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] ...@@ -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]. 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. 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 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. 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. 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} ...@@ -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 $\omega_i \in \Omega = x_i \in \Omega'$. 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]. 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 ## A Stochastical Sampling Model for DfE
...@@ -159,16 +159,23 @@ S = ...@@ -159,16 +159,23 @@ S =
\right\}^{\mathbb{N}} \right\}^{\mathbb{N}}
= =
\left\{ \left\{
\frac{\overline{X}} {\overline{Y}} \frac{\overline{X}_{N_X}} {\overline{Y}_{N_Y}}
\right\}^{\mathbb{N}} \right\}^{\mathbb{N}}
\; , \; ,
\end{equation}$$ \end{equation}$$
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 To indicate that the comparison of prospects on the ordinal scale is of primary interest, we define
$$\begin{equation} $$\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}$$ \end{equation}$$
as a set of subsets of $S$ and the decision variable as the measure as a set of subsets of $S$ and the decision variable as the measure
...@@ -182,21 +189,75 @@ with the mapping ...@@ -182,21 +189,75 @@ with the mapping
$$\begin{equation} $$\begin{equation}
D:= D:=
\left( \left(
\frac{\overline{X}} {\overline{Y}} \frac{\overline{X}_{N_X}} {\overline{Y}_{N_Y}}
\right) \right)
\in S : \in S :
f f
\left( \left(
\frac{\overline{X}} {\overline{Y}} \frac{\overline{X}_{N_X}} {\overline{Y}_{N_Y}}
\right) \right)
= =
\begin{cases} \begin{cases}
1 & if & \frac{\overline{X}}{\overline{Y}} > 0 \in \mathscr{D} \\ 1 & \text{if} & \frac{\overline{X}_{N_X}}{\overline{Y}_{N_Y}} > 0 \in \mathscr{D} \\
0 & if & \frac{\overline{X}}{\overline{Y}} \leq 0 \in \mathscr{D} 0 & \text{else}.
\end{cases} \end{cases}
\; .
\end{equation}$$ \end{equation}$$
## 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 # Method
......
Markdown is supported
0% or .
You are about to add 0 people to the discussion. Proceed with caution.
Finish editing this message first!
Please register or to comment