Demographic Forecasts Using the Game Theory

International Journal of Environmental Research and Public Health

Article Demographic Forecasts Using the Game Theory

Marek Ogryzek 1,* , Krzysztof Rz ˛asa 1 and Edita Šarkiene˙ 2 1 Faculty of Geodesy, Geospatial and Civil Engineering, Institute of Geography and Land Management, University of Warmia and Mazury, 15 Prawochenskiego Street, 10-720 Olsztyn, Poland; [email protected] 2 Department of Roads, Faculty of Environmental Engineering, Vilnius Gediminas Technical University, Sauletekio al. 11, LT-10223 Vilnius, Lithuania; [email protected] * Correspondence: [email protected]

Received: 25 January 2019; Accepted: 12 April 2019; Published: 18 April 2019

Abstract: This paper offers certain predictions concerning the demographic population of the cities Vilnius and Olsztyn. The authors used a method of analyzing and synthesizing data sources, and comparing the actual data with the forecast between the years 1997–2014. Each prediction was prepared in connection with its use in various areas of life, particularly for all studies involving spatial planning. The data collected on the basis of the forecasts were used by spatial planners to devise strategies for local development at the city, municipality, and provincial levels. In this sense, they created basic documents for the sustainable planning of space. The process of forecasting is a difficult and complex issue, and its accuracy determines both the choice of methods and the quality of the output. Our study sets out predictions concerning the demographic processes over the coming years in the two cities mentioned. Given that all long-range forecasts are characterized by high risk, especially taking into account the unstable political situation in Europe, the steadily deteriorating situation in the labor market and rising social discontent are of relevance, as they are causing the ongoing dynamics of the population to change, making statistical errors more likely and more serious. This has meant that organizations like Poland’s Central Statistical Office, Eurostat, and the United Nations have to adjust their demographic projections at least every two years, and the methods for making demographic forecasts which are used by governmental institutions have proven to be less than satisfactory. The main purpose of the article, therefore, is to present the authors’ method of making demographic projections by using elements of game theory. The results obtained in this method were compared with the results of the forecasting methods currently used by the governments of Poland and Lithuania. The developed method, based on the same input data and analogous coefficients, brings more probable results.

Keywords: urban design planning; urban development; demographic forecasts; game theory; land management

1. Introduction Deﬁning the concept of locating the objects on the earth’s surface is one of the basic tasks of Urban Planning (Land Management). This is accomplished by implementing geodata into already existing or created spatial information systems supporting the planning processes. Diagnosing the current socio-economic situation and the conditions of urban development leads to the determination of changes in the projected factors (development poles), on the basis of which directions of spatial development and spatial policies in the area are determined. One of these poles of development is the forecast of demographic growth in the study area. Based on reasonable predictions about demographic development, urban planners can better oversee the necessary conditions of providing basic human needs, as well as deploying investments. According to [1], predictions of the Lithuanian

Int. J. Environ. Res. Public Health 2019, 16, 1400; doi:10.3390/ijerph16081400 www.mdpi.com/journal/ijerph Int. J. Environ. Res. Public Health 2019, 16, 1400 2 of 13 population pose a particular challenge. The main difficulty in predicting the long-term population is the socio-economic situation of the country, as well as political events, such as accession to the EU. Matysiak et al. [2] confirmed similar difficulties in forecasting Polish demographic developments. In 1990, Poland experienced a fundamental change in the system, and Lithuania announced its independence. In 2004, the two countries joined the European Union. These are similar situations, and the political, economic, and social changes that accompanied the transformation have had a great impact on the demographic structure of both countries. When comparing the actual situation in the Polish and Lithuanian demographies, there was a decrease in population in both cases—by 37.6 per − 1000 inhabitants in Poland, and 28.4 per 1000 inhabitants in Lithuania (Eurostat, 2014). Negative − population growth was due to emigration to Western Europe to find jobs and improve living conditions. International migration is one of the effects of Polish and Lithuanian accession to the EU, and is expected to increase in the future. According to Holzer [3], the main problem in population forecasting is the period of forecasted time and the evaluation system of the parameter’s distribution in space. Changes in the shape and structure of the population are long-term processes; therefore, the long-term forecast does not explicitly provide real value, but only determines the potential development of the study area. Already in 1985, Alho and Spencer [4] pointed out that, "It is possible to express the uncertainty of the jump-off population in probabilistic terms. This has not been done, because the current version of the PEP does not support this source of uncertainty”. Forecasting the population is an important element of urban planning practices and land use. When managing space, it is important to include, among other things, urban greenspace, as it has a positive impact on reducing psychological stress among adolescents [5–7]. The uncontrolled development of urbanization has a negative impact on the urban population [8–10] (Nisbet et al., 2009, Cox et al., 2018, Barton et al., 2010). Larger population growth creates three solution models. The first one is the construction of housing estates with high multi-family housing, using a smaller area and limiting the decrease in the rural areas. The second is increasing cities at the expense of rural areas, using single-family houses. The second model can be considered better in the context of urban development, due to the beneficial effects on the health of the population thanks to home gardens and rural-urban green spaces [11–13]. Another important design element is the distance a person has to overcome to arrive at a natural area. The availability and ecological indicators are equally important [14,15]. It is therefore reasonable to combine accessibility to services and green space planning with the quantity and health of the population [16]. The results of research conducted by the European Union are a rich source of information about the European Union area and offer a diversity of instruments, such as multilevel analysis, typologies, scenarios, and interactive tools, which are useful in spatial planning and territorial development. The demographic changes shown in Figure1 may be analyzed as an example of the use of a multilevel approach and typology studies. Seven types of region, experiencing many of the various effects of demographic and migration flows, are distinguished here. The maps show that all Lithuanian and Polish regions, as well as other regions of Central and Eastern Europe, have been classified as an area of potential challenges for the workforce. Compared to the EU average, they have a higher proportion of people at a young age. However, the main problem here is a mismatch between the number of economically active people, their education, and aspirations, as well as the opportunities of finding suitable employment in the regional labor market. Therefore, despite the great potential of the quantity of labor, such regions are characterized by a decrease in population, due to lower birth rates and outflows. Int. J. Environ. Res. Public Health 2019, 16, 1400 3 of 13 Int. J. Environ. Res. Public Health 2019, 16, x 3 of 13

