Return just a sample equilibrium for any set of realisation-equivalent equilibria.

Author: tturocy

doc/tools.enumpoly.rst

@@ -24,6 +24,19 @@ supports which have the fewest strategies in total.  For many classes
 of games, this will tend to lower the average time until finding one equilibrium,
 as well as finding the second equilibrium (if one exists).
 
+For extensive games, a support of actions equates to allowing positive
+probabilities over a subset of terminal nodes.  The indifference conditions
+used are those for the sequence form defined on the projection of the game
+to that support of actions.  A solution to these equations implies a probability
+distribution over terminal nodes.  The algorithm then searches for
+a profile which is a Nash equilibrium which implements that probability
+distribution.  If there exists such a profile, a sample one is returned.
+Note that for probability distributions which assign zero probability to some terminal
+nodes, it is generally the case that there are (infinitely) many such profiles.
+Subsequent analysis of unreached information sets can yield alternative
+profiles which specify different choices at unreached information sets
+while satisfying the Nash equilibrium conditions.
+
 When the verbose switch `-v` is used, the program outputs each support
 as it is considered. The supports are presented as a comma-separated
 list of binary strings, where each entry represents one player. The

src/solvers/enumpoly/efgpoly.cc

@@ -151,79 +151,37 @@ std::map<GameSequence, double> ToSequenceProbs(const ProblemData &p_data, const
   return x;
 }
 
-/// Compute the set of information sets which are not reachable given the actions in
-/// @p p_support, but are reachable via a *single* deviation to an action at a
-/// reachable information set.
-std::set<GameInfoset> FindDeviationInfosets(const BehaviorSupportProfile &p_support)
+std::optional<MixedBehaviorProfile<double>>
+FindNashExtension(const MixedBehaviorProfile<double> &p_baseProfile, double p_maxRegret)
 {
-  struct SingleDeviationReachableVisitor {
-    const BehaviorSupportProfile &m_support;
-    std::set<GameInfoset> m_deviationReachable;
-
-    explicit SingleDeviationReachableVisitor(const BehaviorSupportProfile &p_support)
-      : m_support(p_support)
-    {
-    }
-    GameRep::DFSCallbackResult OnEnter(const GameNode &p_node, int)
-    {
-      const auto infoset = p_node->GetInfoset();
-      if (!infoset) {
-        return GameRep::DFSCallbackResult::Continue;
-      }
-      if (p_node->GetPlayer()->IsChance()) {
-        return GameRep::DFSCallbackResult::Continue;
-      }
-      if (m_support.IsReachable(infoset)) {
-        return GameRep::DFSCallbackResult::Continue;
-      }
-      m_deviationReachable.insert(infoset);
-      return GameRep::DFSCallbackResult::Prune;
-    }
-    GameRep::DFSCallbackResult OnAction(const GameNode &, const GameNode &, int)
-    {
-      return GameRep::DFSCallbackResult::Continue;
+  const Game &game = p_baseProfile.GetGame();
+  std::list<GameInfoset> extensionInfosets;
+  for (const auto &infoset : game->GetInfosets()) {
+    if (!p_baseProfile.IsDefinedAt(infoset)) {
+      extensionInfosets.push_back(infoset);
     }
-    GameRep::DFSCallbackResult OnExit(const GameNode &, int)
-    {
-      return GameRep::DFSCallbackResult::Continue;
-    }
-    void OnVisit(const GameNode &, int) {}
-  };
-
-  SingleDeviationReachableVisitor visitor(p_support);
-  const Game game = p_support.GetGame();
-  GameRep::WalkDFS(game, game->GetRoot(), TraversalOrder::Preorder, visitor);
-  return visitor.m_deviationReachable;
-}
-
-/// Produce the set of mixed behavior profiles which extend @param p_baseProfile
-/// to complete profiles by specifying a pure action at each information set which
-/// is reachable by a single deviation from the profile, and the centroid at all
-/// information sets which are reachable only by two deviations.
-std::list<MixedBehaviorProfile<double>>
-ExtendWithDeviations(const MixedBehaviorProfile<double> &p_baseProfile)
-{
-  const auto deviationInfosets = FindDeviationInfosets(p_baseProfile.GetSupport());
-  std::list<MixedBehaviorProfile<double>> result;
-  Array<int> firstIndex(deviationInfosets.size());
+  }
+  Array<int> firstIndex(extensionInfosets.size());
   std::fill(firstIndex.begin(), firstIndex.end(), 1);
-  Array<int> lastIndex(deviationInfosets.size());
-  std::transform(deviationInfosets.begin(), deviationInfosets.end(), lastIndex.begin(),
+  Array<int> lastIndex(extensionInfosets.size());
+  std::transform(extensionInfosets.begin(), extensionInfosets.end(), lastIndex.begin(),
                  [](const auto &infoset) { return infoset->GetActions().size(); });
   CartesianIndexProduct indices(firstIndex, lastIndex);
   for (const auto &index : indices) {
     auto extension = p_baseProfile.ToFullSupport();
-    for (auto [i, infoset] : enumerate(deviationInfosets)) {
+    for (auto [i, infoset] : enumerate(extensionInfosets)) {
       extension[infoset->GetAction(index[i + 1])] = 1.0;
     }
-    extension.UndefinedToCentroid();
-    result.push_back(extension);
+    if (extension.GetMaxRegret() < p_maxRegret) {
+      return extension;
+    }
   }
-  return result;
+  return std::nullopt;
 }
 
 std::list<MixedBehaviorProfile<double>> SolveSupport(const BehaviorSupportProfile &p_support,
-                                                     bool &p_isSingular, int p_stopAfter)
+                                                     bool &p_isSingular, int p_stopAfter,
+                                                     double p_maxRegret)
 {
   ProblemData data(p_support);
   PolynomialSystem<double> equations(data.space);
@@ -252,7 +210,10 @@ std::list<MixedBehaviorProfile<double>> SolveSupport(const BehaviorSupportProfil
   for (const auto &root : roots) {
     const MixedBehaviorProfile<double> sol(
         data.m_support.ToMixedBehaviorProfile(ToSequenceProbs(data, root)));
-    solutions.splice(solutions.end(), ExtendWithDeviations(sol));
+    auto extended = FindNashExtension(sol, p_maxRegret);
+    if (extended.has_value()) {
+      solutions.push_back(extended.value());
+    }
   }
   return solutions;
 }
@@ -277,12 +238,11 @@ EnumPolyBehaviorSolve(const Game &p_game, int p_stopAfter, double p_maxregret,
   for (auto support : possible_supports->m_supports) {
     p_onSupport("candidate", support);
     bool isSingular = false;
-    for (const auto &solution : SolveSupport(
-             support, isSingular, std::max(p_stopAfter - static_cast<int>(ret.size()), 0))) {
-      if (solution.GetMaxRegret() < p_maxregret) {
-        p_onEquilibrium(solution);
-        ret.push_back(solution);
-      }
+    for (const auto &solution :
+         SolveSupport(support, isSingular, std::max(p_stopAfter - static_cast<int>(ret.size()), 0),
+                      p_maxregret)) {
+      p_onEquilibrium(solution);
+      ret.push_back(solution);
     }
     if (isSingular) {
       p_onSupport("singular", support);

tests/test_nash.py

@@ -2122,8 +2122,7 @@ def test_nash_strategy_solver_w_start(test_case: EquilibriumTestCaseWithStart, s
             factory=functools.partial(games.read_from_file, "3_player.efg"),
             solver=functools.partial(gbt.nash.enumpoly_solve, stop_after=None),
             expected=[
-                [[d(1, 0), d(1, 0)], [d(1, 0), d("1/2", "1/2")], [d(1, 0), d(1, 0)]],
-                [[d(1, 0), d(1, 0)], [d(1, 0), d("1/2", "1/2")], [d(1, 0), d(0, 1)]],
+                [[d(1, 0), d(1, 0)], [d(1, 0), d(1, 0)], [d(1, 0), d(1, 0)]],
             ],
             regret_tol=TOL,
             prob_tol=TOL,
@@ -2136,8 +2135,7 @@ def test_nash_strategy_solver_w_start(test_case: EquilibriumTestCaseWithStart, s
             factory=functools.partial(games.read_from_file, "3_player_with_nonterm_outcomes.efg"),
             solver=functools.partial(gbt.nash.enumpoly_solve, stop_after=None),
             expected=[
-                [[d(1, 0), d(1, 0)], [d(1, 0), d("1/2", "1/2")], [d(1, 0), d(1, 0)]],
-                [[d(1, 0), d(1, 0)], [d(1, 0), d("1/2", "1/2")], [d(1, 0), d(0, 1)]],
+                [[d(1, 0), d(1, 0)], [d(1, 0), d(1, 0)], [d(1, 0), d(1, 0)]],
             ],
             regret_tol=TOL,
             prob_tol=TOL,
@@ -2170,10 +2168,8 @@ def test_nash_strategy_solver_w_start(test_case: EquilibriumTestCaseWithStart, s
                 # candidate,10,10,1000,10000
                 [[d(1, 0)], [d(1, 0), d(1, 0, 0, 0)], [d(1, 0, 0, 0, 0)]],
                 # candidate,01,00,0000,00000
-                [[d(0, 1)], [d(1, 0), d("1/4", "1/4", "1/4", "1/4")],
-                 [d("1/5", "1/5", "1/5", "1/5", "1/5")]],  # only 1 off path
-                [[d(0, 1)], [d(0, 1), d("1/4", "1/4", "1/4", "1/4")],
-                 [d("1/5", "1/5", "1/5", "1/5", "1/5")]],
+                [[d(0, 1)], [d(1, 0), d(1, 0, 0, 0)],
+                 [d(1, 0, 0, 0, 0)]],
             ],
             regret_tol=TOL,
             prob_tol=TOL,
@@ -2191,13 +2187,8 @@ def test_nash_strategy_solver_w_start(test_case: EquilibriumTestCaseWithStart, s
                 [[d(1, 0)], [d(1, 0), d(1, 0, 0, 0)], [d(1, 0, 0, 0, 0)]],
                 [
                     [d(0, 1)],
-                    [d(1, 0), d("1/4", "1/4", "1/4", "1/4")],
-                    [d("1/5", "1/5", "1/5", "1/5", "1/5")],
-                ],
-                [
-                    [d(0, 1)],
-                    [d(0, 1), d("1/4", "1/4", "1/4", "1/4")],
-                    [d("1/5", "1/5", "1/5", "1/5", "1/5")],
+                    [d(1, 0), d(1, 0, 0, 0)],
+                    [d(1, 0, 0, 0, 0)],
                 ],
             ],
             regret_tol=TOL,
@@ -2216,13 +2207,8 @@ def test_nash_strategy_solver_w_start(test_case: EquilibriumTestCaseWithStart, s
             expected=[
                 [
                     [d(0, 1)],
-                    [d(1, 0), d("1/4", "1/4", "1/4", "1/4")],
-                    [d("1/5", "1/5", "1/5", "1/5", "1/5")],
-                ],
-                [
-                    [d(0, 1)],
-                    [d(0, 1), d("1/4", "1/4", "1/4", "1/4")],
-                    [d("1/5", "1/5", "1/5", "1/5", "1/5")],
+                    [d(1, 0), d(1, 0, 0, 0)],
+                    [d(1, 0, 0, 0, 0)],
                 ],
             ],
             regret_tol=TOL,
@@ -2239,8 +2225,7 @@ def test_nash_strategy_solver_w_start(test_case: EquilibriumTestCaseWithStart, s
             ),
             solver=functools.partial(gbt.nash.enumpoly_solve, stop_after=None),
             expected=[
-                [[d(0, 1), d("1/5", "1/5", "1/5", "1/5", "1/5")], [d(1, 0)]],
-                [[d(0, 1), d("1/5", "1/5", "1/5", "1/5", "1/5")], [d(0, 1)]],
+                [[d(0, 1), d(1, 0, 0, 0, 0)], [d(1, 0)]],
             ],
             regret_tol=TOL,
             prob_tol=TOL,