Interdependent Decisions In The Game Theory

Transcription

1 Katarzyna Włodarczyk Śpiewak PhD. The University of Szczecin Interdependent Decisions In The Game Theory Keywords Game theory, dominant strategy, Nash equilibrium, prisoner s dilemma 1. Introduction Transformation of the world economy requires that companies gain competitive advantage on the market due to the strong competition. In today s economy, one of the most important resources and factors, which have an influence on strength, and competitiveness of a company is information and the way it is used in the decision-making process. The objective of this study is to show how one of the theories of firm the game theory may be put into practice to gain competitive advantage by a company. In this study, basic information on the game theory will be described as well as decision-making processes and their examples. This study may be helpful to students of microeconomics and to popularize the game theory. In the 30 s of the 20th century, the companies put focus on taking the right decisions to achieve the set goals. This resulted in creating a new branch of science the theory of firm. In its very beginnings the theory described the company activity in the conditions of perfect competition or monopoly, and the main task was to set a production level, which would maximize profits. The marginalization account was used in order to determine the company s activity. It was based on marginal cost and marginal 58

2 revenue. The optimal level of production was set by making equal the level of marginal revenue and cost. Contemporary economics literature describes alternative theories and models, which deny the basic principles of the neoclassic theory. According to the above theories, the main drawbacks of the neoclassic theory are the following principles: aim of a company is to make profit; entrepreneur is identified with the owner; entrepreneur s actions are rational; during the decision-making process only economic variables are taken into account; variability of the environment is not taken into account. The development of the alternative theories resulted from the criticism of the above principles. Nowadays trends in the development of the theory of firm more often take into consideration the influence of the variable environment on development of basic conditions, principles and conclusions. It stems from the fact that external and internal factors have an influence on neoclassic economics. The external factors, recognized by most of new theories are the possibility to shape prices and market ties with consumers, the government and other companies. The internal factors mainly reflect the workforce potential. As practice becomes more important in theory and a well functioning company in economic life, the company management arouses more interest than creating new theories (Galbraith, 1992). Specific theories of firm, traditional ones and those created in the 20 th century may be divided into groups. The basis for this division is the discipline, which was the cause of their creation. Thus, there are three main groups of theories: 1) theories based on economics (classic and neoclassic theories of firm), 2) theories based on the basics of organisation and management (managerial, behavioural, praxeological theories, model of techno structure, corporation model, transaction cost theory, agency theory), 3) theories based on cybernetics (system theories, theories based on linear programming, game theory). 59

3 Scientists are trying to classify the contemporary theories of firm. It is necessary to remember that the earliest alternative theories were not only the first attempts at criticizing neoclassic theory but they were also used in the company management. Among the oldest alternative theories, we can count managerial, behavioural, biological and life cycle theory. As for the younger ones, we can distinguish theory of innovative enterprise and institutional theories: i.e. transaction cost theory, contractual theory, agency theory and theory of property rights. Contemporary theories of firm distinguish, classify and divide the above concepts in numerous ways. Some divisions or distinctions of the alternative theories may not necessarily base on their amount or time of creation but on reference to the basic principles of the theory and its origin The economics literature treats about many concepts on how to classify alternative theories of firm. Thus, there is a problem how to name a specific theory; after its author or maybe it is better to group works of a few authors with similar conclusions and assumptions. The first way carries the risk of creating too many theories and methodological inconsistencies. Thus, the second one may be applied in many more cases. Polish economists, e.g. M.Goryń, use the second method and distinguish managerial and behavioural theories, agency theory, theory of property rights and transaction cost theory. U. Grzelońska, who also uses this method, separates three main groups: continuation of traditional neoclassic theory, school of industrial organization and institutional theory of firm. (M. Gorynia 2000, U. Grzelońska 1999, T. Gruszecki 2002). 2. Game theory One of the most common contemporary theories of firm is the game theory. Its basic rules were published in 1947 by J. von Neumann and O. Morgenstern (J. von Neumann, O. Morgenstern 1947, W. Stankiewicz, 2000). The game theory treats about behaviours in conflict situations. This discipline is better known thanks to J. Nash, J. Harsnyi and R. Selten, who got Nobel Prize in economics for their pioneering 60

