To assess whether or not you have mastered the skill of taking a verbal description and converting it to a game tree, DRAW game trees with payoffs at terminal nodes for the following games:
1:Blotto Variant:Two player game. One player is A -- attacker. Second player is D - defender. Defender has two defend city with two gates North and South. Both sides have THREE army units, which they can position at either gate. So the Attacker can make the following moves: N-3, S-0 or N-2 S-1 0r N-1,S-2 or N-0 S-3 -- that is, all three units at North Gate, or two at North and one at South or one at North and three at south, or all three at South Gate. The defender has the same four choices. After both sides position their armies, the outcome is determined as follows. If Attacker has MORE units than defender at EITHER north gate or at south gate, then attacker wins, gaining payoff +1, and defender loses with payoff -1. OTHERWISE, defender wins with payoff +1 to defender and -1 to attacker. Draw the game tree if attacker moves first. How does the tree change if we make it a simultaneous move game?
2: Sharing Game: 6 Notes of PKR 1000 are on the table. Players A & B take turns picking up notes. Each player gets to keep what he picks up. A is first to move and can choose to pick one note, two notes or three notes. At any time, when either player picks up three notes, the game ends. Any money remaining on the table goes back to the experimenter, and the two players keep whatever they have picked up, up to this move. If A picks up only one or two note then the game continues and the turn passes to B. The same rules apply to B. He can pick up one, two or three notes. The game ends if he picks up three, and continues if he picks up one or two note. The game continues until either one of the two picks up three notes, or all the notes on the table have been picked up. Draw the game tree and the payoffs.
Nim Variant:
3. There are two piles, first pile contain 3 beads while second pile contains 4 beads, Now player B has to move first. Both of them can pick one or two bead from single pile at a time. The one who pick the last bead lose the game.Draw Game tree and tell which player will win the game if both play their strategies optimally.
4.There are three piles, first pile contain 2 beads, second pile contains 3 beads and the third pile contains 5 beads. Player A have to move first. Both of them can pick one or two bead from single pile at a time. The one who pick the last bead WIN the game. Draw the game tree and tell which player will win the game if both played their strategies optimally.
5. There is one heap, which contains eight pieces of stones. Each piece has some number written on it:
100, 400, 50,600,150,50,1000, 300
player A moves first and both player pick stones in sequence at their turn, but at a time they either pick one stone or two. game ends when last stone is reached, score is what each player pick. Draw game tree and their payoffs.