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Basics of Game theory

- 1. Game Theory in Economics Introduction Game theory seeks to analyse competing situations which arise out of conflicts of interest. There is conflict of interest between animals and plants in the consumption of natural resources. Animals compete among themselves for securing food. Man competes with animals to earn his food. A man also competes with another man. Every intelligent and rational participant in a conflict wants to be a winner but not all participants can be the winners at a time. The situations of conflict gave birth to Darwin’s theory of the ‘survival of the fittest’. Game theory is a tool to examine situations of conflict so as to identify the courses of action to be followed and to take appropriate decisions in the long run. Thus this theory assumes importance from managerial perspectives. In the present day world, business organizations compete with each other in getting the market share. In today’s world, decisions about many practical problems are made in a competitive situation, where two or more opponents are involved under the conditions of competition and conflict situations. The outcome does not depend on the decision alone but also the interaction between the decision-maker and the competitor. The objective, in theory, of games is to determine the rules of rational behaviour in game situations, in which the outcomes are dependent on the actions of the interdependent players. The results obtained by the application of this theory can serve as an early warning to the top level management in meeting the threats from the competing business organizations and for the conversion of the internal weaknesses and external threats into opportunities and strengths, thereby achieving the goal of maximization of profits. While this theory does not describe any procedure to play a game, it will enable a participant to select the appropriate strategies to be followed in the pursuit of his goals. Definition: Game is defined as an activity between two or more persons involving moves by each person according to a set of rules, at the end of which each person receive some benefit or satisfaction or suffers loss. In a game, there are two or more opposite parties with conflicting interests. They know the objectives and the rules of the game. An experienced player usually predicts with accuracy how his opponent will react if a particular strategy is adopted. When one player wins, his opponent losses. The games are classified based on the following characteristics. 1. Chance or strategy: If in a game the moves are determined by chance, we call it a game of chance, if they are determined by skill; it is a game of strategy. In general a game may involve partly strategy and partly chance. 2. Number of Persons: A game is called an n-person game if the number of persons playing it is n. 3. Number of moves: The number of moves may be finite or infinite. 4. Number of alternative available to each person per move: These also may be finite of infinite. A finite game has a finite number of moves, each involving a finite number alternative. Otherwise the game is infinite. 5. Information available to players of the past moves of the other players: The two extreme cases are, (a) no information at all (b) complete information available. There can be cases in between in which information is partly available. 6. Pay off: It is a quantitative measure of satisfaction a person gets at the end of the play. It is real a valued function of the variables in the game. Let Pi be the payoff to the person, Pi, i = 1, 2, …, n, in an n-person game. Then if ∑ Pi = 0, the game is said to be a zero-sum game. Types of Games Games can be of several types. Important ones are as follows: (1) Two-person games and n-person games. In two-person games the players may have many possible choices open to them for each play of the game but the number of players remains only two. But games can as well involve many people as active participants, each with his own set of choices for each play of the
- 2. game. In case of three players, it can be named as three-person game. Thus, in case of more than two persons, the game is generally named as n person game. (2) Zero sum and non-zero sum game. A zero sum game is one in which the sum of the payments to all the competitors is zero for every possible outcome of the game. In other words, in such a game the sum of the points won equals the sum of the points lost i.e., one player wins at the expense of the other (others). Two-person matrix game is always zero sum game since one player loses what the other wins. But in a non-zero sum game the sum of the payoffs from any play of the game may be either positive or negative but not zero. (3) Games of perfect information and games of imperfect information: A game of perfect information is the one in which each player can find out the strategy that would be followed by his opponent. On the other hand, a game of imperfect information is the one in which no player can know in advance what strategy would be adopted by the competitor and a player has to proceed in his game with his guess works only. (4) Games with finite number of moves / players and games with unlimited number of moves: A game with a finite number of moves is the one in which the number of moves for each player is limited before the start of the play. On the other hand, if the game can be continued over an extended period of time and the number of moves for any player has no restriction, then we call it a game with unlimited number of moves. (5) Constant-sum games: If the sum of the game is not zero but the sum of the payoffs to both players in each case is constant, then we call it a constant sum game. It is possible to reduce such a game to a zero sum game. Assumptions for a Competitive Game Game theory helps in finding out the best course of action for a firm in view of the anticipated countermoves from the competing organizations. A competitive situation is a competitive game if the following properties hold: 1. The number of competitors is finite, say N. 2. A finite set of possible courses of action is available to each of the N competitors. 3. A play of the game results when each competitor selects a course of action from the set of courses available to him. In game theory we make an important assumption that all the players select their courses of action simultaneously. As a result, no competitor will be in a position to know the choices of his competitors. 4. The outcome of a play consists of the particular courses of action chosen by the individual players. Each outcome leads to a set of payments, one to each player, which may be either positive, or negative, or zero. Managerial Applications of the Theory of Games The techniques of game theory can be effectively applied to various managerial problems as detailed below: 1) Analysis of the market strategies of a business organization in the long run. 2) Evaluation of the responses of the consumers to a new product. 3) Resolving the conflict between two groups in a business organization. 4) Decision making on the techniques to increase market share. 5) Material procurement process. 6) Decision making for transportation problem. 7) Evaluation of the distribution system. 8) Evaluation of the location of the facilities. 9) Examination of new business ventures and 10) Competitive economic environment.
