Download Chapter 4 Sequential Games

Chapter 4 Sequential Games
1
Extensive Form Games
T
H
H
(1,2)
T
(2,1)
H
(2,1)
T
Any finite game of
perfect information has a
pure strategy Nash
equilibrium. It can be
found by backward
induction.
(4,0)
Chess is a finite game of perfect information.
Therefore it is a “trivial” game from a game
theoretic point of view.
2
Extensive Form Games - Intro
• A game can have complex temporal structure.
• Information
–
–
–
–
–
set of players
who moves when and under what circumstances
what actions are available when called upon to move
what is known when called upon to move
what payoffs each player receives
• Foundation is a game tree.
3
• Big Monkey and Little Monkey eat warifruit, which dangle from the
extreme tip of a lofty branch of the waritree.
• A waritree produces only one fruit. To get the warifruit, at least one
of the monkeys must climb the tree and shake the branch bearing
the fruit until the fruit comes loose and falls to the ground.
• A warifruit is worth 10 calories of energy. Climbing the tree uses 2
calories for Big Monkey, but uses no energy for Little Monkey, who is
smaller. If Little Monkey climbs the tree and shakes it down, Big
Monkey will eat 90% of the fruit (or 9 calories) before Little Monkey
climbs back down, and Little Monkey will get only 10% of the fruit (or
1 calorie).
• If Big Monkey climbs the tree and Little Monkey waits, Little Monkey
will get 40% of the fruit and Big Monkey will get 60%. If both
monkeys climb the tree, Big Monkey will get 70% of the fruit and
Little Monkey will get 30%. Assume each monkey is simply
interested in maximizing his caloric intake.
• Each monkey can decide to climb the tree or wait at the bottom.
• a. What is likely to happen if Big Monkey makes his decision first?
• b. What is likely to happen if Little Monkey must decide first?
• c. What if they both decide simultaneously?
4
Fundamental Tools
•
•
•
•
Big Monkey (BM) – Little Monkey (LM)
Warifruit from waritree (only one per tree) = 10 Calories
Climb the tree to get the fruit
Cost to get the fruit :
– 2 Calories for Big Monkey
– zero for Little Monkey
• Payoff :
– Both climb : BM 7 Calories – LM 3 Calories
– BM climbs : BM 6 Calories – LM 4 Calories
– LM climbs : BM 9 Calories – LM 1 Calories
• What will they do to maximize payoff taking into account
cost?
5
Fundamental Tools
Extensive form games--Definition
• An extensive form game G consists of :
– Players
– Game tree
– Payoffs
• Terminal node t : i(t)   (payoffs)
• G has tree property : only 1 path from root to any
terminal node
• Occurrence of stochastic event : fictitious player
Nature
probability assigned to each
branch of which Nature is head node
6
Fundamental Tools
Extensive form games—Illustration (BM-LM)
• 3 possibilities :
– BM decides first what to do
– LM decides first what to do
– Both decide simultaneously
• BM decides first :
Big Monkey
w
c
Little Monkey
Little Monkey
w
c
w
c
0,0
9,1
4,4
5,3
7
Fundamental Tools
Extensive form games—Illustration (BM-LM)
• Strategies :
– BM :
• Wait (w)
• Climb (c)
– LM : Actions are ordered, depending on (w,c) of BM
•
•
•
•
Climb no matter what BM does (cc)
Wait no matter what BM does (ww)
Do the same thing BM does (wc)
Do the opposite of what BM does (cw)
• A series of actions that fully define the behavior of a
player = strategy.
• A strategy for a player is a complete plan of how to
plan the game and prescribes his choices at every
information set (in this case, node).
8
Fundamental Tools
Extensive form games—Illustration (BM-LM)
• LM decides first :
Little Monkey
w
c
Big Monkey
w
Big Monkey
c
w
c
0,0
4,4
1,9 3,5
– The strategies are conversed Utility=(LM, BM)
9
Fundamental Tools
Extensive form games—Illustration (BM-LM)
• They choose simultaneously :
LM
Big Monkey
w
BM
Little Monkey
w
c
c
w
c
5,3
4,4
w
9,1
0,0
c
w
c
0,0
9,1
4,4 5,3
• Information Set : a set of nodes at which :
• The same player chooses
• The player choosing does not know which node represents the
actual choice node – represented by dotted line
10
• The key to representing information in a
game tree is realizing the connection
between nodes and history.