4 analysis of equilibria in the theory of non-cooperative games. (J. Harsanyi, R. Selten 1988). In economics, game theory there are market players i.e. at least two competing entities. Each of them makes independent decisions that are not reliant on the competitor s behaviour. Every decision carries a specific risk, thus the players make assumptions about the competitor s decision. Each of the players has a set of moves or actions he can follow and he knows their results. The players have specific information and within that information, they act rationally to solve the market conflict and maximize profits. During basics of microeconomics course students assume that these entities are two companies, which act in the condition of oligopoly, and to be more precise in the condition of duopoly. These enterprises compete with each other and try to gain dominant market position or to eliminate the competitor from the market. In order to explain the decisions made within the game theory the following terms has been set: GAME a specific conflict situation on the market; PLAYER entity on the market; STRATEGY defines a set of moves or actions a player will follow in a given game; PAYOFF = VICTORY profit gained by the player after choosing a specific strategy Below, the payoff matrix will be presented, thus more assumptions have to be made for further analysis: there are two entities on the market two players; the players choose between two strategies; the players know the results of their own decisions; the way the competitor takes his decision is unknown; the basis for decision-making process is a payoff matrix. There are eight numbers in the matrix. They can reflect profits, turnover, costs, etc., depending on a specific data. The payoff matrix may be as follows: 61

5 PLAYER II Strategy C Strategy D Strategy A PLAYER I Strategy B I A II C I A II D I B II C I B II D The Player I can choose either strategy A or strategy B. The Player II also has two choices: strategy C or strategy D. Individual numbers indicate: I A payoff of the Player I using the strategy A; I B payoff of the Player I using the strategy B; II C payoff of the Player II using the strategy C; II D payoff of the Player II using the strategy D. On the basis of the matrix payoffs, the players take decision to choose the best strategy. The Player I considers choosing the strategy A or B and compares the payoffs stemming from their usage. Firstly the Player I assumes that his competitor the Player II chose the strategy C and he, the Player I, compares payoffs I A and I B. Then, assuming that the Player II chose the strategy D he compares his payoffs for using A or B. Thus the Player I compares payoffs that are in the first places in the matrix fields. The Player II does likewise. If he assumes that his competitor chose the strategy A he compares payoffs C and D. If he assumes that the Player II chose the strategy B, he also compares the payoffs C or D. Thus the Player II compares payoffs that are in the second places in the matrix fields. Microeconomics distinguishes three main game strategies: dominant strategies, Nash equilibrium, prisoner s dilemma. In dominant strategy, each of the players on the market takes decisions independently from the competitor s decisions. Each of the players has a set of moves, which maximises his payoffs. 62

6 PLAYER II Strategy C Strategy D PLAYER I Strategy A Strategy B The Player I: - assumes that the Player II chose the strategy C and he compares the payoff A 4 with the payoff B -2. The payoff A (4) is bigger than the payoff B (2). Thus the Player I chooses the strategy A; mathematical notation: A v B => 4 v 2 => 4 > 2=> 4 => A - assumes that the Player II chose the strategy D and he compares the payoff A 6 with the payoff B 0. The payoff A (6) is bigger than the payoff B (0). Thus the Player I chooses the strategy A; A v B => 6 v 0 => 6 > 0 => 6 => A Conclusion: regardless of the choice of the Player II the Player I will always choose the strategy A, which is a dominant strategy for him. The Player II: - assumes that the Player I chose the strategy A and he compares the payoff C 2 with the payoff D 0. The payoff C (2) is bigger than the payoff D (0). Thus, the Player II chooses the strategy C; C v D => 2 v 0 => 2 > 0 => 2 => C - assumes that the Player I chose the strategy B and he compares the payoff C 4 with the payoff D 2. The payoff C (4) is bigger than the payoff D (2). Thus, the Player II chooses the strategy C; C v D => 4 v 2 => 4 >2 => 4 => C Conclusion: regardless of the choice of the Player I the Player II will always choose the strategy C, which is a dominant strategy for him. 63

7 The Nash equilibrium is a set of strategies where a specific strategy chosen by the first player is optimal only for a specific strategy chosen by the second player. This game theory has three main situations connected with the Nash equilibrium: a) basic Nash equilibrium (single Nash equilibrium), b) two Nash equilibrium c) no Nash equilibrium. Basic Nash equilibrium is when one of the players has a dominant strategy and makes decisions regardless of the competitor s actions. The Player II does not have a dominant strategy and is reliant on the competitor s decisions; he chooses the best strategy for him, assuming that the Player I has first chosen a strategy. PLAYER II Strategy C Strategy D PLAYER I Strategy A Strategy B The Player I: - assumes that the Player II chose the strategy C, thus: A v B => 8 v 4 => 8 > 4 => 8 = > A - assumes that the Player II chose the strategy D, thus: A v B => 12 v 0 => 12 > 0 => 12 => A Conclusion: regardless of the choice of the Player II the Player I will always chose the strategy A; it is a dominant strategy for him. The Player II: - assumes that the Player I chose the strategy A, thus: C v D => 4 v 0 => 4 > 0 => 4 => C - assumes that the Player I chose the strategy A, thus: 64