- 3. Key concepts in the Theory of Games Several of the key concepts used in the theory of games are described below: Players: The competitors or decision makers in a game are called the players of the game. Strategies: The alternative courses of action available to a player are referred to as his strategies. Pay off: The outcome of playing a game is called the payoff to the concerned player. Optimal Strategy: A strategy by which a player can achieve the best pay off is called the optimal strategy for him. Zero-sum game: A game in which the total payoff to all the players at the end of the game is zero is referred to as a zero -sum game. Non-zero sum game: Games with “less than complete conflict of interest” are called non-zero sum games. The problems faced by a large number of business organizations come under this category. In such games, the gain of one player in terms of his success need not be completely at the expense of the other player. Payoff matrix: The tabular display of the payoffs to players under various alternatives is called the payoff Matrix of the game. Pure strategy: If the game is such that each player can identify one and only one strategy as the optimal strategy in each play of the game, then that strategy is referred to as the best strategy for that player and the game is referred to as a game of pure strategy or a pure game. Mixed strategy: If there is no one specific strategy as the ‘best strategy’ for any player in a game, then the game is referred to as a game of mixed strategy or a mixed game. In such a game, each player has to choose different alternative courses of action from time to time. N-person game: A game in which N-players take part is called an N-person game. Maximin-Minimax Principle : The maximum of the minimum gains is called the maximin value of the game and the corresponding strategy is called the maximin strategy. Similarly the minimum of the maximum losses is called the minimax value of the game and the corresponding strategy is called the minimax strategy. If both the values are equal, then that would guarantee the best of the worst results. Negotiable or cooperative game: If the game is such that the players are taken to cooperate on any or every action which may increase the payoff of either player, then we call it a negotiable or cooperative game. Non-negotiable or non-cooperative game: If the players are not permitted for coalition then we refer to the game as a non-negotiable or non-cooperative game. Saddle point: A saddle point of a game is that place in the payoff matrix where the maximum of the row minima is equal to the minimum of the column maxima. The payoff at the saddle point is called the value of the game and the corresponding strategies are called the pure strategies. Dominance: One of the strategies of either player may be inferior to at least one of the remaining ones. The superior strategies are said to dominate the inferior ones. Value of the game (V): The value of the game is the maximum guaranteed gain to the maximizing player A, if both the players use their best strategies. It is the expected pay off of a play when all the players of the game follow their optimal strategies. The amount of payoff, at an equilibrium point is known as the value of the game. Fair game: The game is said to be fair if the value of the game (V) = 0. Two-Person Zero-Sum Game In a game with two players, if the gain of one player is equal to the loss of another player, then the game is a two person zero-sum game. Thus a game with only two players, say player A and player B, and if the gain of the player A is equal to the loss of the player B, so that the total sum is zero, then this is a two person zero-sum game. Such a game is also called rectangular game. The two person zero sum game may be pure strategy game or mixed strategy game.