• If you know which node you have reached,
you know precisely the history of the play.
• To express uncertainty, we use concept of
information set (set of nodes you could be
in at a given time).
11
Composition of information sets
• Each decision node is in exactly one information
set
• all nodes of an information set must belong to
same player
• every node of an information set must have
exactly the same set of available actions.
• If every information set of every player is a
singleton, we have a game of perfect
information.
12
Fundamental Tools
Normal form games--Definition
The n-player normal form game consists of :
– Players i = 1,…,n
– A set Si of strategies for player i = 1,…,n.
We call s = (s1, …, sn) where si  Si for i =
1,…,n, a strategy profile for the game.
Each si is a strategy for player i.
– A function i : S   for player i = 1,…,n,
where S is the set of strategy profiles, so i(s)
is player i’s payoff when strategy profile s is
chosen.
13
Fundamental Tools
Normal form games--Illustration
• Another way to depict the BM-LM game (where BM
chooses first) :
LM : Actions are ordered, depending on (w,c) of BM
• Climb no matter what BM does (cc)
• Wait no matter what BM does (ww)
• Do the same thing BM does (wc)
• Do the opposite of what BM does (cw)
LM
BM
w
cc
9,1
cw
9,1
wc
0,0
ww
0,0
c
5,3
4,4
5,3
4,4
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Fundamental Tools
Normal form games--Illustration
• Don’t get rid of weakly dominated, as lose
equilibrium
LM
BM
w
cc
9,1
cw
9,1
wc
0,0
ww
0,0
c
5,3
4,4
5,3
4,4
15
Sequential games
• If players take turns to move then we have
a sequential game (sometimes called a
dynamic game)
• We model a sequential game by using a
‘game tree’ (or an ‘extensive form
representation’)
16
• It can be shown that every strategic form game
can be represented by an extensive game form
game and vice versa
• But strategies that are in equilibrium in strategic
form games are not necessary equilibrium
strategies in extensive form games games. Ex.
Monopolist. Made sense to fight as an
equilibrium, not not if other has already entered.
• – We need to define the concept of an
equilibrium in extensive form games
17
Problems with Nash equilibrium
• Sequential nature of the game is lost when
representing extensive form games in strategic
form
• Some Nash equilibria rely on playing actions
that are not rational once that action node has
been reached. In other words, a choice only
makes sense if you know what the opponent will
do.
• Nash equilibrium does not distinguish between
credible and non-credible threats
18
Solving sequential games
• To solve a sequential game we look for the
‘subgame perfect Nash equilibrium’
• For our purposes, this means we solve the game
using ‘rollback’
– To use rollback, start at the end of each branch and
work backwards, eliminating all but the optimal choice
for the relevant player
(Technical point – you can only use this trick if there are
no information sets. If you don’t know where you are,
it may be too difficult to decide.)
19
20
Subgame
• Its game tree is a branch of the original
game tree
• The information sets in the branch
coincide with the information sets of the
original game and cannot include nodes
that are outside the branch.
• The payoff vectors are the same as in the
original game.
21
Subgame perfect equilibrium & credible threats
• Proper subgame = subtree (of the game
tree) whose root is alone in its information
set
• Subgame perfect equilibrium
– Strategy profile that is in Nash equilibrium in
every proper subgame (including the root),
whether or not that subgame is reached
along the equilibrium path of play
22
•
•
•
•
•
On October 22, 1962, after reviewing newly acquired intelligence,
President John F. Kennedy informed the world that the Soviet Union
was building secret missile bases in Cuba, a mere 90 miles off the
shores of Florida.
After weighing such options as an armed invasion of Cuba and air
strikes against the missiles, Kennedy decided on a less dangerous
response.
In addition to demanding that Russian Premier Nikita S. Khrushchev
remove all the missile bases and their deadly contents, Kennedy
ordered a naval quarantine (blockade) of Cuba in order to prevent
Russian ships from bringing additional missiles and construction
materials to the island.
In response to the American naval blockade, Premier Khrushchev
authorized his Soviet field commanders in Cuba to launch their tactical
nuclear weapons if invaded by U.S. forces.
