docs/documentation/valuing_actions/index.rst

Valuing actions
================

Once you've :doc:`collected the data </documentation/data/index>` and
:doc:`converted it to the SPADL format </documentation/spadl/index>`, you can
start valuing the contributions of soccer players. This document gives
a general introduction to action-valuing frameworks and links to a detailed
discussion of the three implemented frameworks.

General idea
------------

When considering event stream data, a soccer match can be viewed as a sequence
of :math:`n` consecutive on-the-ball actions :math:`\left[a_1, a_2, \ldots, a_n\right]`
(e.g., [*pass*, *dribble*,..., *interception*]). Action-valuing frameworks aim
to assign a numeric value to each of these individual actions that quantifies
how much the action contributed towards winning the game. This value should
reflect both the circumstances under which it was performed as well as its
longer-term effects. This is illustrated in the figure below:

.. image:: ../../actions_bra-bel.png
   :width: 600
   :alt: a sequence of actions with action values
   :align: center

However, rather than directly assigning values to actions, the existing
approaches all start by assigning values to game states. To illustrate the
underlying intuition, consider the pass below:

.. image:: action.gif
   :alt: example action
   :align: center

|

The effect of the pass was to change the game state:

.. image:: action_changes_gamestate.png
   :alt: example action changes gamestate
   :align: center

|

The figure on the left shows the game in state :math:`S_{i−1}
= \{a_1,\dots,a_{i−1}\}`, right before Benzema passes to Valverde and the one
on the right shows the game in state :math:`S_i = \{a_1, \ldots, a_{i−1},
a_i\}` just after Valverde successfully controlled the pass.

Consequently, a natural way to assess the usefulness of an action is to assign
a value to each game state. Then an action’s usefulness is simply the
difference between the post-action game state :math:`S_i` and pre-action game
state :math:`S_{i-1}`. This can be expressed as:

.. math::
  U(a_i) = V(S_i) - V(S_{i-1}),

where :math:`V` captures the value of a particular game state.

The differences between different action-valuing frameworks arise in terms of
(1) how they represent a game state :math:`S_i`, that is, define features such
as the ball's location or score difference that capture relevant aspects of
the game at a specific point in time; and (2) assign a value :math:`V` to
a specific game state.


Implemented frameworks
----------------------

The socceraction package implements three frameworks to assess the impact of the
individual actions performed by soccer players: Expected Threat (xT), VAEP and
Atomic-VAEP.

.. toctree::

  xT
  vaep
  atomic_vaep