- 4. A Game in a competitive situation possesses the following properties i. The number of players is finite. ii. Each player has finite list of courses of action or strategy. iii. A game is played when each player chooses a course of action (strategy) out of the available strategies. No player is aware of his opponent’s choice until he decides his own. iv. The outcome of the play depends on every combination of courses of action. Each outcome determines the gain or loss of each player. Assumptions for Two-Person Zero Sum Game For building any model, certain reasonable assumptions are quite necessary. Some assumptions for building a model of two-person zero sum game are listed below. a) Each player has available to him a finite number of possible courses of action. Sometimes the set of courses of action may be the same for each player. Or, certain courses of action may be available to both players while each player may have certain specific courses of action which are not available to the other player. b) Player A attempts to maximize gains to himself. Player B tries to minimize losses to himself. c) The decisions of both players are made individually prior to the play with no communication between them. d) The decisions are made and announced simultaneously so that neither player has an advantage resulting from direct knowledge of the other player’s decision. e) Both players know the possible payoffs of themselves and their opponents. Payoff Matrix In a game, the gains and the losses, resulting from different moves and countermoves, when represented in the form of a matrix, is known as a pay off matrix. Each element of the pay off matrix is the gain of the maximizing player when a particular course of action is chosen by him as against the course faction chosen by the opponent. Suppose we consider the decision problem in which two rational players, Player A and Player B, each have a set of possible actions available to them. Suppose A has m strategies and B has n strategies. Suppose players A and B have m actions a1, a2, …, am and n actions b1, b2, …, bn available, respectively. The consequence of this decision by both players is a specific return or pay off. If the payoff is non-zero, it represents a gain or loss, to player. We will consider the case where Player A’s gain is Player B’s loss and conversely. No units enter or leave the game. This type of game is called a two person zero sum game. In general, we will let aij represent the pay off from Player B to Player A if Player A chooses action ai and Player B chooses action bj. If Player A chooses action ai, we will say that he plays row i. Likewise, if Player B chooses action bj, we will say that he plays column j. Player A wishes to gain as large a payoff aij as possible while player B will do his best to reach as small a value aij as possible where the gain to player B and loss to player A be (-aij).
- 5. Pure & Mixed Strategies If both players use their optimal minimax strategies, the resulting expected pay off is called the value of the game. When the optimal minimax strategies are pure strategies, then the expected pay off using these optimal minimax strategies is just the value of the game. So a pure strategy is a decision in advance of all players, always to choose a particular course of action., it is a predetermined course of action. The player knows it in advance. The Saddle Point The saddle point in a payoff matrix is one which is the smallest value in its row and the largest value in its column. A saddle point of a game is that place in the payoff matrix where the maximum of the row minima is equal to the minimum of the column maxima. The payoff at the saddle point is called the Value of the Game and the corresponding strategies are called the Pure Strategies. The saddle point is also known as equilibrium point in the theory of games. An element of a matrix that is simultaneously minimum of the row in which it occurs and the maximum of the column in which it occurs is a saddle point of the matrix game. In a game having a saddle point optimum strategy for player I is always to play the row containing a saddle point and for the prayer II to play the column that contains a saddle point. Saddle point also gives the value of such a game. Saddle point in a payoff matrix concerning a game may be there and may not be there. If there is a saddle point we can easily find out the optimum strategies and the value of the game but when saddle point is not there we have to use algebraic methods for working out the solutions concerning game problems. If there is more than one saddle point there will be more than one solution. When there is no saddle point for a game problem, the minimax-maximin principle cannot be applied to solve the problem. In those cases the concept of chance move is introduced. Here the choice among a number of strategies is not the decision of the player but by some chance mechanism. That is, predetermined probabilities are used for deciding the course of action. The strategies thus made are called mixed strategies. Solution to a mixed strategy problem can be arrived at by any of the methods like probabilities method or principle of dominance. Pure Strategies: Game with Saddle Point The selection of an optimal strategy by each player without the knowledge of the competitor’s strategy is the basic problem of playing games. The aim of the game is to determine how the players must select their respective strategies such that the pay-off is