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Introduction to to theory theory of of control control in in organizations organizations Introduction Introduction to theory of control in organizations Introduction to theory of control in organizations for kids kids via via interactive interactive games games for for kids via interactive games for kids via interactive games
N.A. Korgin* Korgin* N.A. N.A. Korgin* N.A. Korgin* ** V.A. of V.A. Trapeznikov Trapeznikov Institute Institute of Control Control Sciences, Sciences, * V.A. Trapeznikov Institute of Control Sciences, 117997, 65 Profsoyuznaya Street, Moscow, Russia, 117997, 65 Profsoyuznaya Street, Moscow, Russia, * V.A. Trapeznikov Institute of Control Sciences, 117997,Skoltech 65 Profsoyuznaya Street, Moscow, Russia, Center for Energy Systems, Skoltech Center for Energy Systems, 117997, 65 Profsoyuznaya Street, Moscow, Russia, Skoltech Center for Energy Systems, Skolkovo Institute of Science Science andSystems, Technology, Skolkovo Institute of and Technology, Skoltech Center for Energy Skolkovo InstituteCenter, of Science and Technology, 143026, Skolkovo Innovation 3 Nobel Street, Moscow, Russia. 143026, Skolkovo Innovation Center, 3 Nobel Street, Skolkovo Institute of Science and Technology, 143026, Skolkovo Innovation Center, 3 Nobel Street, Moscow, Moscow, Russia. Russia. (Tel: +7495-335-60-37; e-mail:
[email protected]). (Tel: +7495-335-60-37; e-mail:
[email protected]). 143026, Skolkovo Innovation Center, 3 Nobel Street, Moscow, Russia. (Tel: +7495-335-60-37; e-mail:
[email protected]). This the of Federation This work work is is supported supported by+7495-335-60-37; the grant grant of of President President of Russian Russian Federation MD-6075.2015.9 MD-6075.2015.9 (Tel:by e-mail:
[email protected]). This work is supported by the grant of project President of Russian Federation MD-6075.2015.9 and RFBR, № 14-07-00875 andgrant RFBR, № of 14-07-00875 This work is supported by the of project President Russian Federation MD-6075.2015.9 and RFBR, project № 14-07-00875 and RFBR, project № 14-07-00875 Abstract: Abstract: In In order order to to present present theory theory of of control control in in organizations organizations for for elementary elementary school school graduates, graduates, aa Abstract: In order todescribes present theory ofofcontrol in organizations foronelementary school graduates, classical game which tragedy commons problem, based Cournot oligopoly model was wasaa classical game whichtodescribes tragedyofofcontrol commons problem, based Cournot oligopoly model Abstract: In order present theory in organizations foronelementary school graduates, classical game which describes tragedy of commons problem, based on Cournot oligopoly model was adopted. game We present present results of oftragedy first conducted conducted games, which show that kids oligopoly familiar model with basic basic adopted. We results first games, which show kids familiar with classical which describes of commons problem, based on that Cournot was adopted. Weoperations present results of first and conducted games, which show that models. kids familiar with basic arithmetical can understand exhibit different decision-making That allows us to to arithmetical operations can understand and exhibit different decision-making models. That allows us adopted. We present results of first conducted games, which show that kids familiar with basic arithmetical operations can understand andmechanisms exhibit different decision-making allowsatusthat to proceed to to design design of examples examples of control control that can can be explained explained models. to young youngThat audience proceed of of that be to audience arithmetical operations can understand andmechanisms exhibit different decision-making models. That allowsatusthat to proceed to of design of examples of control mechanisms that can be explained to young audience at that basic level level mathematical knowledge. basic of mathematical knowledge. proceed to of design of examples of control mechanisms that can be explained to young audience at that basic level mathematical knowledge. basic level of mathematical knowledge. © 2015, IFAC (International Federation of Automatic Control) Hosting by Game Elseviertheory, Ltd. All rights reserved. Keywords: Social and behavioural behavioural sciences, Control in Organization, Organization, Game theory, Mechanism Design, Keywords: Social and sciences, Control in Mechanism Design, Keywords: Social and behavioural sciences, Control in Organization, Game theory, Mechanism Design, Nash games Nash gamesSocial and behavioural sciences, Control in Organization, Game theory, Mechanism Design, Keywords: Nash games Nash games younger schoolchildren schoolchildren with with elements elements of of this this theory theory (see (see younger 1. younger schoolchildren with its elements of this theory (see 1. INTRODUCTION INTRODUCTION Burkov, 1989), 1989), which showed showed fruitfulness and relevance. relevance. Burkov, which its fruitfulness and 1. INTRODUCTION younger schoolchildren with elements of this theory (see Burkov, 1989), which showed its fruitfulness and relevance. 