diff --git a/bin/ltlsynt.cc b/bin/ltlsynt.cc
index 78571fe50..09bbe02fb 100644
--- a/bin/ltlsynt.cc
+++ b/bin/ltlsynt.cc
@@ -184,55 +184,8 @@ ARGMATCH_VERIFY(solver_args, solver_types);
 static solver opt_solver = SPLIT_DET;
 static bool verbose = false;
 
-
 namespace
 {
-
-  // Ensures that the game is complete for player 0.
-  // Also computes the owner of each state (false for player 0, i.e. env).
-  // Initial state belongs to Player 0 and the game is turn-based.
-  static std::vector<bool>*
-  complete_env(spot::twa_graph_ptr& arena)
-  {
-    unsigned sink_env = arena->new_state();
-    unsigned sink_con = arena->new_state();
-
-    auto um = arena->acc().unsat_mark();
-    if (!um.first)
-      throw std::runtime_error("game winning condition is a tautology");
-    arena->new_edge(sink_con, sink_env, bddtrue, um.second);
-    arena->new_edge(sink_env, sink_con, bddtrue, um.second);
-
-    std::vector<bool> seen(arena->num_states(), false);
-    unsigned init = arena->get_init_state_number();
-    std::vector<unsigned> todo({init});
-    auto owner = new std::vector<bool>(arena->num_states(), false);
-    (*owner)[init] = false;
-    (*owner)[sink_env] = true;
-    while (!todo.empty())
-      {
-        unsigned src = todo.back();
-        todo.pop_back();
-        seen[src] = true;
-        bdd missing = bddtrue;
-        for (const auto& e: arena->out(src))
-          {
-            if (!(*owner)[src])
-              missing -= e.cond;
-
-            if (!seen[e.dst])
-              {
-                (*owner)[e.dst] = !(*owner)[src];
-                todo.push_back(e.dst);
-              }
-          }
-        if (!(*owner)[src] && missing != bddfalse)
-          arena->new_edge(src, sink_con, missing, um.second);
-      }
-
-    return owner;
-  }
-
   static spot::twa_graph_ptr
   to_dpa(const spot::twa_graph_ptr& split)
   {
@@ -548,8 +501,7 @@ namespace
       nb_states_dpa = dpa->num_states();
       if (want_time)
         sw.start();
-      auto owner = complete_env(dpa);
-      dpa->set_named_prop("state-player", owner);
+      propagate_players(dpa, false, true);
       if (want_time)
         bgame_time = sw.stop();
       if (verbose)
diff --git a/spot/misc/game.cc b/spot/misc/game.cc
index a21feface..f4a4cd21a 100644
--- a/spot/misc/game.cc
+++ b/spot/misc/game.cc
@@ -247,4 +247,61 @@ namespace spot
 
     solve_rec(arena, owner, states_, m.max_set(), w, s);
   }
+
+  void propagate_players(spot::twa_graph_ptr& arena,
+                         bool first_player, bool complete0)
+  {
+    auto um = arena->acc().unsat_mark();
+    if (!um.first)
+      throw std::runtime_error("game winning condition is a tautology");
+
+    unsigned sink_env = 0;
+    unsigned sink_con = 0;
+
+    std::vector<bool> seen(arena->num_states(), false);
+    unsigned init = arena->get_init_state_number();
+    std::vector<unsigned> todo({init});
+    auto owner = new std::vector<bool>(arena->num_states(), false);
+    (*owner)[init] = first_player;
+    while (!todo.empty())
+      {
+        unsigned src = todo.back();
+        todo.pop_back();
+        seen[src] = true;
+        bdd missing = bddtrue;
+        for (const auto& e: arena->out(src))
+          {
+            bool osrc = (*owner)[src];
+            if (complete0 && !osrc)
+              missing -= e.cond;
+
+            if (!seen[e.dst])
+              {
+                (*owner)[e.dst] = !osrc;
+                todo.push_back(e.dst);
+              }
+            else if ((*owner)[e.dst] == osrc)
+              {
+                delete owner;
+                throw
+                  std::runtime_error("propagate_players(): odd cycle detected");
+              }
+          }
+        if (complete0 && !(*owner)[src] && missing != bddfalse)
+          {
+            if (sink_env == 0)
+              {
+                sink_env = arena->new_state();
+                sink_con = arena->new_state();
+                owner->push_back(true);
+                owner->push_back(false);
+                arena->new_edge(sink_con, sink_env, bddtrue, um.second);
+                arena->new_edge(sink_env, sink_con, bddtrue, um.second);
+              }
+            arena->new_edge(src, sink_con, missing, um.second);
+          }
+      }
+
+    arena->set_named_prop("state-player", owner);
+  }
 }
diff --git a/spot/misc/game.hh b/spot/misc/game.hh
index 91c3fbb70..b6cf30d22 100644
--- a/spot/misc/game.hh
+++ b/spot/misc/game.hh
@@ -31,6 +31,22 @@
 
 namespace spot
 {
+
+  /// \brief Transform an automaton into a parity game by propagating
+  /// players
+  ///
+  /// This propagate state players, assuming the initial state belong
+  /// to \a first_player, and alternating players on each transitions.
+  /// If an odd cycle is detected, a runtime_exception is raised.
+  ///
+  /// If \a complete0 is set, ensure that states of player 0 are
+  /// complete.
+  SPOT_API
+  void propagate_players(spot::twa_graph_ptr& arena,
+                         bool first_player = false,
+                         bool complete0 = true);
+
+
   typedef std::unordered_set<unsigned> region_t;
   typedef std::unordered_map<unsigned, unsigned> strategy_t;
 
diff --git a/tests/core/ltlsynt.test b/tests/core/ltlsynt.test
index 86da4626d..663994c00 100644
--- a/tests/core/ltlsynt.test
+++ b/tests/core/ltlsynt.test
@@ -23,7 +23,7 @@
 set -e
 
 cat >exp <<EOF
-parity 18;
+parity 16;
 0 1 0 1,2 "INIT";
 2 1 1 3;
 3 2 0 4,5;