Natalie is a teacher and holds an MA in English Education and is in progress on her PhD in psychology.
Walter owns a company that makes bumper stickers. His old bumper sticker printer is not very efficient anymore, and Walter thinks it might be time to buy a new one. But he's not sure if he should go ahead and buy a new one, or keep the old one a little longer. And if he does buy a new one, he's not sure which one he should buy. Walter is faced with a dilemma, and he has to make a business decision. There's a lot for him to consider. For example, what if the cost of ink suddenly goes up? What if bicycles have a comeback and there are fewer cars on the road? Will people buy fewer bumper stickers?
A state of nature is something that might affect the profitability of a company and over which the company has no control. For example, if people buy fewer cars, and as a result fewer bumper stickers, that will drastically affect Walter's company. At the same time, he can't control when or if that will happen. But states of nature can affect Walter's decision. Replacing the printer might make sense if one thing happens, but not if another thing happens. So how can Walter make the best decision possible while taking into consideration possible states of nature?
One type of decision making tries to assign probabilities to different states of nature. What if Walter really doesn't know how likely it is that something will happen? To help him out, let's look at three ways of making business decisions without using probabilities: the optimistic approach, the conservative approach and the minimax approach.
Okay. Walter needs to know whether to replace his bumper sticker printer or not. He has three choices: keep the current printer, replace it with new printer A or replace it with new printer B. But what's the best option?
One way to make this decision is the optimistic approach, also called the maximax approach. This involves choosing the option with the largest possible payoff or the smallest possible cost. To figure out which one has the largest possible payoff, a payoff table is constructed. Here's Walter's payoff table:
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Along the left side you can see the different options: keep the current printer, replace with printer A or replace with printer B. Along the top, you see different possible states of nature. In Walter's case, state one is that everything remains as it is, state two is that people buy 25 percent fewer bumper stickers over the next five years and state three is that people buy 50 percent fewer bumper stickers over the next five years. Inside the table, Walter is able to calculate how much profit he will make for every 10 bumper stickers for each option in each state of nature. This takes into consideration the cost of printing and ink, other costs and how much he could sell the bumper stickers for. Negative numbers mean that he'll be losing money instead of making it.
For the optimistic approach, Walter will want to compare the maximum possible outcome for each of the three options. In this case, keeping his current printer will result in four dollars profit for every 10 bumper stickers in the best-case scenario. Switching to printer A will result in three dollars profit in the best-case scenario and switching to printer B will result in five dollars profit for every 10 bumper stickers in the best-case scenario. According to the optimistic approach, Walter should replace his current printer with printer B.
Looking at this example, it's not hard to see why this is called the optimistic approach. Walter is being optimistic here, considering only the best possible outcome. And he's choosing the maximum number among the maximum numbers for each option, so it's also called the maximax approach.
But Walter's a little nervous with the optimistic approach. What if something goes wrong and he doesn't get the best possible scenario? Another approach is known as the conservative approach to decision making, or the maximin approach. This involves choosing the option with the largest payoff (or lowest cost) among the lowest set of numbers. In other words, instead of comparing the best-case scenario as we did with the optimistic approach, we are comparing the worst-case scenario and seeing which option is best in that case.
Let's go back to Walter's table. Instead of comparing the largest number for each option, he now wants to compare the three lowest numbers for each option. In this instance, keeping his current printer will result in breaking even for every 10 bumper stickers, losing one dollar for every 10 bumper stickers if he switches to printer A and losing three dollars for every 10 bumper stickers if he switches to printer B. According to the conservative approach, Walter should keep his current printer because in a worst-case scenario, he will lose the least amount of money with that one.
Like the optimistic approach, the conservative approach is named well. In involves trying to conserve the money you have by looking at the worst-case scenario and going from there. It also involves looking for the maximum number among the minimum numbers for each option, so it's called the maximin approach.
So far, Walter has tried two pretty simple and straightforward approaches. But, because they're so simple, he feels like they aren't a very accurate judge of what he should do. He wonders if there's a better way.
The third way to approach this problem is the minimax approach, which involves figuring out which option has the best chance across different states of nature. To do this, you must find the difference between the highest possible payoff for each state of nature and all the other payoffs for that state of nature. Then, you choose the option with the lowest maximum value. As you probably already guessed, this is a bit more complicated than the other systems. Let's help Walter figure this one out.
Now, he has to figure out, for each state of nature, what the difference is between the highest value and the other values. So let's make a new table for him.
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In this new table, he subtracts the values for each state of nature from the highest value for that state of nature. For example, for the first state of nature (people keep buying bumper stickers as they do now) the highest possible profit is five if he switches to printer B. So he needs to subtract each number in that column from five. Then, he can move on to the second state of nature and do the same thing, subtracting each number in that column from four, which is the highest value for that state.
When he's done all of the subtracting for each state of nature, he has a new table that shows the difference between the highest possible payoff, and that option's possible payoff.
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Now, he chooses the highest number for each option. In Walter's case, the highest number for option one, keeping his current printer, is one. The highest number for option two, printer A, is two and the highest number for option three, printer B, is three. Finally, Walter will choose the lowest number from this last column: the option with the minimum of the maximum difference, hence its name, the minimax approach. In this case, the minimum is one, which corresponds to option one, keeping his current printer.
There are three major ways of making business decisions without using probabilities, but while still considering states of nature, or things that might affect the profitability of a company, and over which the company has no control.
The optimistic approach, also called the maximax approach, involves choosing the option with the largest possible payoff or the smallest possible cost. The conservative approach to decision making, or the maximin approach, involves choosing the option with the largest payoff (or lowest cost) among the lowest set of numbers. Finally, the minimax approach involves figuring out which option has the best chance across different states of nature. To do this, you must find the difference between the highest possible payoff for each state of nature and all the other payoffs for that state of nature. Then, you choose the option with the lowest maximum value.
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