1. INTRODUCTION Theory of control in organizations is an integrated theory Burkov, 1989), which showed its fruitfulness and relevance. Theory of control in organizations is an integrated theory To improve students’ understanding and feedback it is To improve students’ and feedback it Theory of control in organizations is an integrated theory combining methodology of analysis and improve students’to understanding understanding andbright feedback it is is combining methodology of systems systems analysis theory and To Theory of control in organizations is an integrated extremely important carefully choose and clear extremely important to carefully choose bright and clear combining methodology of systems analysis and To improve students’ understanding and feedback it is mathematical methods of control theory and operations extremely important to carefully choose bright and clear mathematical methods of control theory and operations combining methodology of systems analysis and examples organizational control problems. Students will ofimportant organizational control choose problems. Students will mathematical methods control theorytheory) and to operations extremely of to carefully bright and clear research (including game of theory and graph graph the area area examples examples ofemploy organizational controlifproblems. Students will research (including game theory and the mathematical methods of control theorytheory) and to operations be able to their intuition the models selected for be able to employ their intuition if the models selected for research (including game theory and graph theory) to the area examples of organizational control problems. Students of organizational organizational management (see Burkov et al., al., 2013b 2013b and be able to employ their intuition if the models selected will for of management (see Burkov et and research (including game theory and graph theory) to the area classes comply with the level of their life experience. On the of organizational management (see Burkov et al., 2013b and classes withtheir the level of their lifemodels experience. On the be able comply to employ intuition if the selected for Novikov, 2013). Formal methods are used to construct classes comply with the level of their life experience. On the Novikov, 2013). Formal methods are used to construct of organizational management (see Burkov et al., 2013b and other not Novikov, 2013). Formal methods areprocedures used to construct other hand, the models studied should not hide hide the classes hand, complythe withmodels the levelstudied of their should life experience. On the robust and efficient decision-making (the, so hand,of the modelseconomic studiedconflicts should innot hide the robust and2013). efficientFormal decision-making (the, so other Novikov, methods areprocedures used to construct complexity real-world organizations complexity real-world organizations robust and efficient decision-making procedures (the, so other hand,of the modelseconomic studiedconflicts should innot hide the called, mechanisms), to and of of real-world economic conflicts in organizations called, mechanisms), to support support all all aspects aspects and stages stages of complexity robust and efficient decision-making procedures (the, so whose resolution demandseconomic sophisticated control mechanisms. resolution demands sophisticated control mechanisms. called, mechanisms), to support all(planning, aspects and stages of whose complexity of real-world conflicts in organizations cycle of management activity organization, whose resolution demands sophisticated control mechanisms. cycle management activity all(planning, organization, called, of mechanisms), to support aspects and stages of cycle of management activity (planning, organization, whose resolution demands sophisticated control mechanisms. motivation, and over decision from of these models, which is used in the educational process motivation, and monitoring), monitoring), over all all(planning, decision horizons, horizons, from One cycle of management activity organization, One of these models, which is used in the educational process motivation, and monitoring), over all decision horizons, from One of these models, which is used in the educational process operational to strategic management. is the Cournot oligopoly model (see, for example, Tirol, operational to strategic management. motivation, and monitoring), over all decision horizons, from One is the Cournot oligopoly model (see, for example, Tirol, of these models, which is used in the educational process operational to strategic management. is the Cournot oligopoly model (see, for example, Tirol, 1989), which describes, in particular, the problem of operational to strategic management. 