Management Studies: Game Theory

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The Prisoners’ Dilemma, an application of game theory analysis

Prisoners’ Dilemma is an application of game theory analysis in which two prisoners both confess to a crime to avoid harsher punishment when not confessing would avoid any punishment. The dilemma emerges because both prisoners are faced with the same choice – confess or not confess – but the outcome of their choice depends on the choice made by the other prisoner.

Unfortunately neither prisoner knows the choice of the other. If neither confesses, then they receive no punishment. If both confess, then they receive limited punishment, such as a year in jail. However, if one confesses and the other doesn't, the confessor receives light punishment, such as six months in jail, and the non confessor receives more severe punishment, such as five years in jail. The result is that both prisoners confess.

The model of prisoner’s dilemma explains how rivals behaving selfishly act contrary to their mutual or common interest. We first explain prisoner’s dilemma with an example given originally while propounding this model.

Let us suppose two persons, A and B have been caught for committing a bank robbery. Further suppose the prosecution has no enough evidence for their committing the crime. In order to obtain confession from them, they are interrogated in two separate rooms so that they cannot communicate with each other. While interrogating each accused, the police offer to A, “If you confess to the crime (that is, cooperate with the police) while the other keeps silent (i.e. does not confess), you will be given imprisonment for only a short period, say, 1 year only but punish the other with 10 years imprisonment. If the other also confesses, both of you would be sentenced to jail for 5 years”. It may however be known that if both prisoners do not confess, each can be jailed only for two years. The choices open to each accused are presented in payoff matrix given in the table and this refers to years of imprisonment.

Prisoner’s Dilemma: Payofff Matrix

B’s Choice

Confess Doesn’t confess

A ’s Choice Confesses B : 5 years B : 10 years

A : 5 years A : 1 year

Doesn’t Confess B : 1 years B : 2 years

A : 10 years A : 2 years

It will be seen that the outcome (i.e. length of sentence to each is determined by the specific strategy, (that is, choice) adopted by each prisoner. The two strategies (choices) refer to;

(i) confess and

(ii) does not confess.

If both B and A confess, each gets 5 years imprisonment. If one confesses, but the other does not, the one who confesses (i.e. cooperate with the police) gets a very light punishment, namely imprisonment for 1 year only and the one who doesn't confess is sentenced for 10 years imprisonment. It will be further seen from the table that if both do not confess (that is, they remain loyal and faithful to each other and do not cooperate with the police), both are sentenced to 2 years imprisonment.

Now, each prisoner faces an uncertainty regarding how the other person will behave, that is, whether or not he will confess. Though each person has to make an independent choice whether to confess or not but the outcome, i.e. payoff depends on what the other does.

Now, under these circumstances what choice will be made by the prisoners when they cannot communicate with each other and have to choose between the two alternatives independently? The model of prisoners’ dilemma suggests that both behaving selfishly and working in self-interest confess to the crime and cheat each other. Since both confess, each will get imprisonment for 5 years. Why do they make this choice and confess can be shown as under. Take B first, most probably, he would confess when he does not know how his co-accused will act. A would reason like this: If I don’t confess it is very likely that I will be imprisoned for 10 years as the other prisoner will most probably confess. If I confess, I will get 5 years imprisonment if the other one also confesses and only one year imprisonment if he does not confess.

So, in the presence of uncertainty about the other person’s choice, and behaving in self-interest, B is likely to confess. A too reasoning similarly would confess. As a result, both prisoners would be sentenced for 5 years, though they would have received a lighter sentence of only two years if they had not confessed and remained loyal to each other. However, it is self-interest which leads each prisoner to confess and prevents them from attaining the best solution for themselves (2 years imprisonment) if both do not confess to the crime and remain loyal to each other. But the decision of each prisoner in favor of confession is quite rational because each person works in self-interest and tries to make the best “best” of the “worst outcomes” in an uncertain situation.

