I am writing a chess engine and I require an efficient way of merging multiple lists into a singular list ordered by the smallest value.

Each list is quintessentially a group of chess pieces that are already ordered in perfect attack order. For example I have a White Queen on a2, White Bishop on b3, White Rook on f1 and White Rook on f2. Now say I have a Black Pawn on f7 then all four White pieces are converging on the f7 square from two different discrete directions - North East (Queen & Bishop) and North (Rooks).

These two groups would be ordered as follows:

Group A) 1st - Bishop (b3); 2nd - Queen (a2)
Group B) 1st - Rook (f2); 2nd - Rook (f1)

Now using the points system below I would expect both lists to be merged into a single list in the following order (lowest value to highest value): Bishop (b3), Rook (f2), Rook (f1) and finally Queen (a2).

Queen = 900 pts
Rook = 500 pts
Bishop = 375 pts
Knight = 350 pts
Pawn = 100 pts

Some code:

using System;
using System.Collections.Generic;
using System.Linq;
using System.Text;
using System.Threading.Tasks;

namespace SO_22015528
{
  public enum eRayDirection
  {
    North,
    NorthEast,
    East,
    SouthEast,
    South,
    SouthWest,
    West,
    NorthWest
  }

  public enum ePieceType
  {
    Empty = 0,
    Pawn = 1,
    Knight = 2,
    Bishop = 3,
    Rook = 4,
    Queen = 5,
    King = 6
  }

  public struct BoardPosition : IComparable<BoardPosition>
  {
    public int File;
    public int Rank;
    public int Square;

    public BoardPosition( int square )
    {
      Square = square;
      File = square % 7;
      Rank = 8 - ( square >> 3 );
    }

    public static Boolean operator >( BoardPosition b1, BoardPosition b2 )
    {
      return ( b1.Rank > b2.Rank ) || ( b1.Rank == b2.Rank && b1.File < b2.File );
    }

    public static Boolean operator <( BoardPosition b1, BoardPosition b2 )
    {
      return ( b1.Rank < b2.Rank ) || ( b1.Rank == b2.Rank && b1.File > b2.File );
    }

    public int CompareTo( BoardPosition obj )
    {
      if ( this < obj ) return 1;
      else if ( this > obj ) return -1;
      else return 0;
    }
  }

  public class ChessPiece
  {
    public int Value { get; set; }
    public ePieceType Type { get; set; }
    public int Square { get; set; }
    public BoardPosition XY
    {
      get
      {
        return new BoardPosition( this.Square );
      }
    }
    public ChessPiece( ePieceType type, int value, int square )
    {
      Value = value;
      Type = type;
      Square = square;
    }
  }

  public class Constraint
  {
    public ChessPiece Piece { get; set; }
    public eRayDirection Ray { get; set; }
    public Constraint( ChessPiece piece, eRayDirection ray )
    {
      Piece = piece;
      Ray = ray;
    }
    public override string ToString()
    {
      return String.Format( "{0} {1} {2}", Piece.Square, Piece.Type, Ray );
    }
  }

}

using System;
using System.Collections.Generic;
using System.Linq;
using System.Text;
using System.Threading.Tasks;

namespace SO_22015528
{
  class Program
  {
    static void Main( string[] args )
    {
      // test code
      ChessPiece a2 = new ChessPiece( ePieceType.Queen, 900, 48 );
      ChessPiece b3 = new ChessPiece( ePieceType.Bishop, 375, 41 );
      ChessPiece f1 = new ChessPiece( ePieceType.Rook, 500, 61 );
      ChessPiece f2 = new ChessPiece( ePieceType.Rook, 500, 53 );

      // This just simulates pieces that attack on the f7 square.
      List<Constraint> f7 = new List<Constraint>();
      f7.Add( new Constraint( b3, eRayDirection.NorthEast ) );
      f7.Add( new Constraint( a2, eRayDirection.NorthEast ) );
      f7.Add( new Constraint( f1, eRayDirection.North ) );
      f7.Add( new Constraint( f2, eRayDirection.North ) );

      // Get all positive ray directions ( use to simplify LINQ orderby )
      List<eRayDirection> positiveRayDirections = new List<eRayDirection>();
      positiveRayDirections.Add( eRayDirection.North );
      positiveRayDirections.Add( eRayDirection.NorthEast );
      positiveRayDirections.Add( eRayDirection.NorthWest );
      positiveRayDirections.Add( eRayDirection.West );

      var groups = f7
        .GroupBy( g => g.Ray )
        .Select( a =>
        new
        {
          Key = a.Key,
          Results = positiveRayDirections.Contains( a.Key ) ? a.OrderBy( x => x.Piece.XY ).ToList() : a.OrderByDescending( x => x.Piece.XY ).ToList()
        } ).ToList();

              // The groups object returns two discrete groups here;
              // NorthEast containing 2 entries (Bishop & Queen) and North
              // also containing to entries (Rook x 2).
      List<Constraint> attackOrder = new List<Constraint>();

      List<Int32> groupIndicies = new List<Int32>( groups.Count() );
      for ( int n = 0; n < groups.Count(); n++ )
        groupIndicies.Add( 0 );

      while ( true )
      {
        Int32 value = Int32.MaxValue;
        Int32 groupIndex = -1;

        for ( int n = 0; n < groups.Count(); n++ )
        {
          var g = groups[ n ];
          int gIndex = groupIndicies[ n ];

          if ( gIndex < g.Results.Count && g.Results[ gIndex ].Piece.Value < value )
          {
            value = g.Results[ gIndex ].Piece.Value;
            groupIndex = n;
          }
        }

        if ( groupIndex < 0 )
          break;

        attackOrder.Add( groups[ groupIndex ].Results[ groupIndicies[ groupIndex ] ] );

        groupIndicies[ groupIndex ]++;

      }

              foreach ( var ao in attackOrder )
                  Console.WriteLine( ao.ToString() );

      Console.ReadKey();
    }
  }
}

I do not think the last bit is very efficient and I would appreciate it if someone could find a much simpler way of doing this.

Order each list individually using quicksort and then sort by insert into the final list.

Order the lists individually.
Create the empty Final list.

do
{
   consider the first item in each of the sorted lists and find the highest ranking candidate.
   remove the item from its sorted list and add it to the final list.
}
until all of the sorted lists are empty.

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