s eco-evolutionary dynamics into states more desirable for the patient via evolutionary therapies. Such therapies offer great promise for increasing patient survival and decreasing drug toxicity, as demonstrated by some recent studies and clinical trials. We discuss clinical relevance of the existing game-theoretic models of cancer and its treatment, and opportunities for future applications. Moreover, we discuss the developments in cancer biology that are needed to better utilize the full potential of game-theoretic models. Ultimately, we demonstrate that viewing tumors with evolutionary game theory has medically useful implications that can inform and create a lockstep between empirical findings and mathematical modeling. We suggest that cancer progression is an evolutionary competition between different cell types and therefore needs to be viewed as an evolutionary game.

Mathematical modeling plays an important role in our understanding of therapy resistance mechanisms in cancer. The polymorphic Gompertzian model, analyzed theoretically by Viossat and Noble, describes a heterogeneous cancer population consisting of therapy sensitive and resistant cells. This theoretically promising model has not previously been validated with real-world data. In this study, we provide this validation. We demonstrate that the polymorphic Gompertzian model successfully captures trends in bothin vitroandin vivodata on non-small cell lung cancer (NSCLC) dynamics under treatment. For thein vivodata of tumor dynamics in patients undergoing treatment, we compare the polymorphic Gompertzian model to the classical oncologic models, which were previously identified as the models fitting this data the best. We showed that the polymorphic Gompertzian model can successfully capture the U-shape trend in tumor size during cancer relapse, which can not be fitted with the classical oncologic models. In general, the polymorphic Gompertzian model corresponds well to bothin vitroandin vivoreal-world data, suggesting it as a candidate for improving the efficacy of cancer therapy, for example through evolutionary/adaptive therapies.

Stackelberg evolutionary game (SEG) theory combines classical and evolutionary game theory to frame interactions between a rational leader and evolving followers. In some of these interactions, the leader wants to preserve the evolving system (e.g. fisheries management), while in others, they try to drive the system to extinction (e.g. pest control). Often the worst strategy for the leader is to adopt a constant aggressive strategy (e.g. overfishing in fisheries management or maximum tolerable dose in cancer treatment). Taking into account the ecological dynamics typically leads to better outcomes for the leader and corresponds to the Nash equilibria in game-theoretic terms. However, the leader’s most profitable strategy is to anticipate and steer the eco-evolutionary dynamics, leading to the Stackelberg equilibrium of the game. We show how our results have the potential to help in fields where humans try to bring an evolutionary system into the desired outcome, such as, among others, fisheries management, pest management and cancer treatment. Finally, we discuss limitations and opportunities for applying SEGs to improve the management of evolving biological systems. This article is part of the theme issue ‘Half a century of evolutionary games: a synthesis of theory, application and future directions’.

The degradation of organophosphorus compounds CVX [O-butyl S-(2-diethylaminoethyl)methylphosphonothioate] and RVX [O-isobutyl S-(2-diethylaminoethyl)methylphosphonothioate] in soils is described. T...

Game theory is a powerful theory to model strategic decision-making, but also complex biological systems, such as cancer. In the past decades, game-theoretical models have helped to understand cancer, its response to various treatments, and to design better therapies. However, to fully utilize the potential of game-theoretical modelling in designing better anti-cancer therapies, we need more information on cancer population (ecological) and strategy (evolutionary) dynamics in response to treatment. Here we explore how transcriptomics data can be utilized as inputs of game-theoretic models for predicting evolutionary response to irradiation using patient-derived glioblastoma organoids. For that purpose, we utilize both supervised and unsupervised methods to disentangle cell mixtures from publicly available datasets and identify proportions of relevant cancer cell types in patient-derived glioblastoma organoids treated with radiotherapy. We then fit these proportions to the replicator dynamics, the most classic evolutionary game dynamics, to predict both transient evolutionary dynamics and evolutionary stable cell proportions in these organoids. Hereby we demonstrate a methodology that can be utilized to design evolutionary therapies, i.e. therapies that anticipate and prevent evolution of therapy resistance in cancer cells. Our predictions in glioblastoma suggest that hypoxia is the most important factor in identifying short-term response to irradiation, while this seems much less relevant for the long-term response. Further, we conclude that supervised methods are the best way to estimate cancer evolutionary dynamics when therapy resistance is a qualitative evolutionary trait. We provide a pipeline for designing better therapies through testing evolutionary responses in patient-derived organoids, while in parallel the ecological response can be tracked through serum biomarkers and imaging in the corresponding patients.

