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Thread: How to deal with unknown distribution?

  1. #1
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    How to deal with unknown distribution?

    I have a game with a player. The player's type is unknown (privately known to him) and distributed within a known interval. I play the payoff-maximizing strategy. When the player's type or its distribution is known the game can be easily solved. Then my equilibrium strategy is some sort of a weighted sum of best-responses to each possible player type realization.
    The problem is what to do if player's type distribution is completely unknown. The only thing I know is that the type lies within a known interval, say, [a,b].


    Any suggestion on how to approach this?

    I tried β-distribution, but it does not completely fit the described setup as it requires some initial parameters to set.

    Thanks in advance.
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  2. #2
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    Re: How to deal with unknown distribution?

    Hey vasyazz.

    If you don't have information, then you should go with the worst case scenario.

    The distribution that has the worst case in my opinion is the uniform distribution where everything is equally likely and you have no advantage any way for any outcome.
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  3. #3
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    Re: How to deal with unknown distribution?

    Thanks for your try, but the uniform distribution is not the correct solution. Assume X is an unknown distribution. Then X^2 is also an unknown distribution. Therefore, if we assume X=U then X^2 also equals U and we obtain U=U^2. A contradiction.
    On the other hand, the solution exists for sure, since we have to play somehow. We cannot play no strategy.
    Last edited by vasyazz; Dec 13th 2016 at 11:40 PM.
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  4. #4
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    Re: How to deal with unknown distribution?

    You need to define your loss function for us.

    A uniform distribution has maximum entropy which is why I suggested it.

    Every distribution in game theory and every estimator has a loss function and you minimize your loss functions to get the best guesses given the information you have [which has a distribution].

    What have you done to describe your loss function and your random variables?
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  5. #5
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    Re: How to deal with unknown distribution?

    I work with a payoff function that is just a negative of the loss function, so it should not matter.
    Basically, if the player's type t is known there is a decreasing function f(t) that defines the payoff for every possible type.
    If payer's type is not known, but its distribution is known, take for example t=a with 50% and t=b with 50%, then the payoff function would be 0.5*f(a)+0.5*f(b). Similarly, when type distribution is a continuous function the payoff function is an integral giving a weighted sum of every possible type realization.

    I'm not sure all this is really important. Bottom line, the payoff (or loss) function is given and it decreases in player's type, i.e. playing against a weak player gives greater payoff.
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  6. #6
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    Re: How to deal with unknown distribution?

    any thoughts?
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  7. #7
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    Re: How to deal with unknown distribution?

    What are you trying to optimize exactly? [In terms of your functions and distributions]
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  8. #8
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    Re: How to deal with unknown distribution?

    I maximize my expected utility.

    An example of the game:

    You go out and play one chess game with the first person you meet. If you win you get an amount of money V. If you lose you get nothing. A draw is not possible for simplicity.
    You can prepare for the game in advance so that for each level x you spend f(x). You win if rival's level is less than x and lose otherwise.

    So, if all citizens in the city play at some level x you can prepare for the level x+epsilon and your payoff would be V-f(x+epsilon). This is obviously an optimal strategy.
    Another example: if people's level distributes according to a cumulative distribution function G then for some level x the payoff would be V*G(x)-f(x). Then the optimal strategy would be argmax(V*G(x)-f(x)).

    The problem is what to do if G is not known...

    The solution should exist since you do prepare somehow; it's impossible to prepare "in no way".
    Last edited by vasyazz; Dec 28th 2016 at 04:40 AM.
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