diff --git a/quantecon/game_theory/normal_form_game.py b/quantecon/game_theory/normal_form_game.py
index 690e5bb3..43cf3af6 100644
--- a/quantecon/game_theory/normal_form_game.py
+++ b/quantecon/game_theory/normal_form_game.py
@@ -909,13 +909,16 @@ def best_response_2p(payoff_matrix, opponent_mixed_action, tol=1e-8):
         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.
+    tol : scalar(float), optional(default=1e-8)
+        Tolerance level used in determining best responses. Must be
+        nonnegative.
 
     Returns
     -------
     scalar(int)
-        Best response action.
+        Best response action. -1 indicates an error condition: no
+        action satisfies the tolerance condition, which occurs only
+        if `tol` < 0.
 
     """
     n, m = payoff_matrix.shape
@@ -932,3 +935,5 @@ def best_response_2p(payoff_matrix, opponent_mixed_action, tol=1e-8):
     for a in range(n):
         if payoff_vector[a] >= payoff_max - tol:
             return a
+
+    return -1  # Unreachable unless tol < 0
diff --git a/quantecon/game_theory/tests/test_normal_form_game.py b/quantecon/game_theory/tests/test_normal_form_game.py
index 1b3bd30a..a5d6afac 100644
--- a/quantecon/game_theory/tests/test_normal_form_game.py
+++ b/quantecon/game_theory/tests/test_normal_form_game.py
@@ -556,3 +556,12 @@ def test_best_response_2p():
             br_computed = \
                 best_response_2p(test_case['payoff_array'], mixed_action)
             assert_(br_computed == br_expected)
+
+
+def test_best_response_2p_negative_tol():
+    # With tol < 0 no action can satisfy the tolerance condition;
+    # -1 is returned rather than falling through to None
+    payoff_array = np.array([[4., 0.], [3., 2.]])
+    mixed_action = np.array([0.5, 0.5])
+    br_computed = best_response_2p(payoff_array, mixed_action, tol=-1e-8)
+    assert_(br_computed == -1)