Tools for Game Theory¶

Daisuke Oyama
Faculty of Economics, University of Tokyo

This notebook demonstrates the functionalities of the game_theory module.

In [1]:
import numpy as np
from quantecon.game_theory import NormalFormGame, Player


Normal Form Games¶

An $N$-player normal form game is a triplet $g = (I, (A_i)_{i \in I}, (u_i)_{i \in I})$ where

• $I = \{0, \ldots, N-1\}$ is the set of players,
• $A_i = \{0, \ldots, n_i-1\}$ is the set of actions of player $i \in I$, and
• $u_i \colon A_i \times A_{i+1} \times \cdots \times A_{i+N-1} \to \mathbb{R}$ is the payoff function of 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, $j = 1, \ldots, N-1$, is player ($i+j$)'s action (modulo $N$).

In our module, a normal form game and a player are represented by the classes NormalFormGame and Player, respectively.

A Player carries the player's payoff function and implements in particular a method that returns the best response action(s) given an action of the opponent player, or a profile of actions of the opponents if there are more than one.

A NormalFormGame is in effect a container of Player instances.

Creating a NormalFormGame¶

There are several ways to create a NormalFormGame instance.

The first is to pass an array of payoffs for all the players, i.e., an $(N+1)$-dimenstional array of shape $(n_0, \ldots, n_{N-1}, N)$ whose $(a_0, \ldots, a_{N-1})$-entry contains an array of the $N$ payoff values for the action profile $(a_0, \ldots, a_{N-1})$.

As an example, consider the following game ("Matching Pennies"):

$\begin{bmatrix} 1, -1 & -1, 1 \\ -1, 1 & 1, -1 \end{bmatrix}$

In [2]:
matching_pennies_bimatrix = [[(1, -1), (-1, 1)],
[(-1, 1), (1, -1)]]
g_MP = NormalFormGame(matching_pennies_bimatrix)

In [3]:
print(g_MP)

2-player NormalFormGame with payoff profile array:
[[[ 1, -1],  [-1,  1]],
[[-1,  1],  [ 1, -1]]]

In [4]:
print(g_MP.players[0])  # Player instance for player 0

Player in a 2-player normal form game with payoff array:
[[ 1, -1],
[-1,  1]]

In [5]:
print(g_MP.players[1])  # Player instance for player 1

Player in a 2-player normal form game with payoff array:
[[-1,  1],
[ 1, -1]]

In [6]:
g_MP.players[0].payoff_array  # Player 0's payoff array

Out[6]:
array([[ 1, -1],
[-1,  1]])
In [7]:
g_MP.players[1].payoff_array  # Player 1's payoff array

Out[7]:
array([[-1,  1],
[ 1, -1]])
In [8]:
g_MP[0, 0]  # payoff profile for action profile (0, 0)

Out[8]:
array([ 1, -1])

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

Consider the following game (symmetric $2 \times 2$ "Coordination Game"):

$\begin{bmatrix} 4, 4 & 0, 3 \\ 3, 0 & 2, 2 \end{bmatrix}$

In [9]:
coordination_game_matrix = [[4, 0],
[3, 2]]  # square matrix
g_Coo = NormalFormGame(coordination_game_matrix)

In [10]:
print(g_Coo)

2-player NormalFormGame with payoff profile array:
[[[4, 4],  [0, 3]],
[[3, 0],  [2, 2]]]

In [11]:
g_Coo.players[0].payoff_array  # Player 0's payoff array

Out[11]:
array([[4, 0],
[3, 2]])
In [12]:
g_Coo.players[1].payoff_array  # Player 1's payoff array

Out[12]:
array([[4, 0],
[3, 2]])

Another example ("Rock-Paper-Scissors"):

$\begin{bmatrix} 0, 0 & -1, 1 & 1, -1 \\ 1, -1 & 0, 0 & -1, 1 \\ -1, 1 & 1, -1 & 0, 0 \end{bmatrix}$

In [13]:
RPS_matrix = [[ 0, -1,  1],
[ 1,  0, -1],
[-1,  1,  0]]
g_RPS = NormalFormGame(RPS_matrix)

In [14]:
print(g_RPS)

2-player NormalFormGame with payoff profile array:
[[[ 0,  0],  [-1,  1],  [ 1, -1]],
[[ 1, -1],  [ 0,  0],  [-1,  1]],
[[-1,  1],  [ 1, -1],  [ 0,  0]]]


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

Let us construct the following game ("Prisoners' Dilemma"):

$\begin{bmatrix} 1, 1 & -2, 3 \\ 3, -2 & 0, 0 \end{bmatrix}$

In [15]:
g_PD = NormalFormGame((2, 2))  # There are 2 players, each of whom has 2 actions
g_PD[0, 0] = 1, 1
g_PD[0, 1] = -2, 3
g_PD[1, 0] = 3, -2
g_PD[1, 1] = 0, 0

In [16]:
print(g_PD)

2-player NormalFormGame with payoff profile array:
[[[ 1.,  1.],  [-2.,  3.]],
[[ 3., -2.],  [ 0.,  0.]]]