8 C v D => 0 v 4 => 0 < 4 => 4 => D Conclusion: The Player II does not have a dominant strategy and he makes decision depending on the choice of the Player I. In the above case the Player I has a dominant strategy A so the Player II will chose the strategy C Two Nash equilibrium is when neither of the players has a dominant strategy and the players make optimal decisions depending on the competitor s choice. PLAYER II Strategy C Strategy D PLAYER I Strategy A Strategy B The Player I: - assumes that the Player II chose the strategy C, thus: A v B => 4 v 0 => 4 > 0 => 4 => A - assumes that the Player II chose the strategy D, thus: A v B => 0 v 2 => 0 < 2 => 2 => B Conclusion: the Player I does not have a dominant strategy; he makes an optimal decision depending on the choice of the Player II The Player II: - assumes that the Player II chose the strategy A, thus: C v D => 2 v 0 => 2 > 0 => 2 => C - assumes that the Player I chose the strategy B, thus: C v D => 0 v 4 => 0 < 4 => 4 => D Conclusion: The Player II does not have a dominant strategy; he makes an optimal decision depending on the choice of the Player I. There are two possible ways of the players actions: 65

9 1) If the Player I chooses the strategy A, the Player II chooses the strategy C (it does not matter whose choice was made in the first place); 2) If the Player I chooses the strategy B, the Player II chooses the strategy D (it does not matter whose choice was made in the first place. In case of no Nash equilibrium the players don t have any dominant strategy; moreover, their actions and strategies are mutually exclusive. PLAYER II Strategy C Strategy D Strategy A PLAYER I Strategy B The Player I: - assumes that the Player II chose the strategy C, thus: A v B => 0 v 4 => 0 < 4 => 4 => B - assumes that the Player II chose the strategy D, thus: A v B => 2 v 0 => 2 > 0 => 2 => A Conclusion: The Player I does not have a dominant strategy; he makes an optimal decision depending on what strategy the Player II has chosen. The Player II: - assumes that the Player I chose the strategy A, thus: C v D => 4 v 0 => 4 > 0 => 4 => C - assumes that the Player I chose the strategy B, thus: C v D => 4 v 8 => 4 < 8 => 8 => D Conclusion: The Player II does not have a dominant strategy; he makes an optimal decision depending on what strategy the Player II has chosen. There are the following possible ways of the players actions: 1) If the Player I chooses the strategy B, the Player II will choose the strategy D; 66

10 2) However, if the Player II chooses the strategy D, the Player I will choose the strategy A; 3) If the Player I chooses the strategy A, the Player II will choose the strategy C; 4) However, If the Player II chooses the strategy C, the Player I will choose the strategy B No co-operation between the players is possible this state of market game is called impossible to determine the Nash equilibrium. The strategy in case of prisoner s dilemma is that the two market players have dominant strategies but the choice they make is not optimal. This method of market game is presented as the example of two prisoners, who together committed a crime, but who haven t been caught red-handed. Detained for further explanations, they are kept in separate cells and they cannot contact each other. They both can be honest and plead guilty or trust each other and plead not guilty. If they trust each other after 48 hours they will be released. If they both plead guilty they ll be sentenced to 3 months in prison. If one pleads guilty and other not guilty then the honest criminal will be released and the other will be sentenced to 6 months in prison. The payoff matrix of this market game is as follows: Plead Prisoner I Plead guilty guilty not Prisoner II Plead guilty Plead not guilty 3* 3* 0* 6* 6* 0* 48h ** 48 h** * months in prison ** under arrest 67

