Commit 7ad48031 authored by linushof's avatar linushof
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Review of probability theoretic definitions

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# Gambles and Prospects
# Prospects
A gamble represents a choice between two or more prospects. Let a prospect be a *probability space* $(\Omega, P)$ and $\Omega$ the *sample space* containing a finite set of possible outcomes $\{\omega_1, ..., \omega_n\}$ [cf. @kolmogorovFoundationsTheoryProbability1950]. P is then a *probability mass function* (PMF) $P: \Omega \mapsto [0,1]$ which assigns each outcome $\omega_i$ a probability of $0 \leq p_i \leq 1$ with $\sum_{i=1}^{n} p(\omega_i) = 1$.
Let a prospect be a *probability space* $(\Omega, P)$ and $\Omega$ the *sample space* containing a finite set of possible outcomes $\{\omega_1, ..., \omega_n\}$ [cf. @kolmogorovFoundationsTheoryProbability1950]. P is then a *probability mass function* (PMF) $P: \Omega \mapsto [0,1]$ which assigns each outcome $\omega_i$ a probability of $0 \leq p_i \leq 1$ with $\sum_{i=1}^{n} p(\omega_i) = 1$.
## Example
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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 prospect's probability space.
So far, we used the probability triple of a prospect and conditions a. and b. 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 treated as the event of interest, it should be justified to state that stochastic model formulated under a. with the restriction b. is abundantly although implicitly assumed to underlie 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 say that the stochastic model formulated under a. with the restriction b. is abundantly although implicitly assumed to underlie the evaluation processes of agents.
We do not contend that this model is not adequate but rather empirically warranted and mathematically convenient, not least because of the measurable nature of the monetary outcomes in $\Omega$.
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 sampling respectively.
We can construct an alternative stochastic sampling model (hereafter SSM) underlying DfE between risky prospects by starting 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 start by defining a decision variable $D$
Thus, we may start rather abstractly by defining a decision variable $D$
$$\begin{equation}
D := f((\Omega, \mathscr{F}, P)_j)
\end{equation}$$
first without any further assumption on which information of the triple $f$ utilizes and how.
first without any further assumption on which information of the probability triple $f$ utilizes and how.
Although in principle many models for $f$ are proposed and tested in the decision literature, in DfE we can restrict the SSM to the case where decisions are based on sequences of single samples generated from the prospect triples.
Since we have already provided a restricted stochastic model of a random variable for this generative process, we can write
Since we have defined the stochastic mechanism for generating such sequences, we write
$$\begin{equation}
D := f((X: (\Omega, \mathscr{F}) \mapsto (\Omega', \mathscr{F'}))_j)
\end{equation}$$
where $\Omega_j = \mathscr{F_j} = \mathscr{F_j'}$.
where $\Omega_j = \Omega'_j$.
For n prospects we can write the above definition as
For n prospects, we write
$$\begin{equation}
D := f(X_1, ..., X_j, ..., X_n)
\end{equation}$$
which reduces to
$$\begin{equation}
D := f(X_1, X_2)
\end{equation}$$
in the case of $n = 2$ prospects, which we now consider further.
Up to this point, we have defined the decision variable $D$ as a function of the random variables associated with the prospects probability spaces.
In summary, we have defined the decision variable $D$ as a function of the random variables associated with the prospects probability spaces.
In the following, we consider the case of a choice between two prospects, their respective random variables we denote as $X$ and $Y$.
Since the decision variable $D$ serves as a measure for the evidence for one prospect over the other, we want $f$ to be a measurable function that maps the comparison of $X_1$ and $X_2$ to the measure space $\mathscr{D}$.
Because $X_1$ and $X_2$ are themselves measurable, we write their sample means as a fraction
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