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tucoopy.transforms.algebra

Algebraic transformations of TU games.

The functions in this module build new games from an existing one by applying simple algebraic operations to the characteristic function:

  • scale_game: multiply all coalition values by a scalar.
  • shift_game: add a constant (with TU conventions preserved).
  • affine_game: combine scale and shift.

These are useful for normalization, unit changes, and testing linearity / scale-equivariance properties of solution concepts.

affine_game 

affine_game(game, factor, constant)

Apply an affine transformation to the characteristic function of a TU game.

The new game is defined by

\[ (a v + b)(S) = \begin{cases} 0, & S = \varnothing \\ a \, v(S) + b, & S \neq \varnothing \end{cases} \]

which combines scaling and translation while preserving the TU convention.

Parameters:

Name Type Description Default
game Game

Original TU game.

 required
factor float

Multiplicative factor \(a\).

 required
constant float

Additive constant \(b\).

 required

Returns:

Type Description
Game

New affine-transformed game.
Notes

Affine transformations define an equivalence class of TU games that share the same strategic structure.

Most central solution concepts are invariant (up to scaling) under affine transformations. Studying games up to affine equivalence is standard in cooperative game theory, particularly in:

  • normalization procedures,
  • theoretical characterizations of values,
  • geometric analysis of the core and related sets.

Examples:

>>> g2 = affine_game(g, 2.0, 3.0)
>>> g2.value(0)
0.0

scale_game 

scale_game(game, factor)

Scale a TU game by a multiplicative factor.

This transformation defines a new game whose characteristic function is

\[ (a v)(S) = a \, v(S), \]

for every coalition \(S\), with the TU convention preserved:

\[ (a v)(\varnothing) = 0. \]

Parameters:

Name Type Description Default
game Game

Original TU game.

 required
factor float

Scaling factor \(a\).

 required

Returns:

Type Description
Game

New scaled game.
Notes

Many solution concepts are scale-equivariant. For example,

  • Shapley value
  • Nucleolus
  • Kernel
  • Core

all scale proportionally when the game is scaled.

This operation is useful for normalization and for comparing games under different units (e.g., cost vs revenue).

Examples:

>>> g2 = scale_game(g, 2.0)
>>> g2.value(g2.grand_coalition) == 2 * g.value(g.grand_coalition)
True

shift_game 

shift_game(game, constant)

Add a constant to all non-empty coalitions of a TU game.

This transformation defines a new game

\[ (v + c)(S) = \begin{cases} 0, & S = \varnothing \\ v(S) + c, & S \neq \varnothing \end{cases} \]

preserving the TU requirement \(v(\varnothing)=0\).

Parameters:

Name Type Description Default
game Game

Original TU game.

 required
constant float

Additive constant \(c\).

 required

Returns:

Type Description
Game

New shifted game.
Notes

This is an affine translation of the game.

Many solution concepts are translation-invariant, meaning their payoff allocations do not change under this transformation. This includes:

  • Shapley value
  • Nucleolus
  • Kernel
  • Core membership

This transformation is central when studying equivalent games and normalizations such as zero-normalized games.

Examples:

>>> g2 = shift_game(g, 5.0)
>>> g2.value(0)  # empty coalition remains 0
0.0
>>> g2.value(g2.grand_coalition) == g.value(g.grand_coalition) + 5
True