Deadlocked in this manner, the two leaders of the world's greatest
nuclear superpowers stared each other down for seven days - until
Khrushchev blinked. On October 28, thinking better of prolonging his
challenge to the United States, the Russian Premier conceded to
President Kennedy's demands by ordering all Soviet supply ships away
from Cuban waters and agreeing to remove the missiles from Cuba's
mainland. After several days of teetering on the brink of nuclear
holocaust, the world breathed a sigh of relief.
23
Example: Cuban Missile Crisis
Kennedy
Khrushchev
Arm
Retract
Nuke
- 100, - 100
Fold
10, -10
-1, 1
Pure strategy Nash equilibria: (Arm, Fold) and
(Retract, Nuke)
Pure strategy subgame perfect equilibria: (Arm, Fold)
Conclusion: Kennedy’s Nuke threat was not credible.
24
Backwards induction
• Start from the smallest subgames containing the terminal
nodes of the game tree
• Determine the action that a rational player would choose
at that action node
– At action nodes immediately adjacent to terminal nodes, the
player should maximize the utility, This is because she no longer
cares about strategic interactions. Regardless of how she
moves, nobody else can affect the payoff of the game.
Replace the subgame with the payoffs corresponding to the
terminal node that would be reached if that action were
played
• Repeat until there are no action nodes left
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(MD,BK) payoff
26
The predation game
• Nasty Guys is an incumbent firm
producing bricks
• SIC (Sweet Innocent Corporation) is a
potential new entrant in the brick market.
• Nasty Guys says that if SIC enters then it
will “squish them like a bug”.
• What should SIC do?
27
The predation game
Don’t
enter
SIC=0, NG=100
Fight
SIC
NG
SIC = -10, NG = -10
Enter
Don’t
fight
SIC = 30, NG = 30
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The predation game
Don’t
enter
SIC=0, NG=100
Fight
SIC
NG
SIC = -10, NG = -10
Enter
If SIC actually enters, then
‘fighting’ is an incredible
threat – it hurts SIC but also
hurts NG. So SIC knows the
threat is just bluff
Don’t
fight
SIC = 30, NG = 30
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The predation game
Don’t
enter
SIC=0, NG=100
Fight
SIC
NG
SIC = -10, NG = -10
Enter
So the equilibrium is:
SIC will enter
Don’t
fight
SIC = 30, NG = 30
NG will not fight
30
Credible commitments
• When Cortes arrived in Mexico he ordered
that his ships should be burnt
• This seems silly
– His troops were vastly outnumbered
– Surely it is better to keep an ‘escape route’
home?
31
Think of Cortes trying to motivate his own soldiers
Fight
Hard
Keep
Ships
C
S
Be
careful
S
Fight
Hard
C = 100, S = 0
C = 0, S = 10
C = 100, S = 0
Burn
ships
Be
careful
C = -100, S = -100
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If no retreat possible, will fight hard or die. But if retreat is possible, may fight
less hard and ‘run away’
Fight
Hard
Keep
Ships
C
S
Be
careful
S
Fight
Hard
C = 100, S = 0
C = 0, S = 10
C = 100, S = 0
Burn
ships
Be
careful
C = -100, S = -100
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So Cortes wants to burn his ships. It is a credible commitment not to retreat
– and this alters how his own troops behave.
Fight
Hard
Keep
Ships
C
S
Be
careful
S
Fight
Hard
C = 100, S = 0
C = 0, S = 10
C = 100, S = 0
Burn
ships
Be
careful
C = -100, S = -100
34
Hold up
• Hold up occurs if one party has to incur sunk
costs before they bargain with another party
• For example, hardware manufacturers and
software developers
– Hardware manufacturers want software
manufacturers to make applications for their
hardware
– But most of the cost of software is sunk
– So if bargain after the software is designed, the
hardware manufacturer can seize most of the
benefits
35
Holdup: in equilibrium, no-one
designs software
payoffs = (software, nintendo)
Bargain hard
(= Pay low price)
(-$50,000; $250,000)
Nintendo
design
Software
Designer
Bargain
“soft”
($100,000; $100,000)
Don’t design
(0, 0)
36
Strategies in extensive form
A strategy in an extensive form game is a
complete description of the actions that player
performs at any action node at which it is her
turn to move turn to move
Key points
–It is not sufficient to specify responses only at
those action nodes that are arrived at via some
particular sequence of plausible play
–A strategy must prescribe an action at any action
node where that player moves. node where that
player
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Definition: The strategy set of agent i is the
cartesian product of the sets of children
nodes of each information set belonging to
i.