11 Prisoner I: - assumes that the Prisoner II pleaded guilty and he compares the payoff plead guilty 3 months in prison with the payoff plead not guilty 6 months in prison. - assumes that the Prisoner II pleaded not guilty and he compares the payoff Plead guilty 0 months in prison and the payoff plead not guilty 48 hours under arrest. Conclusion: It is better for the Prisoner I to confess because he will get lower or no sentence. Prisoner II: - assumes that the Prisoner I pleaded guilty and he compares the payoff plead guilty 3 months in prison with the payoff plead not guilty 6 months in prison - assumes that the Prisoner I pleaded not guilty and he compares the payoff plead guilty 0 months in prison with the payoff plead not guilty 48 hours under arrest. Conclusion: It is better for the Prisoner II to plead guilty. Each of the prisoners has the dominant strategy plead guilty. In such a case both will be sentenced to 3 months in prison. However, if they both risked and pleaded not guilty, after 48 hours they would be released and this strategy would be much more beneficial for the prisoners. On the basis of the above-mentioned information student should be able to use the payoff matrix in game theory (examples 1-3) and in decision-making process by the companies (examples 4-6) EXAMPLE 1 Define the strategy, used by companies X and Y, having given their payoff matrix. 68

12 COMPANY X Strategy A Strategy B COMPANY Y Strategy C Strategy D The Company X: - assumes that the company Y chose the strategy C, thus: A v B => 4 v 2 => 4 > 2 => 4 => A - assumes that the company Y chose the strategy D, thus: A v B => 20 v10 => 20 >10 => 20 => A Conclusion: The Company X will always choose the strategy A The Company Y: - assumes that the company X chose the strategy A, thus: C v D => 4 v 2 => 4 > 2 => 4 => C - assumes that the company X chose the strategy B, thus: C v D => 20 v 10 => 20 > 10 => 20 => C Conclusion: The Company Y will always choose the strategy C. Solution: X chooses the strategy A and Y the strategy C. However, looking at the payoff matrix we can observe that it would be better if X chose the strategy B and Y chose the strategy D. In the latter case the companies would gain higher profits. This payoff matrix is an example of the prisoner s dilemma there are dominant strategies for both companies but other solutions are more advantageous. EXAMPLE 2 Define the strategy, used by companies X and Y, having given their payoff matrix. 69

13 COMPANY X Strategy A Strategy B COMPANY Y Strategy C Strategy D The Company X: - assumes that the company Y chose the strategy C, thus: A v B => 20 v 10 => 20 > 10 => 20 => A - assumes that the company Y chose the strategy D, thus: A v B => 15 v8 => 15 >8 => 15 => A Conclusion: The Company X will always choose the strategy A. The Company Y: - assumes that the company X chose the strategy A, thus: C v D => 12 v 20 => 12 < 20 => 20=> D - assumes that the company X chose the strategy B, thus: C v D => 15 v 10 =>15 > 10 => 15 => C Conclusion: The Company Y does not have a dominant strategy. Solution: X chooses the strategy A and Y the strategy D. The Company X will always choose the strategy A and the company Y is dependent on X s choice (it is a single Nash equilibrium). EXAMPLE 3 Define the strategy, used by companies X and Y, having given their payoff matrix. 70

14 COMPANY X Strategy A Strategy B COMPANY Y Strategy C Strategy D The Company X: - assumes that the company Y chose the strategy C, thus: A v B => 20 v 40=> 20 < 40 => 40 => B - assumes that the company Y chose the strategy D, thus: A v B => 20 v10 => 20 >10 => 20 => A Conclusion: The Company X does not have a dominant strategy. The Company Y: - assumes that the company X chose the strategy A, thus: C v D => 40 v 20 => 40 > 20 => 40 => C - assumes that the company X chose the strategy B, thus: C v D => 40 v 50 => 40 < 50 => 50 => D Conclusion: The Company X does not have a dominant strategy. Solution: X will choose the strategy B if Y chooses the strategy C or X will choose the strategy A if Y chooses the strategy D. However, if X chooses the strategy B then Y will choose the strategy D. No cooperation between the companies is possible (it is a No Nash Equilibrium case). EXAMPLE 4 Two producers of goods X concluded a secret agreement concerning a division of market in order to increase profits. They agreed to supply 1600 pieces of the goods (800 each). After signing the agreement, each of the companies may abide by it or break it, assuming that the competitor will keep to the rules. By breaking the agreement, each of the companies increases the production in order 71