Figure 1. Map typology of demographic status. Source: www.espon.eu[17]. Figure 1. Map typology of demographic status. Source: www.espon.eu [17]. Spatial modeling based on demographic projections is burdened with the risk of adopting forecast variantsSpatial that modeling are too optimisticbased on ordemographic pessimistic; therefore,projections it isis essential burdened to verifywith the forecasts risk of annually adopting forecastand to variants update them.that are A reliancetoo optimistic on long-term or pessimistic forecasts; therefore bring with, it them is essential the potential to verify danger forecasts of annuallyfuture economicand to update losses, them. because A reliance the accuracy on long of-term any long-termforecasts bring prediction with them will naturally the potential decrease danger of overfuture time. economic To counter losses, this, because the forecasters the accuracy would of usuallyany long prepare-term prediction various variants will naturally of the forecast decrease over(minimum, time. To maximum, counter this, and intermediatethe forecasters variant), would though usually this prepare solution increasesvarious variants the cost of of the the studies forecast (minimum,and makes maximum the process, moreand time-consuming.intermediate variant), Ogryzek though et al. [18 ]this drew solution up an algorithm increases of demographicthe cost of the studiesdevelopment and makes for one thevariant—one process more that time was-consuming. the most likely Ogryzek and at et the al. lowest [18] drew risk of up using an aalgorithm random of factor. He incorporated game theory to obtain a prediction of the future population with the lowest demographic development for one variant—one that was the most likely and at the lowest risk of risk of inaccuracy. Game theory is a mathematical theory dealing with the study of optimal behavior using a random factor. He incorporated game theory to obtain a prediction of the future population in the event of a conflict of interest. It was derived from studies on gambling, and continues to uses with the lowest risk of inaccuracy. Game theory is a mathematical theory dealing with the study of that terminology [19]. However, the theory has come to find applications mainly in economics, biology optimal(especially behavior in sociobiology), in the event sociology, of a conflict and computerof interest. science It was (artificial derived intelligence). from studies In on 2005, gambling Thomas,, and continuesSchelling, to anduses Aumann that terminology received the [19 Nobel]. However, Prize in the economics theory has for come the application to find application of game theorys mainly in in economics,the social sciencesbiology and (especially in microeconomics in sociobiology), (for the behavior sociology of individuals, and computer and conflict science resolution) (artificial [20]. intelligence).The mechanisms In 2005, used Thomas in economic, Schelling solutions, and wereAumann used received as the archetype the Nobel of mathematicalPrize in economics solutions. for the applicationOgryzek [18of] extendedgame theory this methodin the tosocial the demographic sciences and forecast in microeconomics of a random factor (for to the heighten behavior the of individualselement of and uncertainty conflict in resolution) decision-making [20]. The events, mechanisms and his study used appears in economic to be accurate solutions in reality-based were used as the archetype of mathematical solutions. Ogryzek [18] extended this method to the demographic forecast of a random factor to heighten the element of uncertainty in decision-making events, and his study appears to be accurate in reality-based modeling on an extensive comparative analysis of results from the classical and random factors. His way of solving the problem is to determine the

Int. J. Environ. Res. Public Health 2019, 16, 1400 4 of 13

Int. J. Environ. Res. Public Health 2019, 16, x 4 of 13 modeling on an extensive comparative analysis of results from the classical and random factors. Hismost way probable of solving variant the problem projections is to determine(preserving the the most future probable direction variant of projections changes (preservingin the basic the futuredemographic direction processes of changes provided in the by basic conventional demographic methods), processes as providedopposed to by an conventional intermediate methods), option asbetween opposed the tomaximum an intermediate and minimum option variant between projections. the maximum The algorithm and minimum was implemented variant projections. for the TheGIS environment algorithm was using implemented the Python for programming the GIS environment language usingin order the to Python automate programming the process. language Saving intime order and to money automate when the forecasting process. Saving the most time likely and money future when population forecasting may thestreamline most likely the futuredata populationacquisition process may streamline of spatial theinformation data acquisition systems, processimprove of adjuvant spatial planning information processes systems,, and improve make adjuvantupdating planningand verification processes, in subsequent and make updatingyears more and efficient. verification In order in subsequent to verify the years utility more algorithm, efficient. Indemographic order to verify forecasts the utility for Vilnius algorithm, and Olsztyn demographic from 1997 forecasts to 2014 for were Vilnius used. and Olsztyn from 1997 to 2014 were used. 2. Materials and Methods 2. MaterialsDemographic and Methods forecasting looks at the population by age and sex, as well as selected characteristicsDemographic (e.g. forecasting, marital status looks and at the education), population or by things age and like sex, labor as wellresources as selected or the characteristics number of (e.g.,households. marital Forecasting status and education),methods can or be things divided like into labor simple resources extrapolation or the number methods of households.based on Forecastingpopulation growth methods trends can be or divided component into simple methods extrapolation where the methodspopulation based number on population is equal to growth the trendsnumber or of component births, deaths, methods and net where migration, the population as well as number the cohort is equal-component to the number method. of There births, are deaths, also andmultistate net migration, models and as well stochastic as the cohort-componentforecasting models. method. A flowchart There of are demographic also multistate forecasting models andis stochasticshown in Fig forecastingure 2. models. A flowchart of demographic forecasting is shown in Figure2.

FigureFigure 2. 2. FlowchartFlowchart of of forecasting forecasting demographic. demographic. Source: Source: [2 [121].].