1989), which describes, in particular, the problem of is the Cournot oligopoly model (see, for example, Tirol, Currently growing growing in in popularity popularity and and scope scope becomes becomes the the 1989), which describes, in particular, the problem of Currently excessive consumption of public resources (see Ostrom et al., excessive consumption of public resources (see Ostrom et al., Currently growing in popularity and scope becomes the 1989), which describes, in particular, the problem of concept of STEM education and (science, technology, excessive consumption of public resources (see Ostrom et al., concept of STEM education (science, technology, Currently growing in popularity scope becomes the 1994), known as the problem or concept ofandSTEM education (science, technology, 1994), known as the the tragedy tragedy ofresources the commons commons problem or excessive consumption of publicof (see Ostrom et al., engineering math, see for example Sanders, 2008 and 1994), known as Hardin, the tragedy of The the commons problem or engineering math, see for example Sanders, 2008 and concept ofand STEM education (science, technology, overgrazing (see 1968). model and its minor engineering and math, see for example Sanders, 2008 and overgrazing (see 1968). model andproblem its minor 1994), known as Hardin, the tragedy of The the commons or Mayo, 2009), one of the main statements of which is the need overgrazing (see Hardin, 1968). The model and its minor Mayo, 2009),and onemath, of the see mainforstatements which is2008 the need engineering example of Sanders, and modifications, being simple1968). enough, allows describing wide modifications, being simple enough, allows describing wide Mayo, 2009), one of the main with statements of which is the need overgrazing (see Hardin, The model and its aaminor to familiarize school children the latest achievements in simple enough, describingsystems a wide to familiarize school children the latest achievements in modifications, Mayo, 2009), one of the main with statements of which is the need assortment of of being control problems in allows organizational assortment control problems in organizational to familiarize school children with the latest achievements in modifications, being simple enough, allows describingsystems a wide fields of technology, engineering and assortment ofexample, control problems in 2009, organizational systems fields of science, science, technology, engineering and math math to familiarize school children with the latest achievements in (see, for Ostrom, Novikov and (see, for example, Ostrom, 2009, Novikov and fields of science, technology, engineering and math assortment of control problems in organizational systems throughout training period. the for (see, for example, Ostrom, 2009, and Novikov and throughout the trainingtechnology, period. For For presenting presenting the material material for Chkhartishvili, fields of the science, engineering and math 2014, Burkov et al. 2015b,) to illustrate Chkhartishvili, 2014, Burkov et al. 2015b,) and to illustrate throughout the training period. For presenting the material for (see, for example, Ostrom, 2009, Novikov and youngest audience is invited to use simple but vivid and Chkhartishvili, 2014, Burkov et al. 2015b,) and to illustrate youngest audience is invited to use simple but vivid and throughout the training period. For presenting the material for the possibilities to solve them with the help of control the possibilities to solve them with the help of control youngest audience is invited to use simple but vivid and Chkhartishvili, 2014, Burkov et al. 2015b,) and to illustrate fascinating examplesisin ininvited conjunction with theirbut realization in the possibilities to solve them with the help of control fascinating examples conjunction their realization in youngest audience to usewith simple vivid and Various modifications the model used mechanisms. Various modifications of the the help modelofare are used fascinating game examples in conjunction with their realization in mechanisms. the possibilities to solve them withof control interactive forms. Various modifications of the model are used interactive game forms. fascinating examples in conjunction with their realization in mechanisms. extensively in educational and experimental games (see interactive game forms. extensively educational and experimental games (see mechanisms. inVarious modifications of the model are used extensively inJanssen educational and2014), experimental games (see interactive game forms. Huck, 1999, at al., including web-based At present, a number of specialized courses on the theory of Huck, 1999,inJanssen at al.,and2014), including games web-based extensively educational experimental (see At present, a number of specialized courses on the theory of Huck, 1999, Janssen at al., 2014), including web-based At present, a number of specialized courses on the theory of systems (see Bodsky, Bodsky, 2014). control in developed and different (see Huck, 1999, Janssen2014). at al., 2014), including web-based control in organizations organizations are developed and taught taught to theory different At present, a number of are specialized courses on theto of systems systems (see Bodsky, 2014). controlaudiences in organizations are developed and and taught to different target (see et Burkov et systems (see 2014).this model as a basis for the setup target (see Burkov Burkov et al., al., 2013a 2013a Burkov et al., al., Therefore, controlaudiences in organizations are developed and and taught to different weBodsky, have chosen chosen Therefore, we have this model as a basis for the setup target audiences (see Burkov et al., 2013a and Burkov et al., 2015a). All of