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Nash Equilibrium or Non Cooperative Equilibrium

Nash equilibrium is a concept from Game theory which establishes that a set of strategies followed by economic agents within a game is in equilibrium if, holding the strategies of all other economic agents are constant, no economic agent can obtain a higher pay-off by choosing a different strategy. For example, when firms operate within an oligopoly, once Nash equilibrium has been reached, none of them will want to change their strategy because by doing it, they cannot obtain a higher profit. In other words, a Nash equilibrium is a solution in which no player can improve his/her pay-off given the other’s strategy. In other words, each player’s strategy is a best response against the other player’s strategy, that is given player A’s strategy, player B can do no better, and given B’s strategy, A can do no better. The Nash equilibrium is also sometimes called the non cooperative equilibrium because each party chooses that strategy which is best for itself, without collusion or cooperation and without regard for the welfare of society or any other party.

In the solution concept of Nash, each player is assumed to know the equilibrium strategies of the other players, and no player has anything to gain by changing only his or her own strategy unilaterally. If each player has chosen a strategy and no player can benefit by changing his or her strategy while the other players keep their unchanged, then the current set of strategy choices and the corresponding pay-offs constitute a Nash equilibrium.

According to the Nash theorem, every game with a finite number of players and a finite number of strategies will have at least one Nash equilibrium. For this to hold, however, there has to be the possibility of some random elements to strategies. A Nash Equilibrium is a set of mixed strategies for finite, non-cooperative games between two or more players whereby no player can improve his or her pay-off by changing their strategy. Each player’s strategy is an ‘optimal’ response based on the anticipated rational strategy of the other players in the game.

The theory of Nash equilibrium has two components:

(i) the players act in accordance with the theory of rational choice, given their beliefs about the other players’ actions (i.e., the player makes rational decision-making in the absence of cooperation), and

(ii) these beliefs are correct. If every player / participant knows the game he / she is playing and faces incentives that correspond to the preferences of the player whose role he / she is taking, then difference / deviation between the observed outcome and a Nash equilibrium can be blamed on a failure of one or both of these two components.

If a Nash equilibrium is established by any means whatsoever, no firm (player) has an incentive to exit / move from it by changing its own behavior. It is self-policing. It is self-policing in the sense that there is no need for group behavior to enforce it. Each firm has self-interest to continue (keep up) it because any move that it can make on its own will not improve its profits, given what other firms are currently doing.

The Nash equilibrium can be illustrated by making some modifications in the pay-off-matrix given in the table. Now we assume that action and counter-action of advertising (Ad) between Firms A and B. It is a regular phenomenon and the pay-off matrix that appears finally is given in table. The only change in the modified pay-off matrix is that neither Firm A nor Firm B increases its ad-expenditure, then pay-offs change from (15, 5) to (25, 5).

Pay-off Matrix of the Game

B’s Options

Increase Ad Don’t Increase

A’s Strategy Increase Ad A B A B

20 10 30 0

Don’t Increase A B A B

10 15 25 5

From the payoffs matrix, we can see that Firm A has no more dominant strategy. Its optimum decision depends now on what Firm B does. If Firm B increases its advertising-expenditure, Firm A has no option but to increase its advertisement expenditure. And, if Firm A reinforces its advertisement, Firm B will have to follow the suit. On the other hand, if Firm B does not increase its advertising-expenditure, Firm A does the best by increasing its ad-expenditure. Under these condition, the conclusion that both the firms arrive at is to increase advertising expenditure if the other firm does so, and ‘don’t increase’, if the competitor ‘does not increase’. In the ultimate analysis, however, both the firms will decide to increase the ad-expenditure.

The reason is that if none of the firms increases advertisement, Firm A gains more in terms of increase in its sales ($ 25 million only). And, if firm B increases advertisement expenditure, its sales increase by $ 10 million. Therefore, Firm B would do best to increase its ad-expenditure. In that case, Firm A will have no option but to increase its ad-expenditure. Thus, the final conclusion that emerges is that both the firms will go for advertisement war. In that case, each firm finds that it is doing the best given what the rival firm in doing. This is the Nash equilibrium.

