Problem solving and understanding



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General approaches to problem solving and understanding

For the most part Responsible Thinking involves recognizing that information we get is deceptive and unreliable, so we should recognize that we are unsure of things that others may become convinced are true.  However there are some general purpose analysis techniques that help us to better understand things so we can improve our ability to deduce what is true when others may be unsure.  We will look at some of them here.  The academic field of "operations research" typically addresses methods of this sort in great depth.

Optimization, objective functions, and local minima

Suppose a problem has many possible solutions, and we are looking for the best one. An example might be that we must visit several cities and we want to choose the order we visit them. The "best" solution is the one with the shortest total distance traveled. This is called an optimization problem. The "objective function" is the method of calculating how good a potential solution is. In this case the objective function is the sum of the distances from each city on the list to the next one on a chosen route. For a large number of cities, it is impractical to try all the possible orderings. One approach to solving such a problem is to start with some solution and see if we can improve on it by making small changes, like switching the order of two cities on the list. If a small change is an improvement, we keep this as our best solution, and try to improve on this. We quit when no small change is better. The final result may or may not be the best possible solution. It is called a "local" minimum because it is better than all the "nearby" solutions.

The law of diminishing returns

As you try to come closer and closer to perfection on some job, progress typically becomes more and more difficult. This is because the easiest parts of the job are normally done first.

Equilibrium and instability

Typically the population of rats, or any other species, is in equilibrium. If at some point in time there is an unusually large population of rats, there won't be enough food for them all of them and many will die, reducing the population. If there is an unusually small number of rats, they will multiply, increasing the number until their food supply cannot support any more. As a result, if the food supply stays constant, the number of rats will stay relatively constant.

Instability is when a change in the quantity of something tends to create a greater change in the future. At present, human population is unstable, since the more people there are, the more babies are born. For at least a little while, food and other resources are sufficient for this to continue in most parts of the world. At such time as some resource prevents further growth, the unstable growth will cease.

These are simple aspects of control theory. They are important concepts in many fields, from chemistry to economics.

Game theory

This is the study of strategies people might use when they are interacting with other people who have somewhat conflicting goals. 

In some cases the goals are in direct conflict, as when a one football team is deciding whether to pass or run, and their opponent is trying to guess which they will do so they can prevent it from being successful.  An outcome that is good for one team is equally bad for the other team.  These are called "zero-sum" games.

Another kind of situation is one where cooperation helps, as in the "prisoner's dilemma" problem. If two partners in crime both confess, they will do worse than if they both keep quiet. However if one confesses when the other doesn't, the one who confesses does better. The best strategy for people not cooperating is not the same as for people who are cooperating.

Forming ideas (brainstorming) vs. evaluating ideas

In a brainstorming session, people are encouraged to suggest all kinds of ideas, even if they may sound silly. Sometimes a silly sounding idea will work or can be modified so it will work, so these should not be discouraged. Once the suggestions have been made, however, they must be evaluated on whether they actually will work. There is nothing particularly great about a "creative" idea if it is wrong.

How logic and goals relate

In choosing a course of action, science and logic (or reasoning) are important, but only if we know what we are trying to achieve. Ultimately, what we want to achieve must be determined by our own preferences, and cannot be deduced from anything else. However, some goals can be derived from other goals by reasoning.

If I want to play golf, I can use my existing knowledge and logic to decide how to go about it. If I don't have golf clubs, I will need to buy or borrow them. I will need to get myself and my equipment to a golf course that I can afford and is open and has a tee time available. I can reason about how to achieve these things. However, doing these things would make no sense if I didn't want to play golf. Assuming I am playing golf for enjoyment (rather than, say, to make a business deal), logic can't tell me whether I want to play. That depends on my own personal preferences. In order to take actions, we need to know what we want to achieve, which may not be something we can arrive at by reasoning, but once that is decided we can use reasoning to decide how to go about achieving it.

Give priority to higher level goals

When a problem is broken down into smaller problems, people working on a smaller problem may take actions that are best for solving that problem, but aren't the best for the overall problem. For example, in order to help achieve the goal of happiness, we may pursue the goal of making money. However if we make extra money working in a job we hate, we could be sacrificing the greater goal in order to achieve the lesser goal.

This problem is well-known in businesses and other large organizations where different departments compete with each other, which may be good for achieving the goals of the individual departments, but hurt the organization as a whole.