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
import quantecon.game_theory as gt

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 = gt.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 = gt.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 = gt.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 = gt.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 = gt.Player([[3, 1],
                     [0, 2]])
player1 = gt.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 = gt.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 = gt.Player(payoff_array)
    return gt.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

There are several algorithms implemented to compute Nash equilibria:

  • Brute force
    Find all pure-action Nash equilibria of an $N$-player game (if any).
  • Sequential best response
    Find one pure-action Nash equilibrium of an $N$-player game (if any).
  • Support enumeration
    Find all mixed-action Nash equilibria of a two-player nondegenerate game.
  • Lemke-Howson
    Find one mixed-action Nash equilibrium of a two-player game.
  • McLennan-Tourky
    Find one mixed-action Nash equilibrium of an $N$-player game.

For more variety of algorithms, one should look at Gambit.

Brute force

For small games, we can find pure action Nash equilibria by brute force, by calling the routine pure_nash_brute. It visits all the action profiles and check whether each is a Nash equilibrium by the is_nash method.

In [34]:
def print_pure_nash_brute(g):
    """
    Print all pure Nash equilibria of a normal form game found by brute force.
    
    Parameters
    ----------
    g : NormalFormGame
    
    """
    NEs = gt.pure_nash_brute(g)
    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]:
print_pure_nash_brute(g_MP)
The game has no pure Nash equilibrium

Coordination game:

In [36]:
print_pure_nash_brute(g_Coo)
The game has 2 pure Nash equilibria:
[(0, 0), (1, 1)]

Rock-Paper-Scissors:

In [37]:
print_pure_nash_brute(g_RPS)
The game has no pure Nash equilibrium

Battle of the Sexes:

In [38]:
print_pure_nash_brute(g_BoS)
The game has 2 pure Nash equilibria:
[(0, 0), (1, 1)]

Prisoners' Dillema:

In [39]:
print_pure_nash_brute(g_PD)
The game has 1 pure Nash equilibrium:
[(1, 1)]

Cournot game:

In [40]:
print_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]:
print_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

Support enumeration

The routine support_enumeration, which is for two-player games, visits all equal-size support pairs and checks whether each pair has a Nash equilibrium (in mixed actions) by the indifference condition. (This should thus be used only for small games.) For nondegenerate games, this routine returns all Nash equilibria.

Matching Pennies:

In [52]:
gt.support_enumeration(g_MP)
Out[52]:
[(array([ 0.5,  0.5]), array([ 0.5,  0.5]))]

The output list contains a pair of mixed actions as a tuple of two NumPy arrays, which constitues the unique Nash equilibria of this game.

Coordination game:

In [53]:
print(g_Coo)
2-player NormalFormGame with payoff profile array:
[[[4, 4],  [0, 3]],
 [[3, 0],  [2, 2]]]
In [54]:
gt.support_enumeration(g_Coo)
Out[54]:
[(array([ 1.,  0.]), array([ 1.,  0.])),
 (array([ 0.,  1.]), array([ 0.,  1.])),
 (array([ 0.66666667,  0.33333333]), array([ 0.66666667,  0.33333333]))]

The output contains three tuples of mixed actions, where the first two correspond to the two pure action equilibria, while the last to the unique totally mixed action equilibrium.

Rock-Paper-Scissors:

In [55]:
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]]]
In [56]:
gt.support_enumeration(g_RPS)
Out[56]:
[(array([ 0.33333333,  0.33333333,  0.33333333]),
  array([ 0.33333333,  0.33333333,  0.33333333]))]

Consider the $6 \times 6$ game by von Stengel (1997), page 12:

In [57]:
player0 = gt.Player(
    [[    9504,     -660,    19976,   -20526,     1776,    -8976],
     [ -111771,    31680,  -130944,   168124,    -8514,    52764],
     [  397584,  -113850,   451176,  -586476,    29216,  -178761],
     [  171204,   -45936,   208626,  -263076,    14124,   -84436],
     [ 1303104,  -453420,  1227336, -1718376,    72336,  -461736],
     [  737154,  -227040,   774576, -1039236,    48081,  -300036]]
)

player1 = gt.Player(
    [[   72336,  -461736,  1227336, -1718376,  1303104,  -453420],
     [   48081,  -300036,   774576, -1039236,   737154,  -227040],
     [   29216,  -178761,   451176,  -586476,   397584,  -113850],
     [   14124,   -84436,   208626,  -263076,   171204,   -45936],
     [    1776,    -8976,    19976,   -20526,     9504,     -660],
     [   -8514,    52764,  -130944,   168124,  -111771,    31680]]
)

g_vonStengel = gt.NormalFormGame((player0, player1))
In [58]:
len(gt.support_enumeration(g_vonStengel))
Out[58]:
75