Definition: An information set I is a
subset of the nodes in a game tree
belonging to player P such that
- All iI belong to P
- For i,jI there is no path from i to j
- All iI have the same number of
outgoing edges
38
Sequential Prisoner’s
Dilemma
dotted line means P2 doesn’t know which state he is in
P1
Confess
Deny
P2
P2
Confess
(-5,-5)
Deny Confess
(0,-10) (-10,0)
Deny
(-1,-1)
39
• With perfect information – each information set is a
singleton (as you always know which state you are in)
• A strategy profile (s1,s2,…sn) determines a unique path
from the root to some terminal node. (where s1 states
what player 1 will do in every situation)
• We say this unique path is supported by the strategy
profile. A path supported by a Nash equilibrium will be
called an equilibrium path.
• A Nash equilbrium in sequential game (perfect or
imperfect):
• U(si*, s-i*) >U(si, s-i*) for all i.
• Note, there can be two different strategy profiles which
have the same path.
• Every path from the root to a terminal node is supported
by at least one strategy profile.
40
Example 4.9
RL” is best path
Stategies ({R}, {R’,L”}) and ({R}, {L’,L”}) both support it
({R}, {R’,L”}) means P1 always takes R, P2 takes R’ if at node B and L” if at node C
[Note: Notation is confusing; you always have to read to get meaning.]
P1
A
L
P2
L’
D
(2,1)
R
P2
B
C
R’
L”
E
F
(0,3) (4,1)
R”
G
(1,0)
41
Theorem (Kuhn): Every sequential game
with perfect information has a Nash
equilibrium (use backwards induction).
P1
A
P2
P2
B
C
D
E
F
(1,0) (0,1)
(2,2)
42
Example 4.12 Stackelberg Duopoly Model
Stackelberg duopoly (like a monopoly, but with exactly two players)
corresponds to a sequential game, first leader (firm 1) chooses
how much to produce, then follower (firm 2) chooses
can be solved by backward induction: for each quantity q1, the
follower chooses its best response q2
i (q1, q2) = qi[p(q) -ci]
where q = q1+q2
p(q) = A-q is the market clearing price when the total output in the
market is q
ci is the marginal cost of the production of the product by firm i.
That is, the profit for each firm is
i (q1, q2) = qi[A-q1-q2 -ci]
43
Solving by backwards induction
• This is a two person game sequential game with two
stages and perfect information.
• Find best response for each choice of q1
2 (q1, q2*) = max q2[A-q1-q2 –c2]
2 (q1, q2) = -(q2)2 + q2[A-q1 –c2]
 2
= -2q2 +A-q1-c2
q 2
Second derivative = -2
So the maximizer is (A-q1-c2)/2
44
Continuing
• Thus, firm 1 should anticipate this result and
choose q1 to maximize
1 (q1, q2*) = q1[A-q1-(A-q1-c2)/2 –c1]
•
= ½(-q12 + (A+c2-2c1)q1)
•

q1
=
-q1
+1/2(A+c2-2c1)
1
• q1= ½((A+c2-2c1) and q2 = ¼*(A+2c1-3c2)
45
Type of games
46
subgame perfect equilibrium
• A strategy profile of a sequential game is a
subgame perfect equilibrium if it is a Nash
equilbrium for every subgame of the
original game. In other words, the strategy
is perfect even if the play never goes to
that part of the tree.
• An imperfect equilibrium is like a strategy
that wouldn’t be optimal if the other player
did something different.
47
Imperfect information
• 􀂄􀂄 SPE is not an appropriate equilibrium concept
because most games with imperfect information have
too few proper subgames to rule out extraneous Nash
equilibria of the game
• Alternative equilibrium concepts
• – Bayesian Equilibrium/Perfect Bayesian Equilibrium
• – Sequentially rational equilibrium
• – Forwards induction
• – Trembling hand equilibrium
Topic of active research
48
• A Nash equilibrium that fails to be
subgame perfect is also known as Nash
equilibrium supported by noncredible
behavior.