15 to increase market share. The following payoff matrix depicts the production volume. * the company that defects from the cartel increases its production by 100 pieces. Company 2 Co-operates Defects Company 1 Co-operates (800, 800) (800, 900) Defects (900, 800) (900, 900) Looking at the production level it is better for the Company 1 to defect from the cartel. Regardless of the second company s choice, the production level of the company 1 amounts to 900. The similar situation is in case of the company 2; it is better to defect from the cartel. Both companies have their dominant strategies (dominant strategy game). EXAMPLE 5 Two companies decided to introduce one of the new products on the market. Neither of them have sufficient funds to introduce both products. The payoff matrix after introduction of the products is as follows: 72

16 Company 2 Product A Product B Product A loss; loss profit; profit Company 1 Product B profit; profit loss; loss If both companies introduce the same product, they will report a loss. The companies want to report a profit and the payoff matrix shows that neither of them has a dominant strategy. Each of the companies waits for the competitor s decision to gain profits. If the company 1 chooses to introduce the product A then the Company 2 decides to produce the product B. However if the Company 1 will choose to produce the product B then the Company 2 will introduce the product A. Analogous situation will take place if the Company 2 make decision as first. (two Nash equilibrium) EXAMPLE 6 Two companies decided to introduce the same product on the market. Each of the companies tries to do it first, to gain competitive advantage. Both companies use advertising campaign to arouse interest in a new drug. The demand for the product of X is 25,000 pieces. The company, which first introduces the product, may increase its sales by 10%. The one that uses advertising campaign will have the same effect 10 % sales increase. If both companies take the same action the production level will stay the same. Decide what decision they should make. 73

17 Company 2 Introduction on the market Advertising campaign Introduction on the market Company 1 Advertising campaign On the basis of the payoff matrix we can state that neither of the companies has a dominant strategy. The same decisions made by two companies will result in the same sales level. Thus, the companies should use different strategies if the Company 1 first introduces the product then the Company 2 should use advertising campaign. If the Company 1 uses the advertising campaign then the Company 2 should introduce the product. Analogous situation will take place if the Company 2 makes decision as first. (two Nash equilibrium). 3. Summary In order to put the game theory into practice one has to know it in depths. It can be applied to such fields as: competition and co-operation. The aim of this study was to introduce the basics of the game theory a branch that is intensively developing. Simplification made in this study may give rise to concerns that the above models do not fit with real life. However, at times when a gap between the practice and the theory is narrowing, the game theory may be treated as an effort to solve the practical problems. The best example is the Nobel Prize in Economics (2005) for R.J Aumann and T.S. Shelling 74

18 for having enhanced our understanding of conflict and cooperation through game-theory analysis 4. Comprehension check: 1. Use the above information about the decision-making process in the market game to answer the question: In case when each of the players acts rationally, what are the chances to anticipate decisions and actions of a competitor? (The companies may run an activity in any branch). 2. In what case two Polish cell phone providers Plus GSM and Orange will establish the Nash equilibrium? 3. If a competitor of your company did not act according to the Nash equilibrium strategy, what market strategy would you choose? (make the analysis in a specific market branch) 4. Describe what the single Nash equilibrium consist in. What is the difference between the conditions in the single Nash equilibrium and in dominant strategies? (show an example from the Polish bank sector). 5. Recommended reading: References H.R. Varian, Mikroekonomia, PWN, Warszawa T.C. Bergstrom, H.R. Varian, Workouts in Intermediate Microeconomics (1990). M. Malawski, A. Wieczorek, H. Sosnowska, Konkurencja i kooperacja. Teoria gier w ekonomii i naukach społecznych, PWN, Warszawa W.F. Samuleson, S.G. Marks, Managerial Economics, 1999 T. Gruszecki, Współczesne teorie przedsiębiorstwa, PWN, Warszawa Galbraith J.K., Economics in perspective: a critical history. 2. Gorynia M., Zachowania przedsiębiorstw w okresie transformacji, Wydawnictwo Akademii Ekonomicznej w 75

19 76 Poznaniu, Poznań 2000; Grzelońska U., O instytucjonalnej i alternatywnych teoriach przedsiębiorstwa, w: Gospodarka w okresie przemian, Oficyna Wydawnicza SGH, Warszawa Gruszecki T., Współczesne teorie przedsiębiorstwa, PWN, Warszawa Harsanayi J., Selten R., A General Theory of Equilibrium Selection in Games, MIT Press, Neumann J., Morgenstern O., Theory of Games and Economic Behaviour, University Press, Princeton Stankiewicz W., Historia myśli ekonomicznej, PWE, Warszawa 2000.