Note that the $n \times n$ game where the payoff matrices are given by the identy matrix has $2^n−1$ equilibria. It had been conjectured that this is the maximum number of equilibria of any nondegenerate $n \times n$ game. The game above, the number of whose equilibria is $75 > 2^6 - 1 = 63$, was presented as a counter-example to this conjecture.

Next, let us study the All-Pay Acution, where, unlike standard auctions, bidders pay their bids regardless of whether or not they win. Situations modeled as all-pay auctions include job promotion, R&D, and rent seeking competitions, among others.

Here we consider a version of All-Pay Auction with complete information, symmetric bidders, discrete bids, bid caps, and "full dissipation" (where the prize is materialized if and only if there is only one bidder who makes a highest bid).

Specifically, each of $N$ players simultaneously bids an integer from $\{0, 1, \ldots, c\}$, where $c$ is the common (integer) bid cap. If player $i$'s bid is higher than all the other players', then he receives the prize, whose value is $r$, common to all players, and pays his bid $b_i$. Otherwise, he pays $b_i$ and receives nothing (zero value). In particular, if there are more than one players who make the highest bid, the prize gets fully dissipated and all the players receive nothing. Thus, player $i$'s payoff function is $$ u_i(b_i, b_{i+1}, \ldots, b_{i+N-1}) = \begin{cases} r - b_i & \text{if $b_i > b_j$ for all $j \neq i$}, \\ - b_i & \text{otherwise}. \end{cases} $$

The following is a script to construct a NormalFormGame instance for the All-Pay Auction game, where we use Numba to speed up the loops:

In [59]:
from numba import jit


def all_pay_auction(r, e, N, dissipation=True):
    """
    Create a `NormalFormGame` instance for the symmetric N-player
    All-Pay Auction game with common reward `r` and common bid cap `e`.

    Parameters
    ----------
    r : scalar(float)
        Common reward value.

    e : scalar(int)
        Common bid cap.

    N : scalar(int)
        Number of players.

    dissipation : bool, optional(default=True)
        If True, the prize fully dissipates in case of a tie. If False,
        the prize is equally split among the highest bidders (or given
        to one of the highest bidders with equal probabilities).

    Returns
    -------
    NormalFormGame
        NormalFormGame instance representing the All-Pay Auction game.

    """
    player = gt.Player(np.empty((e+1,)*N))
    populate_APA_payoff_array(r, dissipation, player.payoff_array)
    return gt.NormalFormGame((player,)*N)


@jit(nopython=True)
def populate_APA_payoff_array(r, dissipation, out):
    """
    Populate the payoff array for a player in an N-player All-Pay
    Auction game.

    Parameters
    ----------
    r : scalar(float)
        Reward value.

    dissipation : bool, optional(default=True)
        If True, the prize fully dissipates in case of a tie. If False,
        the prize is equally split among the highest bidders (or given
        to one of the highest bidders with equal probabilities).

    out : ndarray(float, ndim=N)
        NumPy array to store the payoffs. Modified in place.

    Returns
    -------
    out : ndarray(float, ndim=N)
        View of `out`.

