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Unformatted text preview: Sequential Move Games Using Backward Induction
(Rollback) to Find Equilibrium Sequential Move Class Game:
Century Mark
• Played by fixed pairs of players taking turns.
• At each turn, each player chooses a number (an
integer) between 1 and 10 inclusive.
• This choice is added to sum of all previous
(
)
choices (the initial sum is 0).
• The first player to take the cumulative sum
greater 100 loses the game
game.
• No talking! Who are my first two volunteers?
cheesy game show music Analysis of the Game
• What is the winning strategy?
• Broadly speaking, bring the total to 89. Then,
your opponent cannot possibly win and you can
win for certain.
• The first mover can guarantee a win!
g
• How to do this: to get to 89, need to get to 78,
which can be done by getting to 67 56 45 34
67, 56, 45, 34,
23, 12, etc.
• Winning strategy: choose 11 minus the number
chosen by the other player – a complete plan of
action, strategy.
action or strategy Sequential Move Games with
Perfect Information
• Models of strategic situations where there is a
strict order of play.
i
d
f l
• Perfect information implies that players know
everything that has happened prior to making a
hi
h h h
d i
ki
decision.
• S
Sequential move games are most easily
ti l
t
il
represented in extensive form, that is, using a
game tree.
tree
• The investment game we played in class was an
example.
example Constructing a sequential move game
• Wh are th players?
Who
the l
?
• What are the action choices/strategies available to
each player.
• When does each player get to move?
p y g
• How much do they stand to gain/lose?
Example 1: The merger game. Suppose an industry has
game
six large firms (think airlines). Denote the largest
firm as firm 1 and the smallest firm as firm 6.
6
Suppose firm 1 proposes a merger with firm 6. Firm
2 must then decide whether to merge with firm 5.
5 The Merger Game Tree
Firm
Fi 1
Buy
Firm 6 Firm 2 Since Firm 1 moves
first, they are placed
Don’t Buy at the root node of
Firm 6
the game tree. Firm 2 Buy
Firm 5 Don’t Buy Buy
Firm 5
Firm 5 1A, 2A 1B, 2B 1C, 2C Don’t Buy
Firm 5
1D, 2D • What payoff values do you assign to firm 1 s payoffs 1A, 1B, 1C, 1D?
1’s
1A 1B 1C
To firm 2’s payoffs 2A, 2B, 2C, 2D? Think about the relative
profitability of the two firms in the four possible outcomes, or
terminal nodes of the tree. Use your economic intuition to rank the
outcomes for each firm. Assigning P ff
A i i Payoffs
Firm 1
Don’t Buy
Firm 6 Buy
Firm 6 Firm 2 Firm 2 Buy
Firm 5 Don’t Buy Buy
Firm 5
Firm 5 1A, 2A 1B, 2B 1C, 2C Don’t Buy
Firm 5
1D, 2D • Fi 1’s Ranking: 1B > 1A > 1D > 1C U 4 3 2 1
Firm 1’ R ki
1C. Use 4, 3, 2,
• Firm 2’s Ranking: 2C > 2A > 2D > 2B. Use 4, 3, 2, 1 The C
Th Completed Game Tree
l dG
T
Firm 1
Don’t Buy
Firm
Fi 6 Buy
Firm 6 Firm 2
Buy
Firm 5
3, 3 Firm 2
Don’t Buy Buy
Firm 5
Firm 5
4, 1 1, 4 • What is the equilibrium? Why? Don’t Buy
Firm 5
2, 2 Example 2: The Senate Race Game
• Incumbent Senator Gray will run for reelection. The
challenger is Congresswoman Green.
• S t G
Senator Gray moves fi t and must decide whether or not
first, d
t d id h th
t
to run advertisements early on.
• The challenger Green moves second and must decide
whether or not to enter the race.
• Issues to think about in modeling the game:
–
–
–
– Players are Gray and Green. Gray moves first.
Strategies for Gray are Ads, No Ads; for Green: In or Out.
g
y
Ads are costly, so Gray would prefer not to run ads.
