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I have a population n as given in the image, a set of strategies for each player and a utility function as per the image

I have a population n as given in the image, a set of strategies for each player and a utility function as per the image

I want to derive the best response function and show that it is a function if x is from a continuous distribution.

I also want to show that if lambda is at least equal to 0, then it exists a least and greatest Nash equilibrium.

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