Predictions of summary strategy

manuscript.Rmd

@@ -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