optimized. The game theory helps to know how these players must select their respective strategies, so that they may optimize their payoffs. Such a criterion of decision making is referred to as Minimax-Maximin principle. The maximizing player arrives at his optimal strategy on the basis of maximin criterion, while minimizing players’ strategy is based on the minimax criterion. The game is solved by equating maximin value with minimax value. In a pay-off matrix, the minimum value in each row represents the minimum gain for player A. Player A will select the strategy that gives him the maximum gain among the row minimum values. The selection of strategy by player A is based on maximin principle. Similarly, the same pay-off is a loss for player B. The maximum value in each column represents the maximum loss for Player B. Player B will select the strategy that gives him the minimum loss among the column maximum values. The selection of strategy by player B is based on minimax principle. If the maximin value is equal to minimax value, the game has a saddle point (i.e., equilibrium point). Thus the strategy selected by player A and player B are optimal. The amount of payoff, i.e. V at an equilibrium point is known as the value of the game. The game is said to be fair if the value of the game V = 0
- 6. We see that the maximum of row minima = the minimum of the column maxima. So the game has a saddle point. The common value is 12. Therefore the value V of the game = 12. Interpretation: In the long run, the following best strategies will be identified by the two players: The best strategy for player A is strategy 4. The best strategy for player B is strategy IV. The game is favourable to player A. Problem 2: Solve the game with the pay-off matrix for player A as given in Table below – Solution: Find the smallest element in rows and largest elements in columns as shown in following Table
- 7. Select the largest element in row and smallest element in column. Check for the minimax criterion, Max Min = Min Max 1 = 1 The value of the game is, V = 1 Therefore, there is a saddle point and it is a pure strategy. Optimum Strategy Player A A2 Strategy Player B B1 Strategy Problem 3: Solve the game with the pay off matrix given in Table below and determine the best strategies for the companies A and B and find the value of the game for them. Solution: The matrix is solved using maximin criteria, as shown in Table below Max Min = Min Max 2 = 2 Therefore, there is a saddle point. Optimum strategy for company A is A1and Optimum strategy for company B is B1or B3 Problem 4: Check whether the following game is given in Table below, determinable and fair. Solution: The game is solved using maximin criteria as shown in Table
- 8. The game is strictly neither determinable nor fair. Strictly Determined Game A game is strictly determined if it has at least one saddle point. The following statements are true about strictly determined games. A. All saddle points in a game have the same payoff value. B. Choosing the row and column through any saddle point gives minimax strategies for both players. In other words, the game is solved via the use of these (pure) strategies. The value of a strictly determined game is the value of the saddle point entry. A fair game has value of zero; otherwise it is unfair or biased. Consider the following game. Column Strategy A B C Row Strategy 1 0 -1 1 2 0 0 2 3 -1 -2 3 In the above game, there are two saddle points. Since the saddle point entries are zero, this is a fair game. Games without Saddle Point: Mixed Strategies For any given pay off matrix without saddle point the optimum mixed strategies are shown in Table below:- We assume that the strategies are not pure strategies (saddle point does not exist). Since this game has no saddle point, the following condition shall hold: In this case, the game is called a mixed game. No strategy of Player A can be called the best strategy for him. Therefore A has to use both of his strategies. Similarly no strategy of Player B can be called the best strategy for him and he has to use both of his strategies. Probability method: This method is applied when there is no saddle point and the pay of matrix has two rows and two columns only. The players may adopt mixed strategies with certain probabilities. Here the problem is to determine probabilities of different strategies of both players and the expected payoff of the game. Let p be the probability of A using strategy A1 and (1-p) is the probability for A using A2. Then we have the equations (i) Expected gain of A if B chooses B1 = ap + c (1-p) (ii) Expected gain of A if B chooses B2 = bp + d (1-p)
- 9. If the expected values in equations (i) and (ii) are different, Player B will prefer the minimum of the two expected values that he has to give to player A. Thus B will have a pure strategy. This contradicts our assumption that the game is a mixed one. Therefore the expected values of the pay-off to Player A in equations (i) and (ii) should be equal. Thus we have the condition ap + c (1-p) = bp + d (1-p) Solving the equation ,we get Similarly, let q and (1-q) be respectively probabilities for B choosing strategies B1 and B2, then When A use his first strategy then we have The expected value of loss to Player B with his first strategy = aq The expected value of loss to Player B with his second strategy = b(1-q) Therefore the expected value of loss to B = aq + b(1-q)………………………………..(iii) When A use his second strategy The expected value of loss to Player B with his first strategy = cq The expected value of loss to Player B with his second strategy = d(1-q) Therefore the expected value of loss to B = cq + d(1-q)……………………………………(iv) If the two expected values are different then it results in a pure game, which is a contradiction. Therefore the expected values of loss to Player B in equations (iii) and (iv) should be equal. Hence we have the condition: aq + b(1-q) = cq + d(1-q) Therefore, the value V of the game is Problem 5: Solve the following game Solution: First consider the row minima Maximum of {2, 1} = 2 Next consider the maximum of each column Minimum of {4, 5}= 4 We see that, Max {Row Minima} ≠ Min {Column Maxima} So the game has no saddle point. Therefore it is a mixed game. Comparing this game with earlier game