• To find subgame perfect equilibrium, use
backward induction on the subgames of
the original problem.
49
Bob and Betty
• Bob and Betty must cook, wash dishes, and vacuum.
Bob can't cook very well and just doesn't like to wash
dishes, so they have concocted the following game for
allocating the tasks. Betty moves first and she chooses
between cooking and doing the dishes.
• If she chooses dishes, then Bob chooses to Go Out or
Cook.
• On the other hand, if Betty chooses to cook, then they
simultaneously choose between the remaining two tasks;
vacuuming and doing the dishes. The payoffs are at the
end of the tree.
• You may conclude whatever you want about the
relationships between the payoffs and the preferences of
Bob and Betty for doing chores and being together.
50
Note, use normal form game to pick what Betty should do at CD
51
•
There are three subgames: There are two proper subgames; one beginning
at node B and one beginning at node A. The game itself is a subgame.
•
•
There are two paths to Nash equilibria.
The first one (Path One) from the root through node A is BettyDishes,
BobOut.
The other one (Path Two) from the root through node B to the information
set containing nodes C and D is BettyCook, BobVacuum, BettyVacuum.
You must use normal form on the subgame beginning at node B in order to
find this second path.
Path Two is supported by the strategy profile ({Cook, Vacuum}, {Out,
Vacuum}). That is, Betty plays Cook at the root and Vacuum at C or D. Bob
plays Out at node A and Vacuum at node B. This path is a subgame perfect
equilibrium.
•
•
•
For the subgame starting at A the proposed strategy profile reduces to ({.},
{Out}), which is a Nash equilibrium. For the subgame starting at node B the
strategy profile reduces to ({.,Vacuum}, {.,Vacuum}), which is a Nash
equilibrium.
•
Are there any other strategy profiles that will support Path Two?
52
The strategy profile ({Dishes, Vacuum}, {Out, Vacuum})
supports Path One, the road to a Nash equilibrium.
At the root node Betty plays Dishes and at node A Bob
plays Out. If they should happen to find themselves at
the subgame starting at node B, then they both play
vacuum, which is a Nash Equilibrium.
But this strategy profile is not the only one that supports
Path One. Path One is also supported by the strategy
profile ({Dishes, Dishes}, {Out, Dishes}).
Betty plays Dishes at the root and Dishes at nodes C or D;
Bob plays Out at node A and Dishes at node B. This is a
Nash equilibrium profile since Out is Bob's best
response to a play of Dishes by Betty at the root node.
But this is not a subgame perfect equilibrium since a play of
Dishes by Bob at node B and a play of Dishes by Betty
at C or D could be improved upon. In this case we have
a strategy profile that involves a Nash equilibrium in one
subgame, but noncredible plays in another subgame.
53
• Consider the strategy profile ({Dishes, Vacuum}, {Out, Dishes}),
which also supports Path One.
• Is this profile a subgame perfect equilibrium? No. This profile does
result in a Nash equilibrium in the subgame beginning at node A, but
there is a hitch. In the subgame that includes the root Betty would
never play Dishes at the root, as called for by the profile.
• Are there any more strategic profiles that support Path One?
Consider ({Dishes, Dishes}, {Out, Vacuum}) and explain why this is
not a subgame perfect equilibrium.
• As in the normal form games we have seen, there may be multiple
Nash equilibria in an extensive form game. The principle of
subgame perfect equilibrium is to eliminate those Nash equilibria
which may be based on non-credible or unreasonable promises or
threats.
• In the analysis of the second Bob and Betty example we eliminated
two strategy profiles that involved a Nash equilibrium in a subgame,
but which also included unreasonable behavior in other subgames.
