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Verbal and statistical models

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Verbal model: description of a market launch
The word-of-mouth spread of knowledge about a product is an example of a simple verbal model.

When a product or service is launched, sales often start slowly until some people (early adopters) become aware of the product.

They buy it and start using it. If they are satisfied with the product, they recommend it to family members and to friends via word-of-mouth communication.

This leads to an acceleration of sales growth, and the producer (importer) is encouraged to place advertisements in various media. At some point, the market potential is approached and growth slows down.

Statistical model: Bayesian decision theory
A statistical model uses systematic statistical reasoning. A numerical example may help to clarify this (Green et al., 1988).

Assume that a company wants to launch a new product. Before the launch, managers want to assess three scenarios.

Based on experience, managers estimate that there is a 30 percent chance of a best possible outcome, 50 percent chance of the modest scenario, and a 20 percent probability of failure. So, should the product be introduced, assumed that the above figures apply?

See Table 1.

Table 1 Three scenarios to be assessed

Scenarios  Market share (%) Profit
1. A best possible case         15    €20 million
2. A modest or “realistic” scenario 5    €5 million
3. A disaster scenario          1  - €10 million
Table 2 Assessing a launch under different scenarios
Scenarios

S1   S2   S3
Decision P(S1) 15% share P(S2) 5% share P(S3) 1%
Share

Introduce 0.3 €20m  0.5 €5m  0.2 -€10m

Don’t Introduce
0.3 0  0.5 0  0.2 0

If we do not introduce the product, the expected income is zero. The expected payoff of introducing product is calculated as €6.5 million:

[0.3 X(20) + 0.5 X(5) + 0.2 X (-10)]

Assume, for a moment, that the company could consult an infallible soothsayer with a crystal ball.

Moreover, presume that the product needs to move along one of a hundred available paths that leads to the future market reality.

Thirty of these paths will provide us a profit of €20 million, fifty paths yield a profit of €5 million while the remaining twenty paths will lead us to a disastrous loss of €10 million.

The problem is that we usually have no idea concerning which of the hundred paths lead us to the profit, and which twenty make us, unknowingly, head for disaster. To make things worse, we only have one shot.

However, the soothsayer is able to identify or label them for us (“Path 1=> profit of 5, Path 2=> loss of 10, Path three => profit of 5, etc.”).

It is easy to see that, possessing this window to the future, we would decide not to launch the product, if we knew that the third scenario (S) would prevail.

By acting thus, we would at least prevent losing money. Consequently, our soothsayer-assisted expected profit would be €8.5 million [0.3 X(20) + 0.5 X(5) + 0.2 X (0)].

Everything else being equal, we would be better off in the long run by paying the soothsayer up to €2 million for the service [8.5 - 6.5]. This net value is called the expected value of perfect information.

Unfortunately, perfect knowledge of the future does not exist. However, we might be able to approach perfect knowledge by using market tests.

Assume that we work with a research agency that operates on a global scale, such as A.C. Nielsen, GfK, Burke or Sofres. Based on an inspection of the agency’s track record, we can establish the data shown below.

The table is to be understood in the following way. In six out of ten situations where a market share of 15 percent turned out once the market conditions became stable, the

Table 3 Possible outcomes of market test by the agency

Probability of forecasting outcome

Scenario Z1 (share > 10%) Z2 (3% < share < 10%) Z3 (share <                 3%)
S, (15% share)  0.6  0.3   0.1
Sz (5% share)  0.3  0.5   0.2
S3 (1 % share)  0.1  0.2   0.7

research agency has, based on a test, forecast a market share of more than 10 percent (cell S1Z1).

However, in three out often cases, the agency underestimated the outcome and forecast a moderate scenario with a share of between 3 percent and 10 percent (cell S1Z1).

Finally, in one out of ten cases the agency failed: it forecast a market share less than 3 percent while the market share turned out to be 15 percent (cell S1Z3).

Usually, a test pitfall of category S3Zi (a “progressive” or “over-optimistic” estimation of market potential) is regarded as worse than SIZ3 (a conservative underestimation).

Some researchers recommend establishing different weights that adjust for different consequences of alternate outcomes.

A performance score, not unlike the one shown above or even better has been reported by researchers and agencies that work with and use test marketing simulators such as TESI (Erichson, 1987) and Assessor (Urban and Hauser, 1993).

Assuming that the data in Tables 2 and 3 are true, it is possible to compute the value of market research, perceived as an upper limit that should not be exceeded (it cannot be justified to pay more for the research).

This amount is called the expected value of additional information. How do we compute this value? Simply by carrying out a preposterior analysis in accordance with Bayes’ theorem.