Green will find it easier to win if Gray does not run ads. Computer Screen View What are the strategies?
•
•
• A pure strategy for a player is a complete plan of action that
specifies the choice to be made at each decision node.
Gray h t pure strategies: Ads or No Ads.
G
has two
t t i Ad
N Ad
Green has four pure strategies:
1.
1
2.
3.
4. •
• If Gray chooses Ads, choose I and if G
G
h
Ad h
In d Gray chooses N Ad choose I
h
No Ads h
In.
If Gray chooses Ads, choose Out and if Gray chooses No Ads choose In.
If Gray chooses Ads, choose In and if Gray chooses No Ads choose Out.
If Gray chooses Ads, choose Out and if Gray chooses No Ads choose Out. Summary: Gray’s pure strategies, Ads, No Ads.
y
y p
g
Greens’ pure strategies: (In, In), (Out, In), (In, Out), (Out, Out). Using Rollback or Backward Induction
to find the Equilibrium of a Game
• Suppose there are two players A and B. A moves first and B moves second.
B
second
• Start at each of the terminal nodes of the game tree. What action will the last
player to move, player B choose starting from the immediate prior decision
node of the tree?
• Compare the payoffs player B receives at the terminal nodes, and assume
player B always chooses the action giving him the maximal payoff.
• Place an arrow on these branches of the tree. Branches without arrows are
“pruned” away.
• Now treat the nexttolast decision node of the tree as the terminal node.
Given player B’s choices, what action will player A choose? Again assume
that player A always chooses the action giving her the maximal payoff.
Place an arrow on these branches of the tree.
• Continue rolling back in this same manner until you reach the root node of
the tree. The path indicated by your arrows is the equilibrium path. Illustration f B k d I d ti i
Ill t ti of Backward Induction in
Senate Race Game: Green’s Best Response
p Illustration of Backward Induction in
Senate Race Game: Gray’s Best Response This is the Nash
equilibrium Is There a First Mover Advantage?
• Suppose the sequence of play in the Senate Race Game is
changed so that Green gets to move first. The payoffs for
first
the four possible outcomes are exactly the same as before,
except now, Green’s payoff is listed first. Whether there i fi t
Wh th th is a first mover advantage
d t
depends on the g
p
game.
• To see if the order matters, rearrange the sequence of
moves as in the senate race game.
• Other examples in which order may matter:
– Adoption of new technology. Better to be first or last?
– Cl presentation of a project. Better to be first or last?
Class
t ti
f
j t B tt t b fi t l t? • Sometimes order does not matter. For example, is there a
first mover advantage in the merger g
g
g game as we have
modeled it? Why or why not?
• Is there such a thing as a second mover advantage?
– Sometimes, for example:
• Sequential biding by two contractors.
• Cakecutting: One person cuts, the other gets to decide how the two
g
p
g
pieces are allocated. Adding more players
• Game becomes more complex.
p
• Backward induction, rollback can still be
used to determine the equilibrium.
equilibrium
• Example: The merger game. There are 6
firms.
– If firms 1 and 2 make offers to merge with
firms 5 and 6, what should firm 3 do?
– Make an offer to merge with firm 4?
– Depends on the payoffs. 3 Player Merger Game
Firm 1
Buy
Firm 6 Don’t Buy
Firm 6 Firm 2 Firm 2 Buy
Firm 5 Don t
Don’t Buy Buy
Firm 5
Firm 5 Firm 3
Buy Firm 4 (2,2,2)
(2 2 2) Firm 3 Don’t
Buy
Buy Firm 4
B
Firm4
(4,4,1)
(4 4 1) (4,1,4)
(4 1 4) Firm 3 Don’t
Buy
Buy Firm 4
Firm 4
(5,1,1)
(5 1 1) Don t
Don’t Buy
Firm 5 (1,4,4)
(1 4 4) Firm 3 Don’t
Buy
Firm 4
Buy
Firm 4
(1,5,1)
(1 5 1) (1,1,5)
(1 1 5) Don’t
Buy
Firm 4
(3,3,3)
(3 3 3) Solving the 3 Player Game
Firm 1
Buy
Firm 6 Don’t Buy
Firm 6 Firm 2 Firm 2 Buy
Firm 5 Don t
Don’t Buy Buy
Firm 5
Firm 5 Firm 3
Buy Firm 4 (2,2,2)
(2 2 2) Firm 3 Don’t
Buy
Buy Firm 4
B
Firm4
(4,4,1)
(4 4 1) (4,1,4)
(4 1 4) Firm 3 Don’t
Buy
Buy Firm 4
Firm 4
(5,1,1)
(5 1 1) Don t
Don’t Buy
Firm 5 (1,4,4)
(1 4 4) Firm 3 Don’t
Buy
Firm 4
Buy
Firm 4
(1,5,1)
(1 5 1) (1,1,5)
(1 1 5) Don’t
Buy
Firm 4
(3,3,3)
(3 3 3) Adding More Moves
dding o e oves
•
•
•
•
•
•
• • Again, the game becomes more complex.