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Payoff Matrix in Game Theory

Game theory summarizes in a tabular way the possible choices available to firms in oligopoly. Game theory applied to oligopoly uses a table that indicates the profit of each firm given the choice of strategy chosen by each. All possible strategies are represented in the table, and so all possible outcomes can be considered at once. This table is called a payoff matrix. The profits listed in the payoff matrix represent / exemplify underlying cost and demand information.

Suppose again that there are two firms. For simplicity, suppose that price and costs are all taken as given and that the only decision for these two firms is how much to spend on advertising and both engage in high levels of advertising, neither will enjoy particularly high profits. If neither firm advertises at a high level, each will keep its respective market share, but both will make larger profits. However, if one advertises while the other does not then the firm that advertises will gain market share and get big increase in profits while the other incurs losses. Assume that advertising might increase on holding the firm’s share of the market but it has little effect in expanding total industry sales. Finally, assume that firms reveal their strategies simultaneously and do not change them. Although it is quite simple, this model contains monetary features of the recognized interdependence of oligopolists.

The model is depicted in the payoff matrix in table.

Payoff Matrix

Firm A’s strategy

High Level of Advertising Low Level of Advertising

High Level of Advertising X gets $ 5,000 A gets $ 2,000

Y gets $ 5,000 B gets $ 2,000

Low Level of Advertising X gets $ 2,000 A gets $ 10,000

Y gets $ 20,000 B gets $ 10,000

Both firms choose high levels of advertising. They then earn profits $ 5,000 each. If both adopt low levels of advertising, they each enjoy profits $ 10,000. But of one firm advertises much and the other little, the firm with the high level of advertising earns profits of $ 20,000 and the other firm losses $ 2,000.

Now put yourself in the place of the Manager of firm B, the choice of B firm will depend precisely on what you think from A will do. If you think firm A will try to do you in, then you will assume that if you try to get the $ 10,000 profit available by going for a low level of advertising, Firm A will choose a high level of advertising in self-protection. This strategy assumes you of at least $ 5, 000.

Firm A has exactly the same choices, and so if Firm A assumes that firm B is not be trusted. Firm A also chooses a high level if advertising for his self-preservation. Thus, the conservative maximum strategy leads both firms to high levels of advertising. As a result, each gets a $ 5,000 profit.

Only of the firms cooperated can earn the $ 10,000. Profits that is available to each. If firm B assumes that firm A is a profit maximizing firm with managers who behave rationally, then firm B concludes that firm A will adopt a low level of advertising. If firm A makes the same assumption about firm B then each attains profits of $ 10,000.

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Meaning of The Game Theory and Importance of Game Theory

The term ‘game’ represents a conflict between two or more parties. A game is a decision situation with multiple decision makers where each person’s welfare depends on his/her own as well as other individuals’ actions. That is, a game is a decision situation with strategic interactions among all decision makers.

Game theory is a theory of individual rational decisions taken under conditions of less than full information concerning the outcomes of those decisions. This theory examines the interaction of individual decisions given certain assumptions concerning decisions made under risk, the general environment, and the cooperative or non-cooperative behavior of other individuals.

In the words of Richard G. Lipsey and K. Alec Chrystal, “Game theory is an approach to analyzing, rational decision-making behavior in interactive or conflict situation.”

According to N. Gregory Mankiw, “Game theory is the study of how people behave in strategic situations." By ‘strategic’, we mean a situation in which each person, when deciding what actions to take, must consider how others might respond to that action. Game theory is a mathematical technique used to show for example, how oligopoly firms play their game of business.