The method of of population population projection projection is is based based on on the the calculation calculation of of successive successive natural natural movement movement and migration based onon populationpopulation numbersnumbers andand ratesrates byby sexsexand andage age for for subsequent subsequent years, years, resulting resulting in prospectivein prospective states states of population of population [22]. [2 These2]. These calculations calculations are performed are performed at the lowestat the levellowest of territoriallevel of division,territorial with division, the results with forthe higherresults levels for higher being levels obtained being by obtained aggregation. by aggregation. The calculations The ofcalculations population sizeof population are carried size out forare the carried permanently out for residing the permanently population. residing The total population current population—presented. The total current aspopulation the condition—presented of the population—is as the condition obtained of the by addingpopulation the constant—is obtained balance by population adding the in timeconstant to the population.balance population The prospective in time to demographic the population. rates The for prospective the lowest-level demographic units necessary rates for for the calculations lowest-level are obtainedunits necessary by taking for thecalculations territorial are diff obtainederentiation by of taking retrospective the territorial coefficients differentiation and the tendency of retrospective resulting fromcoefficients prospective and the national tendency coe ffiresultingcients into from account. prospective It is clear national that coefficients the territorial into di ffaccounterentiation. It is factorsclear dothat not the need territorial to persist differentiation in the same factors shape asdo the not baseline need toin persist the future. in theIn same the caseshape of as partial the baseline coefficients in ofthe births future. and In deaths, the case the tendencyof partial tocoefficients blur social of di ffbirthserences and associated deaths, withthe tendency fertility and to healthblur social trends differences associated with fertility and health trends as a result of the media of mass culture and the growth of social communication, is justified. In the case of internal migration, one can expect

Int. J. Environ. Res. Public Health 2019, 16, 1400 5 of 13 as a result of the media of mass culture and the growth of social communication, is justified. In the case of internal migration, one can expect dominant trends in favor of more diversified flows, which will be associated with a greater tendency to move without attachment to the place of residence [3]. There are three methods of producing predictive distributions for fertility, mortality, and migration: time-series extrapolation, expert judgement, and an analysis of past forecast errors [4,23]. In practice, a combination of at least two of these methods is used to produce predictive distributions [2]. Cohort-indicator methods are used in forecasts for CSO (GUS) [24], Eurostat [25–28], the UN [29], with multistate models for Poland being developed by Kotowska [30] and Strzelecki [31], and a stochastic forecast by Matysiak et al. [2]. The demographic forecast by using a random factor, developed by Ogryzek et al. [18], is a modification of the cohort-component method described above, involving the inclusion of mechanisms for solving problems in game theory, with particular emphasis on "playing with nature". Games with nature, according to Trzaskalik [32] and Kukula [33], are double games in which the opponent is nature. The opponent is not interested in the result of the game, so the game is resolved from the point of view of one of the players. The optimal strategy can be achieved using one of the rules of decision-making:

Wald Criterion (max–min rule) • Hurwicz Criterion • Bayesian Criterion • Optimistic Criterion (max–max rule) • Savage Criterion (rule of minimum regret). • Getting the most anticipated attributes of natural increase and migration, at their lowest risk of occurrence, will enable the most probable forecast of the future population. The projected population for a given year is the sum of the population from the previous year and the future rate of natural increase and migration. The state of the population at time t = state population at time t 1 + birth in the period (t 1, t) − − − deaths in the period (t 1, t) + immigration in the period (t 1, t) emigration in the period (t 1, t). − − − L (t) = L (t 1) + B (t 1, t) D (t 1, t) + I (t 1, t) E (t 1, t) − − − − − − − Aging: L (x + h, s, t + h) = L (x, p, t) p (x, s, t + h) L (x s, t)—the number of people s sex at the age x at the time t, wherein x = 0, 1, ..., z. (source: [34]) Time t is set as the beginning of (1.01) for the period. On this basis, you can determine the population for the next period, and so forth. p (x, s, t + h) the probability of living till age x + h year period t + h by a person of a certain − gender taken from life table for h = 1 ... n, etc.; the expected value of the parameter index kB, kD, kI, kE (kB—birth rate, kD - death rate, kI—population inflow rate, kE—emigration rate). This indicator is the number of live births, deaths, inflow, and outflow of the population of a thousand people for a given year, and is inversely proportional to the population from the previous year. B = LD * 1/kB; D = LD * 1/kD; I = LD * 1/kI; E = LD * 1/kE; (own source) Index k has been determined by computer simulations of the basic demographic processes (rate of natural increase and migration). The simulation performs the search for each attribute in 1000 draws, from 0 to 1000 (singles). Each group of 1000 people plays with population growth and migration, and the prize in the game depends on a number selected at random by the simulator, from 0–1000. If the drawn number is within the confidence interval, a single game is completed successfully (live birth or death, the inflow or outflow of the population). For most of the expected values of the parameter (at the lowest risk), a single game is repeated n times (the number of games is determined by the planner). Int. J. Environ. Res. Public Health 2019, 16, 1400 6 of 13

The value of parameter k is determined by us, using a modiﬁcation of the formula for the expected value of the game [18].

EV = kU = kZ = kN = kO = (w1,w2, p1, p2) = p1 w1 + p2 w2 (1)

where: w1, w2 May—value parameter p1, p2—likelihood parameter For a more precise measurement of risk, the variances of the game need to be calculated. The greater the deviation from the results, the riskier the game is [35] (Kaminska 2006). The software selects this parameter, that is, the lowest risk, derived from the formula for the variance of the game [18,35].

Xn WG = ps (ws EV)2, (2) − s = 1

where: ws—result of the game ps—probability of occurrence For each of the attributes, confidence intervals of the parameter were defined (ranges of numbers). These intervals are selected based on mathematical formulas using coefficient attributes (Table1), where the minimum limit range is the number zero, and the maximum limit used the formula:

Birth

U97 = L96/WU97/1000 U98 = L97/WU98/1000 ......

U25 = L97/WU24/1000 Deaths

Z97 = L96/WZ97/1000 Z98 = L97/WZ98/1000 ......

Z25 = L97/WZ24/1000 Population inﬂow

N97 = L96/WN97/1000 N98 = L97/WN98/1000 ......