- 10. We have a = 2, b = 5, c = 4 and d = 1. Let p be the probability that player X will use his first strategy. We have Let r be the probability that Player Y will use his first strategy. Then the probability that Y will use his second strategy is (1-r). We have Therefore, out of 2 trials, player X will use his first strategy once and his second strategy once. Again Therefore, out of 3 trials, player Y will use his first strategy twice and his second strategy once. The Principle Of Dominance The principle of dominance states that if the strategy of a player dominates over another strategy in all conditions, then the latter strategy can be ignored because it will not affect the solution in any way. A strategy dominates over the other only if it is preferable in all conditions. In case of pay-off matrices larger than (2×2) size, the dominance property can be used to reduce the size of the pay-off matrix by eliminating the strategies that would never be selected. Principle of dominance is applicable to pure strategy and mixed strategy problems. Mathematically speaking,
- 11. In a given pay -off matrix A, we say that the ith row dominates the kth row if And In such a situation player A will never use the strategy corresponding to kth row, because he will gain less for choosing such a strategy. And In this case, the player B will loose more by choosing the strategy for the qth column than by choosing the strategy for the pth column. So he will never use the strategy corresponding to the qth column. When dominance of a row (or a column) in the pay-off matrix occurs, we can delete a row (or a column) from that matrix and arrive at a reduced matrix. This principle of dominance can be used in the determination of the solution for a given game. Conditions 1. If all the elements in a row of a pay off matrix are less than or equal to the corresponding elements of another row, then the latter dominates and so former is ignored. Example 1 3 4 5 -2 1 4 0 Here every element of second row is less than or equal to the corresponding elements of row 1.Therefore first row dominates and so second row can be ignored. 2. If all the elements in a column of a pay off matrix are greater than or equal to the corresponding elements of another column, then the former dominates and so latter is ignored. 2 6 -3 2 4 4 2 3 Here the elements of second column are greater than or equal to the corresponding elements of first column. So second column dominates and first column can be ignored. 3. If the linear combination of two or more rows or columns dominates a row or column, then the latter is ignored. If all the elements of a row are less than or equal to average of the corresponding elements of two other rows, then the former is ignored. For example, 2 0 2 3 -1 3 1 -2 1 Here the elements of third column are greater than or equal to the average of corresponding elements of two other columns, so third column dominates, and other columns can be ignored. Problem 6: Solve the game given below in Table below after reducing it to 2 × 2 game Solution: Reduce the matrix by using the dominance property. In the given matrix for player A, all the elements in Row 3 are less than the adjacent elements of Row 2. Strategy 3 will not be selected by player A, because it gives less profit for player A. Row 3 is dominated by Row 2. Hence delete Row 3, as shown in Table below –
- 12. For Player B, Column 3 is dominated by column 1 (Here the dominance is opposite because Player B selects the minimum loss). Hence delete Column 3. We get the reduced 2 × 2 matrix as shown below Now, solve the 2 × 2 matrix, using the maximin criteria as shown below Therefore, there is no saddle point and the game has a mixed strategy. Applying the probability formula, Problem 7: Solve the game whose pay off matrix is given by Solution: Applying principle of dominance, B3 is dominated by B1. So ignore B3 and the reduced matrix is
- 13. Now A3 is dominated by A2. So ignore A3. The resulting matrix is Let p and 1-p be the probabilities for A choosing A1 and A2. Let q and 1-q, be the probabilities for B choosing B1 and B2. Then, Probability that A choosing A1 = 2/5 Probability that A choosing A2 = 3/5 Probability that B choosing B1 = 1/2 Probability that B choosing B2 = ½ A’s mixed strategy = 2/5, 3/5 B’s mixed strategy = 1/2, ½ Problem 8: Solve the game whose pay off matrix is given by Solution: First consider the minimum of each row. Maximum of {2, 3, 3} = 3 Next consider the maximum of each column Minimum of {6, 4, 7, 6} = 4 The following condition holds: Max {row minima} ≠ min {column maxima} Therefore we see that there is no saddle point for the game under consideration. Compare columns II and III.
