Abstract
This study identifies a new optimal decision rule in a general pairwise choice framework, taking into account behavioral aspects where subjective probabilities are assumed. The optimal rule under such a setting is compared to the one identified where behavioral considerations are not incorporated into the model.
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Notes
Individuals might have different “degrees” of strategic considerations [see Baharad and Nitzan (2002)]. For identifying the optimal decision rule see, for example, Baharad and Nitzan (2007), Ben-Yashar and Danziger (2011), Ben-Yashar et al. (2001), Ben-Yashar and Kraus (2002), Ben-Yashar and Nitzan (1997, 1998), Buchanan and Tullock (1962), Dietrich and List (2013), Ladha (1995), Nitzan and Paroush (1982) and Sah and Stiglitz (1986, 1988).
Baharad and Nitzan (2008) consider these behavioral aspects in an economic context of contests.
In this context, we use the term "subjective probability" to describe the distorted probabilities presented in behavioral findings. This should not be confused with the concept of subjective probabilities in the context of subjective expected utility maximization introduced by Savage (1954) and his followers.
Clearly, \((1 - p_{i}^{1} )\) and \((1 - p_{i}^{2} )\) can be interpreted as the two types of errors probabilities that are associated with individual i's decision.
Under our setting we do not assume different γ values for gains and losses. For such a symmetric setting see Tversky and Wakker (1995).
For consistency, when possible, we use the same notations as Ben-Yashar and Nitzan (1997).
A similar problem for objective probabilities appears in Ben-Yashar and Nitzan (1997).
Note that in the symmetric case w(α) = w(1−α), which implies that α = ½ yet w(α) < ½.
This is due to monotonicity of the weighting function w(pi) with respect to pi under the relevant values of γ (see Table 1 in Baharad and Kliger (2013) for a short literature review of estimated γ values, that were obtained by experiments).
The proof of this claim is based on the following condition: \(\frac{\beta (x) + \beta (y)}{2}\mathop = \limits_{ < }^{ > } 0 \Leftrightarrow x + y\mathop = \limits_{ < }^{ > } 1\). This latter condition is proved in Ben-Yashar and Nitzan (1997).
The effect of the individual's decision on the collective decision is represented by: \(\gamma \frac{{\beta (p_{i}^{1} ) + \beta (p_{i}^{2} )}}{2}x_{i} + \gamma \psi_{i} + \frac{{1}}{\gamma }\theta_{i} .\) Suppose that \(\gamma \frac{{\beta (p_{i}^{1} ) + \beta (p_{i}^{2} )}}{2}\) is positive. Thus, if \(\gamma \psi_{i} + \frac{{1}}{\gamma }\theta_{i}\) is positive as well, the individual is more influential when xi = 1, comparing to the case when xi = -1 (that is \(\left| {\gamma \frac{{\beta (p_{i}^{1} ) + \beta (p_{i}^{2} )}}{2} + \gamma \psi_{i} + \frac{{1}}{\gamma }\theta_{i} } \right|\) and \(\left| { - \gamma \frac{{\beta (p_{i}^{1} ) + \beta (p_{i}^{2} )}}{2} + \gamma \psi_{i} + \frac{{1}}{\gamma }\theta_{i} } \right|\), respectively). Note that in the absence of the bias factor, \(\gamma \psi_{i} + \frac{{1}}{\gamma }\theta_{i}\) = 0, the individual affects the collective decision equally when xi = 1 and xi = -1.
That is \( \alpha = 1/2{\text{,}}\,B(1) = B( - 1),p^{1} = p^{2} \).
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Baharad, E., Ben-Yashar, R. Judgment Aggregation by a Boundedly Rational Decision-Maker. Group Decis Negot 30, 903–914 (2021). https://doi.org/10.1007/s10726-021-09740-3
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DOI: https://doi.org/10.1007/s10726-021-09740-3