Importance of Game Theory

Game theory is an analysis that illustrates how choices between two plays affect the outcome of a “game”. Game theory which sounds playful/laughing in its terminology is filled with significance. It has been used by economists to study the interaction of oligopolistic markets, union-management bargaining disputes, countries’ trade policies, international environmental agreements, reputations, conflicts such as games and war and a large number of other situations. Game theory offers insights for politics, warfare, and everyday life as well.

Game theory is commonly used in economics to illustrate interdependent decision-making among oligopoly firms. It illustrates that one firm makes a decision based on the decision expected from the other firm. One key conclusion from the game theory analysis is that firms often make decisions that are “second best” or the “lesser of two evils”. The classic example of such a decision is the prisoners’ dilemma, in which two prisoners both confess to a crime to avoid harsher punishment when not confessing would avoid any punishments.
Thus, game theory has proved to be useful in analyzing suspects of economic behavior such as natural resource depletion and public goods. The theory of cooperative games which allows collaboration between individuals has been used to analyze cartel formation and industrial and labor market collusion.
Game theory is a body of knowledge which is concerned with the study of decision-making in situations where two or more rational opponents are involved under conditions of competition and conflicting interests. It deals with human processes in which an individual decision making unit who can be an individual, a group, a formal or informal organization, or a society, is not in complete control of the other decision making units, the opponents, and is addressed to problems involving conflict, co-operation or both at various levels.
The main objective in the theory of games is to determine the rules of rational behavior in the game situations, in which the outcomes are dependent on the actions of the interdependent players. A game refers to a situation in which two or more players are competing it.
Game theory is quite useful for understanding the behavior of oligopolies. When game theory is applied to oligopoly, the players are firms. Their game is played in the markets, their strategies are their price/output decisions, and the payoffs are their profits. Because the number of firms in an oligopolistic market is small, each firm must act strategically. Each firm knows that its profit depends not only on how much it produces but also on how much the other firms produce. In making its production decision, each firm in an oligopoly should consider how its decision might affect the production decisions of all the other firms.

In summary, a game theory framework can often help us understand the strategic choices available but it does not always help to predict which of many possible outcomes may occur.

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Concept of Oligopoly and Kinked Demand Curve Model

Price rigidity under oligopoly in terms of kinked demand curve

Price rigidity in the oligopoly market is best explained by the kinked demand curve.

The oligopoly is a reduced form of monopolistic competition. The term oligopoly has a Greek base and means few sellers, oligopoly as such, refers to markets with small number of large firms, each selling either differentiated or homogeneous product.

A few sellers imply a number so small or a few market share of each firm in so large that it can influence the market price. It also implies that each seller commands a sizeable proportion of the total market supply. The products traded by the oligopolists may be differentiated or homogeneous. Accordingly, the oligopoly market may be a heterogeneous oligopoly or a homogeneous (or pure) oligopoly. It seems the following features:

Sellers are few in number.
Any of them is of such a size that can increase and decrease in his output will appreciably affect the market price. In fact, the size of each seller’s output in relation to the total supply is the test.
Each seller knows his competitors individually in each market.

Each oligopolist realizes that any change in his price and advertising policy may lead rivals to change their policies. Hence, an individual firm must consider the possible reaction of the other firms to its own policies. The smaller the number of firms, the more interdependent are their policies. The reactions of rivals will generally be immediate and strong, and tendencies to close collaboration in price determination are appeared.

It is the fewness of sellers that introduces interactions into the price and output decision problem under oligopoly a special form of oligopoly in duopoly, under which only two firms produce a particular product.

Kinked Demand Curve Model

The kinked demand curve model developed by Paul M. Sweezy, has features common to most of oligopoly pricing models. The kinked demand curve analysis does not deal with price and output determination. It seeks to establish that once a price-quantity combination is determined, an oligopoly firm will not find it profitable to change its price in response to a moderate change in cost of production. An oligopoly form believes that if it reduces the price of its product, rival firms would follow and neutralize the expected gain from price reduction. But, if it raises its price, rival firms would either maintain their prices or may even cut their price down. In either case, the price rising firm stands to lose, at least a part of its share in the market. This behavioral assumption is made by all the firms in respect of others. The oligopoly firms would therefore, find it more desirable to maintain their price and output at the existing level.