Here we analyze Darwinian dynamics of cancer introduced in [1], extended by including a competition matrix, and evaluate (i) when the eco-evolutionary equilibrium is positive and (ii) when the eco-evolutionary equilibrium is asymptotically stable.

We extend a two-step lottery model of Craft et al. to test the hypothesis that oak trees pursue a form of within-flower female choice to increase the diversity of fathers. Oak trees produce six ovules per flower while maturing just one acorn. When assuming a random ovule selection - which is a natural assumption in the absence of other hypotheses - observed fertilization patterns in oaks cannot be explained: long-distance fertilization is unusually common, even as nearby oak trees may be absent as pollen donors. Our model demonstrates how producing multiple ovules per flower permits selection for rare, distant fathers. The number of ovules per flower that maximizes paternal diversity increases with the number of trees. We introduce a cost function for ovule production for which six ovules per flower balance these costs with the benefits of diversifying fathers. Using data from two published field studies, 7 of 8 investigated maternal oaks had actual paternal diversity indices (average diversity index of 15.42) that fit the female choice hypothesis (estimated diversity of 14.66) significantly better than assuming a random selection from the six available ovules (estimated diversity of 7.649). A third field study permitted us to compare paternity by distance classes for two maternal trees. Both fit the female choice model better than random ovule selection.

Early Access; International audience; The application of evolutionary and ecological principles to cancer prevention and treatment, as well as recognizing cancer as a selection force in nature, has gained impetus over the last 50 years. Following the initial theoretical approaches that combined knowledge from interdisciplinary fields, it became clear that using the eco-evolutionary framework is of key importance to understand cancer. We are now at a pivotal point where accumulating evidence starts to steer the future directions of the discipline and allows us to underpin the key challenges that remain to be addressed. Here, we aim to assess current advancements in the field and to suggest future directions for research. First, we summarize cancer research areas that, so far, have assimilated ecological and evolutionary principles into their approaches and illustrate their key importance. Then, we assembled 33 experts and identified 84 key questions, organized around nine major themes, to pave the foundations for research to come. We highlight the urgent need for broadening the portfolio of research directions to stimulate novel approaches at the interface of oncology and ecological and evolutionary sciences. We conclude that progressive and efficient cross-disciplinary collaborations that draw on the expertise of the fields of ecology, evolution and cancer are essential in order to efficiently address current and future questions about cancer.

Fish populations subject to heavy exploitation are expected to evolve over time smaller average body sizes. We introduce Stackelberg evolutionary game theory to show how fisheries management should be adjusted to mitigate the potential negative effects of such evolutionary changes. We present the game of a fisheries manager versus a fish population, where the former adjusts the harvesting rate and the net size to maximize profit, while the latter responds by evolving the size at maturation to maximize the fitness. We analyze three strategies: i) ecologically enlightened (leading to a Nash equilibrium in game-theoretic terms); ii) evolutionarily enlightened (leading to a Stackelberg equilibrium) and iii) domestication (leading to team optimum) and the corresponding outcomes for both the fisheries manager and the fish. Domestication results in the largest size for the fish and the highest profit for the manager. With the Nash approach the manager tends to adopt a high harvesting rate and a small net size that eventually leads to smaller fish. With the Stackelberg approach the manager selects a bigger net size and scales back the harvesting rate, which lead to a bigger fish size and a higher profit. Overall, our results encourage managers to take the fish evolutionary dynamics into account. Moreover, we advocate for the use of Stackelberg evolutionary game theory as a tool for providing insights into the eco-evolutionary consequences of exploiting evolving resources.