them are designed for higher school graduate Therefore, we have chosen this model as be a basis forduring the setup 2015a). All of them designed higher school graduate target audiences (see are Burkov et al.,for2013a and Burkov et al., of the educational game, which could used the of the educational game, which could used the 2015a). All of them are designed for higher school graduate Therefore, we have chosen this model as be a basis forduring the setup and graduate students. However, early as in the educational game,elementary which could be used during the and post postAll graduate students. However, as earlyschool as 1980s 1980s in the the of 2015a). of them are designed for as higher graduate interactive lessons with school graduets (about interactive lessons with elementary school graduets (about and post graduate students. However, as early as 1980s in the of the educational game, which could be used during the Soviet Union some attempts were made to familiarize the interactive lessons with elementary school graduets (about Soviet some attempts were made to as familiarize and postUnion graduate students. However, as early 1980s in the ten years years old). old). Soviet Union some attempts were made to familiarize the ten interactive lessons with elementary school graduets (about Soviet Union some attempts were made to familiarize the ten years old). ten years old). 2405-8963 © 2015, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Copyright 2015 IFAC 289 Peer review© of International Federation of Automatic Copyright ©under 2015 responsibility IFAC 289Control. Copyright © 2015 IFAC 289 10.1016/j.ifacol.2015.11.250 Copyright © 2015 IFAC 289
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2. THE GAME
2.2 Order of play All game sessions were organized in the following way. 9 pupils were selected from a class and divided into 3 groups (each consisting of three participants). Each participant was provided with a brief instruction (see Appendix A) and playing cards (see Table 1). The whole lesson was divided in two sessions: the training session (without prizes) and the competition session (when participants were earning real marmalade). Total length of the lesson was about 90 minutes. Training session lasts 60 minutes approximately, the competition session – the rest of time (30 minutes).
2.1 The model The story behind the game is as follows. There are three farmers whose cows share a common pasture ground. Every morning each farmer chooses how many cows to put on grass. Cows eat and trample grass, and so, the more cows browse the ground, the less grass is left to each, and, therefore, the less milk is produced by each cow. We turned pasture grounds into fabulous glades and milk into magic juice which magically turns into marmalade (see Appendix A). And we simplified mathematical model itself.
Table 1. Playing card
The game was designed for small groups (3 players) with simplified utility function for each player (farmer) − ui = (S - X )x i , where i Î N − index of player, N − is set
Player’s name Player’s colour Group’s title Round of game
of players ( # N = 3 ), S – is size of playing field (12 for three players), X – is total number of cows on playing field, x i Î [0, 9] − is the decision of the i-th player: the number of
Orange/Blue/Green Number of cows
Number of free places
Amount of marmalade
cows he send to graze, x − is the profile of decisions of all players. This model illustrates the problem of inconsistency of individual interests and the interests of society as a whole.
Training session was organized in following way. After short introduction of the game we conducted three learning rounds. During the introduction it is important to explain to kids, how their outcome depends from free places on playing field.
There is the sole Nash equilibrium in this game: " i Î N x i* = 3 , ui* = 9 . It gives the society welfare
U* =
å
u i* = 27 .
This equilibrium corresponds to the
All the actions of the participants are displayed on a magnetic board with magnetic cards, depicting cows, that placed by players themselves on the playing board, which makes the gameplay more enthralling. Figure 1 illustrates how playing field and cows were visualised for each group of players.
iÎ N
ideology of individual rationality in the sense that each player maximizes her utility by solving the nonlinear program u i (x i , X - i ) ® max, xi
(1)
where X - i − is number of cows for all players except i . Solution of the program (1) is usually referred to as the best response − bri (X - i ). But the global maximum of social welfare is Uˆ = 36 , which can be obtained with a family of efficient solutions (those strategy profiles x for which X = 6 ). There is the sole symmetric efficient solution " i Î N xˆ i = 2 and uˆ i = 12 . This solution is sometimes referred to as the fair solution. Unfortunately, it is not stable, i.e., it is not the solution of the program (1): " i Î N bri (4) = 4 . Using as an example of the control problem the task to ensure stability (in game-theoretic sense) of effective solution, this model can be successfully applied to illustrate the diversity of control mechanisms that allow to solve this problem, their capabilities and limitations.
