1. Consider the game in extensive form shown in Figure 1 in which Player 1 owns vertices a,d,e,f, and g, and Player 2 owns vertices b and c.

a   P1

P2

P1        g 

(2, −1)     (0, 3)  (1, −1)    (−2, 3)      (1, 1)     (−2, 4)    (−3, 1)

Figure 1: Game in extensive form for Question 1

For vertex x ∈ {a,b,c,d,e,f,g}, use the label L北  if the left hand edge is chosen, R北  if the right hand edge is chosen, and C北  if there is only one possible move.

(a) Redraw the game in extensive form labelling all the edges accordingly.

(b) Describe the strategy set for each player, give the normal form of the game, and

find any pairs of pure strategies in equilibrium if the game is of

(i) perfect information;

(ii) imperfect information and Player 2 cannot distinguish between vertices b and

c, and Player 1 cannot distinguish between vertices d and e.

2. Consider the game with chance moves in extensive form shown in Figure 2 in which Player 1 cannot distinguish between the two vertices s/he owns. Describe the strategy set for each player and give the normal form of the game where the payoffs are expected payoffs.

C represents Chance or Nature.

Find all pairs of Nash equilibria in pure strategies, if they exist.

C

0.3                          0.7

P1                                                                                     P2

2                                                                                     1 0.5        0.5                                     0.6         0.4

(0 ,0)        (1 ,−2) (−2,1)       (3,1)    (2 ,−2)      (3,0)   (− 1,2)      (1 ,− 1)

Figure 2: Game with chance moves in extensive form 

3. Consider the following two-person zero-sum game       V  =  ] .

(a) Determine the value of the game, as well as a pair of optimal strategies (北 1(*) , y* ).

(b) Show that x2(*)  = (  , ) is an optimal mixed strategy for player 1, and that x = (  , ) is not an optimal mixed strategy for player 1.

xt(*)  = t x 1(*) + (1 − t) x2(*) ,        0 ≤ t ≤ 1

of the two optimal strategies x 1(*)  and x2(*)  is also an optimal strategy for player 1. Using this result, provide a third optimal strategy for player 1.

4. Consider the following two-person constant-sum game      l   」 「     .

Use the linear programming method, possibly in combination with other methods (eg. saddle points, dominance elimination, 2×2 formulae, etc.) when necessary, to de- termine the values and optimal strategies for both players of this two-person constant- sum game.