Evolutionary game theory mathematically conceptualizes and analyzes biological interactions where one’s fitness not only depends on one’s own traits, but also on the traits of others. Typically, the individuals are not overtly rational and do not select, but rather, inherit their traits. Cancer can be framed as such an evolutionary game, as it is composed of cells of heterogeneous types undergoing frequency-dependent selection. In this article, we first summarize existing works where evolutionary game theory has been employed in modeling cancer and improving its treatment. Some of these game-theoretic models suggest how one could anticipate and steer cancer’s eco-evolutionary dynamics into states more desirable for the patient via evolutionary therapies. Such therapies offer great promise for increasing patient survival and decreasing drug toxicity, as demonstrated by some recent studies and clinical trials. We discuss clinical relevance of the existing game-theoretic models of cancer and its treatment, and opportunities for future applications. We discuss the developments in cancer biology that are needed to better utilize the full potential of game-theoretic models. Ultimately, we demonstrate that viewing tumors with an evolutionary game theory approach has medically useful implications that can inform and create a lockstep between empirical findings, and mathematical modeling. We suggest that cancer progression is an evolutionary game and needs to be viewed as such.Competing Interest StatementThe authors have declared no competing interest.Funding StatementThis research was supported by two European Union's Horizon 2020 research and innovation programs under the Marie Sklodowska-Curie grant agreement No's 690817 and 955708, the Dutch National Foundation (NWO) Grant number OCENW.KLEIN.277, the James S. McDonnell Foundation grant, Cancer therapy: Perturbing a complex adaptive system, a V Foundation grant, NIH/National Cancer Institute (NCI) R01CA170595, Application of Evolutionary Principles to Maintain Cancer Control (PQ21), NIH/NCI U54CA143970-05 Physical Science Oncology Network (PSON) Cancer as a complex adaptive system and Austrian Science Fund (FWF): DK W1225-B20. The last author wants to thank to her daughter Julia for keeping her awake during nights, which allowed her to work on this paper.Author DeclarationsI confirm all relevant ethical guidelines have been followed, and any necessary IRB and/or ethics committee approvals have been obtained.YesThe details of the IRB/oversight body that provided approval or exemption for the research described are given below:For this paper we have not conducted any clinical trials.All necessary patient/participant consent has been obtained and the appropriate institutional forms have been archived.YesI understand that all clinical trials and any other prospective interventional studies must be registered with an ICMJE-approved registry, such as ClinicalTrials.gov. I confirm that any such study reported in the manuscript has been registered and the trial registration ID is provided (note: if posting a prospective study registered retrospectively, please provide a statement in the trial ID field explaining why the study was not registered in advance).YesI have followed all appropriate research reporting guidelines and uploaded the relevant EQUATOR Network research reporting checklist(s) and other pertinent material as supplementary files, if applicable.YesThere was no original data created for this manuscript.

In the absence of curative therapies, treatment of metastatic castrate-resistant prostate cancer (mCRPC) using currently available drugs can be improved by integrating evolutionary principles that govern proliferation of resistant subpopulations into current treatment protocols. Here we develop what is coined as an ‘evolutionary stable therapy’, within the context of the mathematical model that has been used to inform the first adaptive therapy clinical trial of mCRPC. The objective of this therapy is to maintain a stable polymorphic tumor heterogeneity of sensitive and resistant cells to therapy in order to prolong treatment efficacy and progression free survival. Optimal control analysis shows that an increasing dose titration protocol, a very common clinical dosing process, can achieve tumor stabilization for a wide range of potential initial tumor compositions and volumes. Furthermore, larger tumor volumes may counter intuitively be more likely to be stabilized if sensitive cells dominate the tumor composition at time of initial treatment, suggesting a delay of initial treatment could prove beneficial. While it remains uncertain if metastatic disease in humans has the properties that allow it to be truly stabilized, the benefits of a dose titration protocol warrant additional pre-clinical and clinical investigations.

The concept of the evolutionary stable strategy (ESS) has been fundamental to the development of evolutionary game theory. It represents an equilibrial evolutionary state in which no rare invader can grow in population size. With additional work, the ESS concept has been formalized and united with other stability concepts such as convergent stability, neighborhood invasion stability, and mutual invisibility. Other work on evolutionary models, however, shows the possibility of unstable and/or non-equilibrial dynamics such as limit cycles and evolutionary suicide. Such textquotedblleftpathologiestextquotedblright remain outside of a well-defined context, especially the currently defined stability concepts of evolutionary games. Ripa et al. (2009) offer a possible reconciliation between work on non-equilibrial dynamics and the ESS concept. They noticed that the systems they analyzed show non-equilibrial dynamics when under-saturated and textquotedblleftfartextquotedblright from the ESS and that getting textquotedblleftclosertextquotedblright to the ESS through the addition of more species stabilized their systems. To that end, we analyzed three models of evolution, two predator-prey models and one competition model of evolutionary suicide, to see how the degree of saturation affects the stability of the system. In the predator-prey models, stability is linked to the degree of saturation. Specifically, a fully saturated community will only show stable dynamics, and unstable dynamics occur only when the community is under-saturated. With the competition model, we demonstrate it to be permanently under-saturated, likely showing such extreme dynamics for this reason. Though not a general proof, our analysis of the models provide evidence of the link between community saturation and evolutionary dynamics. Our results offer a possible placement of these evolutionary textquotedblleftpathologiestextquotedblright into a wider framework. In addition, the results concur with previous results showing greater evolutionary response to less biodiversity and clarifies the effect of extrinsic vs. intrinsic non-equilibrial evolutionary dynamics on a community.

