ug_delta parameter bounds

[Momo]

Hi all,

I'm trying to use ug_delta to fit behavior data from a modified ultimatum game. If I understand it correctly, all three parameters (alpha, tau, and epsilon) should be bounded between 0 and 1. However, I constantly got alphas larger than 1. Further inspection of the output showed that some generated initials of alpha were larger than 1 (I tried both "vb" and "random" and the problem persisted). I went on to check ug_delta.R and found the upper bounds of alpha and tau seem to have been set at 20 and 10, respectively (bold):

  ug_delta <- hBayesDM_model(

  task_name       = "ug",
  model_name      = "delta",
  model_type      = "",
  data_columns    = c("subjID", "offer", "accept"),
  parameters      = list(
    "alpha" = c(0, 1, 20),
    "tau" = c(0, 1, 10),
    "ep" = c(0, 0.5, 1)
  ),
 ...

Did I miss something? How to make sure initials generated by inits = "vb" or inits = "random" and the fitted parameters are bounded between 0 to 1?

Thank you!

Best,
Momo

[wooyou...@gmail.com]

Hi Momo,

Apologies for the delayed response. I believe it was coded based on Xiang et al. (2013) Journal of neuroscience (https://www.jneurosci.org/content/33/3/1099.short), where alpha (envy) > 0. I'm not entirely sure why it was constrained to [0, 1] in Gu et al. (2015), but I believe alpha can exceed 1 in certain cases (e.g., when large envy causes the subject value of winning a positive amount to become negative). Tau (inverse temperature) was also set between 0 and 1 in Gu et al. (2015). However, I don't think it necessarily needs to remain within that range. To allow more flexibility and avoid it being constrained by an upper bound, I adjusted it to vary between 0 and 10. 

Best,

Young

[Momo]

Hi Young,

Thank you for the reply! It makes sense to allow higher upper bound for both alpha and tau since in theory they have no upper limit.

My follow-up question is how were the numbers 20 (modified tau upper bound) and 10 (modified alpha upper bound) decided? They look both arbitrary and deliberate at the same time. They look deliberate because I noticed that in the stan code (attached) the initial norm f is hard coded as 10 (ug_delta.stan line 56), which is exactly the 50% point of the endowment of 20 Chinese Yuan in the paper. Were the modified bounds chosen based on task? Or were they decided for some other reasons and the numeric similarity is just coincidence?

I have another question regarding the initial norm as well. According to the paper, they tested both fixed initial norm model and variable initial norm model, and the variable initial norm model was the superior one among all models they tested. Why does ug_delta adopt the less superior fixed initial norm model?

Thank you!

Best,
Momo

[Jeongyeon Shin]

Hi Momo,

Sorry for the delayed response.

Regarding the upper bounds for alpha and tau, they weren't derived from strict theoretical constraints. Rather, they were set heuristically to balance flexibility and numerical stability during model fitting. The similarity between the alpha upper bound (10) and the fixed initial norm value (10) is coincidental.

The fixed-norm was adopted in ug_delta primarily for simplicity and ease of generalization. But we agree that allowing the initial norm to vary would make the model more flexible and better reflect individual differences.

Please let me know if you have any further questions!

Best,

Jeongyeon

2025년 1월 6일 월요일 오전 11시 23분 18초 UTC+9에 Momo님이 작성: