Utility Theory
****************

.. article-info::
    :avatar: UCLA_Suit.jpg
    :avatar-link: https://www.decisionneurosciencelab.com/elijahgalvan
    :author: Elijah Galvan
    :date: September 1, 2023
    :read-time: 5 min read
    :class-container: sd-p-2 sd-outline-muted sd-rounded-1

.. _Tverksy & Kahneman, 1992: https://link.springer.com/article/10.1007/bf00122574
.. _Levy & Glimcher, 2012: https://pubmed.ncbi.nlm.nih.gov/22766486/
.. _Bartra et al. 2013: https://pubmed.ncbi.nlm.nih.gov/23507394/
.. _Ruff & Fehr, 2014: https://pubmed.ncbi.nlm.nih.gov/24986556/

Assumptions
===========

Utility Theory relies on the following assumptions.

.. dropdown:: 1) People act in a way which maximizes their Subjective Utility
    
    ``Subjective Utility`` - personally experienced satisfaction

    .. dropdown:: Cumulative Prospect Theory (`Tverksy & Kahneman, 1992`_) disproved this, right?

        The short answer is yes. Prospect Theory advanced 4 notions which are problematic for our purposes:

        1. The carrier of value is gains and losses, not net value

        2. Perceptions of probabilities reflect diminishing sensitivity.

        3. Risk attitudes change as a function of probability and outcome (gains or losses)

        4. The framing of choices affects preferences


        Then why do we still endorse this premise?

        1. Prospect Theory advances the notion that people do act in what they *perceive* to be in their best interest, but this perception is warped by several psychological factors - thus, the underlying idea is the same but the mathematical implementation differs.
        
        2. The decision-making tasks used often lead to model predictions are not meaningfully different between a Subjective Utility Theory and Prospect Theory - thus, we ought to adopt the more parsimonious model.

    .. Note:: 
        Although this is suited for most social decision making paradigms we will discuss, we recommend that whenever you use paradigms involving uncertainty, 
        you should strongly consider using the probability weighting function from `Tverksy & Kahneman, 1992`_.

.. dropdown:: 2) Subjective Utility is derived from both economic and social goods 
    
    ``Economic Goods`` - goods which are objectively useful to people but require effort to obtain (money, food, water, shelter, etc.)

    ``Social Goods`` - goods which are subjectively experienced via real or imagined social interactions (fairness, equality, self-validation, etc.)

    .. Note::
        When referencing this, consider citing `Levy & Glimcher, 2012`_, `Bartra et al. 2013`_, and `Ruff & Fehr, 2014`_ who all roughly state that social and economic goods are represented in a common neural currency in the brain.

.. dropdown:: 3) Economic and social goods can be quantified

    Economic goods are already quantified. 

    Social goods are also quantifiable, but via experimental variables. 
    Norm adherance or violation is expressed as a function of the choice (or dependant variable) **and** any situational factors (or indepedent variables).    


Logical Implications
=========

If we adopt these assumptions, it must be true that:

.. dropdown:: 1) Subjective Utility can be estimated via the Decisions people make.

    .. tab-set::

        .. tab-item:: Plain English

            Since we have adopted the premise that people make decisions with the purpose of maximizing their Subjective Utility, 

            then the decision that a person has made **must** have the Highest Subjective Utility out of all of the possible choices that that person could have made.
            
        .. tab-item:: R

            :bdg-primary:`Since` 

            ``Decision = Possible_Choices[Maximizes_Utility]``           
            
             :bdg-primary-line:`where` 

            ``Maximizes_Utility = which(U(Possible_Choices)) == max(U(Possible_Choices))``

            :bdg-secondary:`It Follows that` 
            
            ``U(Decision) == max(U(Possible_Choices))`` 

            .. Note::
                ``U()`` generically refers to a Subjective Utility Function: in other words, the Subjective Utility derived from making a given decision is given as ``U(Decision)``.

        .. tab-item:: MatLab

            :bdg-primary:`Since` 

            ``Decision = Possible_Choices(Maximizes_Utility);``
            
             :bdg-primary-line:`where` 

            ``[~, Maximizes_Utility] = max(U(Possible_Choices)) == max(U(Possible_Choices));``

            :bdg-secondary:`It Follows that` 
            
            ``U(Decision) == max(U(Possible_Choices));``

            .. Note::
                ``U()`` generically refers to a Subjective Utility Function: in other words, the Subjective Utility derived from making a given decision is given as ``U(Decision)``.

        .. tab-item:: Python

            :bdg-primary:`Since` 

            ``Decision = Possible_Choices[Maximizes_Utility]``
            
             :bdg-primary-line:`where` 

            ``import numpy as np``
            ``utilities = U(Possible_Choices)``
            ``max_utility_index = np.argmax(utilities)`` 
            ``Maximizes_Utility = Possible_Choices[max_utility_index]``

            :bdg-secondary:`It Follows that` 
            
            ``U(Decision) == max(utilities)``

            .. Note::
                ``U()`` generically refers to a Subjective Utility Function: in other words, the Subjective Utility derived from making a given decision is given as ``U(Decision)``.


.. dropdown:: 2) Different choices by different people are caused by different preferences for Economic and Social Goods.

    Since we have adopted the assumption that people make decisions which maximize their Subjective Utility, 
    
    if some people make different decisions than others, 
    
    then it **must** be true that they value these goods differently from one another.