    """
    nums_actions = out.shape
    N = out.ndim
    for bids in np.ndindex(nums_actions):
        out[bids] = -bids[0]
        num_ties = 1
        for j in range(1, N):
            if bids[j] > bids[0]:
                num_ties = 0
                break
            elif bids[j] == bids[0]:
                if dissipation:
                    num_ties = 0
                    break
                else:
                    num_ties += 1
        if num_ties > 0:
            out[bids] += r / num_ties
    return out

Consider the two-player case with the following parameter values:

In [60]:
N = 2
e = 5  # odd
r = 8
In [61]:
g_APA_odd = all_pay_auction(r, e, N)
print(g_APA_odd)
2-player NormalFormGame with payoff profile array:
[[[ 0.,  0.],  [ 0.,  7.],  [ 0.,  6.],  [ 0.,  5.],  [ 0.,  4.],  [ 0.,  3.]],
 [[ 7.,  0.],  [-1., -1.],  [-1.,  6.],  [-1.,  5.],  [-1.,  4.],  [-1.,  3.]],
 [[ 6.,  0.],  [ 6., -1.],  [-2., -2.],  [-2.,  5.],  [-2.,  4.],  [-2.,  3.]],
 [[ 5.,  0.],  [ 5., -1.],  [ 5., -2.],  [-3., -3.],  [-3.,  4.],  [-3.,  3.]],
 [[ 4.,  0.],  [ 4., -1.],  [ 4., -2.],  [ 4., -3.],  [-4., -4.],  [-4.,  3.]],
 [[ 3.,  0.],  [ 3., -1.],  [ 3., -2.],  [ 3., -3.],  [ 3., -4.],  [-5., -5.]]]

Clearly, this game has no pure-action Nash equilibrium. Indeed:

In [62]:
gt.pure_nash_brute(g_APA_odd)
Out[62]:
[]

As pointed out by Dechenaux et al. (2006), there are three Nash equilibria when the bid cap e is odd (so that there are an even number of actions for each player):

In [63]:
gt.support_enumeration(g_APA_odd)
Out[63]:
[(array([ 0.5 ,  0.  ,  0.25,  0.  ,  0.25,  0.  ]),
  array([ 0.  ,  0.25,  0.  ,  0.25,  0.  ,  0.5 ])),
 (array([ 0.  ,  0.25,  0.  ,  0.25,  0.  ,  0.5 ]),
  array([ 0.5 ,  0.  ,  0.25,  0.  ,  0.25,  0.  ])),
 (array([ 0.125,  0.125,  0.125,  0.125,  0.125,  0.375]),
  array([ 0.125,  0.125,  0.125,  0.125,  0.125,  0.375]))]

In addition to a symmetric, totally mixed equilibrium (the third), there are two asymmetric, "alternating" equilibria (the first and the second).

If e is even, there is a unique equilibrium, which is symmetric and totally mixed. For example:

In [64]:
e = 6  # even
g_APA_even = all_pay_auction(r, e, N)
gt.support_enumeration(g_APA_even)
Out[64]:
[(array([ 0.125,  0.125,  0.125,  0.125,  0.125,  0.125,  0.25 ]),
  array([ 0.125,  0.125,  0.125,  0.125,  0.125,  0.125,  0.25 ]))]

Lemke-Howson

The routine lemke_howson implements the Lemke-Howson algorithm (Lemke and Howson 1964), which returns one Nash equilibrium of a two-player normal form game. For the details of the algorithm, see, e.g., von Stengel (2007).

Matching Pennies:

In [65]:
gt.lemke_howson(g_MP)
Out[65]:
(array([ 0.5,  0.5]), array([ 0.5,  0.5]))

Coordination game:

In [66]:
gt.lemke_howson(g_Coo)
Out[66]:
(array([ 1.,  0.]), array([ 1.,  0.]))

The initial pivot can be specified by init_pivot, which should be an integer $k$ such that $0 \leq k \leq n_1 + n_2 - 1$ (default to 0), where $0, \ldots, n_1-1$ correspond to player 0's actions, while $n_1 \ldots, n_1+n_2-1$ to player 1's actions.

In [67]:
gt.lemke_howson(g_Coo, init_pivot=1)
Out[67]:
(array([ 0.,  1.]), array([ 0.,  1.]))

