How to interpret this psychological construct mathematically?

Question

I'm looking for some guidance as to how a psychological construct I have learned about could be rigorously formulated.

The idea is that when a decision maker begins to favor a particular choice early on, this tentative preference often shifts the evaluation of subsequent information in a manner that benefits on the early leader (called predecisional information distortion). There is what is called proleader distortion, a positive distortion of the tentatively leading option, and antitrailer distortion, the tendency to negatatively distort information about the trailing option.

To be clear:

1. Feedback loop:
   • Interpretation of new information biased to favor emerging preference
   • Emerging preference is biased in the direction of interpretation
2. Information arrives and individuals evaluate the information and subconsciously update their preferences.

A "dynamical probability system" is a tuple $S$ $(X,P,F)$ where $S_X$ is a set of outcomes, $X_P$ is a probability distribution over all events in $X$, and $S_F$ is an update rule for $S_P(x_i\in X)$.

As mentioned, we have two probability trajectory systems. The first is the interpretation system($I$) which describes how interpretation probabilities evolve over time, and the second is the ultimate choice system ($C$)which describes how the probability distribution over ultimate choices evolves.

1. We can begin with some initial probability distributions $I_P$ and $C_P$ which should be close to each-other.
2. Stipulate that $\sum{P(x_i)}=1$

3. The probability that any choice will be made is responsive to how information at time $n$ is interpreted.

   • We have some update rule $I_F$ for $I_P$ as a function of $C_P$ and new information corresponding to each choice $x \in C_X$ that arrives the environment.

   • We then use a second update rule $C_F \iff C_P(x \in C_X)_{n+1}=k_iI_P(x_i)-\sum_{j\ne i}{q_jI(x_j)}$ where $1<k_i,0<q_j<1 \in \mathbb{N}$ are, respectively, proleader and antitrailer learning rates relating each $C_P(x \in C_X)$ with each $I_P(x \in I_X)$. This allows for nuanced relationships between all the updates. I hope, but have no idea how to set things up well enough for a proof, that eventually $C_P$ will converge into a long tail.

This seems a little clunky and not well defined. There is no guarantee that $\sum {C_{Pi}}=1$, etc.

I'm not an expert and have not studied probability theory. That said, I will appreciate a simple mathematical formulation of this idea to a great degree, but not too complex a formulation. And I know I may have committed some great fallacy. Making definitions of things is very easy. Really understanding them is hard and rigorous.

Answer 1

Sounds like there are three main players in your framework: (1) the set of the possible choices (with an evolving probability distribution on them), (2) the set of the symbols that are used to convey the incoming information, and (3) some kind of process for "evaluating" a piece of incoming information. Let's try to formulate them mathematically:

(1) A probability distribution $P_{C}$ on the set $C$ of the possible choices $c \in C$. Distribution $P_{C}$ changes with time as the subject's preferences shift. So, what we are really thinking of is a collection of all the possible distributions $P_{C}$.

(2) A finite set $S$ of the possible symbols, which act as the "alphabet" to convey the incoming information. A message conveying a piece of new information is a sequence $s_{1} s_{2} \ldots s_{n}$ of $n$ symbols (taken from $S$). Symbols are allowed to repeat in a message, like letters in a sentence.

(3) The "evaluation", for the purposes of mathematical modeling, is a process whereby the subject receives a pair

(the current $P_{C}$, the current $s_{1} s_{2} \ldots s_{n}$),

and uses that pair to update $P_{C}$. Mathematically, this is a mapping that carries every such pair to some (generally new) $P_{C}$.

Time proceeds in discrete time steps. At each step, the subject receives a new message and updates $P_{C}$.

Your psychological statement can now be made mathematical: If two choices $c_{1}, c_{2}$ are such that the preference $P_{C}(c_{1}) \geq P_{C}(c_{2})$ (i.e., subject favors $c_{1}$ over $c_{2}$) holds currently, then it continues to hold at all later time steps.

The above mapping need to be such that this statement holds no matter what $P_{C}$ we start with.

Answer 2

Coincidentally, I just read this question about a robot that flips coins and thought that it might provide an insight.

Someone walks into your room and dumps a huge bag of quarters all over the floor. They spread them out so no quarters are on top of any other quarters. a robot then comes into the room and is programmed such that if it sees a head, it flips it to tails. If it sees a tail, it throws it in the air.

My thinking is that the robot resembles a decision maker who has developed a bias towards tails. Think of the decisionmaker receiving a sequence of coins;

• if it is tails (aligning with their bias) they accept it and move onto the next packet of information.
• If it is heads (against their bias) they reject it.

The modification to your application could be if the decisionmaker has a slightly different rule:

• If it is tails I will accept with choice probability $p_{\text{Accept T}}$ otherwise reject (and new coin flip)
• If it is head I will accept with choice probability $p_{\text{Accept H}}$ otherwise reject (and new coin flip).

Those choice probabilities would initially both start at 1/2, but would adjust based on the coins that they initially see. In particular, if they were calculated from the proportion of tails observed in a sample. My conjecture is that any local variations in the sample proportion would cause the choice probabilities to move away from 1/2.