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Utility Function Models

utility.Rd

Fits utility models.

  • utility_pow_c() fits a power utility for continuous responses.

  • utility_pow_d() fits a power utility for discrete respoonses.

Usage

utility_pow_d(
  formula,
  data,
  choicerule,
  fix = list(),
  discount = 0,
  options = list(),
  ...
)

utility_pow_c(formula, data, fix = list(), discount = 0, options = list(), ...)

Arguments

formula

A formula, the variables in data to be modeled. For example, y ~ x1 models response y as function of one stimulus value x1.

data

A data frame, the data to be modeled.

choicerule

A string, the choice rule. Allowed values, see cm_choicerules(): "none" is no choice rule, "softmax" is soft-maximum, "luce" is Luce's axiom.

fix

(optional) A list with parameter-value pairs of fixed parameters. If missing all free parameters are estimated. If set to "start" all parameters are fixed to their start values. Model parameter names are rp, rn (see details - model parameters).

  • list(rp = 5.40) sets parameter rp equal to 5.40.

  • list(rp = "rn") sets parameter rp equal to parameter rn (estimates rn).

  • list(rn = "rp", rp = 5.40) sets parameter rn equal to parameter rp and sets rp equal to 5.40 (estimates none of the two).

  • list(rp = NA) omits the parameter rp, if possible.

  • "start" sets all parameters equal to their initial values (estimates none). Useful for building a first test model.

discount

A number, how many initial trials to not use during parameter fitting.

options

(optional) A list, list entries change the modeling procedure. For example, list(lb = c(k=0)) changes the lower bound of parameter k to 0, or list(fit_measure = "mse") changes the goodness of fit measure in parameter estimation to mean-squared error, for all options, see cm_options.

...

other arguments, ignored.

Value

Returns a cognitive model object, which is an object of class cm. A model, that has been assigned to m, can be summarized with summary(m) or anova(m). The parameter space can be viewed using pa. rspace(m), constraints can be viewed using constraints(m).

Details

The power utility \(U(x)\) of positive inputs, \(x > 0\), is \(x^r\) if \(r > 0\), and is \(log(x)\) if \(r = 0\), and is \(-x^r\) if \(r < 0\). The power utility of negative inputs \(x\) is \(-U(-x)\) with a separate exponent r (Wakker, 2008). To fit a power utility with one single exponent for positive and negative x, set fix = list(rp = "rn"), not recommended for mixed input.

Model Parameters

The model has between 1 and 3 free parameters, depending on model and data (see npar()):

  • rp is the power utility exponent for positive data \(x \ge\) 0 (omitted if all \(x <\) 0).

  • rn is the exponent for negative data \(x < 0\) (omitted if all \(x \ge\) 0).

  • In utility_pow_c(): sigma is the standard deviation of the normally-distributed loglikelihood of the responses.

  • In utility_pow_d(): If choicerule = "softmax": tau is the temperature or choice softness, higher values cause more equiprobable choices. If choicerule = "epsilon": eps is the error proportion, higher values cause more errors from maximizing.

References

Wakker, P. P. (2008). Explaining the characteristics of the power (CRRA) utility family. Health Economics, 17(12), 1329-1344. doi:10.1002/hec.1331

Tversky, A. (1967). Utility theory and additivity analysis of risky choices. Journal of Experimental Psychology, 75(1), 27-36. doi:10.1037/h0024915

See also

Other cognitive models: baseline_const_c(), bayes(), choicerules, cpt, ebm(), hm1988(), shift(), shortfall, threshold()

Author

Jana B. Jarecki, jj@janajarecki.com

Examples

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