Commit 7969ba18 authored by linushof's avatar linushof
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Probability theoretic definition of sample integration in DfE

parent 14c42a99
......@@ -42,7 +42,7 @@ Provide short summary of simulation results.
## Prospects as Probability Spaces
Let a prospect be a *probability space* $(\Omega, \mathscr{F}, P)$ [@kolmogorovFoundationsTheoryProbability1950; @georgiiStochasticsIntroductionProbability2008, for an more accessible introduction].
Let a prospect be a *probability space* $(\Omega, \mathscr{F}, P)$ [@kolmogorovFoundationsTheoryProbability1950; @georgiiStochasticsIntroductionProbability2008, for an accessible introduction].
$\Omega$ is the *sample space* containing a finite set of possible outcomes
......@@ -64,7 +64,7 @@ $$\begin{equation}
$\mathscr{P}(\Omega)$ denotes the power set of $\Omega$.
$P$ is a *probability mass function* (PMF) which maps the event space to the set of real numbers in $[0, 1]$:
$P$ is a *probability mass function* (PMF) which maps the event space to the set of real numbers in $[0, 1]$
$$\begin{equation}
P: \mathscr{F} \mapsto [0,1]
......@@ -112,15 +112,27 @@ Given this restriction, we define a realization of the described random variable
Because for a sufficiently large number of single samples from a given prospect the relative frequencies of $\omega_{i}$ approximate their probabilities in $p_i \in P$, sampling in principle allows to explore a prospects probability space.
So far, we used the random variable defined on a prospect's triple $(\Omega, \mathscr{F}, P)$ and the restriction $\Omega = \mathscr{F'}$ solely to provide a probability theoretic definition of a single sample.
However, since in the decision literature the stochastic occurrence of the raw outcomes in $\Omega$ is often the event of interest, it should be justified to state that this restricted stochastic model is abundantly but implicitly assumed to underlay the evaluation processes of agents.
However, since in the decision literature the stochastic occurrence of the raw outcomes in $\Omega$ is often treated as the event of interest, it should be justified to state that this restricted stochastic model is abundantly but implicitly assumed to underlay the evaluation processes of agents.
We do not contend that this model is not adequate but rather empirically warranted and mathematically convenient because of the measurable nature of monetary outcomes.
However, in line with the literature that deviates from utility models and its derivatives [@heOntologyDecisionModels2020, for an ontology of decision models], we propose that the above restricted model is not the only suitable for describing the random processes agents are interested in when building a preference between risky prospects.
However, in line with the literature that deviates from utility models and its derivatives [@heOntologyDecisionModels2020, for an ontology of decision models], we propose that the above restricted model is not the only suitable for describing the random processes agents are interested in, when building a preference between risky prospects, from experience or sampling respectively.
How to construct alternative stochastic models underlying choices between risky prospects?
The above definition of a random variable allows us to depart from the standard model of the random process
How to construct alternative stochastic models underlying DfE between risky prospects?
We start from the assumption that agents do not make random choices but base their decisions on the information provided by the prospects, which is readily described by their probability triples.
Thus, we may define a decision variable $D$ as
$$\begin{equation}
D := f((\Omega, \mathscr{F}, P)_j)
\end{equation}$$
irrespective of the information $f$ utilizes and how.
Although in principle many models for $f$ were proposed and tested in the decision literature, in DfE we can restrict the model to the case where decisions are based on single samples generated from the prospect triples.
Although in principle many models for $f$ were proposed and tested in the decision literature, we can use the general (unrestricted) model of the random variable from above as a starting point but now interpret the image $(\Omega', \mathscr{F'})$ as a decision variable.
This allows us to depart from the standard model of the random process
...
use the general (unrestricted) model of the random variable from above as a starting point but now interpret the image $(\Omega', \mathscr{F'})$ as a decision variable.
# Method
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