Lemke on trees stops after finding one eq; stop_after/max_depth not used here

Author: rahulsavani

doc/tools.lcp.rst

@@ -6,13 +6,20 @@
 
 :program:`gambit-lcp` reads a two-player game on standard input and
 computes Nash equilibria by finding solutions to a linear
-complementarity problem. For extensive games, the program uses the
+complementarity problem.
+
+For extensive games, the program uses the
 sequence form representation of the extensive game, as defined by
 Koller, Megiddo, and von Stengel [KolMegSte94]_, and applies the
-algorithm developed by Lemke. For strategic games, the program uses
-the method of Lemke and Howson [LemHow64]_.  There exist strategic
-games for which some equilibria cannot be located by this method; see
-Shapley [Sha74]_.
+algorithm developed by Lemke. In that case, the method will find
+one Nash equilibrium.
+
+For strategic games, the program uses the method of Lemke and Howson
+[LemHow64]_. In this case, the method will find all "accessible"
+equilibria, i.e., those that can be found as concatenations of Lemke-Howson
+paths that start at the artificial equilibrium.
+There exist strategic-form games for which some equilibria cannot be found
+by this method, i.e., some equilibria are inaccessible; see Shapley [Sha74]_.
 
 In a two-player strategic game, the set of Nash equilibria can be expressed
 as the union of convex sets. This program will find extreme points
@@ -53,9 +60,11 @@ game.
 
 .. cmdoption:: -e EQA
 
-   By default, the program will find all equilibria accessible from
-   the origin of the polytopes.  This switch instructs the program
-   to terminate when EQA equilibria have been found.
+   By default, when working with the reduced strategic game, the program
+   will find all equilibria accessible from the origin of the polytopes.
+   This switch instructs the program to terminate when EQA equilibria
+   have been found. This has no effect when using the extensive representation
+   of a game, in which case the method always only returns one equilibrium.
 
 .. cmdoption:: -h
 

src/pygambit/nash.pxi

@@ -74,15 +74,15 @@ def _enummixed_strategy_solve_rational(game: Game) -> list[MixedStrategyProfileR
 
 
 def _lcp_behavior_solve_double(
-        game: Game, stop_after: int, max_depth: int
+        game: Game
 ) -> list[MixedBehaviorProfileDouble]:
-    return _convert_mbpd(LcpBehaviorSolve[double](game.game, stop_after, max_depth))
+    return _convert_mbpd(LcpBehaviorSolve[double](game.game))
 
 
 def _lcp_behavior_solve_rational(
-        game: Game, stop_after: int, max_depth: int
+        game: Game
 ) -> list[MixedBehaviorProfileRational]:
-    return _convert_mbpr(LcpBehaviorSolve[c_Rational](game.game, stop_after, max_depth))
+    return _convert_mbpr(LcpBehaviorSolve[c_Rational](game.game))
 
 
 def _lcp_strategy_solve_double(

src/pygambit/nash.py

@@ -184,12 +184,13 @@ def lcp_solve(
         representation even if the game's native representation is extensive.
 
     stop_after : int, optional
-        Maximum number of equilibria to compute.  If not specified, computes all
-        accessible equilibria.
+        Maximum number of equilibria to compute when using the strategic representation.
+        If not specified, computes all accessible equilibria.
 
     max_depth : int, optional
-        Maximum depth of recursion.  If specified, will limit the recursive search,
-        but may result in some accessible equilibria not being found.
+        Maximum depth of recursion when using the strategic representation.
+        If specified, will limit the recursive search, but may result in some accessible
+        equilibria not being found.
 
     Returns
     -------
@@ -200,7 +201,19 @@ def lcp_solve(
     ------
     RuntimeError
         If game has more than two players.
+
+    ValueError
+        If stop_after or max_depth are supplied for use on the tree representation.
     """
+    if game.is_tree and not use_strategic:
+        if stop_after:
+            raise ValueError(
+                "lcp_solve(): stop_after can only be used on the strategic representation"
+            )
+        if max_depth:
+            raise ValueError(
+                "lcp_solve(): max_depth can only be used on the strategic representation"
+            )
     if stop_after is None:
         stop_after = 0
     elif stop_after < 0:
@@ -215,9 +228,9 @@ def lcp_solve(
         else:
             equilibria = libgbt._lcp_strategy_solve_double(game, stop_after or 0, max_depth or 0)
     elif rational:
-        equilibria = libgbt._lcp_behavior_solve_rational(game, stop_after or 0, max_depth or 0)
+        equilibria = libgbt._lcp_behavior_solve_rational(game)
     else:
-        equilibria = libgbt._lcp_behavior_solve_double(game, stop_after or 0, max_depth or 0)
+        equilibria = libgbt._lcp_behavior_solve_double(game)
     return NashComputationResult(
         game=game,
         method="lcp",