Fig. 1. Glade and cows.
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At first round participants act according to instruction in order to get acquainted with game.
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3. RESULTS OF CONDUCTED GAMES Up to the time of writing of this text, 3 games were conducted. Figure 2 illustrates the game process. A brief video report from one of these games is available at https://youtu.be/n0R25jQGfoY (English subtitles should be turned on). Results of competition sessions of these games are provided in appendix B. Also
After first round pupils are explained equation, how to count total amount of marmalade that is collected from playing field, depending from number of cows that graze. And how to count number of cows that produce maximum amount of marmalade from one playing field – that is how to maximize total utility of society. Second round of game finalize this explanation in order to illustrate, how pupils understand it and how this explanation affects their actions. After second round idea of best response and individual utility maximization should be explained to pupils. Exactly, problem (1) is explained to them. For illustration, there actions (number of cows) from round two should be used. This part of training session is essential because it is necessary to explain to audience, how actions of one player affect utilities of others. And it turns out to be quite difficult, because for audience it may become clear that such actions may lead to worse result for all players comparing to actions that lead to maximization of total utility of society. Also it is highly desirable to explain to pupils, that result of the game may be negative in case when total number of cows is bigger than number of places on playing field.
Fig. 2. Game in progress.
Depending of the time, left third round may be held in order to understand, how pupils react on this explanation.
In all three cases pupils demonstrated high involvement in game’s process (Figure 3 provides good example) and good understanding of mathematical model of the game. All participants caught the idea how to solve simple optimization problems (utility maximization) and understand problem of interaction between them.
Competition session consists of three rounds of game. In all three groups all players participate in first and second rounds and earn real marmalade. Result for each group is determined from their playing fields and groups do not interact with each other. It is forbidden for players to interact in any way with each other during first and second rounds. Additionally it should be emphasized, that any correction in playing cards is not allowed during main session – player with correction in playing card will not receive payment for the round of game in which he makes such correction. After first and second rounds total amount of marmalade earned by each player is counted. From each group players with maximal amount proceed to final, third round. These representatives of each group act like one player determining number of cows they will send to graze and dividing marmalade earned in third round equally among them. Participants from one group are allowed to interact with each other in order to determine their collective action. But no interaction among groups is allowed. There is only one playing field in third round, were representatives from each group interact with each other.
Fig. 3. Ready for action. During playing rounds they exhibit different behavioral models – cooperative behaviour (maximizing utility of whole group), individual rational behaviour (utilizing best response strategies), altruistic behaviour (placing zero cows on playing field in order to transfer all utility to other members, player G12P3, see Table 2) and complex reflexive behaviour (see Korepanov and Novikov, 2012 and Crawford, 2013) (trying to forecast behaviour models of other players, player G13P2, see Table 2).
After third, final round, all players should receive all marmalade they earned during main session. And discussion about results of the main session should take place. It is important to encourage all participants to tell the audience about his impressions of the game and try to explain how he made decisions during the rounds of main session.
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During competition sessions from 9 groups in total 5 groups exhibit cooperative behaviour that resulted in efficient solutions (6 cows on playing field) both in first and second rounds. But only in two groups (G11 and G33, see Tables 2 and 4) it was symmetric solution (all players selected 2 cows). Only one group did not exhibit collective behaviour during this round at all (G23, see Table 3).
game that was realized makes it available to schools with low levels of IT equipment that can be quite actual for the developing countries. Besides, conduction of the game via software may take additional time to familiarize pupils with the system (and additional efforts from them) that could adversely affect the possibilities of presenting the game itself. Nowadays the most suitable for such purposes platform is MOBLAB (https://www.moblab.com/) but it is still not oriented on such young audience. Withal during last presentation of this game during MEETmeTONIGHT event by Politecnico di Milano (http://www.meetmetonight.it/) elder audience (about 14-15 years old) exhibited active interest to this game also.
