⚡️ Speed up method PolymatrixGame.range_of_payoffs by 1,749%

[misrasaurabh1]

📄 1,749% (17.49x) speedup for PolymatrixGame.range_of_payoffs in quantecon/game_theory/polymatrix_game.py

⏱️ Runtime : 3.83 seconds → 207 milliseconds (best of 5 runs)

📝 Explanation and details

The optimization replaces the inefficient double-pass approach in range_of_payoffs() with a single-pass vectorized operation using NumPy.

Key changes:

• Original approach: Two separate list comprehensions calling min([np.min(M) for M in ...]) and max([np.max(M) for M in ...]), which iterate through all matrices twice and involve Python's built-in min/max functions on a list of scalar values.
• Optimized approach: Single concatenation of all flattened matrices using np.concatenate([M.ravel() for M in ...]), then applying np.min() and np.max() directly on the combined array.

Why this is faster:

• Eliminates redundant iterations: Instead of scanning all matrices twice (once for min, once for max), we flatten and concatenate once, then perform both min/max operations on the same contiguous array.
• Vectorized operations: NumPy's min and max functions are highly optimized C implementations that operate on contiguous memory, compared to Python's built-in functions working on lists.
• Reduces function call overhead: The original code calls np.min() once per matrix, while the optimized version calls it once total.

Performance characteristics:
The optimization shows dramatic speedup especially for larger games - achieving 651% to 2010% improvements on large-scale test cases with many players/matchups, while maintaining 9-30% improvements on smaller cases. The single-pass approach scales much better as the number of matrices increases.

✅ Correctness verification report:

Test	Status
⚙️ Existing Unit Tests	🔘 None Found
🌀 Generated Regression Tests	✅ 68 Passed
⏪ Replay Tests	🔘 None Found
🔎 Concolic Coverage Tests	🔘 None Found
📊 Tests Coverage	100.0%

🌀 Generated Regression Tests and Runtime

from collections.abc import Iterable, Mapping, Sequence
from math import isqrt

# function to test
import numpy as np
# imports
import pytest  # used for our unit tests
from numpy.typing import NDArray
from quantecon.game_theory.polymatrix_game import PolymatrixGame


# Dummy _nums_actions2string for repr
def _nums_actions2string(nums_actions):
    return str(nums_actions)
from quantecon.game_theory.polymatrix_game import PolymatrixGame

# unit tests

# -----------------
# BASIC TEST CASES
# -----------------

def test_basic_single_payoff():
    # Single matchup, single action, single payoff
    pmg = PolymatrixGame({(0, 1): [[5.0]], (1, 0): [[-3.0]]}, nums_actions=[1, 1])
    # Should return the min and max over both matrices
    codeflash_output = pmg.range_of_payoffs() # 30.3μs -> 35.5μs (14.7% slower)

def test_basic_multiple_players_actions():
    # Two players, two actions each, all payoffs positive
    pmg = PolymatrixGame({
        (0, 1): [[1, 2], [3, 4]],
        (1, 0): [[5, 6], [7, 8]]
    }, nums_actions=[2, 2])
    # min: 1, max: 8
    codeflash_output = pmg.range_of_payoffs() # 21.2μs -> 17.7μs (20.2% faster)

def test_basic_negative_and_positive():
    # Two players, negative and positive payoffs
    pmg = PolymatrixGame({
        (0, 1): [[-10, 0], [5, 2]],
        (1, 0): [[-7, 4], [8, -1]]
    }, nums_actions=[2, 2])
    # min: -10, max: 8
    codeflash_output = pmg.range_of_payoffs() # 18.4μs -> 15.2μs (21.2% faster)

def test_basic_zero_payoffs():
    # Two players, all payoffs zero
    pmg = PolymatrixGame({
        (0, 1): [[0, 0], [0, 0]],
        (1, 0): [[0, 0], [0, 0]]
    }, nums_actions=[2, 2])
    codeflash_output = pmg.range_of_payoffs() # 17.4μs -> 15.4μs (13.2% faster)

def test_basic_float_and_int_mix():
    # Payoffs are a mix of floats and ints
    pmg = PolymatrixGame({
        (0, 1): [[1, 2.5], [3, 4]],
        (1, 0): [[-2, 7.1], [0, 1]]
    }, nums_actions=[2, 2])
    # min: -2, max: 7.1
    codeflash_output = pmg.range_of_payoffs() # 18.1μs -> 15.8μs (14.6% faster)

