Note on Game Theory

When participants arrived at hunt headquarters, a game of Kill Dr. Lucky began, with Dr. Lucky beginning in room 2. As the game proceeded, instead of moving in his usual fashion, Dave moved Dr. Lucky successively from room 2 to 7 to 17 to 8 to 18 to 19 to 14 to 15 to 7 to 4 and then back to 2, looping through these rooms again. If a player used a card to move Dr. Lucky, then at the end of the turn Dr. Lucky would jump back to room 2.

At the end of the game, the winner was given the following hint.