Question

2. consider the following simultaneous move game.

		Player B
		LEFT	RIGHT
Player A	UP	4,1	1,4
	DOWN	2,3	3,2

a. If there is a Nash equilibrium in pure strategies, what is it and what are the payoffs?

b. If there is a Nash equilibrium in mixed strategies, what is it and what are the expected payoffs?

3. Continue with the previous game but suppose this was a sequential game where Player A got to go first.

a. Diagram the game in extensive form.

b. What are the Sub game perfect Nash equilibrium and what are the pay offs?

Explain all the answers.

Answer 1

Plagu B Left Right Playut 4,1 1,4 () ยาง 2,3 3,2 a when playn ? when Playn Lett? A choose Lupo, Playn B. Chooss Right? À choosu 'Downs playn o will choose Down. plays When Plyn A chooses 'Lett's Playn A will choose 'Up' When play" B chosn "Right' , Playn A will choose Huse there is no prue strategy wash equilibria. by Now to find mined strategis, suppose plays B - Left with Probility Probability a and Right with probability (1-2). thrice Play Playn A's eaputed papoft from "up' against [2, (2-2)] = q (4) + (1-2) (1) =4qt l-q - 34+1 and Playn A's expected pay off from Down' against [ q, (1-21] = 9(2)+(1-2 ) (3) 2 2q + 3-34 11 Scanned with CamScanner a 3-q

but these expected payo lsts me to be equal fince 39+1 = 3-4 .-) 4q=2 = q = 2 2 4 = Ź and (1-2) = (1+) = Ź Also suppose Player A plays 'up' with Probability P and 'Downs with probability (1-P). Auce, plays B's expected payable from Left against [ P, (1-P] = P(1) + (1-P)(3) - P + 3-3P 3- 2p and Playn B's enpected pay off from Right' against TP/l-p13 is = P(4) + (1-P) (2) - 4P +2-20 = 2972 . But these enpected payofts me to be equal ? hance 3-2p= 2 Pt2 ! 2) -4P = -1 =) P = 4 and (1-P) = (1-4) = tince mined stategy hash equilibria. LTP(1-P)}ſa.creys] - [(+ >)(+5) Scanned with CamScanner

3 A az Now, huce, suppose dayn A mous brist. entrusive form of the que will be Playn. A Up / Down (14) PlayuB playn B (2,3) Left Right Left / \ Right (401) (1,4) (263) (3, 2) - b2 Now, by backward induction method , plays B chooses 'Right'. When playn A chooses up". and Playu B choosn Left' when playn A Choose Down' Now Playn A will choose betway the two outcomes that playn B to have chosm and will choose' Downs as of us manimises plagu As pay-off. . Hace, the sub que putect Nash equilibrium is (Down, Left) = (2,3) Scanned with CamScanner