# -----------------
# EDGE TEST CASES
# -----------------

def test_edge_missing_matchup_filled_with_zeros():
    # Missing (1,0) matchup, should be filled with zeros
    pmg = PolymatrixGame({
        (0, 1): [[3, 4], [5, 6]]
    }, nums_actions=[2, 2])
    # (1,0) is filled with zeros, so min: 0, max: 6
    codeflash_output = pmg.range_of_payoffs() # 18.0μs -> 13.8μs (30.5% faster)

def test_edge_missing_matchup_and_actions():
    # 3 players, missing some matchups
    pmg = PolymatrixGame({
        (0, 1): [[1, 2], [3, 4]],
        (1, 2): [[-5, 0], [2, 3]],
        (2, 0): [[10, 20], [30, 40]]
    }, nums_actions=[2, 2, 2])
    # Other matchups filled with zeros, min: -5, max: 40
    codeflash_output = pmg.range_of_payoffs() # 33.3μs -> 16.0μs (108% faster)

def test_edge_unspecified_actions_filled_with_neg_inf():
    # 2 players, 3 actions, but only 2x2 matrix provided
    pmg = PolymatrixGame({
        (0, 1): [[1, 2], [3, 4]],
        (1, 0): [[-1, -2], [-3, -4]]
    }, nums_actions=[3, 3])
    # Extra actions have -np.inf payoff
    min_expected = min(1, 2, 3, 4, -1, -2, -3, -4, -np.inf)
    max_expected = max(1, 2, 3, 4, -1, -2, -3, -4, -np.inf)
    codeflash_output = pmg.range_of_payoffs() # 17.3μs -> 15.8μs (9.46% faster)

def test_edge_all_neg_inf():
    # All payoffs are -np.inf
    pmg = PolymatrixGame({
        (0, 1): [[-np.inf, -np.inf], [-np.inf, -np.inf]],
        (1, 0): [[-np.inf, -np.inf], [-np.inf, -np.inf]]
    }, nums_actions=[2, 2])
    codeflash_output = pmg.range_of_payoffs() # 16.6μs -> 15.1μs (9.36% faster)

def test_edge_large_negative_and_positive():
    # Large magnitude negative and positive payoffs
    pmg = PolymatrixGame({
        (0, 1): [[-1e9, 0], [1e9, -1e8]],
        (1, 0): [[1e8, -1e8], [0, -1e9]]
    }, nums_actions=[2, 2])
    codeflash_output = pmg.range_of_payoffs() # 17.7μs -> 14.5μs (21.6% faster)

def test_edge_single_player_no_matchups():
    # One player: no matchups
    pmg = PolymatrixGame({}, nums_actions=[1])
    # No matchups, so nothing in polymatrix; should raise ValueError
    with pytest.raises(ValueError):
        pmg.range_of_payoffs() # 2.29μs -> 3.29μs (30.4% slower)


def test_edge_non_square_matrix():
    # Non-square payoff matrices
    pmg = PolymatrixGame({
        (0, 1): [[1, 2, 3], [4, 5, 6]],
        (1, 0): [[-1, -2], [-3, -4], [-5, -6]]
    }, nums_actions=[2, 3])
    # min: -6, max: 6
    codeflash_output = pmg.range_of_payoffs() # 27.4μs -> 25.4μs (7.57% faster)

# -----------------
# LARGE SCALE TEST CASES
# -----------------

def test_large_scale_10_players_10_actions():
    # 10 players, 10 actions each, all matchups present
    n_players = 10
    n_actions = 10
    # Each payoff matrix is filled with its matchup sum
    polymatrix = {}
    for i in range(n_players):
        for j in range(n_players):
            if i != j:
                matrix = np.full((n_actions, n_actions), i + j)
                polymatrix[(i, j)] = matrix
    pmg = PolymatrixGame(polymatrix, nums_actions=[n_actions]*n_players)
    # min: 1+0=1, max: 9+8=17
    codeflash_output = pmg.range_of_payoffs() # 341μs -> 45.5μs (651% faster)

def test_large_scale_sparse_matchups():
    # 100 players, 2 actions, only one matchup present
    n_players = 100
    n_actions = 2
    polymatrix = {
        (0, 1): [[-1, 2], [3, 4]]
    }
    pmg = PolymatrixGame(polymatrix, nums_actions=[n_actions]*n_players)
    # All other matchups filled with zeros, min: -1, max: 4
    codeflash_output = pmg.range_of_payoffs() # 36.5ms -> 1.73ms (2010% faster)

def test_large_scale_random_payoffs():
    # 20 players, 5 actions, random payoffs
    n_players = 20
    n_actions = 5
    rng = np.random.default_rng(12345)
    polymatrix = {}
    min_payoff = float('inf')
    max_payoff = float('-inf')
    for i in range(n_players):
        for j in range(n_players):
            if i != j:
                mat = rng.integers(-100, 100, size=(n_actions, n_actions)).astype(float)
                polymatrix[(i, j)] = mat
                mat_min = np.min(mat)
                mat_max = np.max(mat)
                if mat_min < min_payoff:
                    min_payoff = mat_min
                if mat_max > max_payoff:
                    max_payoff = mat_max
    pmg = PolymatrixGame(polymatrix, nums_actions=[n_actions]*n_players)
    codeflash_output = pmg.range_of_payoffs() # 1.39ms -> 86.4μs (1515% faster)

def test_large_scale_missing_matchups_and_actions():
    # 50 players, 3 actions, only a few matchups present
    n_players = 50
    n_actions = 3
    polymatrix = {
        (0, 1): [[1, 2, 3], [4, 5, 6], [7, 8, 9]],
        (2, 3): [[-10, -20, -30], [-40, -50, -60], [-70, -80, -90]],
        (4, 5): [[100, 200, 300], [400, 500, 600], [700, 800, 900]]
    }
    pmg = PolymatrixGame(polymatrix, nums_actions=[n_actions]*n_players)
    # All other matchups filled with zeros, min: -90, max: 900
    codeflash_output = pmg.range_of_payoffs() # 9.02ms -> 435μs (1973% faster)

def test_large_scale_unspecified_actions_filled_with_neg_inf():
    # 10 players, 10 actions, but only 2x2 matrices provided for a few matchups
    n_players = 10
    n_actions = 10
    polymatrix = {
        (0, 1): [[1, 2], [3, 4]],
        (2, 3): [[-1, -2], [-3, -4]]
    }
    pmg = PolymatrixGame(polymatrix, nums_actions=[n_actions]*n_players)
    # All other matchups filled with zeros, but (0,1) and (2,3) have -np.inf in unspecified actions
    min_expected = min(1, 2, 3, 4, -1, -2, -3, -4, 0, -np.inf)
    max_expected = max(1, 2, 3, 4, -1, -2, -3, -4, 0, -np.inf)
    codeflash_output = pmg.range_of_payoffs() # 340μs -> 44.1μs (672% faster)
# codeflash_output is used to check that the output of the original code is the same as that of the optimized code.
#------------------------------------------------
from collections.abc import Iterable, Mapping, Sequence
from math import isqrt

# function to test
import numpy as np
# imports
import pytest  # used for our unit tests
from numpy.typing import NDArray
from quantecon.game_theory.polymatrix_game import PolymatrixGame


def _nums_actions2string(nums_actions):
    return str(nums_actions)
from quantecon.game_theory.polymatrix_game import PolymatrixGame

# unit tests

# ----------- BASIC TEST CASES -----------

def test_basic_two_players_positive_payoffs():
    # 2 players, 2 actions each, all payoffs positive
    pm = {
        (0, 1): [[1, 2], [3, 4]],
        (1, 0): [[5, 6], [7, 8]]
    }
    game = PolymatrixGame(pm, nums_actions=[2, 2])
    codeflash_output = game.range_of_payoffs(); result = codeflash_output # 21.5μs -> 18.5μs (16.5% faster)

def test_basic_two_players_negative_payoffs():
    # 2 players, 2 actions each, all payoffs negative
    pm = {
        (0, 1): [[-1, -2], [-3, -4]],
        (1, 0): [[-5, -6], [-7, -8]]
    }
    game = PolymatrixGame(pm, nums_actions=[2, 2])
    codeflash_output = game.range_of_payoffs(); result = codeflash_output # 19.1μs -> 16.5μs (15.8% faster)

def test_basic_mixed_payoffs():
    # 2 players, 2 actions each, mixed positive and negative payoffs
    pm = {
        (0, 1): [[-1, 2], [3, -4]],
        (1, 0): [[5, -6], [7, 0]]
    }
    game = PolymatrixGame(pm, nums_actions=[2, 2])
    codeflash_output = game.range_of_payoffs(); result = codeflash_output # 17.5μs -> 14.7μs (18.7% faster)

