?????Nash??

1
?????Nash??

• ????,????????????Nash?????????,??????????????,??
  ????????????
• ?????G????g???????,??g??????????????,????G???????g
  ???,??????g?
• ??Nash??????????,????? ????????Nash???

2
?????Nash??

• ??1 (?????)?????????,????????????????????????????
  ????????(a),?????(0,1)??????????,????????(A)???(B
  ),????????(-1,-1),????????(1,0)?????

3
??
a
b
??
(0,1)
A
B
(-1,-1)
(1,0)
4
?????Nash??

• ??????????

??
A
B
a
0, 1
0, 1
??
-1, -1
b
1, 0
5
?????Nash??

• ??1(?)??????,??????????Nash??(a,A),(b,B)????(a,A),
  ????????????,A????????,????????????NE??(a,A)????S
  PNE????A??????????(b,B),????,B?????????????NE,??(b
  ,B)????????

6
?????Nash??

• ???????????????,?????????????????????????????????
  ?????????????????????,?????????????????,????????H
  ????,?????????????H???????????????????????????????
  ?????????????????????????,???????????????????????

7
?????Nash??

• ???????????,?????????Nash?????????
• ?????????????????????????????Nash???
• ?????????????????,?????????????????,??????????(??
  ?)???(??)????????????????????????????,?????(??)??
  ???????????????????????

8
?????Nash??

• ????????????????????????????????????,????????????
  ?????
• ????????????????,???????????????????????Nash???
• ???????Nash??(SPNE)????????????SPNE??????2?
• ????????????????

9
Game Tree Examples

• In last Lectures we analyzed games in normal form
  All the dynamic aspects have been stripped
• Sometimes it is valuable to analyze games in
  extensive form with dynamics intact
• Example. Consider the following two-person
  non-zero sum game in extensive form to minimize
  costs

• Q. How to solve it?
• Two methods
• M1 Convert to normal form
• M2 Deal directly in extensive form

10

• Normal form analysis
• DM1 3 strategies, L, M, and R
• DM2 23 8 strategies
• Game in normal form

DM1\2 LLL LLR LRL LRR RLL RLR RRL RRR
L (0, -1) (0, -1) (0, -1) (0, -1) (-2, 1) (-2, 1) (-2, 1) (-2, 1)
M (3, 2) (3, 2) (0, 3) (0, 3) (3, 2) (3, 2) (0, 3) (0, 3)
R (2, 1) (-1, 0) (2, 1) (-1, 0) (2, 1) (-1, 0) (2, 1) (-1, 0)

• Q. Major difficulties?
• Dimensionality can be very large (Recall the DP
  example)
• Dynamic aspects are not appropriately considered
  
• Which of the four Nash solutions will actually
  happen?

11

• To overcome the difficulties, we shall analyze
  the extensive form directly. How?

• The solution is unique
• (0, -1) is not a solution since DM1 who acts
  first will not select L

• Extensive form is a reasonable approach for this
  problem

Q. If DM1 selected L, what should DM2 do? How
about if DM1 selected M or R?

• The solution process is backward induction
• Starting from leaf nodes and work backward until
  the root node is reached, each time solve a
  simple problem
• Then moving forward from the root to obtain the
  solution

12

• Example. With a slight variation

• If DM1 selected M or R, DM2 does not know how DM1
  acted

Q. How to solve it?

• Again there are two methods
• M1 Convert to a normal form
• M2 Deal directly in extensive form
• Normal form analysis How?
• DM1 3 strategies, L, M, and R
• DM2 22 4 strategies

13

• Game in normal form

• Q. Major difficulties?
• Same as before
• Dimensionality can be very large
• Dynamic aspects are not appropriately considered

14

• Q. To analyze the extensive form directly. How?

Q. At information set 2A, what should DM2 do?
How about at 2B? What should DM1 select?

• At 2A, DM2 should select L with costs (0, -1)

• At 2B, DM2 faces the following normal game

Q. What problem does DM1 face? How should he
select?

• The solution process is backward induction

15
An exercise on backward induction
16
Subgames and Subgame Perfection

• Subgames
• for any non-terminal history h is the part of the
  game that remains after h occurred.

Subgames

• Subgame perfect equilibrium No subgame can any
  player do better by choosing a different strategy

17
Some examples that is not subgames
2A
2B
18
Location GameExample Dynamic Game of Perfect
InformationGrocery Shopping on Market Street

• Market Street is a one-way street.
  

One-Way Market Street
1
100
Two firms locate grocery stores on Market Street
sequentially. That is, first firm 1 locates and
then firm 2.
19
Consumers live along streets 1-100.
N
W
i
1
2
3
99
100
Consumers drive to market street, then drive west
on market street (there are no left turns onto
market street) until they reach a grocery store.

20
Payoffs An example Firm 1 locates at 15
and 2 locates at 47.
Consumers 1 consumer uniformly distributed on
each street.
Firm 1
Firm 2
15
47
1
100
Since firm 1 gets all consumers who live on 15th
St, 16th St, 45th St and 46th St.

• ?1(15, 47) 47-15 32.
• ?2 (15, 47) 101-47 54

21
Now lets use backward induction to find all
subgame perfect Nash equilibrium.
Recall that subgame perfection is an equilibrium
refinement concept. If SGPNE then NE.
1
1
2
100
i
2
2
2
2
100
1
j
2
What are the pay-offs to Player 1? Player 2?
22
Payoffs ?i (i,j) 101-i if igtj
(101-i)/2 if ij
j-i
if iltj
i-j
(101-i)/2
101-i
Firm j
Firm i
Firm j
Firm i
ij
1
j
i
100
1
100
Payoffs ?j (i,j) i-j if igtj
(101-i)/2 if ij
101-j if
iltj
23
Backward Induction

• Fix a player 2 node i0 (player 1 has located at
  i0). What maximizes player 2s payoff?
• First note that player 2 will always want to be
  at 1, i0, or i0 1.
• For example suppose player 1 has located at 4
  (i.e. player 2 is at node 4). Where will 2 want
  to locate?
• Suppose player 1 has located at 75. Where will
  player 2 want to locate?

24
Solving the game via backward induction

• At node i , firm 2 plays j i1 if 1? i
  ? 50
• 
  j 1 if 51? i ? 100
• Back at firm is node
• ?1(i, j ) i1-i if 1? i ? 50
• 101-i if 51? i ? 100
• Therefore unique subgame perfect equilibrium is
• firm 1 plays 51 i
• firm 2 plays ji1 if 1? i ? 50 and
  j1 if 51? i ? 100 j
• Note the way in which the strategies are stated.

25
The end of market street

• Equilibrium path firm 1 plays 51, firm 2 plays
  1.
• Payoffs ?1(i, j ) 50 and ?2 (i, j) 50
• Also note that there is no first mover advantage.