Metastatic prostate cancer is initially treated with androgen deprivation therapy (ADT). However, re- sistance typically develops in about 1 year –a clinical condition termed metastatic castrate-resistant prostate cancer (mCRPC). We develop and investigate a spatial game (agent based continuous space) of mCRPC that considers three distinct cancer cell types: (1) those dependent on exogenous testosterone ( T + ), (2) those with increased CYP17A expression that produce testosterone and provide it to the envi- ronment as a public good ( T P ), and (3) those independent of testosterone ( T −). The interactions within and between cancer cell types can be represented by a 3 ×3 matrix. Based on the known biology of this cancer there are 22 potential matrices that give roughly three major outcomes depending upon the absence (good prognosis), near absence or high frequency (poor prognosis) of T −cells at the evolution- arily stable strategy (ESS). When just two cell types coexist the spatial game faithfully reproduces the ESS of the corresponding matrix game. With three cell types divergences occur, in some cases just two strategies coexist in the spatial game even as a non-spatial matrix game supports all three. Discrepancies between the spatial game and non-spatial ESS happen because different cell types become more or less clumped in the spatial game – leading to non-random assortative interactions between cell types. Three key spatial scales influence the distribution and abundance of cell types in the spatial game: i. Increasing the radius at which cells interact with each other can lead to higher clumping of each type, ii. Increasing the radius at which cells experience limits to population growth can cause densely packed tumor clusters in space, iii. Increasing the dispersal radius of daughter cells promotes increased mixing of cell types. To our knowledge the effects of these spatial scales on eco-evolutionary dynamics have not been explored in cancer models. The fact that cancer interactions are spatially explicit and that our spatial game of mCRPC provides in general different outcomes than the non-spatial game might suggest that non-spatial models are insufficient for capturing key elements of tumorigenesis.

This paper deals with integration of energy storage systems into electricity markets. We explain why the energy storage systems increase flexibility of both power systems and energy markets and why such flexibility is desirable, particularly when variable renewable energy sources are being used in existing power systems. As opposed to the existing literature, our model includes a dual technology energy storage system, acting in two different markets. We introduce a mathematical formulation for this model applied to two Dutch electricity markets. Adopting optimal control approach with the goal to maximize the yearly benefit, we show that the dual energy storage system can be profitable already when the same buying/selling strategies are adopted for the working days and weekends. We show that the profitability (slightly) increases with different buying/selling strategies for the weekdays and weekends. Finally, we demonstrate how the yearly benefit varies with size and efficiency of the devices chosen and market prices.

We propose a model with two types of cancer cells differentiated by their defense mechanisms against the immune system. “Selfish”cancer cells develop defense mechanisms that benefit the individual cell, whereas “cooperative”cells deploy countermeasures that increase the chance of survival of every cell. Our phenotypes capture the two main features of the tumor’s efforts to avoid immune destruction, crypticity against immune cells for the selfish cells, and tumor-induced immunosuppression for the cooperative cells. We identify steady states of the system and show that only homogeneous tumors can be stable in both size and composition. We show that under generic parameter values, a tumor of selfish cells is more benign than a tumor of cooperative cells, and that a treatment against cancer crypticity may promote immunosuppression and increase cancer growth.

In this paper we put forward a simple game-theoretical model of pollution control, where each country is in control of its own pollution, while the environmental effects of policies do not stop at country borders. In our noncooperative differential game, countries as players minimize the present value of their own costs defined as a linear combination of pollution costs and costs of environmentally friendly policies, where the state vector of the system consists of the pollution stock per country. A player’s time-varying decision is her investment into clean policies, while her expected costs include also pollution caused by her neighbors. We analyze three variants of this game: (1) a Nash game in which each player chooses her investment into clean policies such that her expected costs are minimal, (2) a game in which the players imitate the investments into clean policies of their neighbors without taking the neighbor’s success concerning their costs into account and (3) a game in which each player imitates her neighbors’ investments into clean policies if this behavior seems to bring a profit. In each of these scenarios, we show under which conditions the countries have incentives to act environmentally friendly. We argue that the different results of these games can be used to understand and design effective environmental policies.