def test_basic_three_players():
    # 3 players, 2 actions each, some zeros
    pm = {
        (0, 1): [[0, 1], [2, 3]],
        (1, 0): [[4, 5], [6, 7]],
        (0, 2): [[-1, -2], [-3, -4]],
        (2, 0): [[8, 9], [10, 11]],
        (1, 2): [[-5, 5], [0, 0]],
        (2, 1): [[-10, 10], [0, 0]]
    }
    game = PolymatrixGame(pm, nums_actions=[2, 2, 2])
    codeflash_output = game.range_of_payoffs(); result = codeflash_output # 32.6μs -> 16.0μs (103% faster)

def test_basic_single_action_each():
    # 2 players, 1 action each
    pm = {
        (0, 1): [[42]],
        (1, 0): [[-42]]
    }
    game = PolymatrixGame(pm, nums_actions=[1, 1])
    codeflash_output = game.range_of_payoffs(); result = codeflash_output # 18.5μs -> 15.3μs (21.0% faster)

# ----------- EDGE TEST CASES -----------

def test_edge_empty_matrix_defaults():
    # 2 players, 2 actions each, only one matchup specified
    pm = {
        (0, 1): [[1, 2], [3, 4]]
        # (1, 0) missing, should default to zeros then filled with -inf
    }
    game = PolymatrixGame(pm, nums_actions=[2, 2])
    codeflash_output = game.range_of_payoffs(); result = codeflash_output # 18.2μs -> 14.5μs (25.4% faster)

def test_edge_matrix_with_inf_values():
    # 2 players, 2 actions each, some payoffs are inf or -inf
    pm = {
        (0, 1): [[np.inf, -np.inf], [0, 1]],
        (1, 0): [[-1, 2], [np.inf, -np.inf]]
    }
    game = PolymatrixGame(pm, nums_actions=[2, 2])
    codeflash_output = game.range_of_payoffs(); result = codeflash_output # 16.9μs -> 15.0μs (12.3% faster)

def test_edge_zeros_and_neg_inf():
    # 2 players, 2 actions each, one matrix all zeros, one all -inf
    pm = {
        (0, 1): [[0, 0], [0, 0]],
        (1, 0): [[-np.inf, -np.inf], [-np.inf, -np.inf]]
    }
    game = PolymatrixGame(pm, nums_actions=[2, 2])
    codeflash_output = game.range_of_payoffs(); result = codeflash_output # 16.9μs -> 13.8μs (22.9% faster)

def test_edge_partial_matrix_missing_actions():
    # 2 players, 3 actions for player 0, 2 actions for player 1
    # Only partial matrix provided for (0,1)
    pm = {
        (0, 1): [[1, 2], [3, 4]]  # should fill missing row with -inf
    }
    game = PolymatrixGame(pm, nums_actions=[3, 2])
    codeflash_output = game.range_of_payoffs(); result = codeflash_output # 15.9μs -> 14.0μs (13.8% faster)

def test_edge_large_negative_and_positive():
    # 2 players, 2 actions each, large magnitude values
    pm = {
        (0, 1): [[-1e10, 1e10], [1e9, -1e9]],
        (1, 0): [[-1e11, 1e11], [0, 0]]
    }
    game = PolymatrixGame(pm, nums_actions=[2, 2])
    codeflash_output = game.range_of_payoffs(); result = codeflash_output # 17.4μs -> 12.8μs (35.7% faster)

def test_edge_unspecified_matchup_filled_with_zeros():
    # 3 players, only one matchup specified
    pm = {
        (0, 1): [[1, 2], [3, 4]]
    }
    game = PolymatrixGame(pm, nums_actions=[2, 2, 2])
    codeflash_output = game.range_of_payoffs(); result = codeflash_output # 32.9μs -> 15.5μs (113% faster)

def test_edge_all_matchups_missing():
    # 2 players, no matchups specified
    pm = {}
    game = PolymatrixGame(pm, nums_actions=[2, 2])
    codeflash_output = game.range_of_payoffs(); result = codeflash_output # 16.3μs -> 14.4μs (13.1% faster)

def test_edge_single_player():
    # 1 player, no matchups possible
    pm = {}
    game = PolymatrixGame(pm, nums_actions=[2])
    codeflash_output = game.range_of_payoffs(); result = codeflash_output
    # No matchups, so no matrices; should raise ValueError due to min/max of empty sequence
    with pytest.raises(ValueError):
        game.range_of_payoffs()

