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Randy Whitehead
What is a Game?

We all know how to play games. Whether
they involve cards, sports equipment,
boards, dice, or a multitude of other things,
but is that the extent of what we can call a
game? A game is defined as an activity
engaged in for activity or amusement. With
this in mind, can we really restrict a game
to an arbitrary set of equipment used? The
reality of the situation is that every
interpersonal reaction can be reduced to a
game with different strategies, outcomes,
and payoffs.
Representation of Games

There are two main types used to
represent a game in the field of game
theory.
Extensive

This method is often used in games where players
alternate moves and you can look back and see all
that has happened before, often called a game of
perfect information, but it can also be used for
simultaneous games as well.
Extensive Example
This example shows the very first move made in a chess
game. As you can see, it can get very large very quickly.
Strategic

This method is favored by most people because it
sets up all possible choices and outcomes into a
single payoff matrix. Game theorists will often try
to reduce a game into one or a collection of 2 x 2
payoff matrices.
Strategic Example
This example shows a simple number calling game
between two players. The payoffs are written with the
corresponding number equivalent to the amount that
player 1 pays player 2, i.e. if the number is negative then
player 2 pays player 1 the absolute value of the number.
General Strategic Form
Let player R have m possible moves
and player C have n possible moves.
 Depending on the two players' moves, a
payoff is made from player C to player
R.

General Strategic Form

For all i = 1,2,...,m, and j = 1,2,...,n, let
us set
 aij= payoff that player C makes to player R if
player R makes move i and player C makes
move j

Therefore, the payoff matrix is:
General Strategic Form

The next natural step is to examine the
probability that a player will make a
certain move
 pi=probability that player R makes move
i.
 qj=probability that player C makes move
j.
General Strategic Form

With the probabilities pi and qj, two
vectors can be formed:

Where the row vector p is called the
strategy of player R and the column
vector q is called the strategy of player
C.
Theorem 1

For a 2 x 2 game that is not strictly
determined, optimal strategies for
players R and C are:

The value of the game is:
Inoculation Strategies

The federal government desires to
inoculate its citizens against a certain flu
virus. The virus has two strains, and the
proportions in which the two strains occur
in the virus population is not known. Two
vaccines have been developed. Vaccine 1
is 85% effective against strain 1 and 70%
effective against strain 2. Vaccine 2 is 60%
effective against strain 1 and 90% effective
against strain 2. What inoculation policy
should the government adopt?
Inoculation Strategies

We may consider this a two-person
game in which player R (the
government) desires to make the payoff
(the fraction of citizens resistant to the
virus) as large as possible, and player C
(the virus) desires to make the payoff as
small as possible. The payoff matrix is:
Inoculation Strategies

This matrix is not strictly determined
because it has no saddle points.
Therefore, Theorem 1 is applicable.
Prisoner’s Dilemma
 Suppose the police have captured two suspects
that they know have committed armed robbery
together but do not have enough evidence to
convict either of the suspects of the armed
robbery charge, but they do have evidence of
the suspects stealing the getaway car. The chief
officer then makes the following offer to each
prisoner: "If you will confess to the robbery,
implicating your partner, and she does not also
confess, then you'll go free and she'll get ten
years. If you both confess, you'll each get 5
years. If neither of you confess, then you'll each
get two years for the auto theft."
Prisoner’s Dilemma

We can model the situation with a payoff
matrix and assign cardinal numbers to
each outcome.




0
2
3
4
10 years in prison
5 years in prison
2 years in prison
0 years in prison
Prisoner’s Dilemma

By using math, which isn’t shown in this
presentation, we arrive at the optimal
strategy for the prisoner’s dilemma.
According to game theory, the prisoner
should defect in all situations regardless
of what the other prisoner does.
Prisoner’s Dilemma
If this example is applied to more
relatable applications, the results
become much more disturbing.
 Let’s take a look at the Cold War for
example.

Cold War

The Cold War between the United
States and the Soviet Union can be
modeled directly as a prisoner’s
dilemma with the following payoff matrix:
Cold War
According to game theory, a rational
player would increase their amount of
arms in all situations. Luckily, the United
States took the more peaceful approach
and the Soviet Union followed suit.
 If both entities had continued to increase
their amount of arms, the world would
be a much different and more
dangerous place.


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