Enke and Shubatt (2023)

Repository Structure

• Jupyter Notebooks:

  • CP Confidence.ipynb: Analysis related to confidence, using subject-level data.
  • CP Confidence No Subject.ipynb: Confidence analysis without subject-specific data.
  • CP Risk.ipynb: Risk analysis using subject-level data.
  • CP Risk No Subject.ipynb: Risk analysis without subject-specific data.
  • Demeaned Confidence.ipynb: Confidence analysis with demeaned data for statistical adjustments.
  • Demeaned Risk.ipynb: Risk analysis with demeaned data.
  • Peterson.ipynb: Analysis or replication related to a study by Peterson (details inside the notebook).
  • Shubatt Replication.ipynb: Replication of a study by Shubatt (details inside the notebook).

• Stata .do File:

  • calc_index.do: A Stata script to calculate complexity indices from Quantifying Lottery Choice Complexity (Enke & Shubatt 2023):

• OPC: Objective Problem Complexity

• SPC: Subjective Problem Complexity

• OAC: Objective Aggregation Complexity

• SAC: Subjective Aggregation Complexity

• OLC: Objective Lottery Complexity

• SLC: Subjective Lottery Complexity

Depending on the Input, the tool calculates these indices and saves them in the output folder, including the necessary features to obtain these indices.

Input

The data I used are provided by Peterson et al., 2021, which could be found at choice13k.

• Max number of lotteries: 2
• Max number of states per lottery: 9
• Payout value is ignored if its probability is 0
• Payouts should be distinct in each lottery
• Problem ID is optional and should be indicated by the column name problem.
• CSV format
• Not existing probabilities and states can be indicated by ,"",, ,"NA", or ,,
• Additional columns can be in the input data and will not be manipulated by code, as long as it does not match the pattern x_, p_, _a_, _a, _b_, _b and cor_

Choice Complexity

If two lotteries are supplied, the column names should be as displayed in the table below. The columns indicating payouts should take the form x_{l}_{i}, where $l \in {a,b}$ indicates which lottery the payout belongs to, and $i$ indexes the lottery states. Similarly, the column names indicating probabilities should take the form p_{l}_{i}. The state index $i$ should take values between 1 and $k_{{l}}$, where $k_{{l}}$ is the maximum number of states in any of the $l$ lotteries. As written, the code can process a maximum of $k_{{l}} = 9$ distinct states. If a payout is repeated in a lottery, they will be treated as separate states. To calculate the complexity-indices in (Enke & Shubatt 2023), we always first collapsed problems to treat same payouts as same state. If both lotteries take on a maximum of two states, then you would include the columns x_a_1, x_a_2, p_a_1, p_a_2, and similarly for $b$.

Lottery Complexity

If only one lottery is supplied, the lottery column names can be either as displayed above (using only the _a_ columns and no _b_ columns). Alternatively, the _a_ segment may be omitted, in which case payoff columns will take the form x_{i} and probability columns will take the form p_{i}. Again, $i$ will range between 0 and $k$, the largest number of distinct states in any lottery; the code can handle a maximum value of $k = 9$. For an example of the correct input format, see sample_just_OLC_SLC_calculation_1.csv or sample_just_OLC_SLC_calculation_2.csv in the sample_data folder.

Output

The results will be saved in output with index_calculated_stata.csv depending on which script you run, including features which are necessary for the calculations. (Additionally .dta, are saved, depending on the executed script).

Choice Complexity

If two lotteries are supplied as above, all 6 indices are automatically calculated. The results are ordered in the CSV as follows: Problem [Optional] | Supplied probabilities and payouts | OPC | SPC | OAC | SAC | OLC_a | SLC_a | OLC_a | SLC_a | then in the same order of the indices the necessary features for their calculations| and in the end any other columns which were also in the input data but not used by the code |. The endings _a or _b at OLC and SLC indicate to which lottery the lottery complexity index is referring. compound is optional.

The features for each index are the following. Please see Section 4 for the development of the indices and appendix Potential Complexity Features for details about feature definition in Quantifying Lottery Choice Complexity.
Consider a choice between two lotteries indexed by $j$ and denoted by letters $A$ and $B$. Each lottery is characterized by payout probabilities $(p_1^j,...p^j_{k_j})$ and payoff $(x_1^j,...x^j_{k_j})$ where $k_j$ denotes the number of distinct payout states of lottery $j$.

Features of OPC and SPC

• Log excess dissimilarity (ln_excess_dissimilarity):
  When $F_A(x)$ and $F_B(x)$ are the CDFs of Lottery A and B with EV(.) indicating the expected value of a lottery then Log excess dissimilarity is defined as

$$log\Big( 1+\int_\mathbb{R} |F_A(x) - F_B(x)|dx - |EV(A) - EV(B)|\Big) $$

• No dominance (no_dominance):
  $$\exists x_1 , x_2: F_A(x_1) < F_B(x_1) \land F_A(x_2) >F_B(x_2) $$
• Average log payout magnitude (ave_ln_payout_magn):
  $$\frac{1}{2} \Big [ log \Big(1 + 1/k_A \sum_{s=1}^{k_A} |x_s^A| \Big) + log \Big (1 + 1/k_B \sum_{s=1}^{k_B} |x_s^B|) \Big) \Big ]$$
• Average log number of states (ave_ln_num_states_a):
  $$\frac{log(1 + k_A) + log(1 + k_B)}{2}$$
• Frac. lotteries involving loss (frac_involves_losses)
  
• If one lottery in the choice is compound (according to definition above)
  
• Absolute expected value difference (abs_ev_diff):
  $$|EV(A) - EV(B)|$$
• Absolute expected value difference squared (abs_ev_diff_sq):
  $$|EV(A) - EV(B)|^2$$

Features of OAC and SAC

As above but without abs_ev_diff and abs_ev_diff_sq features.

Lottery Complexity

If just one lottery is supplied the lottery complexity is calculated (OLC/SLC). In principle, the ordering of the output is the same as for the Choice Complexity output. However, payouts and probabilities are now named x_1, x_2, ... p_1, p_2. Additionally, the features and indices don't have the appendix _a as just one lottery is in the dataset.

Features of OLC_a/b and SLC_a/b

The following defines features for both lotteries indicated with $j\in {A,B }$.

• Log Variance (ln_variance_a/b):
  $$log\Big ( 1+ \sum_{s=1}^{k_j} p_s^j(x_i^j)^2 -( \sum_{s=1}^{k_j} p_s^jx_s^j)^2 \Big)$$

• Log payout magnitude (ln_payout_magn_a/b):
  $$log\Big( 1 + 1/k_j\sum_{s=1}^{k_j}|x_s^j| \Big)$$

• Log number of states (ln_num_states_a/b):
  

$$ log \Big ( 1 + k_j \Big )$$

• 1 if involves loss (involves_loss_a/b)
  
• 1 if involves compound probability(compound)
   

Running the Python Script

See Shubatt Replication.ipynb for the replication code.