N25 = L97/WN24/1000 Emigration

O97 = L96/WO97/1000 O98 = L97/WO98/1000 ......

O25 = L97/WO24/1000 L—population, WU—birth rate, WZ—deaths rate, WN—population inﬂow, WO—emigration rate

In order to obtain the most expected value of a given parameter (at its lowest risk), a single game is repeated n times. N in this study was 1000. Int. J. Environ. Res. Public Health 2019, 16, 1400 7 of 13

Table 1. List of rates.

Year Birth Deaths Population Inﬂow Emigration 1997 1.4 1.18 0.74 0.07 1998 1.4 1.18 0.74 0.07 1999 1.4 1.18 0.74 0.07 2000 1.5 1.18 0.74 0.07 2001 1.5 1.18 0.74 0.07 2002 1.5 1.18 0.74 0.07 2003 1.5 1.18 0.74 0.07 2004 1.5 1.18 0.74 0.07 2005 1.5 1.16 0.74 0.07 2006 1.6 1.16 0.74 0.07 2007 1.6 1.16 0.74 0.07 2008 1.6 1.16 0.85 0.07 2009 1.6 1.16 0.95 0.07 2010 1.6 1.16 1.05 0.07 2011 1.58 1.16 1.15 0.07 2012 1.58 1.16 1.15 0.07 2013 1.58 1.16 1.15 0.07 2014 1.58 1.16 1.15 0.07 2015 1.58 1.14 1.15 0.07 2016 1.58 1.14 1.15 0.07 2017 1.58 1.14 1.15 0.07 2018 1.58 1.14 1.15 0.07 2019 1.58 1.14 1.15 0.07 2020 1.58 1.14 1.15 0.07 2021 1.58 1.14 1.16 0.07 2022 1.58 1.14 1.17 0.07 2023 1.58 1.14 1.18 0.07 2024 1.58 1.14 1.19 0.07 2025 1.58 1.12 1.2 0.07 Source: [34].

The main sources of dates used in the article were: the Central Statistical Office (CSO) and Lietuvos Statistikos Departamente (LSD). Statistics Poland (formerly known in English as the Central Statistical Office) is Poland’s chief government executive agency charged with collecting and publishing statistics related to the country’s economy, population, and society, at the national and local levels. Lietuvos Statistikos Departamente (formerly known in English as Statistics Lithuania), is an institution in Lithuania which is managed by the government. It was founded in 1990 by a resolution during the process of Lithuanian re-establishment of independence, and codified by the Law on Statistics of the Republic of Lithuania. The institution became a member of the European Statistical System in 2004. These are the most reliable statistical institutions that collect and share national data for different purposes (e.g., for EUROSTAT). These data are used for various types of analyses: demographic, sociological, economic, and suchlike. Every formal institution that wants to prepare this type of analysis should use that data. The source of data for our simulator, obtained from CSO and LSD, were: Olsztyn population in 1996: 168,711 people, Poland in 1999: 38,260 people, and Vilnius population in 1996: 578,327 people, Lithuania in 1999: 3536 people. In the next step, we used rates from Table1 and repeated the calculations year by year. That way, we could predict the number of people in the following years.

3. Results and Discussion Our preliminary study summarizes demographic data for Poland and Lithuania. Figures3 and4 show the real and projected populations by Eurostat, and the result of the estimation made by the authors performed using our simulator according to the methodology described in Section2: Materials and Methods. Int. J. Environ. Res. Public Health 2019, 16, 1400 8 of 13 Int. J. Environ. Res. Public Health 2019, 16, x 8 of 13 Int. J. Environ. Res. Public Health 2019, 16, x 8 of 13

41004100 39003900 37003700 35003500 33003300 31003100 29002900 27002700 25002500 1998 2000 2002 2004 2006 2008 2010 2012 2014 1998 2000 2002real 2004forecast2006 simulation2008 2010 2012 2014 real forecast simulation

Figure 3. The demographic forecast of Lithuania (population in thousands). Source: own study based Figure 3. TheFigure demographic 3. The demographic forecast forecast of Lithuania of Lithuania (population (population in in thousands). thousands). Source: Source: own own study study based based on CSO, EUROSTAT, LSD [24–28,36]. on CSO, EUROSTAT,on CSO, EUROSTAT, LSD [24–28 LSD,36 [2].4–28,36].

43000 43000 42000 42000 41000 41000 40000 40000 39000 39000 38000 38000 37000 370001998 2000 2002 2004 2006 2008 2010 2012 2014 1998 2000 2002real 2004forecast2006 simulation2008 2010 2012 2014 real forecast simulation

Figure 4. The demographic forecast of Poland (population in thousands). Source: own study based on CSO, EUROSTAT, LSD [24–28,36]. Figure 4. TheFigure demographic 4. The demographic forecast offorecast Poland of Poland (population (population in thousands). in thousands). Source: Source: own own study study based based on CSO, EUROSTAT,Theon CSO,data LSD EUROSTAT,shown [24– in28 Figures,36 LSD]. [2 34 and–28,3 46 ]in. blue (real) are the actual state from the Statistical Offices, and was not subject to modeling. These are statistical data made available at the request of the The datagovernmental shownThe data in Figuresinstitutionsshown in3 Figures andboth 4in 3in Poland and blue 4 inand (real) blue in (real)Lithuania. are are the the actualThe actual next state stateelement from from in the the the Statistical chart Statistical is the Offices O, ﬃces, and was not subject to modeling. These are statistical data made available at the request of the and was notforecast, subject which tomodeling. is carried out Theseby governmental are statistical institutions data of Poland made and available Lithuania atfor theeach requestcity of the abovegovernmental 100,000. Suchinstitutions forecasting both should in Poland guarantee and rational,in Lithuania. sustainable The next development element inin the areachart of is the governmentallandforecast, institutions development, which bothis demographycarried in Poland out ,by and andgovernmental spatial in Lithuania.planning institutions for a The period nextof Polandof element10 to and30 years. Lithuania in the These chart forare eachthe isthe city forecast, which is carriedlegalabove out requirements 100,000. by governmental Such of forecastsforecasting performed institutions should guaranteein Poland of Poland andrational, Lithuania, and sustainable Lithuania and the development conclusions for each cityresultingin the above area of 100,000. Such forecastingfromland should themdevelopment, form guarantee the basisdemography for rational, all spatial, and sustainable activities,spatial planning suchdevelopment as forchanging a period the ofform in 10 the ofto development area30 years. of landThese fromdevelopment, are the forestlegals requirements into a housing of estate forecasts with theperformed provision in of Poland basic social and functions.Lithuania, By and analyzing the conclusions the data from resulting demography,Figuresfrom and them spatial3 and form 4, planningwe the can basis conclude for for all athat,spatial period for activities, Polish of 10 and such to Lithuania, 30 asyears. changing our Theseoriginal the form aresimulation of the development legal method requirements from of forecasts performedestimatesforests into the a infuture housing Poland size estate ofand the with population Lithuania, the provision more and ofaccurately. basic the conclusionssocial However, functions. decision resulting By analyzing-making from basedthe data them on from form the basis for all spatialeitherFigures method activities,3 and (prediction 4, we such can and conclude as simulation) changing that, for for the such Polish form a large and of area Lithuania, development is risky. ourOur originalstudy from confirms simulation forests that into themethod a housing estimates the future size of the population more accurately. However, decision-making based on estate with the provision of basic social functions. By analyzing the data from Figures3 and4, we can either method (prediction and simulation) for such a large area is risky. Our study confirms that the conclude that, for Polish and Lithuania, our original simulation method estimates the future size of the population more accurately. However, decision-making based on either method (prediction and simulation) for such a large area is risky. Our study conﬁrms that the demographic forecasts compiled using a simulation with elements of game theory can help us to predict the population better than by using traditional forecasting methods. Figures5 and6 contain the results of the comparison of demographic changes in Vilnius and Olsztyn in the years 1997–2014, that is, the dynamics of change, with the total amount in thousands. Int. J. Environ. Res. Public Health 2019, 16, x 9 of 13 Int.demographic J. Environ. Res. forecasts Public Health compiled 2019, 16, x using a simulation with elements of game theory can help9 usof 13to demographicpredict the population forecasts bettercompiled than using by using a simulation traditional with forecasting elements methods. of game theory can help us to Int. J. Environ. Res. Public Health 2019, 16, 1400 9 of 13 predictFigures the population 5 and 6 contain better thanthe results by using of thetraditional comparison forecasting of demographic methods. changes in Vilnius and OlsztynFigures in the 5 yearsand 6 1997contain–2014, the that results is, the of dynamicsthe comparison of change, of demographic with the total changes amount in in Vilniusthousands. and Olsztyn in the years 1997–2014, that is, the dynamics of change, with the total amount in thousands.

30 30 20 20 10 10 0 01997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 -101997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 -10 -20 -20 -30 -30 -40 -40 -50 -50 the dynamics of demographic change simulation of demographic change the dynamics of demographic change simulation of demographic change Figure 5. The dynamics of demographic change in Vilnius in the years 1997–2014. Source: own study Figure 5. The dynamics of demographic change in Vilnius in the years 1997–2014. Source: own studyCommented [M2]: Please check if there is Figure 5 citation in Figurebased on 5. TheCSO, dynamics EUROSTAT, of demographic LSD [24–28,3 change6]. in Vilnius in the years 1997–2014. Source: own study Commentedthis manuscript [M2]: Please check if there is Figure 5 citation in based on CSO,based EUROSTAT, on CSO, EUROSTAT, LSD [ 24LSD–28 [24,36–28].,36]. this manuscript 20 20 10 10 0 0 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 -10 -10 -20 -20 the dynamics of demographic change simulation of demographic change the dynamics of demographic change simulation of demographic change

Figure 6. The dynamics of demographic change in Olsztyn in the years 1997 –2014. Source: own study Commented [M3]: Please check if there is Figure 6 citation in Figurebased on 6. TheCSO, dynamics EUROSTAT, of demographic LSD [24–28 ,3change6]. in Olsztyn in the years 1997 –2014. Source: own study Figure 6. The dynamics of demographic change in Olsztyn in the years 1997–2014. Source: own studyCommentedthis manuscript [M3]: Please check if there is Figure 6 citation in based on CSO, EUROSTAT, LSD [24–28,36]. based on CSO,The EUROSTAT, dynamics of changes LSD [24 in– the28 ,population36]. of Vilnius resemble a sinusoid curve that regularly this manuscript rises Theand dynamicsfalls, but does of changes not exceed in the the population population of of Vilnius 1997 until resemble after 2008. a sinusoid However, curve in thatOlsztyn, regularly with The dynamicsrisesthe exception and offalls, changes of but a slightdoesin not decline the exceed population between the population 2001 ofand Vilnius of2002, 1997 until until resemble 2009 after there 2008. a is sinusoid However,generally ain curve positive Olsztyn, that trend. with regularly rises and falls, buttheHowever, does exception not from of exceed 2010a slight, strong thedecline populationdynamics between of 2001 change of and 1997 can 2002, be until until noticed. 2009 after This there 2008. situation is generally However, in Poland a positive inhappened Olsztyn, trend. with the However,due to the fromfact that2010 there, strong was dynamics an economic of change crisis canin 2001 be noticed. and 2002, This meaning situation that in Polandthe population happened of exception ofdueOlsztyn a slightto the migrated fact decline that to there larger between was cities an economicor 2001 emigrated and crisis abroad. 2002, in 2001 untilAn and important 2002, 2009 meaning thereelement that is that generally the was population possible a positive ofto trend. However, fromOlsztynpredict 2010, was migrated Poland’s strong to larger dynamicsaccession cities to orthe of emigrated European change abroad. Union, can be whichAn noticed. important broughtThis elementwith situationit the that legal was opportunity inpossible Poland to happened due to the factpredictto work that was in there the Poland’s EU. was A accessionlabor an outflow economic to the of European 50,000 crisis people Union, in 2001was which recorded and brought 2002, in Vilnius, with meaning it theand legal 13,000 that opportunity people the populationin of toOlsztyn. work inThe the simulated EU. A labor and outflowreal data of in 50 Figures,000 people 5 and was6 show recorded the dynamics in Vilnius, of demographic and 13,000 people changes in Olsztyn migratedOlsztyn. toThe larger simulated cities and real or emigrateddata in Figures abroad. 5 and 6 show An the important dynamics of elementdemographic that changes was possible to predict was Poland’s accession to the European Union, which brought with it the legal opportunity to work in the EU. A labor outflow of 50,000 people was recorded in Vilnius, and 13,000 people in Olsztyn. The simulated and real data in Figures5 and6 show the dynamics of demographic changes from 1997 to 2014. All the abrupt breaks and increases were treated by the forecast simulator as extremes and affected by the smoothing function. It is obvious that, in the simulation process, the authors performed tests using what is known as try n = 1000; that is, the simulation was repeated 1000 times. The larger the sample, the greater the smoothening of results (averaging). Figure7 shows a comparison of: the real population (real), state forecast (forecast), and the simulator (simulation) for Olsztyn in 1997–2014. Similarly to Figures3 and4, where the national results were presented, better results were obtained for the city of Olsztyn using our simulator in comparison with state forecasts. In this chart, it is even clearer that forecasting changes in the population for a period longer than five years, regardless of the method used, does not bring satisfactory results. In conclusion, such forecasts should be performed for a maximum period of three to five years. Int. J. Environ. Res. Public Health 2019, 16, x 10 of 13