All-Pay Auction:

In [68]:
gt.lemke_howson(g_APA_odd, init_pivot=0)
Out[68]:
(array([ 0.5 ,  0.  ,  0.25,  0.  ,  0.25,  0.  ]),
 array([ 0.  ,  0.25,  0.  ,  0.25,  0.  ,  0.5 ]))
In [69]:
gt.lemke_howson(g_APA_odd, init_pivot=1)
Out[69]:
(array([ 0.  ,  0.25,  0.  ,  0.25,  0.  ,  0.5 ]),
 array([ 0.5 ,  0.  ,  0.25,  0.  ,  0.25,  0.  ]))

Additional information is returned if the option full_output is set True:

In [70]:
NE, res = gt.lemke_howson(g_APA_odd, full_output=True)
res
Out[70]:
        NE: (array([ 0.5 ,  0.  ,  0.25,  0.  ,  0.25,  0.  ]), array([ 0.  ,  0.25,  0.  ,  0.25,  0.  ,  0.5 ]))
 converged: True
      init: 0
  max_iter: 1000000
  num_iter: 6

lemke_howson runs fast, with a reasonable time amount for games with up to several hundreds actions. (In fact, this is the only routine among the Nash equilibrium computation routines in the game_theory submodule that scales to large-size games.) For example:

In [71]:
N = 2
e = 200  # 201 actions
r = 500
g_APA200 = all_pay_auction(r, e, N)
In [72]:
gt.lemke_howson(g_APA200)
Out[72]:
(array([ 0.002,  0.002,  0.002,  0.002,  0.002,  0.002,  0.002,  0.002,
         0.002,  0.002,  0.002,  0.002,  0.002,  0.002,  0.002,  0.002,
         0.002,  0.002,  0.002,  0.002,  0.002,  0.002,  0.002,  0.002,
         0.002,  0.002,  0.002,  0.002,  0.002,  0.002,  0.002,  0.002,
         0.002,  0.002,  0.002,  0.002,  0.002,  0.002,  0.002,  0.002,
         0.002,  0.002,  0.002,  0.002,  0.002,  0.002,  0.002,  0.002,
         0.002,  0.002,  0.002,  0.002,  0.002,  0.002,  0.002,  0.002,
         0.002,  0.002,  0.002,  0.002,  0.002,  0.002,  0.002,  0.002,
         0.002,  0.002,  0.002,  0.002,  0.002,  0.002,  0.002,  0.002,
         0.002,  0.002,  0.002,  0.002,  0.002,  0.002,  0.002,  0.002,
         0.002,  0.002,  0.002,  0.002,  0.002,  0.002,  0.002,  0.002,
         0.002,  0.002,  0.002,  0.002,  0.002,  0.002,  0.002,  0.002,
         0.002,  0.002,  0.002,  0.002,  0.002,  0.002,  0.002,  0.002,
         0.002,  0.002,  0.002,  0.002,  0.002,  0.002,  0.002,  0.002,
         0.002,  0.002,  0.002,  0.002,  0.002,  0.002,  0.002,  0.002,
         0.002,  0.002,  0.002,  0.002,  0.002,  0.002,  0.002,  0.002,
         0.002,  0.002,  0.002,  0.002,  0.002,  0.002,  0.002,  0.002,
         0.002,  0.002,  0.002,  0.002,  0.002,  0.002,  0.002,  0.002,
         0.002,  0.002,  0.002,  0.002,  0.002,  0.002,  0.002,  0.002,
         0.002,  0.002,  0.002,  0.002,  0.002,  0.002,  0.002,  0.002,
         0.002,  0.002,  0.002,  0.002,  0.002,  0.002,  0.002,  0.002,
         0.002,  0.002,  0.002,  0.002,  0.002,  0.002,  0.002,  0.002,
         0.002,  0.002,  0.002,  0.002,  0.002,  0.002,  0.002,  0.002,
         0.002,  0.002,  0.002,  0.002,  0.002,  0.002,  0.002,  0.002,
         0.002,  0.002,  0.002,  0.002,  0.002,  0.002,  0.002,  0.002,  0.6  ]),
 array([ 0.002,  0.002,  0.002,  0.002,  0.002,  0.002,  0.002,  0.002,
         0.002,  0.002,  0.002,  0.002,  0.002,  0.002,  0.002,  0.002,
         0.002,  0.002,  0.002,  0.002,  0.002,  0.002,  0.002,  0.002,
         0.002,  0.002,  0.002,  0.002,  0.002,  0.002,  0.002,  0.002,
         0.002,  0.002,  0.002,  0.002,  0.002,  0.002,  0.002,  0.002,
         0.002,  0.002,  0.002,  0.002,  0.002,  0.002,  0.002,  0.002,
         0.002,  0.002,  0.002,  0.002,  0.002,  0.002,  0.002,  0.002,
         0.002,  0.002,  0.002,  0.002,  0.002,  0.002,  0.002,  0.002,
         0.002,  0.002,  0.002,  0.002,  0.002,  0.002,  0.002,  0.002,
         0.002,  0.002,  0.002,  0.002,  0.002,  0.002,  0.002,  0.002,
         0.002,  0.002,  0.002,  0.002,  0.002,  0.002,  0.002,  0.002,
         0.002,  0.002,  0.002,  0.002,  0.002,  0.002,  0.002,  0.002,
         0.002,  0.002,  0.002,  0.002,  0.002,  0.002,  0.002,  0.002,
         0.002,  0.002,  0.002,  0.002,  0.002,  0.002,  0.002,  0.002,
         0.002,  0.002,  0.002,  0.002,  0.002,  0.002,  0.002,  0.002,
         0.002,  0.002,  0.002,  0.002,  0.002,  0.002,  0.002,  0.002,
         0.002,  0.002,  0.002,  0.002,  0.002,  0.002,  0.002,  0.002,
         0.002,  0.002,  0.002,  0.002,  0.002,  0.002,  0.002,  0.002,
         0.002,  0.002,  0.002,  0.002,  0.002,  0.002,  0.002,  0.002,
         0.002,  0.002,  0.002,  0.002,  0.002,  0.002,  0.002,  0.002,
         0.002,  0.002,  0.002,  0.002,  0.002,  0.002,  0.002,  0.002,
         0.002,  0.002,  0.002,  0.002,  0.002,  0.002,  0.002,  0.002,
         0.002,  0.002,  0.002,  0.002,  0.002,  0.002,  0.002,  0.002,
         0.002,  0.002,  0.002,  0.002,  0.002,  0.002,  0.002,  0.002,
         0.002,  0.002,  0.002,  0.002,  0.002,  0.002,  0.002,  0.002,  0.6  ]))