Consider, as an illustration, the Game of Nim
id
ill
i
h
f i
Two players, move sequentially.
Initially th
I iti ll there are two piles of matches with a certain number
t
il
f t h
ith
t i
b
of matches in each pile.
Players take turns removing any number of matches from a
single pile.
The winner is the player who removes the last match from
either pile.
Suppose, for simplicity that there are 2 matches in the first
pile and 1 match in the second pile. We will summarize the
pile
initial state of the piles as (2,1), and call the game Nib (2,1)
What does the game look like in extensive form?
g Nib (2 1) in Extensive Form
(2,1) i E
i F How
H reasonable i rollback/backward
bl is llb k/b k
d
induction as a behavioral principle?
p
p
• May work to explain actual outcomes in simple games,
with few players and moves.
• More difficult to use in complex sequential move games
such as Chess.
– We can’t draw out the game tree because there are too many
possible moves, estimated to be on the order of 10120.
– Need a rule for assigning payoffs to nonterminal nodes – a
intermediate valuation function.
i t
di t
l ti f ti • May not always predict behavior if players are unduly
concerned with “fair” behavior by other players and do not
fair
act so as to maximize their own payoff, e.g., they choose
to punish “unfair” behavior. Existence of a Solution to Perfect
Information Games
Games of perfect information are ones where every information set consists of a
single node in the tree.
Kuhn’s Theorem: Every game of perfect information with a finite number of
nodes, n, has a solution to backward induction.
Corollary: If the payoffs to players at all terminal nodes are unequal,
(no ties) then the backward induction solution is unique.
Sketch of Proof: Consider a game with a maximum number of n nodes. Assume
the game with just n1 steps has a backward induction solution. (Think e.g. n=2).
g
j
p
(
g
)
Figure out what the best response of the last player to move at step n, taking into
account the terminal payoffs. Then prune the tree, and assign the appropriate
terminal payoffs to the n1 node. Since the game with just n1 steps has a
n1
n1
solution, by induction, so does the entire nstep game. “Nature” as a Player
Nature
• Sometimes we allow for special type of player,  nature to make
random decisions. Why?
y
• Often there are uncertainties that are inherent to the game, that do not
arise from the behavior of other players.
– e g whether you can find a parking place or not.
e.g.,
not • A simple example: 1 player game against Nature. With probability ½
Nature chooses G and with probability ½ Nature chooses B. Nature
G B Player
y
l
4 r
5 Player
y
l r 3 1 • In this sequential move game, nature moves first. Equilibria are G,r
and B,l Playing Against Nature, Cont’d
ay g ga s Na u e, Co d
• Suppose the game is changed to one of simultaneous moves:
Nature
G B Player
l
4 r
5 Player
l r 3 1 • Player doesn’t know what nature will do as symbolized by
doesn t
do,
the ’d line.
• What is your strategy for playing this game if you are the
player?
• A risk neutral player treats expected payoffs the same as
certain payoffs: Expected payoff from left=½*4+½*3=7/2;
left ½*4+½*3 7/2;
Expected payoff from right = ½*5+½*1=3: Choose left (l). ...
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 Fall '08
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