# normal_form_game¶

Authors: Tomohiro Kusano, Daisuke Oyama

Tools for normal form games.

## Definitions and Basic Concepts¶

An $$N$$-player normal form game $$g = (I, (A_i)_{i \in I}, (u_i)_{i \in I})$$ consists of

• the set of players $$I = \{0, \ldots, N-1\}$$,
• the set of actions $$A_i = \{0, \ldots, n_i-1\}$$ for each player $$i \in I$$, and
• the payoff function $$u_i \colon A_i \times A_{i+1} \times \cdots \times A_{i+N-1} \to \mathbb{R}$$ for each player $$i \in I$$,

where $$i+j$$ is understood modulo $$N$$. Note that we adopt the convention that the 0-th argument of the payoff function $$u_i$$ is player $$i$$‘s own action and the $$j$$-th argument is player ($$i+j$$)’s action (modulo $$N$$). A mixed action for player $$i$$ is a probability distribution on $$A_i$$ (while an element of $$A_i$$ is referred to as a pure action). A pure action $$a_i \in A_i$$ is identified with the mixed action that assigns probability one to $$a_i$$. Denote the set of mixed actions of player $$i$$ by $$X_i$$. We also denote $$A_{-i} = A_{i+1} \times \cdots \times A_{i+N-1}$$ and $$X_{-i} = X_{i+1} \times \cdots \times X_{i+N-1}$$.

The (pure-action) best response correspondence $$b_i \colon X_{-i} \to A_i$$ for each player $$i$$ is defined by

$b_i(x_{-i}) = \{a_i \in A_i \mid u_i(a_i, x_{-i}) \geq u_i(a_i', x_{-i}) \ \forall\,a_i' \in A_i\},$

where $$u_i(a_i, x_{-i}) = \sum_{a_{-i} \in A_{-i}} u_i(a_i, a_{-i}) \prod_{j=1}^{N-1} x_{i+j}(a_j)$$ is the expected payoff to action $$a_i$$ against mixed actions $$x_{-i}$$. A profile of mixed actions $$x^* \in X_0 \times \cdots \times X_{N-1}$$ is a Nash equilibrium if for all $$i \in I$$ and $$a_i \in A_i$$,

$x_i^*(a_i) > 0 \Rightarrow a_i \in b_i(x_{-i}^*),$

or equivalently, $$x_i^* \cdot v_i(x_{-i}^*) \geq x_i \cdot v_i(x_{-i}^*)$$ for all $$x_i \in X_i$$, where $$v_i(x_{-i})$$ is the vector of player $$i$$‘s payoffs when the opponent players play mixed actions $$x_{-i}$$.

## Creating a NormalFormGame¶

There are three ways to construct a NormalFormGame instance.

The first is to pass an array of payoffs for all the players:

>>> matching_pennies_bimatrix = [[(1, -1), (-1, 1)], [(-1, 1), (1, -1)]]
>>> g = NormalFormGame(matching_pennies_bimatrix)
>>> print(g.players[0])
Player in a 2-player normal form game with payoff array:
[[ 1, -1],
[-1,  1]]
>>> print(g.players[1])
Player in a 2-player normal form game with payoff array:
[[-1,  1],
[ 1, -1]]


If a square matrix (2-dimensional array) is given, then it is considered to be a symmetric two-player game:

>>> coordination_game_matrix = [[4, 0], [3, 2]]
>>> g = NormalFormGame(coordination_game_matrix)
>>> print(g)
2-player NormalFormGame with payoff profile array:
[[[4, 4],  [0, 3]],
[[3, 0],  [2, 2]]]


The second is to specify the sizes of the action sets of the players, which gives a NormalFormGame instance filled with payoff zeros, and then set the payoff values to each entry:

>>> g = NormalFormGame((2, 2))
>>> print(g)
2-player NormalFormGame with payoff profile array:
[[[ 0.,  0.],  [ 0.,  0.]],
[[ 0.,  0.],  [ 0.,  0.]]]
>>> g[0, 0] = 1, 1
>>> g[0, 1] = -2, 3
>>> g[1, 0] = 3, -2
>>> print(g)
2-player NormalFormGame with payoff profile array:
[[[ 1.,  1.],  [-2.,  3.]],
[[ 3., -2.],  [ 0.,  0.]]]


The third is to pass an array of Player instances, as explained in the next section.

## Creating a Player¶

A Player instance is created by passing a payoff array:

>>> player0 = Player([[3, 1], [0, 2]])
>>> player0.payoff_array
array([[3, 1],
[0, 2]])


Passing an array of Player instances is the third way to create a NormalFormGame instance.

>>> player1 = Player([[2, 0], [1, 3]])
>>> player1.payoff_array
array([[2, 0],
[1, 3]])
>>> g = NormalFormGame((player0, player1))
>>> print(g)
2-player NormalFormGame with payoff profile array:
[[[3, 2],  [1, 1]],
[[0, 0],  [2, 3]]]


Beware that in payoff_array[h, k], h refers to the player’s own action, while k refers to the opponent player’s action.

class quantecon.game_theory.normal_form_game.NormalFormGame(data, dtype=None)[source]

Bases: object

Class representing an N-player normal form game.

Parameters: data : array_like of Player, int (ndim=1), or float (ndim=2 or N+1) Data to initialize a NormalFormGame. data may be an array of Players, in which case the shapes of the Players’ payoff arrays must be consistent. If data is an array of N integers, then these integers are treated as the numbers of actions of the N players and a NormalFormGame is created consisting of payoffs all 0 with data[i] actions for each player i. data may also be an (N+1)-dimensional array representing payoff profiles. If data is a square matrix (2-dimensional array), then the game will be a symmetric two-player game where the payoff matrix of each player is given by the input matrix. dtype : data-type, optional(default=None) Relevant only when data is an array of integers. Data type of the players’ payoff arrays. If not supplied, default to numpy.float64.

