DIFFERENTIAL GAMES AND NUCLEAR WAR

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DIFFERENTIAL GAMES AND NUCLEAR WAR
Dan Beamish
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GAME THEORY IS THE MATHEMATICS OF CONFLICT

• Game theory is a kind of optimization.

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GAME THEORY IS THE MATHEMATICS OF CONFLICT

• Game theory is a kind of optimization.
• In regular optimization problem, we are trying
  to choose x which maximize/minimize some function
  f(x) where x is chosen from set. If x is a real
  number, its just a regular calculus problem.

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GAME THEORY IS THE MATHEMATICS OF CONFLICT

• Game theory is a kind of optimization.
• In regular optimization problem, we are trying
  to choose x which maximize/minimize some function
  f(x) where x is chosen from set.
• But we can also have x range over some other set
  such as a set of functions (e.g. minimal surface
  problem, etc.), the set of all possible routes
  through a city (travelling salesman problem),
  etc.

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GAME THEORY IS THE MATHEMATICS OF CONFLICT
In Game Theory, two players choose from a set of
possible strategies with the goal of maximizing
the payoff for themselves.
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GUARDING A TARGET
Aircraft A wishes to drop an atomic bomb as
close as possible to target city T. Aircraft B
wishes to intercept Aircraft A before it can
reach the target. The two aircraft are free to
move in the plane and have the same speed.
T
B
A
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GUARDING A TARGET
How should Aircraft A move in order to reach
the target while avoiding Aircraft B?
Aircraft A wishes to drop an atomic bomb as
close as possible to target city T. Aircraft B
wishes to intercept Aircraft A before it can
reach the target. The two aircraft are free to
move in the plane and have the same speed.
T
B
A
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GUARDING A TARGET
How should Aircraft B move in order to
intercept A and prevent him from reaching the
target?
Aircraft A wishes to drop an atomic bomb as
close as possible to target city T. Aircraft B
wishes to intercept Aircraft A before it can
reach the target. The two aircraft are free to
move in the plane and have the same speed.
T
B
A
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GUARDING A TARGET
How should Aircraft B move in order to
intercept A and prevent him from reaching the
target?
Aircraft A wishes to drop an atomic bomb as
close as possible to target city T. Aircraft B
wishes to intercept Aircraft A before it can
reach the target. The two aircraft are free to
move in the plane and have the same speed.
T
B
WHAT ARE WE OPTIMIZING?
A
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GUARDING A TARGET (MATHEMATICAL FORMULATION)
State Variables. The state variables tell us the
state of the game what we need to know to
describe the progress of the game.
T
B
A
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GUARDING A TARGET (MATHEMATICAL FORMULATION)
State Variables. The state variables tell us the
state of the game what we need to know to
describe the progress of the game. How are the
pieces are arranged on the board? In this case,
the state variables are the positions
xA(t),yA(t), xB(t),yB(t) of the two aircraft.
xB(t),yB(t)
T
B
A
xA(t),yA(t)
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GUARDING A TARGET (MATHEMATICAL FORMULATION)
State Variables. The state variables tell us the
state of the game what we need to know to
describe the progress of the game. How are the
pieces are arranged on the board? In this case,
the state variables are the positions
xA(t),yA(t), xB(t),yB(t) of the two
aircraft. As the game progresses, the state
variables will change.
xB(t),yB(t)
T
B
A
xA(t),yA(t)
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GUARDING A TARGET (MATHEMATICAL FORMULATION)
Control Variables. The two players (Aircraft A
and B) are free to choose whichever direction
they want to fly in, ?A(t), ?B(t).
T
B
?B(t)
?A(t)
A
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GUARDING A TARGET (MATHEMATICAL FORMULATION)
Control Variables. The two players (Aircraft A
and B) are free to choose whichever direction
they want to fly in, ?A(t), ?B(t). These are
called the control variables because they are
under the control of the players.
T
B
?B(t)
?A(t)
A
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GUARDING A TARGET (MATHEMATICAL FORMULATION)
Control Variables. The two players (Aircraft A
and B) are free to choose whichever direction
they want to fly in, ?A(t), ?B(t). These are
called the control variables because they are
under the control of the players. The choices