# ----------- LARGE SCALE TEST CASES -----------

def test_large_scale_many_players_and_actions():
    # 10 players, 5 actions each
    N = 10
    A = 5
    pm = {}
    # Fill every matchup with a matrix of random values between -100 and 100
    rng = np.random.default_rng(123)
    for i in range(N):
        for j in range(N):
            if i != j:
                pm[(i, j)] = rng.integers(-100, 101, size=(A, A)).tolist()
    game = PolymatrixGame(pm, nums_actions=[A]*N)
    codeflash_output = game.range_of_payoffs(); result = codeflash_output # 337μs -> 35.2μs (861% faster)

def test_large_scale_sparse_matchups():
    # 20 players, 3 actions each, only a few matchups specified
    N = 20
    A = 3
    pm = {}
    # Only specify (i, i+1) for i in range(N-1)
    for i in range(N-1):
        pm[(i, i+1)] = [[i, -i, 0], [i+1, -(i+1), 0], [0, 0, 0]]
    game = PolymatrixGame(pm, nums_actions=[A]*N)
    codeflash_output = game.range_of_payoffs(); result = codeflash_output # 1.39ms -> 92.0μs (1415% faster)

def test_large_scale_maximum_fill():
    # 50 players, 2 actions each, all matchups filled with same values
    N = 50
    A = 2
    pm = {}
    for i in range(N):
        for j in range(N):
            if i != j:
                pm[(i, j)] = [[1, 2], [3, 4]]
    game = PolymatrixGame(pm, nums_actions=[A]*N)
    codeflash_output = game.range_of_payoffs(); result = codeflash_output # 9.05ms -> 441μs (1950% faster)

def test_large_scale_partial_matrix_fill():
    # 100 players, 2 actions each, only diagonal matchups filled
    N = 100
    A = 2
    pm = {}
    # Only (i, (i+1)%N) filled
    for i in range(N):
        pm[(i, (i+1)%N)] = [[i, -i], [i+1, -(i+1)]]
    game = PolymatrixGame(pm, nums_actions=[A]*N)
    codeflash_output = game.range_of_payoffs(); result = codeflash_output # 36.6ms -> 1.77ms (1969% faster)

To edit these changes git checkout codeflash/optimize-PolymatrixGame.range_of_payoffs-mgh46ign and push.

[coveralls]

coverage: 92.58% (+0.001%) from 92.579%
when pulling e3a5e39 on codeflash-ai:codeflash/optimize-PolymatrixGame.range_of_payoffs-mgh46ign
into 29cda3a on QuantEcon:main.

[oyamad]

@misrasaurabh1 Thanks for your contribution.

For this PR, the line np.concatenate([M.ravel() for M in ...]) makes a large amount of memory allocation for large games. I am not totally sure if this is justified in terms of cost/benefit balance.

An alternative approach is to use Numba with explicit loops (even for obtaining the max and the min) over the Dict (which has to be converted to a Numba typed Dict).

[misrasaurabh1]

I agree the concatenation will use more memory, even though ravel helps here reduce it. We currently don't measure memory unfortunately. Although the speedup achieved is a lot. I'll leave it to your judgement

[misrasaurabh1]

Hi @oyamad, can i ask you for a resolution on this PR? I would recommend accepting the changes since the performance gains are so large.

[oyamad]

@misrasaurabh1 Here's an "alternative approach" that uses Numba: 66c4037
This is much faster than your Python/NumPy implementation.

Sample timing on my machine:

import quantecon.game_theory as gt
seed = 123
N = 100
M = 2
pmg = gt.random_polymatrix_game((M,)*N, random_state=seed)
%timeit pmg.range_of_payoffs()

• Current main (original):

  17.1 ms ± 104 μs per loop (mean ± std. dev. of 7 runs, 100 loops each)
  

• Your version:

  1.03 ms ± 11.3 μs per loop (mean ± std. dev. of 7 runs, 1,000 loops each)
  

• Numba version (66c4037):

  36.7 μs ± 1.23 μs per loop (mean ± std. dev. of 7 runs, 10,000 loops each)
  

[misrasaurabh1]

awesome! these gains look great. I will look into how I can make codeflash automatically convert code to use numba.