Finals of all three games turn out to be quite dramatic. None of finals resulted with efficient solutions. Moreover, in game 1 all participants received zero outcome (see Table 2) and in game 3 only due to fact of penalization of representatives from one group all participants didn’t received negative payments (see Table 4). A more detailed analysis of the game is undoubtedly of interest in terms of behavioral and psychological theories (see Fehr et al., 2008), but it is not the main purpose of the experiment, what will be discussed in the conclusion. 4. CONCLUSIONS The main conclusion from the conducted games is that the experience with the attempt to adapt the model used in experimental and educational games with the adult audience for children audience can be considered successful. Children demonstrated an understanding of the model and exhibited the same behavioral models that are inherent to adult players. This allows us to proceed to the next stage - an attempt to illustrate to young audiences control mechanisms in organizational systems. As noted above, this model can be used for this purpose rather fruitfully. To ensure the sustainability of effective solution in the model can be applied various control mechanisms - motivational, institutional, information, etc. (see Burkov et al. 2013a)
Fig. 4. Giulia Cesari and Giulia Bernardi successfully introduced the game to the Italian audience. REFERENCES Bodsky, R. (2014). MOBLAB: a new technology for running interactive and strategic games in social science classes. EDULEARN14 Proceedings, 171-178. Burkov V. (1989) Chelovek. Upravlenie. Matematika. 162. Prosveshchenie (in Russian). Burkov V., Goubko M., Kondrat’ev V., Korgin N., Novikov D. (2013) Mechanism Design and Management: Mathematical Methods for Smart Organizations, 163. Nova Publishers. Burkov, V. N., Goubko, M. V., Korgin, N. A., & Novikov, D. A. (2013). Mechanisms of Organizational Behavior Control: A Survey. Advances in Systems Science and Application, 13(1), 1-20. Burkov, V. N., Goubko, M., Korgin, N., & Novikov, D. (2015). Introduction to Theory of Control in Organizations, 395. CRC Press. Burkov V., Novikov D., Shchepkin A. (2015) Control Mechanisms for Ecological-Economic Systems, 174. Springer, 2015. Crawford, V. P. (2013). Boundedly rational versus optimization-based models of strategic thinking and learning in games. Journal of Economic Literature, 51(2), 512-527. Fehr, E., Bernhard, H., & Rockenbach, B. (2008). Egalitarianism in young children. Nature, 454(7208), 1079-1083.
The second direction of development of this model can be considered its complication by including such important issues from the point of view of the decision-making and control theories, as a nature uncertainty, different possibilities of players, incomplete mutual awareness, public good issues (free-rider problem), etc. Such extensions are also actively used in the practice of teaching in universities, however, for their adaptation to the school audience before it is necessary to assess the level of mathematical training, which the participants of games must have for the familiarization of the proposed material. We should also discuss the prospects of implementation of software systems, including web-based systems to support the process of conduction of the game based on this model and its extensions, which is quite actual in face of STEM concept (see Mayo, 2009). As mentioned earlier, more complex version of this model are implemented for older audience in web-based systems for remote conducting of experimental and educational games. In addition, the implementation of modern technological opportunities that are available in schools, such as interactive whiteboards, tablets, etc., allows to combine the process of mastering the basics of theories of decision-making and control in organization with the development of children's skills of working with IT technologies. However, the format of the 292
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Janssen, M.A., Lee, A., Waring, T.M., & Galafassi, D. (2014). Experimental platforms for behavioral experiments on social-ecological systems. Ecology and Society, 19(4), 20. Hardin, G. (1968). The tragedy of the commons. Science, 162(3859), 1243-1248. Huck, S., Normann, H. T., & Oechssler, J. (1999). Learning in Cournot oligopoly: An experiment. The Economic Journal, 109, 80-95. Korepanov, V. O., & Novikov, D. A. (2012). The reflexive partitions method in models of collective behavior and control. Automation and Remote Control, 73(8), 14241441. Mayo, M. J. (2009). Video games: a route to large-scale STEM education? Science, 323(5910), 79-82. Novikov, D. (2013). Theory of Control in Organizations, 341. Nova Publishers. Novikov D., Chkhartishvili A.. (2014) Reflexion and Control: Mathematical Models, 298. CRC Press. Ostrom, E. (2009). A general framework for analyzing sustainability of social-ecological systems. Science 325, 419-422. Ostrom, E., Gardner, R., & Walker, J. (1994). Rules, games, and common-pool resources. University of Michigan Press. Sanders, M. (2008). STEM, STEM Education, STEMmania: A Series of Circumstances Has Once More Created an Opportunity for Technology Educators to Develop and Implement New Integrative Approaches to STEM Education Championed by STEM Education Reform Doctrine over the Past Two Decades. The Technology Teacher, 68(4), 20. Tirole, J. (1988). The theory of industrial organization, 479. MIT press.