Attributes

 players (tuple(Player)) Tuple of the Player instances of the game. N (scalar(int)) The number of players. nums_actions (tuple(int)) Tuple of the numbers of actions, one for each player.

Methods

 is_nash(action_profile[, tol]) Return True if action_profile is a Nash equilibrium.
is_nash(action_profile, tol=None)[source]

Return True if action_profile is a Nash equilibrium.

Parameters: action_profile : array_like(int or array_like(float)) An array of N objects, where each object must be an integer (pure action) or an array of floats (mixed action). tol : scalar(float) Tolerance level used in determining best responses. If None, default to each player’s tol attribute value. bool True if action_profile is a Nash equilibrium; False otherwise.
payoff_profile_array
class quantecon.game_theory.normal_form_game.Player(payoff_array)[source]

Bases: object

Class representing a player in an N-player normal form game.

Parameters: payoff_array : array_like(float) Array representing the player’s payoff function, where payoff_array[a_0, a_1, ..., a_{N-1}] is the payoff to the player when the player plays action a_0 while his N-1 opponents play actions a_1, ..., a_{N-1}, respectively.

Attributes

 payoff_array (ndarray(float, ndim=N)) See Parameters. num_actions (scalar(int)) The number of actions available to the player. num_opponents (scalar(int)) The number of opponent players. dtype (dtype) Data type of the elements of payoff_array. tol (scalar(float), default=1e-8) Default tolerance value used in determining best responses.

Methods

 best_response(opponents_actions[, ...]) Return the best response action(s) to opponents_actions. is_best_response(own_action, opponents_actions) Return True if own_action is a best response to opponents_actions. payoff_vector(opponents_actions) Return an array of payoff values, one for each own action, given a profile of the opponents’ actions. random_choice([actions, random_state]) Return a pure action chosen randomly from actions.
best_response(opponents_actions, tie_breaking='smallest', payoff_perturbation=None, tol=None, random_state=None)[source]

Return the best response action(s) to opponents_actions.

Parameters: opponents_actions : scalar(int) or array_like A profile of N-1 opponents’ actions, represented by either scalar(int), array_like(float), array_like(int), or array_like(array_like(float)). If N=2, then it must be a scalar of integer (in which case it is treated as the opponent’s pure action) or a 1-dimensional array of floats (in which case it is treated as the opponent’s mixed action). If N>2, then it must be an array of N-1 objects, where each object must be an integer (pure action) or an array of floats (mixed action). tie_breaking : str, optional(default=’smallest’) str in {‘smallest’, ‘random’, False}. Control how, or whether, to break a tie (see Returns for details). payoff_perturbation : array_like(float), optional(default=None) Array of length equal to the number of actions of the player containing the values (“noises”) to be added to the payoffs in determining the best response. tol : scalar(float), optional(default=None) Tolerance level used in determining best responses. If None, default to the value of the tol attribute. random_state : int or np.random.RandomState, optional Random seed (integer) or np.random.RandomState instance to set the initial state of the random number generator for reproducibility. If None, a randomly initialized RandomState is used. Relevant only when tie_breaking=’random’. scalar(int) or ndarray(int, ndim=1) If tie_breaking=False, returns an array containing all the best response pure actions. If tie_breaking=’smallest’, returns the best response action with the smallest index; if tie_breaking=’random’, returns an action randomly chosen from the best response actions.
is_best_response(own_action, opponents_actions, tol=None)[source]

Return True if own_action is a best response to opponents_actions.

Parameters: own_action : scalar(int) or array_like(float, ndim=1) An integer representing a pure action, or an array of floats representing a mixed action. opponents_actions : see best_response tol : scalar(float), optional(default=None) Tolerance level used in determining best responses. If None, default to the value of the tol attribute. bool True if own_action is a best response to opponents_actions; False otherwise.
payoff_vector(opponents_actions)[source]

Return an array of payoff values, one for each own action, given a profile of the opponents’ actions.

Parameters: opponents_actions : see best_response. payoff_vector : ndarray(float, ndim=1) An array representing the player’s payoff vector given the profile of the opponents’ actions.
random_choice(actions=None, random_state=None)[source]

Return a pure action chosen randomly from actions.

Parameters: actions : array_like(int), optional(default=None) An array of integers representing pure actions. random_state : int or np.random.RandomState, optional Random seed (integer) or np.random.RandomState instance to set the initial state of the random number generator for reproducibility. If None, a randomly initialized RandomState is used. scalar(int) If actions is given, returns an integer representing a pure action chosen randomly from actions; if not, an action is chosen randomly from the player’s all actions.
quantecon.game_theory.normal_form_game.best_response_2p[source]

Numba-optimized version of Player.best_response compilied in nopython mode, specialized for 2-player games (where there is only one opponent).

Return the best response action (with the smallest index if more than one) to opponent_mixed_action under payoff_matrix.

Parameters: payoff_matrix : ndarray(float, ndim=2) Payoff matrix. opponent_mixed_action : ndarray(float, ndim=1) Opponent’s mixed action. Its length must be equal to payoff_matrix.shape[1]. tol : scalar(float), optional(default=None) Tolerance level used in determining best responses. scalar(int) Best response action.
quantecon.game_theory.normal_form_game.pure2mixed(num_actions, action)[source]

Convert a pure action to the corresponding mixed action.

Parameters: num_actions : scalar(int) The number of the pure actions (= the length of a mixed action). action : scalar(int) The pure action to convert to the corresponding mixed action. ndarray(float, ndim=1) The mixed action representation of the given pure action.