A Statistical Analysis of Games with No Certain Nash Equilibrium Make Many Results Doubtful

Some games may have a Nash equilibrium if the parameters (e.g. probabilities for success) take certain values but no equilibrium for other values. So there is a transition from Nash equilibrium to no Nash equilibrium in parameter space. However, in real games in business and economics, the input parameters are not given. They are typically observed in several similar occasions of the past. Therefore they have a distribution and the average is used. Even if the averages are in an area of Nash equilibrium, some values may be outside making the entire result meaningless. As the averages are sometimes just guessed, the distribution cannot be known. The main focus of this article is to show this effect in an example, and to explain the surprising result by topological explanations. We give an example of two players having three strategies each (e.g. player and keeper in penalty shooting) where we demonstrate the effect explicitly. As the transition of Nash equilibrium to no equilibrium is sharp, there may be a special form of chaos which we suggest to call topological chaos.