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Scale Lemke-Howson tableau rows by per-player payoff max #980

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Scale Lemke-Howson tableau rows by per-player payoff max #980

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34 changes: 24 additions & 10 deletions quantecon/game_theory/lemke_howson.py
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Original file line number Diff line number Diff line change
Expand Up @@ -247,6 +247,12 @@ def _initialize_tableaux(payoff_matrices, tableaux, bases):
`abs(payoff_matrices[i].min()) + 1` (to ensure for the tableau not
to have a negative entry or a column identically zero).

The shifted payoffs of each player are additionally divided by their
maximum, so that the payoff block of each tableau lies in a fixed
range and the absolute tolerances of the pivoting routine act as
tolerances relative to the payoff scale. Nash equilibria are
invariant to such per-player positive affine transformations.

Suppose that the players 0 and 1 have m and n actions, respectively.

* `tableaux[0]` has n rows and m+n+1 columns, where columns 0, ...,
Expand Down Expand Up @@ -294,30 +300,38 @@ def _initialize_tableaux(payoff_matrices, tableaux, bases):
>>> bases = (np.empty(n, dtype=int), np.empty(m, dtype=int))
>>> tableaux, bases = _initialize_tableaux((A, B), tableaux, bases)
>>> tableaux[0]
array([[3., 2., 3., 1., 0., 1.],
[2., 6., 1., 0., 1., 1.]])
array([[0.5 , 0.33333333, 0.5 , 1. , 0. ,
1. ],
[0.33333333, 1. , 0.16666667, 0. , 1. ,
1. ]])
>>> tableaux[1]
array([[1., 0., 0., 4., 4., 1.],
[0., 1., 0., 3., 6., 1.],
[0., 0., 1., 1., 7., 1.]])
array([[1. , 0. , 0. , 0.57142857, 0.57142857,
1. ],
[0. , 1. , 0. , 0.42857143, 0.85714286,
1. ],
[0. , 0. , 1. , 0.14285714, 1. ,
1. ]])
>>> bases
(array([3, 4]), array([0, 1, 2]))

"""
nums_actions = payoff_matrices[0].shape

consts = np.zeros(2) # To be added to payoffs if min <= 0
scales = np.ones(2) # To divide the shifted payoffs by their maximum
for pl in range(2):
min_ = payoff_matrices[pl].min()
if min_ <= 0:
consts[pl] = min_ * (-1) + 1
scales[pl] = (payoff_matrices[pl] + consts[pl]).max()

for pl, (py_start, sl_start) in enumerate(zip((0, nums_actions[0]),
(nums_actions[0], 0))):
for i in range(nums_actions[1-pl]):
for j in range(nums_actions[pl]):
tableaux[pl][i, py_start+j] = \
payoff_matrices[1-pl][i, j] + consts[1-pl]
(payoff_matrices[1-pl][i, j] + consts[1-pl]) / \
scales[1-pl]
for j in range(nums_actions[1-pl]):
if j == i:
tableaux[pl][i, sl_start+j] = 1
Expand Down Expand Up @@ -381,12 +395,12 @@ def _lemke_howson_tbl(tableaux, bases, init_pivot, max_iter):
>>> _lemke_howson_tbl(tableaux, bases, 1, 10)
(True, 4, False)
>>> tableaux[0]
array([[ 0.875 , 0. , 1. , 0.375 , -0.125 , 0.25 ],
[ 0.1875, 1. , 0. , -0.0625, 0.1875, 0.125 ]])
array([[ 0.875 , 0. , 1. , 2.25 , -0.75 , 1.5 ],
[ 0.1875, 1. , 0. , -0.375 , 1.125 , 0.75 ]])
>>> tableaux[1]
array([[ 1. , -1.6 , 0.8 , 0. , 0. , 0.2 ],
[ 0. , 0.4667, -0.4 , 1. , 0. , 0.0667],
[ 0. , -0.0667, 0.2 , 0. , 1. , 0.1333]])
[ 0. , 3.2667, -2.8 , 1. , 0. , 0.4667],
[ 0. , -0.4667, 1.4 , 0. , 1. , 0.9333]])
>>> bases
(array([2, 1]), array([0, 3, 4]))

Expand Down
34 changes: 25 additions & 9 deletions quantecon/game_theory/tests/test_lemke_howson.py
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Original file line number Diff line number Diff line change
Expand Up @@ -134,21 +134,37 @@ def test_lemke_howson_invalid_init_pivot_float():


@pytest.mark.parametrize('scale', [1e15, 1e16])
def test_lemke_howson_large_payoffs_report_breakdown(scale):
# With payoffs of this size, the ratios in the lexico-minimum ratio
# test all tie within the (absolute) `tol_ratio_diff`, and the tie
# breaking fails. The routine must then report non-convergence
# rather than return a wrong "equilibrium" with `converged=True`.
def test_lemke_howson_large_payoffs_converge(scale):
# gh-951: with the payoff blocks normalized in `_initialize_tableaux`,
# the absolute pivoting tolerances act relative to the payoff scale,
# so the routine converges to a valid equilibrium instead of
# reporting a breakdown.
A = scale * np.array([[3., 1.],
[1., 3.]])
B = scale * np.array([[1., 3.],
[3., 1.]])
g = NormalFormGame((Player(A), Player(B)))
NE, res = lemke_howson(g, full_output=True)
assert_(not res.converged)
# Without capping, no other initial pivot is tried on breakdown
assert_(res.init == 0)
assert_(res.num_iter == 0)
assert_(res.converged)
assert_(g.is_nash(NE))


@pytest.mark.parametrize('scale', [1e-12, 1e-10, 1e12, 1e15])
def test_lemke_howson_payoff_scale_invariance(scale):
# gh-951: Nash equilibria are invariant to per-player positive affine
# transformations of the payoffs. With the payoff blocks normalized
# in `_initialize_tableaux`, the routine must converge to an
# equilibrium that is valid for the unit-scale game at any scale.
# Uniform payoffs are used as in the issue: all values are positive,
# so only the rescaling (not the positivity shift) is exercised.
rng = np.random.default_rng(0)
for _ in range(10):
base = rng.random(size=(4, 4, 2))
g_unit = NormalFormGame(base)
NE, res = lemke_howson(NormalFormGame(base * scale),
full_output=True)
assert_(res.converged)
assert_(g_unit.is_nash(tuple(NE)))


def test_lemke_howson_tbl_breakdown():
Expand Down