McLennan-Tourky

The routine mclennan_tourky computes one approximate Nash equilibrium of an $N$-player normal form game by the fixed-point algorithm of McLennan and Tourky (2006) applied to the best response correspondence

Consider the symmetric All-Pay Auction with full dissipation as above, but this time with three players:

In [73]:
N = 3
r = 16
e = 5
g_APA3 = all_pay_auction(r, e, N)

Run mclennan_tourky:

In [74]:
NE = gt.mclennan_tourky(g_APA3)
NE
Out[74]:
(array([ 0.24986561,  0.10338418,  0.07959071,  0.06705212,  0.05909946,
         0.44100792]),
 array([ 0.24986561,  0.10338418,  0.07959071,  0.06705212,  0.05909946,
         0.44100792]),
 array([ 0.24986561,  0.10338418,  0.07959071,  0.06705212,  0.05909946,
         0.44100792]))

This output is an $\varepsilon$-Nash equilibrium of the game, which is a profile of mixed actions $(x^*_0, \ldots, x^*_{N-1})$ such that for all $i$, $u_i(x^*_i, x^*_{-i}) \geq u_i(x_i, x^*_{-i}) - \varepsilon$ for all $x_i$, where the value of $\varepsilon$ is specified by the option epsilon (default to 1e-3).

In [75]:
g_APA3.is_nash(NE, tol=1e-3)
Out[75]:
True
In [76]:
epsilon = 1e-4
NE = gt.mclennan_tourky(g_APA3, epsilon=epsilon)
NE
Out[76]:
(array([ 0.2500157 ,  0.1035519 ,  0.07944073,  0.06699354,  0.05902238,
         0.44097575]),
 array([ 0.2500157 ,  0.1035519 ,  0.07944073,  0.06699354,  0.05902238,
         0.44097575]),
 array([ 0.2500157 ,  0.1035519 ,  0.07944073,  0.06699354,  0.05902238,
         0.44097575]))
In [77]:
g_APA3.is_nash(NE, tol=epsilon)
Out[77]:
True

Additional information is returned by setting the full_output option to True:

