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Currently, I am modelling the results of game theory experiments where two parties negotiate to divide a fixed sum of money between themselves. Y is recorded as the percentage of the money which the first party takes (e.g. when Y = 40, the first party takes 40% and the second party is implied to take 60%).

When applying a simple linear regression model to this problem, the outputs of the model does not make sense. For instance, Y may be predicted to be 110, but there is no 110% of that fixed sum in reality.

I understand that logistic regression outputs a range which may be interpreted as between 0 - 100. But my understanding is that logistic regression does not apply here because:

  1. this is not a classification problem, and
  2. logistic regression requires the values of Y in the training set to be binary (which is not the case here)

Hence, what should I do to constrain the linear regression model to output a range between 0 to 100, or are there other models that I should be looking at? Also, is there a name for this problem I am facing? My vocabulary is lacking here.

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  • $\begingroup$ Why not 110%? One player gets 100% of the pot plus the other player has to give them 10%, ha ha. On a serious note, you haven't provided us much insight as to what the input variables are which will be used to model the output, nor what relationship they should logically have to that output. $\endgroup$
    – Mark L. Stone
    Commented Jun 4, 2017 at 4:11
  • $\begingroup$ You might try beta regression. $\endgroup$
    – GeoMatt22
    Commented Jun 4, 2017 at 4:13
  • $\begingroup$ 1. logistic regression is NOT only - or even mainly - for classification. If you see a book claim otherwise, throw it away, because it's harming your understanding and the person who is telling you that doesn't appear to comprehend logistic regression - who knows what other nonsense they might peddle at the same time? ... It's a model for estimating/predicting probabilities in terms as a function of predictors. ... 2. logistic regression does not require 0/1 data. (Some implementations do but that's not the fault of logistic regression - should work with $p_i$ & $n_i$ just fine) ... ctd $\endgroup$
    – Glen_b
    Commented Jun 4, 2017 at 4:34
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    $\begingroup$ ctd... ... 3. That said it sounds like you have a continuous proportion, so perhaps beta regression might be more suitable (though quasi-binomial might do in a pinch). 4. Do you have any exact 0's and 1's or are the proportions all strictly away from the endpoints? $\endgroup$
    – Glen_b
    Commented Jun 4, 2017 at 4:34
  • $\begingroup$ Thank you @Glen_b. I am trying out beta regression in R now. $\endgroup$
    – Misnomer_N
    Commented Jun 26, 2017 at 5:26

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I guess that in real life you'd have very few 0s and 1s, but since it seems that you're using an ultimatum/divide-the-dollar type of game theoretic model, the 0s and 1s should at least be of theoretical interest (since there are equilibria with these allocations). Only for that reason I'd suggest you use a fractional logit (or fractional response model - frmpackage in R) otherwise a beta regression would be fine.

Also I suggest you check this out, it looks like it might be of interest to you.

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