Commit 1354223e authored by linushof's avatar linushof
Browse files

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}$$
where $\mathbb{N}$ is the number of comparisons, $x_i$ and $y_j$ are the realizations of the respective random variables, i.e., the single samples, and $N_X$ and $N_Y$ are the numbers of single samples within a comparison.
For the elements of $S$ to be defined, however, the condition
$$\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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