each player makes for her control variable(s) is
called the strategy for that player.
T
B
?B(t)
?A(t)
A
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GUARDING A TARGET
Termination. Play of the game terminates if
Aircraft B intercepts Aircraft A, i.e. when their
positions are the same, xA(t),yA(t)
xB(t),yB(t).
T
B
A
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GUARDING A TARGET
Termination. Play of the game terminates if
Aircraft B intercepts Aircraft A, i.e. when their
positions are the same, xA(t),yA(t)
xB(t),yB(t).
T
B
Payoff. The score of the game is how close
Aircraft A gets to the target when termination
occurs. Note It is not necessarily true that
termination occurs, but we will assume it does.
A
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GUARDING A TARGET
Strategy and Play. If we are given initial
positions xA(0),yA(0) and xB(0),yB(0) for the
two aircraft, and their choices for the control
variables ?A(t), ?B(t) the outcome of the game
will be completely determined.
T
B
A
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GUARDING A TARGET
Strategy and Play. If we are given initial
positions xA(0),yA(0) and xB(0),yB(0) for the
two aircraft, and their choices for the control
variables ?A(t), ?B(t) the outcome of the game
will be completely determined. Question Is it
always possible B can always reach the target no
matter what?
T
B
A
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GUARDING A TARGET
Strategy and Play. If we are given initial
positions xA(0),yA(0) and xB(0),yB(0) for the
two aircraft, and their choices for the control
variables ?A(t), ?B(t) the outcome of the game
will be completely determined. Question Is it
always possible B can always reach the target no
matter what? Maybe it is impossible. If so,
how close can B get to the target before
interception?
T
B
A
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GUARDING A TARGET
Complication What B does to guard the target
depends on how A attacks the target and vice
versa.
T
B
A
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GUARDING A TARGET
Complication What B does to guard the target
depends on how A attacks the target and vice
versa. Example Suppose B has anticipated that A
will head straight to the target and tried to
head him off. A, knowing this is Bs strategy,
will go around him.
T
B
A
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GUARDING A TARGET
Complication What B does to guard the target
depends on how A attacks the target and vice
versa. Example Likewise, suppose A does head
directly to the target. If B knows that A is
going to do this, he can intercept A much earlier
than other possible strategies.
T
B
A
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GUARDING A TARGET
What does it mean to play optimally? We make
an observation about this particular game.
T
B
A
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GUARDING A TARGET
What does it mean to play optimally? If we draw
a line midpoint between the starting point of
both aircraft, then regardless of what strategy
B chooses to guard, A can always reach any point
on his side of the line before being
intercepted. Likewise, regardless of what A does,
B can always intercept A at any point on his side
of the line.
T
B
A
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GUARDING A TARGET
What does it mean to play optimally? As
Strategy Move in a straight line towards
whichever point (on his side of the line) is
desired. Bs strategy Move in a straight line
towards A.
T
B
A
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GUARDING A TARGET
Optimal Strategies The closest point A can get to
the target is the point P on the midpoint line
closest to the target. A should drop his bomb at
the point on his side of the line closest to the
target. The payoff will be this minimum distance.
T
B
P
A
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GUARDING A TARGET
In What Sense is this Optimal? A can never do
any better than this payoff, since if B plays
optimally he can always intercept before the
bomber gets this close.
T
B
P
A
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GUARDING A TARGET
In What Sense is this Optimal? A can never do
any better than this payoff, since if B plays
optimally he can always intercept before the
bomber gets this close. Likewise, B can never
force interception further from the target than
this because if A plays optimally then A can
always reach the target within this distance.
T
B
P
A
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GUARDING A TARGET
In What Sense is this Optimal? The strategies
?A(t), ?B(t) of each player for which this occurs
are called optimal strategies.
T
B
P
A
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GUARDING A TARGET
