博弈论习题集.docVIP

PROBLEMSETIOFGAMETHEORY

Statewhetherthefollowinggameshaveuniquepurestrategysolutions,andifsowhattheyareandhowtheycanbefound.

Player2

Left

Middle

Right

Up

4,3

2,7

0,4

Down

5,5

5,-1

-4,-2

Player1

Player2

Left

Middle

Right

Up

4,10

3,0

1,3

Down

0,0

2,10

1,3

Player1

Player2

Left

Middle

Right

Up

10,10

4,3

7,2

Down

5,6

8,10

6,12

Player1

Drawthenormalformgameforthefollowinggameandidentifyboththepure-andmixed-strategyequilibria.Inthemixed-strategyNashequilibriumdetermineeachfirm’sexpectedprofitlevelifitentersthemarket.

Therearetwofirmsthatareconsideringenteringanewmarket,andmustmaketheirdecisionwithoutknowingwhattheotherfirmhasdone.Unfortunatelythemarketisonlybigenoughtosupportoneofthetwofirms.Ifbothfirmsenterthemarket,thentheywilleachmakealossof￡onlyonefirmentersthemarket,thatfirmwillearnaprofitof￡50m,andtheotherfirmwilljustbreakeven.

Convertthefollowingextensiveformgameintoanormalformgame,andidentifytheNashequilibriaandsubgameperfectNashequilibria.Finally,whatistheNashequilibriumifbothplayersmaketheirmovessimultaneously

Consideraneconomyconsistingofonegovernmentandtwopeople.Letxibethechoiceofthepeople,wherexi∈X={xL,xM,xH},andi=1,2，andythechoiceofthegovernment,wherey∈Y={yL,yM,yH}.Thepayoffstothegovernment-householdaregivenbythevaluesofu1(x1,x2,y)andu2(x1,x2,y)=u1(x2,x1,y).Thesepayoffsareenteredinthefollowingtable:

One-periodpayoffstothegovernment-household(valuesofu(x1,x2,y))

(x1,x2,)

y

(xL,xL)

(xM,xM)

(xH,xH)

(xL,xH)

(xH,xL)

(xL,xM)

(xM,xL)

(xM,xH)

(xH,xM)

yL

3*

7

9

YM

5

10*

20

yH

0

4

15*

Suppose(*)denotesthat(x1,x2)isaNashequilibriumgiveny,thegovernment’spolicy.Entertheblankwithvaluerangessuch