### Predictions of summary strategy

parent 40ea6b02
 ... ... @@ -70,7 +70,7 @@ P: \mathscr{F} \mapsto [0,1] that assigns each $\omega_i \in \Omega$ a probability of $0 \leq p_i \leq 1$ with $P(\Omega) = 1$ [cf. @kolmogorovFoundationsTheoryProbability1950, pp. 2-3]. In such a choice paradigm, agents are asked to evaluate the prospects and build a preference for either one of them. It is common to make a rather crude distinction between two variants of this evaluation process [cf. @hertwigDescriptionExperienceGap2009]. It is common to make a rather crude distinction between two variants of this evaluation process [cf. @hertwigDescriptionexperienceGapRisky2009]. For decisions from description (DfD), agents are provided a full symbolic description of the triples $(\Omega, \mathscr{F}, P)_j$, where j denotes a prospect. For decisions from experience [DfE; e.g., @hertwigDecisionsExperienceEffect2004], the probability triples are not described but must be explored by the means of sampling. To provide a formal definition of sampling in risky or uncertain choice, we make use of the mathematical concept of a random variable. ... ... @@ -103,7 +103,7 @@ A'_i \in \mathscr{F'} \Rightarrow X^{-1}A'_i \in \mathscr{F} B) The image $X: \Omega \mapsto \Omega'$ must be such that $\omega_i \in \Omega = x_i \in \Omega'$. Given conditions A and B, we denote any realization of a random variable defined on the triple $(\Omega, \mathscr{F}, P)$ as a *"single sample"* of the respective prospect and any systematic approach to generate a sequence of single samples from multiple prospects as a sampling strategy [see also @hillsInformationSearchDecisions2010]. Because for a sufficiently large number of single samples *n* from a given prospect, i.e., $\lim_{n \to \infty}$, the relative frequencies of $\omega_{i}$ approximate their probabilities in $p_i \in P$ [@bernoulliOpusPosthumumAccedit1713], sampling in principle allows to explore a prospect's probability space. Because for a sufficiently large number of single samples *n* from a given prospect, i.e., $\lim_{n \to \infty}$, the relative frequencies of $\omega_{i}$ approximate their probabilities in $p_i \in P$ [@bernoulliArsConjectandiOpus1713], sampling in principle allows to explore a prospect's probability space. ## A Stochastical Sampling Model for DfE ... ... @@ -159,16 +159,23 @@ S = \right\}^{\mathbb{N}} = \left\{ \frac{\overline{X}} {\overline{Y}} \frac{\overline{X}_{N_X}} {\overline{Y}_{N_Y}} \right\}^{\mathbb{N}} \; , \end{equation}$$where \mathbb{N} is the number of comparisons, x_i and y_j are the realizations of the respective random variables, i.e., the single samples, and N_X and N_Y are the numbers of single samples within a comparison. where \mathbb{N} is the number of comparisons, x_i and y_j are the realizations of the respective random variables, i.e., the single samples, and N_X and N_Y are the numbers of single samples within a comparison. For the elements of S to be defined, however, the condition$$\begin{equation} P(\overline{Y} = 0) = 0 \; . \end{equation}$$must be fulfilled. To indicate that the comparison of prospects on the ordinal scale is of primary interest, we define$$\begin{equation} \mathscr{D} = \left\{\frac{\overline{X}}{\overline{Y}} > 0, \frac{\overline{X}}{\overline{Y}} \leq 0 \right\} \mathscr{D} = \left\{\frac{\overline{X}_{N_X}}{\overline{Y}_{N_Y}} > 0, \frac{\overline{X}_{N_X}}{\overline{Y}_{N_Y}} \leq 0 \right\} \end{equation}$$as a set of subsets of S and the decision variable as the measure ... ... @@ -182,21 +189,75 @@ with the mapping$$\begin{equation} D:= \left( \frac{\overline{X}} {\overline{Y}} \frac{\overline{X}_{N_X}} {\overline{Y}_{N_Y}} \right) \in S : f \left( \frac{\overline{X}} {\overline{Y}} \frac{\overline{X}_{N_X}} {\overline{Y}_{N_Y}} \right) = \begin{cases} 1 & if & \frac{\overline{X}}{\overline{Y}} > 0 \in \mathscr{D} \\ 0 & if & \frac{\overline{X}}{\overline{Y}} \leq 0 \in \mathscr{D} 1 & \text{if} & \frac{\overline{X}_{N_X}}{\overline{Y}_{N_Y}} > 0 \in \mathscr{D} \\ 0 & \text{else}. \end{cases} \; . \end{equation}$$## Predicting Choices From the SSM Hills and Hertwig [-@hillsInformationSearchDecisions2010] proposed the two different sampling strategies in combination with the respective decision strategies, i.e., piecewise sampling and round-wise comparison vs. comprehensive sampling and summary comparison, as an explanation for different choice patterns in DfE. How does the current version of the SSM support this proposition? Given prospects X and Y, the sample spaces S = \left\{\frac{\overline{X}_{N_X}} {\overline{Y}_{N_Y}}\right\}^{\mathbb{N}} can be varied by changes on three parameters, i.e., the number of comparisons \mathbb{N} and the sample sizes N_X and N_Y on which these comparisons are based. First, only considering the pure cases formulated by the above authors, the following restrictions are put the parameters:$$\begin{equation} \mathbb{N} = \begin{cases} 1 & \text{if} & \text{Summary} \\ \geq 1 & \text{if} & \text{Round-wise} \end{cases}\\ \end{equation}$$and$$\begin{equation} N_X \, \text{and} \, N_Y = \begin{cases} \geq 1 & \text{if} & \text{Summary} \\ 1 & \text{if} & \text{Round-wise} \end{cases}\\ \end{equation}$$For the summary strategy, the following prediction is obtained: Given that$$\begin{equation} P\left(\lim_{N_X \to \infty} \overline{X}_{N_X} = E_X \right) = P\left(\lim_{N_Y \to \infty} \overline{Y}_{N_Y} = E_X \right) = 1 \; , \end{equation}$$we obtain that$$\begin{equation} \left( \frac{\overline{X}_{N_X}} {\overline{Y}_{N_Y}} \right) \in S : P\left(\lim_{N_X \to \infty} \lim_{N_Y \to \infty} \frac{\overline{X}_{N_X}} {\overline{Y}_{N_Y}} = \frac{E(X)} {E(Y)} \right ) = 1 \; , \end{equation} I.e., for the summary strategy, we assume that for increasing sample sizes $N_X$ and $N_Y$, the prospect with the larger expected value is chosen almost surely. For the round-wise strategy, the following prediction is obtained: # Method ... ...
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