54
In Class Exercise
Ask the students to choose partners from the other side of the room or have
them imagine that each is playing with one person who is sitting on the
other side of the room. Each student will eventually be asked to write either
pink or purple. If both students in the real or imaginary pair write pink, the
person on the right-hand side of the room gets 50 points and the person on
the left-hand side of the room gets 40 points. ("Right-hand" and "left-hand"
are defined from the students’ point of view.) If both write purple, the person
on the lefthand side of the room gets 50 points and the person on the righthand side of the room gets 40 points. If the answers don't match, neither
player gets anything. To play without the delay tactic, simply ask the
students to choose a color and write the choice. Then play again,
immediately, but explain that you will flip a coin first. If it comes up heads,
those on the right-hand side of the room get to write their answers first;
otherwise those on the left-hand side of the room write first.
What happens if we delay?
55
• Games like the battle of the two cultures or
chicken have first-mover advantages in
their sequential move versions; the tennispoint example has a second-mover
advantage in its sequential-move version.
Other games show no change in
equilibrium as a result of the change in
rules; games like the prisoners’ dilemma,
in which both players have dominant
strategies, fall into this category.
56
Prove or disprove!
• When both players have a dominant
strategy, the dominant-strategy equilibrium
will hold in both the simultaneous and the
sequential versions of the game.
57
Sequential Monopolist View
What are Nash equilibria?
Are they subgame perfect?
58
Thought Question
• How do we change a game to our
advantage?
• Use commitment, threats, and promises to
change the nature of a game.
59
Commitment
•
•
•
•
Reduce freedom of action by commitment.
Thereby forcing a “first mover advantage.”
This move has to be “observable.”
Move has to be credible (believable).
– Reputation leads to credibility.
60
Commitment – An Example
STUDENT
TEACHER
Punctual Late
Weak
4, 3
2, 4
Tough
3, 2
1, 1
61
Commitment – An Example
For those that intend to teach…
STUDENT
TEACHER
Punctual Late
Weak
4, 3
Tough
3, 2
2, 4
N.E.
1, 1
62
Commitment – An Example
But if we announce we are tough
STUDENT
TEACHER
Punctual Late
Weak
4, 3
2, 4
Tough
3, 2
1, 1
63
Commitment – An Example
Get different NE
STUDENT
TEACHER
Punctual Late
Weak
4, 3
2, 4
Tough
3, 2
N.E.
1, 1
64
Strategic Moves and Threats!
• Deterrence and Compellence achieved through either a
threat with an implied promise or a promise with an
implied threat.
• Either deterrence or compellence requires credibility.
– Credibility involves some cost to the threatener as well.
– If threatener preferred the threat then she would carry it out anyway and
the promise part of the threat would never be carried out.
– This cost is problematic because it might tempt the threatener to avoid
actually carrying out the threat, thus making it less credible!
– Thus imposing a cost on herself is a necessary condition for a threat to
be successful, but is not a sufficient condition.
– CAREFUL!!!!!
65
Trade Negotiation
Japan
U.S.
Open
Closed
Open
4, 3
3, 4
Closed
2, 1
1, 2
66
Trade Negotiation
Japan
U.S.
Open
Closed
Open
4, 3
3, 4
Closed
2, 1
1, 2
67
Changing the game – A threat!
• A threat -- “We will close our market if you
close yours.”
• And a promise – “ We will open market if you
open yours.”
• Effectively reduces Japans options to
– If Japan stays open then the U.S. stays open,
giving Japan 3 and the U.S., 4 (favorable to the
U.S. over the no threat scenario)
– If Japan stays closed then U.S. closes as well,
thus giving Japan 2 and the U.S., 1.
68
Trade Relations – Threats in Action.
•
Credible partly because the threat is costly to the US. If it were not costly
to the US to play close then Japan would not believe that US would play
open otherwise and therefore would never open her own borders.
•
If Japan calls US bluff then US has the temptation to not carry out the
threat! – cutting off freedom of action in the future (as threat will not be
believed) That fact may keep threat credible.
•
If Japanese market is already open then threat is part of deterrence
strategy
•
If Japanese market is closed then threat is part of compellence strategy.
• A good threat is one that does not need to be carried out.
NOTE: If the size of the threat is too big to be credible then a probabilistic
element may be introduced. (Gamble on whether will be carried out and on
what amount of penalty will be.)
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Threats in action (cont)
• Japan may respond by agreeing in principle and then stalling –
banking on the US’s lack of desire to impose a cost on itself, a
technique called salami tactics.