Value of the Game. If both players play optimally
then the bomb always gets dropped at the point P
on the midpoint-line between the starting points
which is closest to the target.
T
B
P
A
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GUARDING A TARGET
Value VxA(0),yA(0),xB(0),yB(0)
Value of the Game. If both players play optimally
then the bomb always gets dropped at the point P
on the midpoint-line between the starting points
which is closest to the target. This payoff if
called the Value of the game. It depends on the
starting points of both players.
T
B
P
A
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GUARDING A TARGET
Value VxA(0),yA(0),xB(0),yB(0)
Value of the Game The Value of the Game
determines which starting conditions are
winnable. In this game, it would be the set of
all initial conditions from which the bomber can
actually reach the target.
T
B
P
A
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MORE GAME-THEORY TERMINOLOGY
If we just care about winning and losing, we
call this a Game of Kind. If the payoff is some
real number, it is called a Game of Degree.
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MORE GAME-THEORY TERMINOLOGY
Zero-Sum Games. The above game belongs to a class
called Zero-Sum Games. This means the payoff to
both players sums to zero, and is the same as
having a single payoff function which one player
wants to minimize, and other maximize
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MORE GAME-THEORY TERMINOLOGY
Zero-Sum Games. The above game belongs to a class
called Zero-Sum Games. This means the payoff to
both players sums to zero, and is the same as
having a single payoff function which one player
wants to minimize, and other maximize. In
zero-sum games, it is not possible for players to
co-operate since the gain of one player is the
loss of the other.
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MORE GAME-THEORY TERMINOLOGY
When co-operation is possible, the whole concept
of what an optimal strategy is, is not at all
clear. Formulating a meaningful definition of
optimal play in this case is what won John Nash
the Nobel prize, and is called a
Nash-Equilibrium. There are also alternative
definitions.
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MORE GAME-THEORY TERMINOLOGY
Perfect Information. We assumed in the above game
that both players have complete knowledge of the
state variables which determine play. This is
called assuming perfect information. There is
currently no satisfactory theory of differential
games with imperfect information.
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GENERALIZATIONS OF GUARDING A TARGET
What if there was more than bomber?
T
B
A
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GENERALIZATIONS OF GUARDING A TARGET
What if there was more than bomber? What if there
was more than one interceptor?
T
B
A
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GENERALIZATIONS OF GUARDING A TARGET
What if there was more than bomber? What if there
was more than one interceptor? What if there was
more than one target?
T
B
A
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GENERALIZATIONS OF GUARDING A TARGET
What if there was more than bomber? What if there
was more than one interceptor? What if there was
more than one target? What if there are targets
on both sides?
T
B
A
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GENERALIZATIONS OF GUARDING A TARGET
What if there was more than bomber? What if there
was more than one interceptor? What if there was
more than one target? What if there are targets
on both sides? What if the game space is a
sphere?
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GAME THEORY, VON NEUMANN, AND RAND
Game theory was (arguably) founded by John Von
Neumann with the publication Theory of Games and
Economic Behavior written together with Oskar
Morgenstern. This revolutionized the field of
economics, although it also had applications to
politics, warfare, psychology, and many other
fields.
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GAME THEORY, VON NEUMANN, AND RAND
From the beginning of World War II, Von Neumann
was confident of the Allies victory. He
sketched out a mathematical model of the conflict
from which he deduced that the Allies would win,
applying some of the methods of game theory to
his predictions.
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GAME THEORY, VON NEUMANN, AND RAND
From the beginning of World War II, Von Neumann
was confident of the Allies victory. He
sketched out a mathematical model of the conflict
from which he deduced that the Allies would win,
applying some of the methods of game theory to
his predictions. Von Neumann used his
game-theory methods to model the cold war
interactions between the US and USSR as two