Finally, a NormalFormGame instance can be constructed by giving an array of Player instances, as explained in the next section.

Creating a Player¶

A Player instance is created by passing an array of dimension $N$ that represents the player's payoff function ("payoff array").

Consider the following game (a variant of "Battle of the Sexes"):

$\begin{bmatrix} 3, 2 & 1, 1 \\ 0, 0 & 2, 3 \end{bmatrix}$

In [17]:
player0 = Player([[3, 1],
[0, 2]])
player1 = Player([[2, 0],
[1, 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.

In [18]:
player0.payoff_array

Out[18]:
array([[3, 1],
[0, 2]])
In [19]:
player1.payoff_array

Out[19]:
array([[2, 0],
[1, 3]])

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

In [20]:
g_BoS = NormalFormGame((player0, player1))

In [21]:
print(g_BoS)

2-player NormalFormGame with payoff profile array:
[[[3, 2],  [1, 1]],
[[0, 0],  [2, 3]]]


More than two players¶

The game_theory module also supports games with more than two players.

Let us consider the following version of $N$-player Cournot Game.

There are $N$ firms (players) which produce a homogeneous good with common constant marginal cost $c \geq 0$. Each firm $i$ simultaneously determines the quantity $q_i \geq 0$ (action) of the good to produce. The inverse demand function is given by the linear function $P(Q) = a - Q$, $a > 0$, where $Q = q_0 + \cdots + q_{N-1}$ is the aggregate supply. Then the profit (payoff) for firm $i$ is given by $$u_i(q_i, q_{i+1}, \ldots, q_{i+N-1}) = P(Q) q_i - c q_i = \left(a - c - \sum_{j \neq i} q_j - q_i\right) q_i.$$

Theoretically, the set of actions, i.e., available quantities, may be the set of all nonnegative real numbers $\mathbb{R}_+$ (or a bounded interval $[0, \bar{q}]$ with some upper bound $\bar{q}$), but for computation on a computer we have to discretize the action space and only allow for finitely many grid points.

The following script creates a NormalFormGame instance of the Cournot game as described above, assuming that the (common) grid of possible quantity values is stored in an array q_grid.

In [22]:
from quantecon import cartesian

def cournot(a, c, N, q_grid):
"""
Create a NormalFormGame instance for the symmetric N-player Cournot game
with linear inverse demand a - Q and constant marginal cost c.

Parameters
----------
a : scalar
Intercept of the demand curve

c : scalar
Common constant marginal cost

N : scalar(int)
Number of firms

q_grid : array_like(scalar)
Array containing the set of possible quantities

Returns
-------
NormalFormGame
NormalFormGame instance representing the Cournot game

"""
q_grid = np.asarray(q_grid)
payoff_array = \
cartesian([q_grid]*N).sum(axis=-1).reshape([len(q_grid)]*N) * (-1) + \
(a - c)
payoff_array *= q_grid.reshape([len(q_grid)] + [1]*(N-1))
payoff_array += 0  # To get rid of the minus sign of -0

player = Player(payoff_array)
return NormalFormGame([player for i in range(N)])


Here's a simple example with three firms, marginal cost $20$, and inverse demand function $80 - Q$, where the feasible quantity values are assumed to be $10$ and $15$.

In [23]:
a, c = 80, 20
N = 3
q_grid = [10, 15]  # [1/3 of Monopoly quantity, Nash equilibrium quantity]

g_Cou = cournot(a, c, N, q_grid)

In [24]:
print(g_Cou)

3-player NormalFormGame with payoff profile array:
[[[[300, 300, 300],   [250, 250, 375]],
[[250, 375, 250],   [200, 300, 300]]],

[[[375, 250, 250],   [300, 200, 300]],
[[300, 300, 200],   [225, 225, 225]]]]

In [25]:
print(g_Cou.players[0])

Player in a 3-player normal form game with payoff array:
[[[300, 250],
[250, 200]],

[[375, 300],
[300, 225]]]

In [26]:
g_Cou.nums_actions

Out[26]:
(2, 2, 2)

Nash Equilibrium¶

A Nash equilibrium of a normal form game is a profile of actions where the action of each player is a best response to the others'.

The Player object has a method best_response.

