sagemathgh-39464: remove deprecated method in normal_form_game
    
found using the scripts in sagemath#39262

### 📝 Checklist

- [x] The title is concise and informative.
- [x] The description explains in detail what this PR is about.
    
URL: sagemath#39464
Reported by: Frédéric Chapoton
Reviewer(s): David Coudert

Author: Release Manager

src/sage/game_theory/normal_form_game.py

@@ -890,7 +890,7 @@ def __len__(self):
         """
         return len(self.utilities)
 
-    def _repr_(self):
+    def _repr_(self) -> str:
         r"""
         Return the strategy_profiles of the game.
 
@@ -909,7 +909,7 @@ def _repr_(self):
         base_str = "Normal Form Game with the following utilities: {}"
         return base_str.format(pformat(self.utilities))
 
-    def _latex_(self):
+    def _latex_(self) -> str:
         r"""
         Return the LaTeX code representing the ``NormalFormGame``.
 
@@ -1194,7 +1194,7 @@ def is_constant_sum(self):
             return False
         m1, m2 = self.payoff_matrices()
         c = m1 + m2
-        t = c[0,0]
+        t = c[0, 0]
 
         for row in c:
             for i in row:
@@ -2021,7 +2021,7 @@ def _solve_enumeration(self, maximization=True):
                                powerset(range(player.num_strategies))]
                               for player in self.players]
 
-        potential_support_pairs = [pair for pair in product(*potential_supports) if len(pair[0]) == len(pair[1])]
+        potential_support_pairs = (pair for pair in product(*potential_supports) if len(pair[0]) == len(pair[1]))
 
         equilibria = []
         for pair in potential_support_pairs:
@@ -2230,7 +2230,7 @@ def _is_NE(self, a, b, p1_support, p2_support, M1, M2):
         p2_payoffs = [sum(v * col[j] for j, v in enumerate(a))
                       for col in M2.columns()]
 
-        #if p1_payoffs.index(max(p1_payoffs)) not in p1_support:
+        # if p1_payoffs.index(max(p1_payoffs)) not in p1_support:
         if not any(i in p1_support for i, x in enumerate(p1_payoffs)
                    if x == max(p1_payoffs)):
             return False
@@ -2240,89 +2240,6 @@ def _is_NE(self, a, b, p1_support, p2_support, M1, M2):
 
         return True
 
-    def _Hrepresentation(self, m1, m2):
-        r"""
-        Create the H-representation strings required to use ``lrsnash``.
-
-        Since lrslib 6.1, this format is referred to as "legacy format".
-
-        This method is deprecated.
-
-        EXAMPLES::
-
-            sage: A = matrix([[1, 2], [3, 4]])
-            sage: B = matrix([[3, 3], [1, 4]])
-            sage: C = NormalFormGame([A, B])
-            sage: print(C._Hrepresentation(A, B)[0])
-            doctest:warning...
-            DeprecationWarning: NormalFormGame._Hrepresentation is deprecated as it
-            creates the legacy input format. Use NormalFormGame._lrs_nash_format instead
-            See https://github.com/sagemath/sage/issues/27745 for details.
-            H-representation
-            linearity 1 5
-            begin
-            5 4 rational
-            0 1 0 0
-            0 0 1 0
-            0 -3 -1 1
-            0 -3 -4 1
-            -1 1 1 0
-            end
-            <BLANKLINE>
-            sage: print(C._Hrepresentation(A, B)[1])
-            H-representation
-            linearity 1 5
-            begin
-            5 4 rational
-            0 -1 -2 1
-            0 -3 -4 1
-            0 1 0 0
-            0 0 1 0
-            -1 1 1 0
-            end
-            <BLANKLINE>
-        """
-        from sage.misc.superseded import deprecation
-        deprecation(27745,
-                    "NormalFormGame._Hrepresentation is deprecated as it "
-                    "creates the legacy input format. "
-                    "Use NormalFormGame._lrs_nash_format instead")
-
-        from sage.geometry.polyhedron.misc import _to_space_separated_string
-        m = self.players[0].num_strategies
-        n = self.players[1].num_strategies
-        midentity = list(matrix.identity(m))
-        nidentity = list(matrix.identity(n))
-
-        s = 'H-representation\n'
-        s += 'linearity 1 ' + str(m + n + 1) + '\n'
-        s += 'begin\n'
-        s += str(m + n + 1) + ' ' + str(m + 2) + ' rational\n'
-        for f in list(midentity):
-            s += '0 ' + _to_space_separated_string(f) + ' 0 \n'
-        for e in list(m2.transpose()):
-            s += '0 ' + _to_space_separated_string(-e) + '  1 \n'
-        s += '-1 '
-        for g in range(m):
-            s += '1 '
-        s += '0 \n'
-        s += 'end\n'
-
-        t = 'H-representation\n'
-        t += 'linearity 1 ' + str(m + n + 1) + '\n'
-        t += 'begin\n'
-        t += str(m + n + 1) + ' ' + str(n + 2) + ' rational\n'
-        for e in list(m1):
-            t += '0 ' + _to_space_separated_string(-e) + '  1 \n'
-        for f in list(nidentity):
-            t += '0 ' + _to_space_separated_string(f) + ' 0 \n'
-        t += '-1 '
-        for g in range(n):
-            t += '1 '
-        t += '0 \n'
-        t += 'end\n'
-        return s, t
-
     def _lrs_nash_format(self, m1, m2):
         r"""
         Create the input format for ``lrsnash``, version 6.1 or newer.
@@ -2346,53 +2263,24 @@ def _lrs_nash_format(self, m1, m2):
             4 3
             <BLANKLINE>
 
-            sage: legacy_format = C._Hrepresentation(A, B)
-            doctest:warning...
-            DeprecationWarning: NormalFormGame._Hrepresentation is deprecated as it
-            creates the legacy input format. Use NormalFormGame._lrs_nash_format instead
-            See https://github.com/sagemath/sage/issues/27745 for details.
-            sage: print('*game: player 1\n', legacy_format[0])
-            *game: player 1
-            H-representation
-            linearity 1 6
-            begin
-            6 5 rational
-            0 1 0 0 0
-            0 0 1 0 0
-            0 0 0 1 0
-            0 -1 0 -4  1
-            0 0 -2 -3  1
-            -1 1 1 1 0
-            end
-            <BLANKLINE>
-            sage: print('*game: player 2\n', legacy_format[1])
-            *game: player 2
-            H-representation
-            linearity 1 6
-            begin
-            6 4 rational
-            0 0 -6  1
-            0 -2 -5  1
-            0 -3 -3  1
-            0 1 0 0
-            0 0 1 0
-            -1 1 1 0
-            end
+        .. NOTE::
+
+            The former legacy format has been removed in :issue:`39464`.
         """
         from sage.geometry.polyhedron.misc import _to_space_separated_string
         m = self.players[0].num_strategies
         n = self.players[1].num_strategies
         s = f'{m} {n}\n\n'
-        for r in m1.rows():
-            s += _to_space_separated_string(r) + '\n'
+        s += '\n'.join(_to_space_separated_string(r) for r in m1.rows())
+        s += '\n\n'
+        s += '\n'.join(_to_space_separated_string(r) for r in m2.rows())
         s += '\n'
-        for r in m2.rows():
-            s += _to_space_separated_string(r) + '\n'
         return s
 
-    def is_degenerate(self, certificate=False):
+    def is_degenerate(self, certificate=False) -> bool:
         """
         A function to check whether the game is degenerate or not.
+
         Will return a boolean.
 
         A two-player game is called nondegenerate if no mixed strategy of

src/sage/game_theory/parser.py

@@ -51,7 +51,7 @@ def __init__(self, raw_string):
         """
         self.raw_string = raw_string
 
-    def format_lrs(self, legacy_format=False):
+    def format_lrs(self):
         r"""
         Parses the output of lrs so as to return vectors
         corresponding to equilibria.
@@ -137,45 +137,14 @@ def format_lrs(self, legacy_format=False):
              [(1/3, 2/3, 0), (1/7, 0, 6/7)],
              [(1, 0, 0), (0, 0, 1)]]
 
-        TESTS:
-
-        An example with the legacy format::
-
-            sage: from sage.cpython.string import bytes_to_str
-            sage: from sage.game_theory.parser import Parser
-            sage: from subprocess import Popen, PIPE
-            sage: A = matrix([[1, 2], [3, 2]])
-            sage: g = NormalFormGame([A])
-            sage: game1_str, game2_str = g._Hrepresentation(A, -A)
-            doctest:warning...
-            DeprecationWarning: NormalFormGame._Hrepresentation is deprecated as it
-            creates the legacy input format. Use NormalFormGame._lrs_nash_format instead
-            See https://github.com/sagemath/sage/issues/27745 for details.
-            sage: g1_name, g2_name = tmp_filename(), tmp_filename()
-            sage: g1_file, g2_file = open(g1_name, 'w'), open(g2_name, 'w')
-            sage: _ = g1_file.write(game1_str)
-            sage: g1_file.close()
-            sage: _ = g2_file.write(game2_str)
-            sage: g2_file.close()
-            sage: from sage.features.lrs import LrsNash
-            sage: process = Popen([LrsNash().absolute_filename(), g1_name, g2_name],        # optional - lrslib
-            ....:                 stdout=PIPE, stderr=PIPE)
-            sage: lrs_output = [bytes_to_str(row) for row in process.stdout]                # not tested, optional - lrslib
-            sage: nasheq = Parser(lrs_output).format_lrs(legacy_format=True)                # not tested, optional - lrslib
-            sage: nasheq                                                                    # not tested, optional - lrslib
-            [[(1/2, 1/2), (0, 1)], [(0, 1), (0, 1)]]
+        The former legacy format has been removed in :issue:`39464`.
         """
         equilibria = []
         from sage.misc.sage_eval import sage_eval
         from itertools import groupby, dropwhile
         lines = iter(self.raw_string)
-        if legacy_format:
-            # Skip until the magic stars announce the beginning of the real output
-            while not next(lines).startswith("*****"):
-                pass
-        else:
-            # Skip comment lines starting with a single star
-            lines = dropwhile(lambda line: line.startswith('*'), lines)
+        # Skip comment lines starting with a single star
+        lines = dropwhile(lambda line: line.startswith('*'), lines)
         for collection in [list(x[1]) for x in groupby(lines, lambda x: x == '\n')]:
             if collection[0].startswith('2'):
                 s1 = tuple([sage_eval(k) for k in collection[-1].split()][1:-1])