First, we compute the marginal probabilities by multiplying the probabilities of the different scenarios by the probabilities of a given test result’s (in)capability to “hit the truth” and add the numbers (Z11, refers to the first cell in Table 3):

S1  x Z11 =(0.3×0.6)      = 0.18
S2 x Z12  =(O.Sx0.3)     = 0.15
S 3  x Z13 = (0.2 X 0.1) = 0.02
?                                 = 0.35

Next, we compute the corresponding posterior probabilities (PS1Z1):
0.18/0.35 = 0.5143
0.15/0.35 = 0.4286
0.02/0.35 = 0.0571
?               = 1.0000

We repeat the calculations with the numbers in the remaining columns.

Once we have done this, we are able to establish the Bayesian decision tree and put in the appropriate profits accompanying the different states of nature as well as the corresponding posterior probabilities (reflecting the uncertainty of the market test). This has been done in Figure 1.

Looking at the far right of the upper branch we see the three posterior probabilities and the accompanying profits.

We now simply multiply these three profit figures by the corresponding probabilities and add the numbers:

(0.5143 X 20) + (0.4286 X 5) + (0.0571 X-10) = 11.858

That is the expected outcome of option A1 (to launch). Not launching would give zero profit (option A2).

But since the expected value of A1 is better than Az, we chose A1. These computations have to be repeated for the middle and the lower of the three main branches.

Finally, we multiply the expected values by the marginal probabilities of the test’s outcome:

(11.858 X 0.35) +(6.972 X 0.38) and (0 X 0.27) _€6.800 (X 1,000)

This figure is called the expected payoff after research. The expected payoff without research was €6.500.

The difference between the two estimates is €300,000. This number is called the expected value of additional information.

Finally, detracting the cost of the test, say, €100,000, from the €300,000 gives the net expected payoff of research.

In this case, the estimate would be €200,000. So, based on the above assumptions, it is better to carry out the market test, and, based on the test, the company should launch the product.

However, this need not be the case universally. Due to differences with regard to the business cycle, consumer expectations, the overall robustness of the economy, etc., the optimal decision may be to launch the product in some markets but not in others.

In some countries it may be recommended to carry out additional research, while in other countries conducting separate research seems not worth the effort (the net expected payoff of research is negative).

Basically, the Bayesian approach has four critical inputs:

? the number of scenarios to be assessed;
? the size of the profit, given the actual level;
? the probability of the given state of nature;
? the historic “performance matrix” of the research agency (Table 8.4).

Table 4 Possible outcomes of market test by a “miserable” agency

Probability of forecasting outcome

State of nature          Z1 (share > 10%) Z2 (3% < share < 10%) Z3              (share < 3%)
51 (15% share) 0.1  0.8  0.1
S2 (5% share) 0.1  0.1  0.8
53 (1 % share) 0.8  0.1  0.1

It is easy to see that a change in any of these inputs will change the expected profit. Assume that the probabilities of the expected scenarios is changed because the company is less optimistic.

With new scenario probabilities of S1 = 0.2, S2 = 0.3 and S3 = 0.5 (and leaving everything else unchanged), the expected value of the upper branch is reduced from €11,858 to €9,034.

The probability of Z, is reduced from 0.35 to 0.26 and the expected value after research becomes €3.298 (formerly €6,800).

Assume that the expected payoffs in Table 2 change to S1 = l0m, S2 = 5m and S3 =-20m.

Also, let the probabilities of the three scenarios be those of Table 2 and let the possible outcome of the market test be the ones of Table 3.

In this case a launch is only profitable given that research is carried out. Why? Because the expected profit without research now is negative (0.3 * 10) + (0.5 *5) + (0.2* -20) = -1.5.

The reader should check if it is correct that in this situation the expected payoff after research is €3.5m implying that the expected value of additional information is €Sm (the range from – 1.5 to 3.5).

Although highly unlikely, situations may appear where the expected payoff after research becomes even lower than the expected payoff without research, implying that the expected value of additional information becomes negative.

In the above example, 3.298 was lower than 6.500, but let us look at another example and assume a research agency has a miserable performance matrix, as in Table 4.

The reader is encouraged to compute the expected value of additional information given that the performance matrix has changed to the one shown in Table 4 and assuming that the probabilities of the states of nature and the attached profits are the same as in the previous example (S1 = 0.2, S2 = 0.3 and S3 = 0.5).

The expected payoff after research is €3,950,000, so the expected value of additional information is €3,950,000 – €6,500,000 =-€2,550,000. Clearly, in this case the company is better off not carrying out the research.

It is very easy to perform a Bayesian sensitivity analysis by using the formula editor in spreadsheet software such as Presentations, Excel or Lotus.

After having done the simple coding, one can change either the scenario probabilities, or the related profits or the figures of the research-performance matrix.

Do not change more than one of the parameters at a time (and remember that the rows as well as the columns of the quality matrix must add up to 1.00).

Comprehensive texts on Bayesian decision theory are Schlaifer (1959), Chernoff and Moses (1987), Viertl (1987) and Carlin and Louis (2000).
Keywords: model, verbal model, Statistical model,

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