players in a zero-sum game.
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GAME THEORY, VON NEUMANN, AND RAND
In 1943, Von Neumann was invited to work on the
Manhattan project. His mathematical models were
also used to plan out the path the bombers
carrying the bombs would take to minimize their
chances of being shot down. He also helped
select the location in Japan to bomb. Among the
potential targets he examined were Kyoto,
Yokohama, and Kokura.
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GAME THEORY, VON NEUMANN, AND RAND
In 1948, Von Neumann became a consultant for the
RAND Corporation. RAND (Research ANd
Development) is an American military-industrial
think tank to think about the unthinkable.
Their main focus was exploring the
possibilities of nuclear war and the possible
strategies for such a possibility.
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GAME THEORY, VON NEUMANN, AND RAND
Von Neumann was a strong supporter of preventive
war. Confident it was only a matter of time
before the Soviet Union became a nuclear power,
he predicted that if they were allowed to develop
a nuclear arsenal, a war with the US would be
inevitable. He recommended a US strategic
nuclear first strike at Moscow, to avoid a more
destructive nuclear war later on.
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GAME THEORY, VON NEUMANN, AND RAND
Two famous Von Neumann quotes are With the
Russians it is not a question of whether but of
when. "If you say why not bomb them tomorrow, I
say why not today? If you say today at 5 o'clock,
I say why not one o'clock?"
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GAME THEORY, VON NEUMANN, AND RAND
Two famous Von Neumann quotes are With the
Russians it is not a question of whether but of
when. "If you say why not bomb them tomorrow, I
say why not today? If you say today at 5 o'clock,
I say why not one o'clock?" By 1953 the Soviet
Union had 300-400 nuclear weapons, and Preemptive
Strike was no longer an option.
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GAME THEORY, VON NEUMANN, AND RAND
In 1954, Von Neumann was appointed to the Atomic
Energy Commission. A year later, he was
diagnosed with bone cancer and confined to a
wheelchair until his death. It has been claimed
that Von Neumann is portrayed as Dr. Strangelove
in the 1963 film Stanley Kubrick film Dr.
Strangelove or How I Learned to Stop Worrying
and Love the Bomb.
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HERMAN KAHN AND WINNABLE ESCALATIONS
However, it was Herman Kahn (who became involved
with Von Neumann in 1952 for the design of the
Hydrogen bomb) who was largely responsible for
developing Game theory as defense policy. It
was also Kahn who proposed the Doomsday
Machine, a computer connected to a stockpile of
hydrogen bombs which it would detonate if the
computer sensed an imminent and intolerable
danger from Soviet attack.
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HERMAN KAHN AND WINNABLE ESCALATIONS
Kahn belonged to a group at RAND working on
nuclear strategy known as the Strategic
Objectives Committee. The Strategic Objectives
Committee recognized that an all out war with an
initial strategy to attack cities was not
feasible. Kahn began working intensely with the
massive computers at RAND's disposal modeling
nuclear wars for the Strategic Operations
Committee.
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HERMAN KAHN AND WINNABLE ESCALATIONS
Kahn proposed a variety of escalation scenarios
simulations that he claimed proved a nuclear war
is winnable. His work had such persuasive force
that it became the basis for the majority of
military strategy during the Cold War. Kahn was
not advocating a preventive war but was calling
for first-use in the face of conflicts that could
not be deterred otherwise.
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HERMAN KAHN AND WINNABLE ESCALATIONS
Criticism of this policy. In his book On
Thermonuclear War (1961), Kahn described the
horrible side effects of radiation on the human
body. Even after graphically describing the
mutations possible, Kahn nonetheless concluded,
"War is a terrible thing, but so is peace. The
difference seems to be a quantitative one of
degree and standards." (p. 228)
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HERMAN KAHN AND WINNABLE ESCALATIONS
Many critics characterized his work as murder.
Even many of his RAND colleagues believe that
with his models as the basis for decision making
a nuclear war became more likely. Two comments as
to why this is entire analysis was wrong. 1)
Game theory makes no distinction between
maximizing winnings and minimizing losses. Its
all just numbers. 2) in a zero-sum game, no
cooperation is possible.
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