• Salami tactics – fail to comply with the others wishes
(particularly in compellence situations) in such a small
way that it is not worth the while of the threatener to
carry out her threat. If that works you transgress a little
more..the way to oppose salami tactics is by using a
graduated threat.
• Used in politics: a gradual process of threats and
alliances as a means of overcoming opposition.
Consequently, the player was able to exert his/her
influence and eventually dominate the political landscape
-- slice by slice.
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Prisoners Dilemma – Promises to
Keep!
• Tit for Tat strategies are examples of
promises that act as a deterrent to
cheating.
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Promises or Threats?
• Deterrence: “Prevent you from doing something”
Threat  “I will punish you if you do it” – requires me to only wait till
you mess up – no monitoring.
Promise  “I will reward you if you don’t do it” – requires me to
monitor you constantly.
Threats are cheaper than promises.
• Compellence: “I want you to do something”
Threat  “I will punish you if you don’t do it” – requires me to
monitor you constantly.
Promise  “I will reward you if you do it” – requires me to only wait
till you do what I want you to do!
Promises are cheaper than threats.
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Countering Threats
• Irrationality: So nuts that any threats
will not have an effect on your behavior.
• Cut of communication so threats don’t
reach you.
• Open escape routes for enemy thus
tempting them to renege on threats.
• Undermine opponents motive to
maintain reputation by promising
secrecy if she does not punish you.
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Credibility Devices
• Since credibility implies the temptation to be “not”
credible certain devices are required to ensure
credibility.
Reducing the freedom of action through:
– automatic fulfillment (doomsday device)
According to a new book exclusively obtained by the Huffington
Post, Saudi Arabia has crafted a plan to protect itself from a
possible invasion or internal attack. It includes the use of a
series of explosives, including radioactive “dirty bombs,” that
would cripple Saudi Arabian oil production and distribution
systems for decades...
– burning bridges,
– cutting off communication so nobody can argue with you
regarding your threat.
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Credibility Devices
• Changing payoffs by using reputation:
• make game part of larger game so that payoffs in the current
game are linked to repurcussions in other game,
• dividing game into little subgames to allow reputation to work,
• reduce monitoring costs and thus change payoffs by allowing
players to monitor each other,
• irrationality – worry I just might Nuke as I’m irrational
• contracts – have way of punishing if deal not kept
• Brinkmanship: the policy or practice, especially in politics and
foreign policy, of pushing a dangerous situation to the brink of
disaster (to the limits of safety) in order to achieve the most
advantageous outcome by forcing the opposition to make
concessions. This might be achieved through diplomatic
maneuvers by creating the impression that one is willing to use
extreme methods rather than concede. During the Cold War, the
threat of nuclear force was often used as such a deterrent.
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Solving Extensive Form Games
• the usual procedure is to convert the extensiveform game to strategic form, and find its
equilibria.
• However, some of these equilibria would have
important drawbacks because they ignore the
dynamic nature of the extensive-form.
• Reinchard Selten was the first to argue that
some Nash equilibria are “more reasonable”
than others in his 1965 article.
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Selten’s Game
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• The strategic form representation has two pure-strategy
Nash equilibria, (D, L) and (U,R). Look closely at the
Nash equilibrium (U,R) and what it implies for the
extensive form. In the profile (U,R), player 2’s
information set is never reached, and she loses nothing
by playing R there.
• But there is something “wrong” with this equilibrium: if
player 2’s information set is ever reached, then she
would be strictly better off by choosing L instead of R. In
effect, player 2 is threatening player 1 with an action that
would not be in her own interest to carry out.
• we are interested in sequencing of moves presumably
because players get to reassess their plans of actions in
light of past moves
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• To anticipate a bit of what follows, the
problem with the (U,R) solution is that it
specifies the incredible action at an
information set that is off the equilibrium
path of play.
• Player 2’s information set is never reached
if player 1 chooses U. Consequently, Nash
equilibrium cannot pin down the optimality
of the action at that information set.
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Little Horsey
• Consider the following simple game.Player 1 gets
to choose between U, M, or D. If he chooses D, the
game ends. If he chooses either U or M, player 2
gets to choose between L and R without knowing
what action player 1 has taken except that it was
not D.
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Where are the NE?
