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<p>The attractiveness parameter, which determines the elevation of the weighting function.</p>
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<p>The attractiveness parameter, which determines the elevation of the weighting function in <code>prelec</code> and <code>gw</code> weighting functions. In <code>tk</code>, it is the probability weighting for losses. Defaults to <code>gamma</code> if not provided.</p>
<p>An optional utility function that takes the magnitude, alpha, and lambda_loss, and returns the utilities of each choice options. The default is a power utility function, see Notes.</p>
<p>where beta is the discriminability parameter of the weighting function;
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alpha_pos and alpha_neg are the risk attitude parameters in the gain and loss domains respectively,
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and lambda is the loss aversion parameter.</p>
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<divclass="arithmatex">\[
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\mathcal{U} = \sum_{i=1}^{n} w(p_i) \cdot u(x_i)
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\]</div>
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<p>where <spanclass="arithmatex">\(w\)</span> is a weighting function of the probability p of a potential outcome,
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and <spanclass="arithmatex">\(u\)</span> is the utility function of the magnitude x of a potential outcome.
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The utility function <spanclass="arithmatex">\(u\)</span> is defined as a power function for both gains and losses. It is implemented
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after Equation 5 in Tversky & Kahneman (1992):</p>
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<divclass="arithmatex">\[
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u(x) =
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\begin{cases}
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x^\alpha & \text{if } x \geq 0 \\
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-\lambda \cdot (-x)^\alpha & \text{if } x < 0
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\end{cases}
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\]</div>
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<p>where <spanclass="arithmatex">\(\alpha\)</span> is the utility curvature parameter, and <spanclass="arithmatex">\(\lambda\)</span> is the loss aversion parameter.
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The weighting function is implemented after Equation 6 in Tversky & Kahneman (1992):</p>
<p>where <code>gamma</code>, denoted via <spanclass="arithmatex">\(\gamma\)</span>, is the discriminability parameter of the weighting function.
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In the original formulation of Tversky & Kahneman (1992), losses are weighted with a different parameter,
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<code>delta</code>, denoted via <spanclass="arithmatex">\(\delta\)</span>, that replaces <spanclass="arithmatex">\(\gamma\)</span> in the weighting function for losses.
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In the current implementation, whether it is a gain or less is determined by the sign of the corresponding
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magnitude.</p>
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<p>Several other definitions of the weighting function have been proposed in the literature,
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most notably in Prelec (1998) and Gonzalez & Wu (1999).
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Prelec (equation 3.2, 1998, pp. 503) proposed the following definition:</p>
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<pre><code>w(p) = exp(-delta * (-log(p))^beta),
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</code></pre>
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<p>where delta and beta are the attractiveness and discriminability parameters of the weighting function.
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<divclass="arithmatex">\[
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w(p) = \exp(-\delta \cdot (-\log(p))^\gamma)
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\]</div>
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<p>where <code>delta</code>, <spanclass="arithmatex">\(\delta\)</span>, and <code>gamma</code>, <spanclass="arithmatex">\(\gamma\)</span>, are the attractiveness and discriminability parameters of the weighting function.
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Gonzalez & Wu (equation 3, 1999, pp. 139) proposed the following definition:</p>
<p>Gonzalez, R., & Wu, G. (1999). On the shape of the probability weighting function. Cognitive psychology, 38(1), 129-166.</p>
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<p>Kahneman, D., & Tversky, A. (1979). Prospect theory: An analysis of decision under risk. <em>Econometrica</em>, 47(2), 263–291.</p>
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<p>Prelec, D. (1998). The probability weighting function. Econometrica, 497-527.</p>
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<p>Tversky, A., & Kahneman, D. (1992). Advances in prospect theory: Cumulative representation of uncertainty. Journal of Risk and uncertainty, 5, 297-323.</p>
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