In [78]:
NE, res = gt.mclennan_tourky(g_APA3, full_output=True)
res
Out[78]:
        NE: (array([ 0.24986561,  0.10338418,  0.07959071,  0.06705212,  0.05909946,
        0.44100792]), array([ 0.24986561,  0.10338418,  0.07959071,  0.06705212,  0.05909946,
        0.44100792]), array([ 0.24986561,  0.10338418,  0.07959071,  0.06705212,  0.05909946,
        0.44100792]))
 converged: True
   epsilon: 0.001
      init: (0, 0, 0)
  max_iter: 200
  num_iter: 127

For this game, mclennan_tourky returned a symmetric, totally mixed action profile (cf. Rapoport and Amaldoss 2004) with the default initial condition (0, 0, 0) (profile of pure actions).

Let's try a different initial condition:

In [79]:
init = (
    [1/2, 0, 0, 1/2, 0, 0], [0, 1/2, 0, 0, 1/2, 0], [0, 0, 0, 0, 0, 1]
)  # profile of mixed actions
gt.mclennan_tourky(g_APA3, init=init)
Out[79]:
(array([ 0.86582198,  0.        ,  0.06955033,  0.        ,  0.06462769,  0.        ]),
 array([ 0.86582198,  0.        ,  0.06955033,  0.        ,  0.06462769,  0.        ]),
 array([ 0.        ,  0.14389389,  0.        ,  0.12286532,  0.        ,
         0.73324079]))

We obtained an asymetric "alternating" mixed action profile. While this is just an approximate Nash equilibrium, it suggests that there is an (exact) Nash equilibrium of the form $((p_0, 0, p_2, 0, p_4, 0), (p_0, 0, p_2, 0, p_4, 0), (0, q_1, 0, q_3, 0, q_5))$. In fact, a simple calculation shows that there is one such that $$ p_0 = \left(\frac{r-4}{r}\right)^{\frac{1}{2}}, p_0 + p_2 = \left(\frac{r-2}{r}\right)^{\frac{1}{2}}, p_4 = 1 - (p_0 + p_2), $$ and $$ q_1 = \frac{2}{r p_0}, q_1 + q_3 = \frac{4}{r (p_0+p_2)}, q_5 = 1 - (q_1 + q_3). $$

To verify:

In [80]:
p0 = ((r-4)/r)**(1/2)
p02 = ((r-2)/r)**(1/2)
p2 = p02 - p0
p4 = 1 - p02
q1 = (2/r)/p0
q13 = (4/r)/(p02)
q3 = q13 - q1
q5 = 1 - q13
a = ([p0, 0, p2, 0, p4, 0], [p0, 0, p2, 0, p4, 0], [0, q1, 0, q3, 0, q5])
a
Out[80]:
([0.8660254037844386, 0, 0.06938894290904674, 0, 0.06458565330651467, 0],
 [0.8660254037844386, 0, 0.06938894290904674, 0, 0.06458565330651467, 0],
 [0, 0.14433756729740646, 0, 0.12292367461501794, 0, 0.7327387580875756])
In [81]:
g_APA3.is_nash(a)
Out[81]:
True

References

  • E. Dechenaux, D. Kovenock, and V. Lugovskyy (2006), "Caps on bidding in all-pay auctions: Comments on the experiments of A. Rapoport and W. Amaldoss," Journal of Economic Behavior and Organization 61, 276-283.

  • C. E. Lemke and J. T. Howson (1964), "Equilibrium Points of Bimatrix Games," Journal of the Society for Industrial and Applied Mathematics 12, 413-423.

  • A. McLennan and R. Tourky (2006), "From Imitation Games to Kakutani."

  • A. Rapoport and W. Amaldoss (2004), "Mixed-strategy play in single-stage first-price all-pay auctions with symmetric players," Journal of Economic Behavior and Organization 54, 585-607.

  • B. von Stengel (1997), "New Lower Bounds for the Number of Equilibria in Bimatrix Games."

  • B. von Stengel (2007), "Equilibrium Computation for Two-Player Games in Strategic and Extensive Form," Chapter 3, N. Nisan, T. Roughgarden, E. Tardos, and V. Vazirani eds., Algorithmic Game Theory.

In [ ]: