Quadraphage is a two-player game played on an infinite chessboard.  One player has a king, which starts on some square on the board, the other player has an unlimited supply of pawns, initially all off the board.  The pawn player places a pawn on any unoccupied square for his/her turn.  The king player moves the king to any adjacent unoccupied square for his/her turn.  The goal for the king player is to prevent the king from being trapped so it cannot move.  The goal for the pawn player is to surround the king so it cannot move.  Which player has the advantage?  What is the winning strategy for the player who has the advantage?

 
Two people can play on the board below, or one person can run both players.  The first step is for calibration purposes.  Click on the upper left corner of the upper left little square below.  The king shows as an x.  Pawns show as circles.  The pawn player clicks on a cell to place a pawn.  The king player uses numbers on the keypad to indicate the direction, 2 - down, 4 - left, 6 - right, 8 - up, 1, 3, 7, 9 for diagonal directions.  Hmm, some keyboards do not have keypads.  You could use the keys q w e a d z x c to indicate directions as well.  Alternatively, the king player can click on a cell to move to that cell.  Or, if the pawn player wishes to type, rather than mouse, the keypad can be used to choose a pawn position relative to the king.  Type digits 1-4 or 6-9 as many times as necessary (or q w e a d z x c) to move a tentative pawn play and then type 5 (or s) to place the pawn.  Either player can click on one of the chevrons around the board to shift the finite viewable portion of the board in one of four directions.  One can type an 'm' before the chevron click to multiply the shift by 2.  Repeating the m repeats the multiplication, up to 32.  This multiplication can also be used to move a tentative pawn play.  If it is the king's turn but the king is not visible, typing any key will shift the king into view.  If you wish to undo a move, type the letter 'u'.