Consider the Matching Pennies game g_MP defined above. For example, player 0's best response to the opponent's action 1 is:

In [27]:
g_MP.players[0].best_response(1)

Out[27]:
1

Player 0's best responses to the opponent's mixed action [0.5, 0.5] (we know they are 0 and 1):

In [28]:
# By default, returns the best response action with the smallest index
g_MP.players[0].best_response([0.5, 0.5])

Out[28]:
0
In [29]:
# With tie_breaking='random', returns randomly one of the best responses
g_MP.players[0].best_response([0.5, 0.5], tie_breaking='random')  # Try several times

Out[29]:
0
In [30]:
# With tie_breaking=False, returns an array of all the best responses
g_MP.players[0].best_response([0.5, 0.5], tie_breaking=False)

Out[30]:
array([0, 1])

For this game, we know that ([0.5, 0.5], [0.5, 0.5]) is a (unique) Nash equilibrium.

In [31]:
g_MP.is_nash(([0.5, 0.5], [0.5, 0.5]))

Out[31]:
True
In [32]:
g_MP.is_nash((0, 0))

Out[32]:
False
In [33]:
g_MP.is_nash((0, [0.5, 0.5]))

Out[33]:
False

Finding Nash equilibria¶

Our module does not have sophisticated algorithms to compute Nash equilibria (yet)... One might look at Gambit, which implements several such algorithms.

Brute force¶

For small games, we can find pure action Nash equilibria by brute force.

In [34]:
def find_pure_nash_brute(g):
"""
Find all pure Nash equilibria of a normal form game by brute force.

Parameters
----------
g : NormalFormGame

"""
NEs = []
for a in np.ndindex(*g.nums_actions):
if g.is_nash(a):
NEs.append(a)
num_NEs = len(NEs)
if num_NEs == 0:
msg = 'no pure Nash equilibrium'
elif num_NEs == 1:
msg = '1 pure Nash equilibrium:\n{0}'.format(NEs)
else:
msg = '{0} pure Nash equilibria:\n{1}'.format(num_NEs, NEs)

print('The game has ' + msg)


Matching Pennies:

In [35]:
find_pure_nash_brute(g_MP)

The game has no pure Nash equilibrium


Coordination game:

In [36]:
find_pure_nash_brute(g_Coo)

The game has 2 pure Nash equilibria:
[(0, 0), (1, 1)]


Rock-Paper-Scissors:

In [37]:
find_pure_nash_brute(g_RPS)

The game has no pure Nash equilibrium


Battle of the Sexes:

In [38]:
find_pure_nash_brute(g_BoS)

The game has 2 pure Nash equilibria:
[(0, 0), (1, 1)]


Prisoners' Dillema:

In [39]:
find_pure_nash_brute(g_PD)

The game has 1 pure Nash equilibrium:
[(1, 1)]


Cournot game:

In [40]:
find_pure_nash_brute(g_Cou)

The game has 1 pure Nash equilibrium:
[(1, 1, 1)]


Sequential best response¶

In some games, such as "supermodular games" and "potential games", the process of sequential best responses converges to a Nash equilibrium.

Here's a script to find one pure Nash equilibrium by sequential best response, if it converges.

In [41]:
def sequential_best_response(g, init_actions=None, tie_breaking='smallest',
verbose=True):
"""
Find a pure Nash equilibrium of a normal form game by sequential best
response.

Parameters
----------
g : NormalFormGame

init_actions : array_like(int), optional(default=[0, ..., 0])
The initial action profile.

tie_breaking : {'smallest', 'random'}, optional(default='smallest')

verbose: bool, optional(default=True)
If True, print the intermediate process.

"""
N = g.N  # Number of players
a = np.empty(N, dtype=int)  # Action profile
if init_actions is None:
init_actions = [0] * N
a[:] = init_actions

if verbose:
print('init_actions: {0}'.format(a))

new_a = np.empty(N, dtype=int)
max_iter = np.prod(g.nums_actions)

for t in range(max_iter):
new_a[:] = a
for i, player in enumerate(g.players):
if N == 2:
a_except_i = new_a[1-i]
else:
a_except_i = new_a[np.arange(i+1, i+N) % N]
new_a[i] = player.best_response(a_except_i,
tie_breaking=tie_breaking)
if verbose:
print('player {0}: {1}'.format(i, new_a))
if np.array_equal(new_a, a):
return a
else:
a[:] = new_a

print('No pure Nash equilibrium found')
return None


A Cournot game with linear demand is known to be a potential game, for which sequential best response converges to a Nash equilibrium.

Let us try a bigger instance:

In [42]:
a, c = 80, 20
N = 3
q_grid = np.linspace(0, a-c, 13)  # [0, 5, 10, ..., 60]
g_Cou = cournot(a, c, N, q_grid)

In [43]:
a_star = sequential_best_response(g_Cou)  # By default, start with (0, 0, 0)
print('Nash equilibrium indices: {0}'.format(a_star))
print('Nash equilibrium quantities: {0}'.format(q_grid[a_star]))

init_actions: [0 0 0]
player 0: [6 0 0]
player 1: [6 3 0]
player 2: [6 3 1]
player 0: [4 3 1]
player 1: [4 3 1]
player 2: [4 3 2]
player 0: [3 3 2]
player 1: [3 3 2]
player 2: [3 3 3]
player 0: [3 3 3]
player 1: [3 3 3]
player 2: [3 3 3]
Nash equilibrium indices: [3 3 3]
Nash equilibrium quantities: [ 15.  15.  15.]