For over 100 years it has been recognized that insect pests evolve resistance to chemical pesticides. More recently, managers have advocated restrained use of pesticides, crop rotation, the use of multiple pesticides, and pesticide-free sanctuaries as resistance management practices. Game theory provides a conceptual framework for combining the resistance strategies of the insects and the control strategies of the pest manager into a unified conceptual and modelling framework. Game theory can contrast an ecologically enlightened application of pesticides with an evolutionarily enlightened one. In the former case the manager only considers ecological consequences whereas the latter anticipates the evolutionary response of the pests. Broader applications of this game theory approach include anti-biotic resistance, fisheries management and therapy resistance in cancer.

We propose an optimal control framework to describe intra-seasonal predator-prey interactions, which are characterized by a continuous-time dynamical model comprising predator and prey density, as well as the energy budget of the prey over the length of a season. The model includes a time-dependent decision variable for the prey, representing the portion of the prey population in time that is active, as opposed to diapausing (a state of physiological rest). The predator follows autonomous dynamics and accordingly it remains active during the season. The proposed model is a generalization of the classical Lotka-Volterra predator-prey model towards non-autonomous dynamics that furthermore includes the effect of an energy variable. The model has been inspired by a specific biological system of predatory mites (Acari: Phytoseiidae) and prey mites (so-called fruit-tree red spider mites) (Acari: Tetranychidae) that feed on leaves of apple trees--its parameters have been instantiated based on laboratory and field studies. The goal of the work is to understand the decisions of the prey mites to enter diapause (a state of physiological rest) given the dynamics of the predatory mites: this is achieved by solving an optimization problem hinging on the maximization of the prey population contribution to the next season. The main features of the optimal strategy for the prey are shown to be that (1) once in diapause, the prey does not become active again within the same season and hence diapause is an irreversible process; (2) for the vast majority of parameter space, the portion of prey individuals entering diapause within the season does not decrease in time; (3) with an increased number of predators, the optimal population strategy for the prey is to start diapause earlier and to enter diapause more gradually. This optimal population strategy will be studied for its ESS properties in a sequel to the work presented in this article.

This paper should be read as addendum to Dieckmann et al. (J Theor Biol 241:370-389, 2006) and Parvinen et al. (J Math Biol 67: 509-533, 2013). Our goal is, using little more than high-school calculus, to (1) exhibit the form of the canonical equation of adaptive dynamics for classical life history problems, where the examples in Dieckmann et al. (J Theor Biol 241:370-389, 2006) and Parvinen et al. (J Math Biol 67: 509-533, 2013) are chosen such that they avoid a number of the problems that one gets in this most relevant of applications, (2) derive the fitness gradient occurring in the CE from simple fitness return arguments, (3) show explicitly that setting said fitness gradient equal to zero results in the classical marginal value principle from evolutionary ecology, (4) show that the latter in turn is equivalent to Pontryagin's maximum principle, a well known equivalence that however in the literature is given either ex cathedra or is proven with more advanced tools, (5) connect the classical optimisation arguments of life history theory a little better to real biology (Mendelian populations with separate sexes subject to an environmental feedback loop), (6) make a minor improvement to the form of the CE for the examples in Dieckmann et al. and Parvinen et al.

Huge numbers of microbes coexist in almost all habitats of our planet. Their interactions are governed by complex mechanisms, where both competition for resources and toxin production play important roles. Our goal is to understand key mechanisms that lead to coexistence. In this chapter we study many possible scenarios of microbial interactions and we analyze whether or not they can lead to coexistence of species. To achieve this we implemented agent- based models that mimic local dynamics of microbes; initially well mixed microbes from different species interact in a grid with a regular structure. Among others, we show that the coexistence rate is negatively correlated with the number of neighbors of each cell in the grid. Another observation is that the order of selection of focal cells in the grid influences the coexistence rate.

