Graduate Game Theory: Are the payoff functions πi continuous?

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The discussion centers on demonstrating that the payoff function πi in game theory is not continuous. Participants suggest finding a specific point where the function fails the continuity condition and proving it. There is a query about the existence of best replies and how to approach identifying discontinuities in large intervals. The importance of checking borders in piecewise functions is highlighted, as these are likely candidates for discontinuities. Overall, the conversation emphasizes the need for a systematic approach to analyze continuity in payoff functions.
vonanka
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  1. How do I show that the payoff function πi isn´t continuous? Why do best replies not always exist?
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vonanka said:
How do I show that the payoff function πi isn´t continuous?
In the same way you show that a function is not continuous for (nearly) every other function. Find a point where it is not continuous and prove that the function does not satisfy the condition for continuity there.

Is this a homework problem?
 
mfb said:
In the same way you show that a function is not continuous for (nearly) every other function. Find a point where it is not continuous and prove that the function does not satisfy the condition for continuity there.

Is this a homework problem?
Okey, but what if the interval was extremely large. Is there a way to find the non continuous part in a smart algorithmic way?
It´s a old exam without answers. Really need a explanation here. Thanks.
 
If you have a piecewise definition, the borders between the pieces are always obvious places to check.
The pieces itself can be discontinuous as well, but if they are simple polynomials like here you know they are continuous.
 

Thread 'Area of Overlapping Squares'

Here is a little puzzle from the book 100 Geometric Games by Pierre Berloquin. The side of a small square is one meter long and the side of a larger square one and a half meters long. One vertex of the large square is at the center of the small square. The side of the large square cuts two sides of the small square into one- third parts and two-thirds parts. What is the area where the squares overlap?
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