Mixed Strategy

In economic reality, pure strategies are rarely used. A pure strategy is an option that must be chosen by rational decision makers because it is evidently the best option for the decision maker. Often, however, the circumstances are not so clear, especially when decisions must be made with imperfect information, as is usually the case in economic practice.

In such cases, game theory replaces the term “uncertainty” with “probabilities” that rational decision makers use to select available options. A probability of p=0 means that an option will certainly not be chosen, while a probability of p=1 means that an option will certainly be chosen.

With imperfect information, the spectrum of probabilities will be between p=0 and p=1, with the individual probabilities of all mutually exclusive options adding up to 1.

Thinking strategically in terms of probabilities systematically opens up access to risks and provides clues to their probabilities of occurrence. Decision makers can adequately prepare for identified risks through mixed, anticipated response strategies.

In this way, game theory provides an important contribution to the corporate foresight concept, an approach that helps to develop scenario-based strategies.

Basically, knowing the uncertainty category of a situation is important for a meaningful decision, in terms of the effects of the respective decision variants, as already suggested by Heinz von Foerster. In this sense, it is advisable to first distinguish whether we know the effects of our decisions with certainty, even though they arise from interactions with decisions of third parties. This case is the most comfortable, although for game theorists also the most boring. If this is not so, we should distinguish whether the probabilities of the possible effects occurring are known or not. If the probabilities are known, an optimized decision recommendation can be found in the form of a mixed strategy. We are then dealing with a calculable, risky decision. However, if the probabilities are not known, we are dealing with a non-calculable decision under uncertainty. In this uncomfortable case, an attempt should be made to make reasonable assumptions about the effects of the decision options. The Laplace Principle can give guidelines. If this is not possible either, decisions should be made in such a way that the decision opens up as many possible courses of action as possible. In such cases, no rational decision is possible. By keeping the room for manoeuvre as wide open as possible, it is possible to decide with higher variety, i.e. with more possibilities for action, at least in the next step.

Finally, a disturbing piece of news: Kurt Gödel recognized that logical systems, even if they are constructed extremely conscientiously (as in the Principia Mathematica of Bertrand Russell and Alfred North Whitehead), are not immune to undecidability.


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