src/solvers/lcp/efglcp.cc

@@ -29,9 +29,8 @@ namespace Gambit::Nash {
 
 template <class T> class NashLcpBehaviorSolver {
 public:
-  NashLcpBehaviorSolver(int p_stopAfter, int p_maxDepth,
-                        BehaviorCallbackType<T> p_onEquilibrium = NullBehaviorCallback<T>)
-    : m_onEquilibrium(p_onEquilibrium), m_stopAfter(p_stopAfter), m_maxDepth(p_maxDepth)
+  NashLcpBehaviorSolver(BehaviorCallbackType<T> p_onEquilibrium = NullBehaviorCallback<T>)
+    : m_onEquilibrium(p_onEquilibrium)
   {
   }
   ~NashLcpBehaviorSolver() = default;
@@ -144,98 +143,23 @@ std::list<MixedBehaviorProfile<T>> NashLcpBehaviorSolver<T>::Solve(const Game &p
   solution.eps = tab.Epsilon();
 
   try {
-    if (m_stopAfter != 1) {
-      try {
-        AllLemke(p_game, solution.ns1 + solution.ns2 + 1, tab, 0, A, solution);
-      }
-      catch (EquilibriumLimitReached &) {
-        // Handle this silently; equilibria are recorded as found so no action needed
-      }
-    }
-    else {
-      tab.Pivot(solution.ns1 + solution.ns2 + 1, 0);
-      tab.SF_LCPPath(solution.ns1 + solution.ns2 + 1);
-      solution.AddBFS(tab);
-      Vector<T> sol(tab.MinRow(), tab.MaxRow());
-      tab.BasisVector(sol);
-      MixedBehaviorProfile<T> profile(p_game);
-      GetProfile(tab, profile, sol, p_game->GetRoot(), 1, 1, solution);
-      profile.UndefinedToCentroid();
-      solution.m_equilibria.push_back(profile);
-      this->m_onEquilibrium(profile, "NE");
-    }
+    tab.Pivot(solution.ns1 + solution.ns2 + 1, 0);
+    tab.SF_LCPPath(solution.ns1 + solution.ns2 + 1);
+    solution.AddBFS(tab);
+    Vector<T> sol(tab.MinRow(), tab.MaxRow());
+    tab.BasisVector(sol);
+    MixedBehaviorProfile<T> profile(p_game);
+    GetProfile(tab, profile, sol, p_game->GetRoot(), 1, 1, solution);
+    profile.UndefinedToCentroid();
+    solution.m_equilibria.push_back(profile);
+    this->m_onEquilibrium(profile, "NE");
   }
   catch (std::runtime_error &e) {
     std::cerr << "Error: " << e.what() << std::endl;
   }
   return solution.m_equilibria;
 }
 
-//
-// All_Lemke finds all accessible Nash equilibria by recursively
-// calling itself.  List maintains the list of basic variables
-// for the equilibria that have already been found.
-// From each new accessible equilibrium, it follows
-// all possible paths, adding any new equilibria to the List.
-//
-template <class T>
-void NashLcpBehaviorSolver<T>::AllLemke(const Game &p_game, int j, linalg::LemkeTableau<T> &B,
-                                        int depth, Matrix<T> &A, Solution &p_solution) const
-{
-  if (m_maxDepth != 0 && depth > m_maxDepth) {
-    return;
-  }
-
-  Vector<T> sol(B.MinRow(), B.MaxRow());
-  MixedBehaviorProfile<T> profile(p_game);
-
-  bool newsol = false;
-  for (int i = B.MinRow(); i <= B.MaxRow() && !newsol; i++) {
-    if (i == j) {
-      continue;
-    }
-
-    linalg::LemkeTableau<T> BCopy(B);
-    // Perturb tableau by a small number
-    A(i, 0) = static_cast<T>(-1) / static_cast<T>(1000);
-    BCopy.Refactor();
-
-    int missing;
-    if (depth == 0) {
-      BCopy.Pivot(j, 0);
-      missing = -j;
-    }
-    else {
-      missing = BCopy.SF_PivotIn(0);
-    }
-
-    newsol = false;
-
-    if (BCopy.SF_LCPPath(-missing) == 1) {
-      newsol = p_solution.AddBFS(BCopy);
-      BCopy.BasisVector(sol);
-      GetProfile(BCopy, profile, sol, p_game->GetRoot(), 1, 1, p_solution);
-      profile.UndefinedToCentroid();
-      if (newsol) {
-        m_onEquilibrium(profile, "NE");
-        p_solution.m_equilibria.push_back(profile);
-        if (m_stopAfter > 0 && p_solution.EquilibriumCount() >= m_stopAfter) {
-          throw EquilibriumLimitReached();
-        }
-      }
-    }
-    else {
-      // Dead end
-    }
-
-    A(i, 0) = static_cast<T>(-1);
-    if (newsol) {
-      BCopy.Refactor();
-      AllLemke(p_game, i, BCopy, depth + 1, A, p_solution);
-    }
-  }
-}
-
 template <class T>
 void NashLcpBehaviorSolver<T>::FillTableau(Matrix<T> &A, const GameNode &n, T prob, int s1, int s2,
                                            Solution &p_solution) const
@@ -341,16 +265,15 @@ void NashLcpBehaviorSolver<T>::GetProfile(const linalg::LemkeTableau<T> &tab,
 }
 
 template <class T>
-std::list<MixedBehaviorProfile<T>> LcpBehaviorSolve(const Game &p_game, int p_stopAfter,
-                                                    int p_maxDepth,
+std::list<MixedBehaviorProfile<T>> LcpBehaviorSolve(const Game &p_game,
                                                     BehaviorCallbackType<T> p_onEquilibrium)
 {
-  return NashLcpBehaviorSolver<T>(p_stopAfter, p_maxDepth, p_onEquilibrium).Solve(p_game);
+  return NashLcpBehaviorSolver<T>(p_onEquilibrium).Solve(p_game);
 }
 
-template std::list<MixedBehaviorProfile<double>> LcpBehaviorSolve(const Game &, int, int,
+template std::list<MixedBehaviorProfile<double>> LcpBehaviorSolve(const Game &,
                                                                   BehaviorCallbackType<double>);
 template std::list<MixedBehaviorProfile<Rational>>
-LcpBehaviorSolve(const Game &, int, int, BehaviorCallbackType<Rational>);
+LcpBehaviorSolve(const Game &, BehaviorCallbackType<Rational>);
 
 } // end namespace Gambit::Nash

src/solvers/lcp/lcp.h

@@ -34,7 +34,7 @@ LcpStrategySolve(const Game &p_game, int p_stopAfter, int p_maxDepth,
 
 template <class T>
 std::list<MixedBehaviorProfile<T>>
-LcpBehaviorSolve(const Game &p_game, int p_stopAfter, int p_maxDepth,
+LcpBehaviorSolve(const Game &p_game,
                  BehaviorCallbackType<T> p_onEquilibrium = NullBehaviorCallback<T>);
 
 } // end namespace Gambit::Nash

src/tools/lcp/lcp.cc

@@ -149,14 +149,14 @@ int main(int argc, char *argv[])
       if (useFloat) {
         auto renderer =
             MakeMixedBehaviorProfileRenderer<double>(std::cout, numDecimals, printDetail);
-        LcpBehaviorSolve<double>(game, stopAfter, maxDepth,
+        LcpBehaviorSolve<double>(game,
                                  [&](const MixedBehaviorProfile<double> &p,
                                      const std::string &label) { renderer->Render(p, label); });
       }
       else {
         auto renderer =
             MakeMixedBehaviorProfileRenderer<Rational>(std::cout, numDecimals, printDetail);
-        LcpBehaviorSolve<Rational>(game, stopAfter, maxDepth,
+        LcpBehaviorSolve<Rational>(game,
                                    [&](const MixedBehaviorProfile<Rational> &p,
                                        const std::string &label) { renderer->Render(p, label); });
       }