There are three possible ways in which rival firms may react:

The rival firms follow the price changes, both cut and hike;
The rival firms do not follow the price changes;
Rival firms do not react to price-hikes but they do follow the price-cuts.

Kinked-demand curve is a demand curve with two distinct segments with different elasticities that join to form a kink. The primary use of the kinked-demand curve is to explain price rigidity in oligopoly. The two segments are:

(i) a relatively more elastic segment for price increase and

(ii) a relatively less elastic segment for price decreases.

The relative elasticities of these two segments are directly based on the interdependent decision-making of oligopolistic firms. Interdependence is the guiding behavioral principle of oligopoly firms in which the decision by one firm is both affected by the decisions of other firms and in turn affects the decisions of other firms. Such interdependence is characteristic of oligopoly firms that practice competition among the few. Interdependence is indicated by the kinked-demand curve, game theory, collusion, and mergers. Merger is the consolidation of two separately-owned businesses under single ownership. This can be accomplished through a mutual, “friendly” agreement by both parties, or through a “Hostile takeover,” in which one business gets ownership without cooperation from the other. Mergers fall into one of three classes –

(i) horizontal – two competing firms in the same industry that sell the same products,

(ii) vertical – two firms in different stages of the production of one good, such that the output of one business is the input of the other, and

(iii) conglomerate – two firms that are in totally, completely separated industries.

According to the kinked demand curve model, firm determines the price and output by intersection of MC and MR. But intersecting point lies on the discontinuous segment of MR. In this model, the demand curve faced by oligopolists has kink at the prevailing price. It means, the upper section of the kinked demand curve has higher price elasticity than lower part. Because, each oligopolist believes that if he reduces his price below the prevailing level, his competitors will follow him, and will accordingly lower their prices. So that an oligopolist firm which lowers the price could not increase its share of the market. Whereas if he raises the price above the prevailing level, his competitors will not follow him and they do not increase their price. So, an oligopolist will lose a considerable part of his customers. Because of this, an oligopolist tends to keep prices constant even if the cost and demand conditions are changed. This model is illustrated in figure.

In the figure, dED is the demand curve faced by an oligopolistic firm and has a kink at point E which represents the prevailing market price. Above this point, demand curve dE is more elastic and below this point, it is less elastic. dABMR is the marginal revenue curve of the firm. MR has two segments; the upper segment dA corresponds to the upper part of the demand curve dE. The lower segment BMR corresponds to lower part of kinked demand curve ED. The kink at point E on the demand curve results in discontinuity ‘AB’ in the MR curve. Oligopolist firm can reach equilibrium position and determine the selling price, and quantity and maximize the profit by equating MC with MR. In the given figure, SMC cuts the discontinued segment of MR at point ‘C’ and the firm determines price QE and selling quantity OQ. This QE level of price will not be changed by firm. If SMC curve rises to SMC1 because of increasing costs and SMC curve goes down to SMC2 because of decreasing cost, this will not affect the pricing decision of the oligopolist. These two curves SMC1 and SMC2 allow the firm to fix the price QE and quantity OQ.

We may conclude that an oligopolist faced with a kinked demand curve will be extremely unwilling to change his price. For a fall in his price will cause no large increase in his sales whereas a price increases will cause a substantial decline in his sales. Thus, neither a price increase nor a price reduction will be an attractive proposition for the oligopolist. During inflationary periods, however, oligopoly firms often follow one another’s price increase, to this extent, the kinked demand curve analysis can be said not to hold true.

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Payoff Matrix

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Importance of Game Theory

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Kinked Demand Curve Model

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