from 1997 to 2014. All the abrupt breaks and increases were treated by the forecast simulator as extremes and affected by the smoothing function. It is obvious that, in the simulation process, the authors performed tests using what is known as try n = 1000; that is, the simulation was repeated 1000 times. The larger the sample, the greater the smoothening of results (averaging). Figure 7 shows a comparison of: the real population (real), state forecast (forecast), and the simulator (simulation) for Olsztyn in 1997–2014. Similarly to Figure 3 and Figure 4, where the national results were presented, better results were obtained for the city of Olsztyn using our simulator in comparison with state forecasts. In this chart, it is even clearer that forecasting changes in the population for a period longer than five years, regardless of the method used, does not bring Int. J. Environ. Res. Public Health 2019, 16, 1400 10 of 13 satisfactory results. In conclusion, such forecasts should be performed for a maximum period of three to five years.

195000

190000

185000

180000

175000

170000

165000 1996 1998 2000 2002 2004 2006 2008 2010 2012 2014 2016 forecast simalation real

Figure 7. The Demographic Forecast of Olsztyn (Years 1997–2014). Source: own study based on CSO Figure 7. The[24]. Demographic Forecast of Olsztyn (Years 1997–2014). Source: own study based on CSO [24]. The use of the simulator requires certain assumptions, and brings with it certain limitations. To The usestart of the the simulation, simulator you requiresneed to have certain reliableassumptions, data from public andsources brings indicating with the itsize certain of the limitations. population. It is also necessary to obtain the rates (Table 1) required for calculations from the state To start thestatistical simulation, institutions. you need In the toabsence have of reliable such rates, data one frommay attempt public to sourcesdevelop itindicating independently, the size of the population.although It is also it would necessary be impossible to obtain to compare the rates the (Tableobtained1 )results required with forthe state calculations forecasts. The from the state statistical institutions.preparing of methodology In the absence for calculating of such such rates, indicators one, mayas well attempt as support to of develop its validity it with independently, specific results of demographic forecasts, may be the subject of separate studies. The simulator, just although it wouldlike the state be impossible forecasts, is not to protected compare from the additional obtained factors results which with are thedifficult state to forecasts.predict, which The preparing of methodologyaffectsfor changes calculating in the population, such indicators, such as the economic as well crisis, as support major changes of its in validitythe state policy, with and specific results of demographicchanges forecasts, in EU membership. may be Forecasts the subject made ofby separategovernmentstudies. institutions The do not simulator, provide different just like the state variants of results, but only an optimal size of the population. In the process of the prepared forecasts, issimulation, not protected you can from not only additional increase or factors decrease which the number are diofffi repetitionscult to predict,of the forecast which (which aff ects changes in the population,affects the such result) as, but the also economic increase (or crisis, decrease) major the share changes of the random in the factor. state The policy, random and factor changes in EU membership.used Forecasts in the process made of game by government theory is actually institutions pseudo-random, donot so the provide simulator di hasfferent to develop variants a of results, certain randomization algorithm based on the assumptions (rates from Table 1). Using the simulator, but only anhowever, optimal it sizeis possible of the to treat population. these assumptions In the as process deterministic of theor stochastic prepared functions, simulation, of which you can not only increasesolution or decrease (results) do thees numbernot have to of be repetitions expected, but ofsimulated the forecast in a certain (which variant. aff ectsIn a stochasti the result),c but also increase (ormodel, decrease) we can the approximate share of the the appearance random of factor. this function The to random any mathematically factor used known in thefunction. process of game This means that the results obtained should be closer to the real values. A limitation of the use of the theory is actuallysimulator pseudo-random, is also the forecast period, so the although simulator similar has restrictions to develop also aapply certain to state randomization forecasts. The algorithm based on the assumptions (rates from Table1). Using the simulator, however, it is possible to treat these assumptions as deterministic or stochastic functions, of which solution (results) does not have to be expected, but simulated in a certain variant. In a stochastic model, we can approximate the appearance of this function to any mathematically known function. This means that the results obtained should be closer to the real values. A limitation of the use of the simulator is also the forecast period, although similar restrictions also apply to state forecasts. The conducted studies indicate a maximum of a three to five year period of the simulator’s applicability, although longer forecasts give better results than state forecasts. Demographic forecasting can be applied both on the macro scale (countries, regions) and on a local scale (city). When creating the assumptions of the development policy of each country, forecasting of populations is a very important element. Authorities need to know the trends of change in this area in order to properly formulate the social policy of the state (e.g., in the case of negative forecasts, to introduce mechanisms that encourage people to have children; or in the opposite situation, to limit social programs). The results of demographic forecasts also justify the need for development (or lack of development needs) of investments in the field of social and technical infrastructure (health care, education, road networks). Demographic forecasting is even more widely used on a local scale. Local governments, when creating strategies for socio-economic development, need to know the results of demographic forecasts. In this way, the demand (or lack of it) can be determined in terms of specific social and infrastructure investments (e.g., the construction of new housing estates, development of road systems, construction of kindergartens, schools, and healthcare facilities). The city Int. J. Environ. Res. Public Health 2019, 16, 1400 11 of 13 must also predict the relevant needs in the relevant documents, such as concerning new investment areas. In Poland, such planning documents are: a municipal development strategy, a study of the conditions and directions of spatial development of the municipality, and local land-use plans. All these documents form the spatial policy of the municipality in terms of spatial development. Without justifying the investment needs, these documents are not justified. A good research tool that will get the most reliable results of demographic forecasts is indispensable here. Thanks to this, there is a good chance that the development of the city will correspond to the actual needs of its current and future residents.