• Convert to normal form
• Use standard techniques
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Giving Gifts
82
• There are two players and player 1 receives a
book which, with probability p is a small game
theory pocket reference, and with probability 1 −
p is a Star Trek data manual. The player sees
the book, wraps it up, and decides whether to
offer it to player 2 as a gift. Player 2 hates Star
Trek and is currently suffering in a graduate
game theory course, so she would prefer to get
one but not the other. Unfortunately, she cannot
know what is being offered until she accepts it.
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• Player 1 observes Nature’s move and offers the wrapped
gift to player 2. If the gift is accepted, then player 1
derives a positive payoff because everyone likes when
their gifts are accepted.
• Player 1 hates the humiliation of having a gift rejected,
so the payoff is −1.
• Player 2 strictly prefers accepting the game theory book
to not accepting it; she is indifferent between not
accepting this book and accepting the Star Trek manual,
but hates rejecting the Star Trek manual more than the
game theory book because while dissing game theory is
cool, dissing Star Trek is embarrassing.
• Let’s construct the strategic form of this game
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The action GN: refers to “give if get game theory, not give if star trek”.
(GG,Y) is a Nash equilibrium for any value of p because p −1 < 0 < p < 1. Player 1 offers
the gift regardless of its type, and player 2 accepts always.
In addition, (NN,N) is a Nash
equilibrium. Player 1 never offers any gifts, and player 2 refuses any gifts if offered.
Two nash equilibruim. GG-Y and NN-N.
But (NN,N) is clearly irrational. If the game ever reaches
player 2’s information set, accepting a gift strictly
dominates non accepting a gift, regardless of the gift.
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• Because the strategic form ignores timing,
Nash equilibrium only ensures optimality
at the start of the game. That is,
equilibrium strategies are optimal if the
other players follow their equilibrium
strategies. But we cannot see whether the
strategies continue to be optimal once the
game begins.
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• We shall require that player 2 form beliefs about the
probability of being at any particular node in her
information set.
• Obviously, if the information set consists of a single
node, then, if that information set is reached, the
probability of being at that node is 1.
• Let’s look at the game in the figure. Let q denote player
2’s belief that she is at the left node in her information
set (given that the set is reached, this is the probability of
player 1 having offered the game theory book), and 1 − q
be the probability that she is at the right node (given that
the set is reached, this is the probability of player 1
having offered the Star Trek book).
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• That is, p is player 2’s initial belief (or the prior) of the
book being a game theory reference; and q is player 2’s
conditional belief (updated belief, or posterior)
• For example, I believe that if you receive a game theory
book, there is a 90% chance you will offer the book as a
gift, but if you receive star trek, there is a 50% chance
you will offer it as a gift.
• Requirement 1 (Beliefs). At each information set, the
player who gets to move must have a belief about which
node in the information set has been reached by the play
of the game.
• The belief will be a probability distribution over the
nodes.
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• A strategy is sequentially rational for
player i at the information set h if player i
would actually want to chose the action
prescribed by the strategy if h is reached.
• A continuation game refers to the
information set and all nodes that follow
from that information set.
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• Take any two nodes x y (that is, y follows x in
the game tree), and consider the mixed strategy
profile σ. Let P(y|σ,x) denote the probability of
reaching y starting from x and following the
moves prescribed by σ.
• That is, P(y|σ,x) is the conditional probability that
the path of play would go through y after x if all
players chose according to σ and the game
started at x.
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• Player i’s expected utility in the
continuation game starting at node x then
is:
• Ui(σ|x) =  zP(z|σ,x)ui(z),
• where Z is the set of terminal nodes in the
game, and ui(·) is the Bernoulli payoff
function specifying player i’s utility from the
outcome z ∈ Z.
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• In the gift game, Player 2 can calculate the expected
payoff from choosing Y, which is q(1)+(1−q)(0) = q,
• the expected payoff from choosing N, which is
q(0)+(1−q)(−1) = q−1.
• Since q > q−1 for all values of q (as q-1 is negative), it is
never optimal for player 2 to choose N regardless of the
beliefs player 2 might hold.
• Therefore, the strategy N is not sequentially rational
because there is no belief that player 2 can have that will
make it optimal at her information set.
• In other words, the unique sequentially rational strategy
is to choose Y with certainty.
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