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vonanka
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- How do I show that the payoff function πi isn´t continuous? Why do best replies not always exist?
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.vonanka said:
Okey, but what if the interval was extremely large. Is there a way to find the non continuous part in a smart algorithmic way?mfb said:
Game theory is a branch of mathematics that studies decision-making in situations where multiple players interact with each other. It uses mathematical models to analyze the strategies and outcomes of these interactions.
Payoff functions are typically represented as mathematical functions that assign a numerical value to each possible outcome in a game. These values represent the utility or satisfaction that each player receives from the outcome.
In game theory, continuity of payoff functions ensures that small changes in strategies or actions will result in small changes in payoffs. This is important because it allows for the analysis and prediction of outcomes based on the players' strategies.
If payoff functions are not continuous, then small changes in strategies or actions may result in large changes in payoffs. This can make it difficult to predict outcomes and can lead to unstable or unpredictable behavior in games.
Scientists can determine if payoff functions are continuous by using mathematical tools such as calculus and limit theory. They can also use experimental methods to test the continuity of payoff functions in real-life situations.