Measurement of Risk by Probability Distribution

The probability of an event is the chance, or odds, that the event will occur, if all possible events or outcomes are listed, and if a probability of occurrence can be assigned to each event, the listing is called a probability distribution. For example, suppose a sales manager observes that there is a 70% chance that a given customer will place a  specific   order   within   the  next   two

weeks, versus a 30% chance that the customer will not. This situation is described by the probability distribution shown in table.

Each possible outcome is listed in column 1, and the probabilities of each outcome, expressed as decimals and percentages, appear in column 2. Notice that the probabilities sum to 1.0 or 100%, as they must if the probability distribution is to be complete (i.e. represent all possible outcomes). In this very simple example, risk can be read from the probability distribution as the 30% chance of neither the firm nor receiving the order. For most managerial decisions, the relative desirability of alternative events or outcomes is not so easily computed. A more general measure of the relation between risk and the probability distribution is measure of the relation to incorporate risk considerations adequately into the decision-making process. The need for a more general measure of risk can be illustrated by the following example.

Suppose that a firm is able to choose only one of two investment projects, each calling for an outlay of $10,000. Assume also that profits earned from the two projects are related to the general level of economic activity during the coming year, as shown in table. This table is known as a payoff matrix since it illustrates the monetary outcomes associated with each possible state of profits from project B very much more as a result the state of the economy than do those from Project A. In a normal economy, both projects return $5,000 in profit. Should the economy be in a recession next year? Project B will produce nothing, whereas Project A still provides a $ 4,000 profit. If the economy is booming next year, Project B’s profit will increase to $ 12,000, but profit for Project A will increase only moderately to $6,000.

Outcomes and Probabilities for Receiving an Order

                                Event                        Probability of Occurrence

                                 (1)                                        (2)

Receive Order                                                   0.7 = 70%

Do not received order                                        0.3 = 30%

Total                                                                  1.0 = 100%

Project A is clearly more desirable if the economy is in recession, whereas Project B is superior in a boom. In a normal economy, the projects offer the same profit potential, and both are equally desirable. To choose the best project, one needs to know the likelihood of a boom, a recession or normal economic conditions. If such probabilities are available, the expected profits and variability of profits for each project can be determined. These measures make it possible to evaluate each project in terms of anticipated or expected returns, and to measure the risk of such returns in terms of the difference between and expected values.

Payoff Matrix for Project A and B

                                                          Profits

State of the Economy         Project A             Project B

Recession                             4,000                     0

Normal                                 5000                  5,000

Boom                                   6,000                 12,000

The expected value is the anticipated receipts from a given payoff matrix with a specified probability distribution. It is the weighted average receipt when the weights are defined by the appropriate probability distribution.

To continue with the previous example, assume that forecasts based on the current trend in economic indicators suggest a 2-in-10 chance of recession, a 6-in-10 chance of normal economy, and a 2-in-10 chance of a boom. As probabilities, the probability of recession is 0.2, or 20%, the probability of normal economic activity is 0.6 or 60% and the probability of a boom is 0.2, or 20%. These probabilities add up to 1.0 (0.2 + 0.6 + 0.2 = 1.0), or 100%, and thereby from a complete probability distribution, as shown in table.

Calculation of Expected Values

	State of the Economy	Probability of this State Occurring	Profit Outcome if this State Occurs	Expected Profit Outcomes ($)
	(1)	(2)	(3)	(4) = (2) x (3)
Project A	Recession	0.2	4,000	800
	Normal	0.6	5,000	3,000
	Boom	0.2	6,000	1,200
		1.0	Expected Profit A
Project B	Recession	0.2	0	0
	Normal	0.6	5,000	3,000
	Boom	0.2	12,000	2,400
		1.0	Expected Profit B

If each possible outcome is multiplied by its probability of occurrence, and the answers are summed, the weighted average outcomes are obtained. In this calculation, the weights are the probabilities of occurrence, and the weighted average is called the expected value. The above mentioned table illustrates the calculation of expected profits for Project A and B. Each possible profit level in column 3 is multiplied by its probability of occurrence from column 2 to obtain weighted values of the possible profits. Summing column 4 of the table for each project gives a weighted average of profits under various states of the economy. This weighted average is the expected profit from the project.

Risk is a complex concept, and some controversy surrounds attempts to define and measure it. Common risk measures that are satisfactory for most purposes are based on the observation that right probability distributions imply low risk. 

The standard deviation is a popular and useful measure of absolute risk. Absolute risk as measured by the standard deviation is the overall dispersion of possible payoff values. The smaller is the standard deviation, the tighter is the probability distribution and therefore the lever is the risk in absolute terms.

 Click on following links for more details: 

• Utility and Risk Aversion in Investment Decision

• Risk and Uncertainty in Managerial Decision Making

• Risk Analysis and Risk Management

• Factor Determining Organizational Buying Decision | Buying Behavior

• Investment Decision under Uncertainty

• Statistics: Uses, Functions and Limitations of the Statistics

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.

  Some Related Links:        

• The Game Theory

• Payoff Matrix in Game Theory

• Meaning of Managerial Economics

• Scope of Managerial Economics

• Relationship of Managerial Economics with Traditional Economics

• Utility and Risk Aversion in Investment Decision

• Investment Decision under Uncertainty

• Risk and Uncertainty in Managerial Decision Making

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.

  Some Related Links:      

• The Game Theory

• The Prisoners’ Dilemma, an application of game theory analysis

• Payoff Matrix in Game Theory

• Scientific Management Theory

• Concept of Oligopoly and Kinked Demand Curve Model

• Measurement of Risk by Probability Distribution

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.

   Some Related Links:        

• The Game Theory

• Nash Equilibrium or Non Cooperative Equilibrium

• The Prisoners’ Dilemma, an application of game theory analysis

• Scientific Management Theory

• Concept of Oligopoly and Kinked Demand Curve Model