- 14. We see that each element in column III is greater than the corresponding element in column II. The choice is for player B. Since column II dominates column III, player B will discard his strategy 3. Now we have the reduced game For this matrix again, there is no saddle point. Column II dominates column IV. The choice is for player B. So player B will give up his strategy 4. The game reduces to the following: This matrix has no saddle point. The third row dominates the first row. The choice is for player A. He will give up his strategy 1 and retain strategy 3. The game reduces to the following
- 15. Prisoner’s Dilemma Prisoners’ dilemma is the best-known game of strategy in social science. The typical prisoner's dilemma is set up in such a way that both parties choose to protect themselves at the expense of the other participant. It helps us understand what governs the balance between cooperation and COMPETITION in business, in politics, and in social settings. Prisoner’s Dilemma is the classic example of application of game theory to oligopolistic market conditions. A classic example of game theory that explains the problem faced by oligopoly is “Prisoner’s Dilemma”. In the traditional version of this game, two Prisoners are accused of a joint crime and they are put in two different jails. They cannot communicate with each other. They are asked to confess and the police are interrogating them in separate rooms. Each can either confess, thereby implicating the other, or keep silent. No matter what the other suspect does, each can improve his own position by confessing. 1. If both confess, both will get 5 years of imprisonment. 2. If no one confesses, both will get only 2 years of jail. 3. If one confesses and the other does not confess, the one who confesses will be jailed for one year and the other - 10 years. In this situation, most likely strategy will be both the prisoners would confess and get 2 years of imprisonment, oligopolistic firms often find themselves in a prisoner’s dilemma.
- 16. They have to decide: a) Whether to compete aggressively & capture larger market share. b) Co-operate and compete more passively. Actually both the firms would do better by co-operating and charging high price. But the firms are in prisoners’ dilemma, where neither can trust its competitors to set a higher price. Other Classic example of prisoner’s dilemma in the real world is encountered when two competitors are battling it out in the marketplace. Many sectors of the economy have two main rivals. In the U.S., for example, the fierce rivalry between Coca-Cola (NYSE:KO) and PepsiCo (NYSE:PEP) in soft drinks. Consider the case of Coca-Cola versus PepsiCo, and assume that the former is thinking of cutting the price of its iconic Coke drink. If it does so, Pepsi may have no choice but to follow suit for its Pepsi Cola to retain its market share. This may result in a significant drop in profits for both companies. A price drop by either company may therefore be construed as defecting, since it breaks an implicit agreement to keep prices high and maximize profits. Thus, if Coca-Cola drops its price but Pepsi continues to keep prices high, the former is defecting while the latter is cooperating (by sticking to the spirit of the implicit agreement). In this scenario, Coca-Cola may win market share and earn incremental profits by selling more Coke drinks. The prisoner’s dilemma can be used to aid decision-making in a number of areas in one’s personal life, such as buying a car, salary negotiations and so on. For example, assume you are in the market for a new car, and you walk into a car dealership. The utility or payoff in this case is a non-numerical attribute, i.e. satisfaction with the deal. You obviously want to get the best possible deal in terms of price, car features, etc. while the car salesman wants to get the highest possible price to maximize his commission. Cooperation in this context means no haggling; you walk in, pay the sticker price (much to the salesman’s delight) and leave with a new car. On the other hand, defecting means bargaining; you want a lower price, while the salesman wants a higher price. Assigning numerical values to the levels of satisfaction, where 10 means fully satisfied with the deal and 0 implies no satisfaction, the payoff matrix is as shown below: Car Buyer vs. Salesman – Payoff Matrix Salesman Cooperate Defect Buyer Cooperate (a) 7, 7 (c) 0,10 Defect (b) 10, 0 (d) 3, 3 What does this matrix tell us? If you drive a hard bargain and get a substantial reduction in the car price, you are likely to be fully satisfied with the deal, but the salesman is likely to be unsatisfied because of the loss of commission (as can be seen in cell b). Conversely, if the salesman sticks to his guns and does not budge on price, you are likely to be unsatisfied with the deal while the salesman would be fully satisfied (cell c). Your satisfaction level may be less if you simply walked in and paid full sticker price (cell a). The salesman in this situation is also likely to be less than fully satisfied, since your willingness to pay full price may leave him wondering if he could have “steered” you to a more expensive model, or added some more bells and whistles to gain more commission. Cell (d) shows a much lower degree of satisfaction for both buyer and seller, since prolonged haggling may have eventually led to a reluctant compromise on the price paid for the car. The prisoner’s dilemma shows us that mere cooperation is not always in one’s best interests. In fact, when shopping for a big-ticket item such as a car, bargaining is the preferred course of action from the consumers' point of view. Otherwise the car dealership may adopt a policy of inflexibility in price negotiations, maximizing its profits but resulting in consumers overpaying for their vehicles. Understanding the relative payoffs of cooperating versus defecting may stimulate you to engage in significant price negotiations before you make a big purchase.