In [44]:
# Start with the largest actions (12, 12, 12)
sequential_best_response(g_Cou, init_actions=(12, 12, 12))

init_actions: [12 12 12]
player 0: [ 0 12 12]
player 1: [ 0  0 12]
player 2: [0 0 6]
player 0: [3 0 6]
player 1: [3 1 6]
player 2: [3 1 4]
player 0: [3 1 4]
player 1: [3 2 4]
player 2: [3 2 3]
player 0: [3 2 3]
player 1: [3 3 3]
player 2: [3 3 3]
player 0: [3 3 3]
player 1: [3 3 3]
player 2: [3 3 3]

Out[44]:
array([3, 3, 3])

The limit action profile is indeed a Nash equilibrium:

In [45]:
g_Cou.is_nash(a_star)

Out[45]:
True

In fact, the game has other Nash equilibria (because of our choice of grid points and parameter values):

In [46]:
find_pure_nash_brute(g_Cou)

The game has 7 pure Nash equilibria:
[(2, 3, 4), (2, 4, 3), (3, 2, 4), (3, 3, 3), (3, 4, 2), (4, 2, 3), (4, 3, 2)]


Make it bigger:

In [47]:
N = 4
q_grid = np.linspace(0, a-c, 61)  # [0, 1, 2, ..., 60]
g_Cou = cournot(a, c, N, q_grid)

In [48]:
sequential_best_response(g_Cou)

init_actions: [0 0 0 0]
player 0: [30  0  0  0]
player 1: [30 15  0  0]
player 2: [30 15  7  0]
player 3: [30 15  7  4]
player 0: [17 15  7  4]
player 1: [17 16  7  4]
player 2: [17 16 11  4]
player 3: [17 16 11  8]
player 0: [12 16 11  8]
player 1: [12 14 11  8]
player 2: [12 14 13  8]
player 3: [12 14 13 10]
player 0: [11 14 13 10]
player 1: [11 13 13 10]
player 2: [11 13 13 10]
player 3: [11 13 13 11]
player 0: [11 13 13 11]
player 1: [11 12 13 11]
player 2: [11 12 13 11]
player 3: [11 12 13 12]
player 0: [11 12 13 12]
player 1: [11 12 13 12]
player 2: [11 12 12 12]
player 3: [11 12 12 12]
player 0: [12 12 12 12]
player 1: [12 12 12 12]
player 2: [12 12 12 12]
player 3: [12 12 12 12]
player 0: [12 12 12 12]
player 1: [12 12 12 12]
player 2: [12 12 12 12]
player 3: [12 12 12 12]

Out[48]:
array([12, 12, 12, 12])
In [49]:
sequential_best_response(g_Cou, init_actions=(0, 0, 0, 30))

init_actions: [ 0  0  0 30]
player 0: [15  0  0 30]
player 1: [15  7  0 30]
player 2: [15  7  4 30]
player 3: [15  7  4 17]
player 0: [16  7  4 17]
player 1: [16 11  4 17]
player 2: [16 11  8 17]
player 3: [16 11  8 12]
player 0: [14 11  8 12]
player 1: [14 13  8 12]
player 2: [14 13 10 12]
player 3: [14 13 10 11]
player 0: [13 13 10 11]
player 1: [13 13 10 11]
player 2: [13 13 11 11]
player 3: [13 13 11 11]
player 0: [12 13 11 11]
player 1: [12 13 11 11]
player 2: [12 13 12 11]
player 3: [12 13 12 11]
player 0: [12 13 12 11]
player 1: [12 12 12 11]
player 2: [12 12 12 11]
player 3: [12 12 12 12]
player 0: [12 12 12 12]
player 1: [12 12 12 12]
player 2: [12 12 12 12]
player 3: [12 12 12 12]

Out[49]:
array([12, 12, 12, 12])

Sequential best response does not converge in all games:

In [50]:
print(g_MP)  # Matching Pennies

2-player NormalFormGame with payoff profile array:
[[[ 1, -1],  [-1,  1]],
[[-1,  1],  [ 1, -1]]]

In [51]:
sequential_best_response(g_MP)

init_actions: [0 0]
player 0: [0 0]
player 1: [0 1]
player 0: [1 1]
player 1: [1 0]
player 0: [0 0]
player 1: [0 1]
player 0: [1 1]
player 1: [1 0]
No pure Nash equilibrium found

In [ ]: