Game Theory. Quizzes. 2014

Quiz 1. Sequential games

1. There are 6 pirates and they need to divide their loot – 100 golden coins. They have to divide their loot. The procedure:

1. The most senior one offers his division.

2. If the half or more agrees then it is a deal.

3. If not they kill the senior one and the new senior pirate offers his division. If they agree then deal, if not they kill him and new leader makes an offer.

4. It continues till the final decision.

How to maximize the profit of the 1^st pirate, if pirates are very rational, greedy and hate being fair or dead? Draw the tree [7] and show the path of the game. [2]

2. Two players take turns choosing a number between 1 and 10 (inclusive), and a cumulative total of their choices is kept. The player who causes the total to equal or exceed 100 is the winner. How to win this game if you begin? [5]

3. Use rollback to find equilibria for the following game [3]. How many pure strategies (complete plans of action) are available to each player? Write out all of the pure strategies for each player.[3]

Solutions.

1.

6^th – YES – [98, 0, 1, 0, 1, 0]

|

NO

|

5^th – YES – [X, 98, 0, 1, 0, 1]

|

NO

…..

2^nd – YES [X, X, X, X, 100, 0]

|

NO – [X, X, X, X, X, 100]

2. Start with 1 and then leave the opponent with 12, 23, 34, 45, 56, 67, 78, 89.

3.

(c) Each player is designated by the rules of the game to move at three nodes, and at

each of these there are 2 choices, so each has 2 ¥ 2 ¥ 2 = 8 available strategies. The lists are:

For Player A, (1) N1, N3, N5, (2) N1, N3, S5, (3) N1, S3, N5, (4) N1, S3, S5, (5) S1, N3,

N5, (6) S1, N3, S5, (7) S1, S3, N5, (8) S1, S3, S5. For Player B, (1) n2, n4, n6, (2) n2, n4, s6,

(3) n2, s4, n6, (4) n2, s4, s6, (5) s2, n4, n6, (6) s2, n4, n6, (7) s2, s4, n6, (8) s2, s4, s6.

c

Quiz 2. Simultaneous games I

1. Consider a game in which there is a prize worth $30. There are three contestants, A, B, and C. Each can buy a ticket worth $15 or $30 or not buy a ticket at all. They make these choices simultaneously and independently. Then, knowing the ticket-purchase decisions, the game organizer awards the prize. If no one has bought a ticket, the prize is not awarded. Otherwise, the prize is awarded to the buyer of the highest-cost ticket if there is only one such player or is split equally between two or three if there are ties among the highest-cost ticket buyers. Show this game in strategic form. [4] {Hint: you need three matrices}. Find all pure-strategy Nash equilibria. [4]

2. Solve this game using dominant strategies if possible, otherwise use iterated elimination of dominated strategies. [5]

		COLUMN
		Left	Middle	Right
ROW	Up	1	2	5
	Straight	2	4	3
	Down	1	3	3

3.Construct payoff matrix for your own two-player game that satisfies the following requirements:

• Each player has three strategies [1/2]

• There are no dominant strategies [1/2]

• The game cannot be solved using minimax [1]

• There are four pure-strategy Nash equilibria [4]

Anwers:

1. There are three ticket buyers, and each ticket buyer can do three things: not purchase a ticket (represented as $0), purchase a $15 ticket, and purchase a $30 ticket. To represent this game, we need three 3-by-3 tables, where each table represents the strategies of the first two players and one strategy of the third. In the table payoffs, the first number represents Larry’s payoff, the second number represents Curly’s payoff, and the third number represents Moe’s payoff. All payoffs are in dollars, with the dollar signs omitted to save space. Best responses are underlined.

Moe $0		Curly
		$0	$15	$30
Larry	$0	0, 0, 0	0, 15, 0	0, 0, 0
	$15	15, 0, 0	0, 0, 0	–15, 0, 0
	$30	0, 0, 0	0, –15, 0	–15, –15, 0

Moe $15		Curly
		$0	$15	$30
Larry	$0	0, 0, 15	0, 0, 0	0, 0, -15
	$15	0, 0, 0	–5, –5, –5	–15, 0, –15
	$30	0, 0, -15	0, –15, –15	–15, –15,–15

Moe $30		Curly
		$0	$15	$30
Larry	$0	0, 0, 0	0, –15, 0	0, –15, –15
	$15	–15, 0, 0	–15, –15, 0	–15, –15, –15
	$30	–15, 0, –15	–15, –15,–15	–20, –20, –20

Best-response analysis shows that there are no pure-strategy Nash equilibria for when any player spends $30 to purchase a ticket. There are six pure-strategy Nash equilibria. Three occur when two purchasers spend nothing, and the other spends $15. The other three have two players spending $15, and the third spends nothing.

2. There are no dominated strategies for Row. For Column, Left dominates Middle and Right. Thus these two strategies may be eliminated, leaving only Left. With only Left remaining, for Row, Straight dominates both Up and Down, so they are eliminated, making the pure-strategy Nash equilibrium (Straight, Left).

3. Many answers possible. An example.

   		Curly
   		X	Y	Z
   Larry	A	10, 12	1, -5	4, 11
   	B	6, 3	7, 8	4, 2
   	C	10, 8	5, 7	4, 8