4. Conclusions The main purpose of the article was to present the author’s method of demographic projections using elements of game theory. The results obtained in this method were compared with the results of the methods currently used for forecasting by the governments of Poland and Lithuania. The obtained results showed that the developed method, based on the same input data and analogous coefficients, gives the possibility of obtaining more probable results. However, the results obtained by taking into account the random factor are more accurate and better than forecasts based on the classic model of reality. The authors noted that such a long forecasting period generates large errors, which is shown in Figures3 and4. In connection with the above, erroneous conclusions may be drawn, and thus may disrupt the sustainable development planned for a given city. To improve mathematical modeling, the forecast period should be shortened to a maximum of three to five years. In addition, a study of the distribution of input data should be performed. Thanks to this, it would be possible to get to know the function of the distribution of this data and carry out modeling in three variants: for the average value and for the simulation of the uncertainty value, minimum, and maximum. However, this was not the subject of our research in this article. In future analyses, the authors plan to use this approach. Additionally, simulations should be preceded by profound expert analysis of demographic change. The choice of factors that characterize the test area should be based on observations of long-term demographic changes. Using the simulator to pick the number of the population is more accurate than according to traditional methods, compared to the real increase in the population. It is necessary to use the same parameters to perform simulations, which are developed according to classical methods. Quantifying the population of cities is a more difficult and complicated process than in small towns. The authors of the article noticed that the use of the forecast method from a developed simulator using game theory by using the same input data models better results, which are closer to reality. However, the method is based on parameters and input data from the Bolesławski method [34] used to predict population numbers in Poland and Lithuania. Therefore, neither of these methods are perfect, though it seems that the method proposed in the article generates better forecast results. In the simulation process, the authors performed tests in the n = 1000 sample. The larger the sample, the greater the smoothening of results (averaging). In the simulator, it is possible to perform tests for a smaller number of attempts, or to indicate the possibility of variants, taking into account only extreme results. In such cases, Figures5 and6 would be closer to each other, but such extremes are not easy to predict (e.g., economic breakdowns, stock market crashes, change of currency to the Euro, accession to the EU, and the opening of labor markets) In 1997, such situations in Poland and Lithuania seemed almost unrealistic, as the countries were still operating in different political and economic systems. Nonetheless, while they may not be sufficient by themselves to form the basis of a decision, they can have a role to play in advanced decision-making support, and can provide reliable information about future populations. There is no specified threshold for statistical error that identifies the truth of this hypothesis, though the mutual comparison of the actual population is a reliable source of information about the method.

Author Contributions: Conceptualization, M.O. and K.R.; methodology, M.O.; software, M.O. and K.R.; validation, M.O. and K.R.; formal analysis, M.O. and K.R.; investigation, M.O. and K.R.; resources, M.O., K.R. and E.Š.; data curation, M.O. and K.R.; writing—original draft preparation, M.O. and K.R.; writing—review and editing, Int. J. Environ. Res. Public Health 2019, 16, 1400 12 of 13

M.O. and K.R.; visualization, M.O. and K.R.; supervision, M.O.; project administration, M.O. and K.R.; funding acquisition, M.O. and K.R. Funding: This research received no external funding. Conﬂicts of Interest: The authors declare no conﬂict of interest.