This work deals with the formulation of a distributed model predictive control scheme as a decision problem in which the decisions of each subsystem affect the decisions of the other subsystems and the performance of the whole system. This decision problem is formulated as a bargaining game. This formulation allows each subsystem to decide whether to cooperate or not depending on the benefits that the subsystem can gain from the cooperation. A solution based on game theory is proposed. The convexity and feasibility properties of the proposed solution are also presented. The proposed control scheme is tested on a case study with a chain of two continuous stirred tank reactors followed by a nonadiabatic flash separator.

Optimal toll design deals with the problem of determining toll which improves performance of a road traffic system. Noncooperative game theory is an excellent tool to investigate possible strategies to analyze such a problem, in which one has to take drivers’ reaction to toll and consequent changes in the traffic flow into account. Depending on the toll structure, the problem may be formulated as a Stackelberg game (when toll is uniform or time-varying) or as an inverse Stackelberg game (when toll is traffic-flow dependent), with the road authority as the leader and drivers as followers. While the inverse Stackelberg approach is more complex to adopt, it has been shown in our previous work that in most situations it brings better outcome for the road authority. This chapter reviews existing results on this topic, discusses our recent case studies within this framework, develops new properties, and brings forward the open issues within this area.

We propose a game-theoretical model to describe intra-seasonal predator–prey interactions between predatory mites (acari: phytoseiidae) and prey mites (also called fruit-tree red spider mites) (acari: tetranychidae) that feed on leaves of apple trees. Its parameters have been instantiated based on laboratory and field studies. The continuous-time dynamical model comprises predator and prey densities, along with corresponding energy levels, over the length of a season. It also includes time-dependent decision variables for the predator and the prey, representing the current portions of the predator and prey populations that are active, as opposed to diapausing (a state of physiological rest).our aim is to find the optimal active/diapausing ratio during a season of interaction between predatory mites and prey mites: this is achieved by solving a dynamic game between predator and prey. We hereby extend our previous work that focused solely on the optimal strategy for the prey. Firstly, we analyze the optimal behavior of the prey. Secondly, we show that the optimal strategy for the predator is to stay active for the entire season. This result corresponds to biological observations.keywordsmathematical modelspredator–prey interactionsdynamic noncooperative game theorydiapausemitesfruit orchard.

Apart from interacting, prey and predators may also avoid each other by moving into refuges where they lack food, yet survive by switching to an energy-saving physiological state. Lotka–Volterra models of predator–prey interactions ignore this option. Therefore, we have modelled this game of ‘joining versus opting out’ by extending Lotka–Volterra models to include portions of populations not in interaction and with different energy dynamics. Given this setting, the prey's decisions to join or to opt out influence those of the predator and vice versa, causing the set of possible strategies to be complex and large. However, using game theory, we analysed and published two models showing (i) which strategies are best for the prey population given the predator's strategy, and (ii) which are best for prey and predator populations simultaneously. The predicted best strategies appear to match empirical observations on plant-inhabiting predator and prey mites. Here, we consider a plausible third model that does not take energy dynamics into account, but appears to yield contrasting predictions. This supports our assumption to extend Lotka–Volterra models with ‘interaction-dependent’ energy dynamics, but more work is required to prove that it is essential and that what is best for the population is also best for the individual.

In this paper we define the European electricity market liberalization problem as a game with electricity producers as players, while the consumers’ electricity demand is exogenous. The model is based on real data. The producers maximize their profit by investing and choosing how much electricity they will produce by available means of electricity production. The aim of the research presented in this paper is to analyze different scenarios: a market with one electricity producer being a Stackelberg leader, a market with two electricity producers being Stackelberg leaders noncooperative among themselves, and a perfectly competitive market. As expected, in our case studies the perfectly competitive market yields the lowest electricity prices for the consumers.

In this paper, we deal with the European electricity market liberalization problem, formulated as a game with electricity producers as players, while the consumers' electricity demand is exogenous. The producers maximize their profit by choosing how much electricity they will produce individually by means of electricity production available to them. The aim of the research presented in this paper is to investigate the differences between the resulting electricity prices with different scenarios: a market with one Stackelberg leading producer, a market with two Stackelberg leading producers being noncooperative among themselves, and a perfectly competitive market. In the case studies the games involving one, two, and eight European countries are played. In the scenarios dealt with in this paper the perfectly competitive market yields the lowest electricity prices for the consumers. However, we also discuss possible drawbacks of liberalization. Our research aims to help understanding the complex process of electricity market liberalization.