- 17. Nash Equilibrium There are many games which don’t have a dominant strategy but still equilibrium can be attained. Here the way is method developed by Nash. In game theory the optimal outcome of a game is one where no player has an incentive to deviate from his chosen strategy after considering an opponent's choice. In other words, each player plays a best-response strategy by assuming the other player’s moves. In the Nash Equilibrium, each player's strategy is optimal when considering the decisions of other players. It is the solution to a game in which two or more players have a strategy, and with each participant considering an opponent’s choice, he has no incentive, nothing to gain, by switching his strategy. Every player wins because everyone gets the outcome they desire. If every player in a game plays his dominant pure strategy (assuming every player has a dominant pure strategy), then the outcome will be Nash equilibrium. Nash Equilibria are self-enforcing; when players are at a Nash Equilibrium they have no desire to move because they will be worse off. When Nash equilibrium is reached, players cannot improve their payoff by independently changing their strategy. For example, in the Prisoner's Dilemma game, confessing is a Nash equilibrium because it is the best outcome, taking into account the likely actions of others. To quickly test if the Nash equilibrium exists, reveal each player's strategy to the other players. If no one changes his strategy, then the Nash Equilibrium is proven. NecessaryConditions The following game has payoffs defined L R T a,b c,d B e,f g,h In order for (T,L) to be an equilibrium in dominant strategies (which is also a Nash Equilibrium), the following must be true: a > e c > g b > d f > h In order for (T,L) to be a Nash Equilibrium, only the following must be true: a > or = e b > or = d Example 1: considering below P2 LEFT RIGHT P1 UP 5, 4 3,10 DOWN 9, 2 0, 1 If P1 goes UP, P2 prefers right since 10>4. But if P1 goes down then P2 prefers left since 2>1. If P2 goes left then P1 goes down since 9>5. If P2 goes right then P1 goes UP since 3>0. There are 2 outcomes which are stable (UP,RIGHT) and (DOWN, LEFT) which are “stable”: neither player would wish to change his action given the action of the other player. This is a NASH equilibrium. Example 2: considering below
- 18. Player 1's best response against each of player 2's strategy = red circle Player 2's best response against each of player 1's strategy = blue circle. Overall considering following - Player 1's best response against each of player 2's strategy = red circle. Player 2's best response against each of player 1's strategy = blue circle. Nash Equillibrium (NE) = (A,B),(B,A) Example 3: Find Nash Equillibrium for the following game P2 A B C P1 A 1,1 10,0 -10,1 B 0,10 1,1 10,1 C 1,-10 1,10 1,1 To start, we find the best response for player 1 for each of the strategies player 2 can play. P2 A B C P1 A 1,1 10,0 -10,1 B 0,10 1,1 10,1 C 1,-10 1,10 1,1 Now we do the same for player 2 by underlining the best responses of the column player P2 A B C P1 A 1,1 10,0 -10,1 B 0,10 1,1 10,1 C 1,-10 1,10 1,1 Now a pure strategy Nash equilibrium is a cell where both payoffs are underlined, i.e., where both strategies are best responses to each other. In the example, the unique pure strategy equilibrium is (A, A).