References

1. Alho, J. Stochastic Forecast of the Lithuanian Population 2001–2050. Res. Rep. 1998, 20, P98-1023-R. 2. Matysiak, A.; Nowok, B. Stochastic forecast of the population of Poland, 2005–2050. Demogr. Res. 2007, 17, 301–333. [CrossRef] 3. Holzer, J. Perspektywy Demograficzne Polski do 2030 r. Projekcje Studialne–Zało˙zenia,Wyniki Liczbowe, Wnioski, Seria Monografie i Opracowani, nr 6/300; ISD SGPiS: Warszawa, Poland, 1990. 4. Alho, J.; Spencer, B. Uncertain population forecasting. J. Am. Stat. Assoc. 1985, 80, 306–314. [CrossRef] [PubMed] 5. Mennisa, J.; Masonb, M.; Ambrusa, A. Urban greenspace is associated with reduced psychological stress among adolescents: A Geographic Ecological Momentary Assessment (GEMA) analysis of activity space. Landsc. Urban Plan. 2018, 174, 1–9. [CrossRef][PubMed] 6. Thompson, C.W.; Roe, J.; Aspinall, P.; Mitchell, R. More green space is linked to less stress in deprived communities: Evidence from salivary cortisol patterns. Landsc. Urban Plan. 2012, 105, 221–229. [CrossRef] 7. Verheij, R.A.; Maas, J.; Groenewegen, P.P. Urban-rural health differences and the availability of green space. Eur. Urban Reg. Stud. 2008, 15, 307–316. [CrossRef] 8. Nisbet, E.K.; Zelenski, J.M.; Murphy, S.A. The nature relatedness scale linking individuals’ connection with nature to environmental concern and behawior. Environ. Behav. 2009, 41, 715–740. [CrossRef] 9. Cox, D.T.C.; Shanahan, D.F.; Hudson, H.L.; Fuller, R.A.; Gaston, K.J. The impact of urbanisation on nature dose and the implications for human health. Landsc. Urban Plan. 2018, 179, 72–80. [CrossRef] 10. Barton, J.; Pretty, J. What is the best dose of nature and green exercise for improving health? A multi-study analysis. Environ. Sci. Technol. 2010, 44, 3947–3955. [CrossRef][PubMed] 11. Dennis, M.; James, P. Evaluating the relative influence on population health of domestic gardens and green space along a rural-urban gradient. Landsc. Urban Plan. 2017, 157, 343–351. [CrossRef] 12. Barthel, S.; Folke, C.; Colding, J. Social-ecological memory in urban gardens: Retaining the capacity for management of ecosystem services. Glob. Environ. Chang. 2010, 20, 255–265. [CrossRef] 13. Bolund, P.; Hunhammar, S. Ecosystem services in urban areas. Ecol. Econ. 1999, 29, 293–301. [CrossRef] 14. Paracchini, M.L.; Zulian, G.; Kopperoinen, L.; Maes, J.; Schägner, J.P.; Termansen, M.; Zandersen, M.; Perez-Soba, M.; Scholefield, P.A.; Bidoglio, G. Mapping cultural ecosystem services: A framework to assess the potential for outdoor recreation across the EU. Ecol. Indic. 2014, 45, 371–385. [CrossRef] 15. Ekkel, E.D.; Vries, S. Nearby green space and human health: Evaluating accessibility metrics. Landsc. Urban Plan. 2017, 157, 214–220. [CrossRef] 16. Van Herzele, A.; de Vries, S. Linking green space to health: A comparative study of two urban neighbourhoods in Ghent, Belgium. Popul. Environ. 2012, 34, 171–193. [CrossRef] 17. Rok, J.; Wojnar, K. Program Epson 2013. Available online: www.epson.eu (accessed on 21 November 2018). 18. Ogryzek, M.; Gał ˛azka, T. Weryfikacja metodyki wyboru optymalnego modelu prognozy rozwoju demograficznego dla potrzeb zarz ˛adzania przestrzeni ˛a planistyczn ˛a. Acta Sci. Pol. UWM Olszt. 2008, 7, 1–15. 19. Schelling, T. Strategy of Conflict. 1960. Available online: http://home.uchicago.edu/~{}rmyerson/research/ stratofc.pdf (accessed on 10 December 2018). 20. Rostański,T.; Drozd, M. Teoria Gier. 2003. Available online: www.giaur.qs.pl (accessed on 22 November 2018). 21. Keilman, N. Uncertainty in National Population Forecasting: Issues, Backgrounds, Analyses, Recommendations; Sweiss & Zeitlinger B.V.: Amsterdam, The Netherlands; Lisse, The Netherlands, 1990; p. 49. 22. Bolesławski, L. Demographic Projection of Poland 2003–2030; Central Statistical Office: Warsaw, Poland, 2004. 23. Keilman, N.; Pham, D.; Hetland, A. Why population forecasts should be probabilistic–Illustrated by the case of Norway. Demogr. Res. 2002, 6, 410–453. [CrossRef] 24. Central Statistical Office. Rocznik Statystyczny 1999, 2003, 2008; CSO (GUS): Warszawa, Poland, 1999. Int. J. Environ. Res. Public Health 2019, 16, 1400 13 of 13

25. European Commission. Eurostat Regional Yearbook2008; European Communities: Maastricht, The Netherlands, 2008. [CrossRef] 26. European Commission. Eurostat Regional Yearbook 2014; European Union: Maastricht, The Netherlands, 2014. [CrossRef] 27. European Commission. Eurostat Yearbook 2004, the Statistical Guide to Europe; European Communities: Maastricht, The Netherlands, 2004. 28. European Commission. Eurostat Yearbook 2010. Europe in Figures; European Union: Maastricht, The Netherlands, 2010. [CrossRef] 29. United Nations Population Information Network. Available online: http://www.un.org/popin/ (accessed on 28 November 2018). 30. Kotowska, I. Zastosowanie Modelu LIPRO do Prognozowania Ludno´sciPolski do Roku 2050; Monografie i opracowania 331, ISiD SGPiS: Warszawa, Poland, 1991. 31. Strzelecki, I. The multi-state projection of Poland’s population by educational level for the years 2003–2030. Studia Demogr. 2007, 2, 23–44. 32. Trzaskalik, T. Wprowadzenie do BadańOperacyjnych z Komputerem; PWE: Warszawa, Poland, 2008. 33. Kukuła, K. Badania Operacyjne w Przykładach i Zadaniach; Wyd. Naukowe PWN: Warszawa, Poland, 2011. 34. Bolesławski, L. Prognoza Demograficzna Miasta Olsztyna Do 2025; GUS: Olsztyn, Poland, 1997. 35. Kaminska, T. Teoria ryzyka. Available online: https://www.google.com/url?sa=t&rct=j&q=&esrc=s&source= web&cd=1&ved=2ahUKEwjgr9jk1dThAhUMtYsKHTZ2D9YQFjAAegQIAhAC&url=https%3A%2F% 2Fekonom.ug.edu.pl%2Fweb%2Fdownload.php%3FOpenFile%3D103&usg=AOvVaw1KW_7NAvnQE_ yyLsxdAmKN (accessed on 12 November 2018). 36. Lietuvos Statistikos Departamente (LSD) 1999, 2003, 2008; Statistics Lithuania: Vilnius, Lithuania, 1999, 2003, 2008.

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).