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Hint: They will bring home 3 * 3 = 9 marmalades. The equation to define your marmalade is M = FP * x 1 How to play During each round of game you should determine number of cows that you would like to send to graze and write it in corresponding field in playing card. Important! Please, do not correct this number in playing card. Let the teacher know, that you are ready to announce your number of cows. Once all classmates from your group announced their numbers, count number of free places that is left on glade and amount of marmalade that your cows will bring to you. Write down these numbers in corresponding fields of playing card. Appendix B. RESULTS OF GAMES Below results of game rounds for competition sessions for three games conducted are presented. Players participated in final round are selected with bold. Total number of cows and amount of marmalade collected are provided for each group during first and second rounds and for all participants of finals Table 2. Results for game 1. Group G11
G12
Appendix A. GAME DESCRIPTION There is a fabulous glade in some fabulous kingdom where are fabulous cows graze. They bring home instead of milk a magic juice which magically turns into marmalade. You and two of your classmates will play role of fairy farmers - you will get 9 cows each, which you can send to graze on this glade. But glade is quite small - only 12 cows can find a place on it: P = 12 .
G13 total
Player G11P1 G11P2 G11P3 total G12P1 G12P2 G12P3 total G13P1 G13P2 G13P3 total
Round 1 2 12 2 12 2 12 6 36 4 24 2 12 0 0 6 36 2 12 2 12 2 12 6 36
Round 2 2 12 2 12 2 12 6 36 2 12 4 24 0 0 6 36 2 12 3 18 1 6 6 36
Final 3
0
3
0
6
0
12
0
In game 1 (see Table 2) all groups obtained efficient solution during first and second round. In final players deviated from cooperative behaviour.
The amount of marmalade that can be produced from juice that one cow can bring home is equal to free places, that will remain on glade after you will decide what number of cows each of you will send to graze. For example you've sent to graze x1 = 3 cows, your classmates x 2 = 2 and x 3 = 4 cows. How many free places are left on glade? Hint: There are 12 - 3 - 2 - 4 = 3 free places left on glade. The equation to define free places is FP = P - x1 - x 2 - x 3 What amount of marmalade your cows will bring home? 293
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Table 3. Results for game 2. Group G21
G22
G23
Player G21P1 G21P2 G21P3 total G22P1 G22P2 G22P3 total G23P1 G23P2 G23P3 total
Round 1 2 12 2 12 2 12 6 36 2 12 2 12 2 12 6 36 3 6 4 8 3 6 10 20
Round 2 3 15 2 10 2 10 7 35 2 10 2 10 3 15 7 35 4 0 3 0 5 0 12 0
total
4
Final 12
3
9
2
6
9
27
In game 2 (see Table 3) efficient solution is observed only during first round for groups G21 and G22. Group G23 didn’t exhibit cooperative behaviour. In final only player G23P2 acted in cooperative way. In game 3 (see Table 4) groups G31 and G33 obtained efficient solution during first and second round. In group G32 player G32P3 exhibited individually rational strategy in round one by announcing 4 cows which is the best response for 4 cows of other to participants from this group. At second round all players from group G32 acted in cooperative way. Group G23 didn’t exhibit cooperative behaviour. In final only player G32P3 acted in cooperative way while other participants exhibited strictly competitive strategies. Table 4. Results for game 3. Group G31
G32
G33 total
Player G31P1 G31P2 G31P3 total G32P1 G32P2 G32P3 total G33P1 G33P2 G33P3 total
Round 1 2 12 2 12 2 12 6 36 2 8 2 8 4 16 8 32 2 12 2 12 2 12 6 36
Round 2 6 36 0 0 0 0 6 36 2 12 2 12 2 12 6 36 2 12 2 12 2 12 6 36
6
Final -12/24
2
-4/8
6/0*
-12/0
14/8
-28/32
*Third group in game 3 was penalized of correction in their playing card, their cows were removed from field for payment count.
294
