Game theory as we know it today came about
in part because of one man’s interest in
poker.
This man was not just your average man on
the street.
He was a mathematician, physicist, and computer
scientist named John von Neumann.
His goal was loftier than becoming a better
poker player.
According to a Forbes article, he “was only
interested in poker because he saw it as a
path toward developing a mathematics of life
itself.”
He “wanted a general theory – he called
it ‘game theory’ – that could be applied
to diplomacy, war, love, evolution or business
strategy.”
He moved closer toward that goal when he collaborated
with economist Oskar Morgenstern on a book
called A Theory of Games and Economic Behavior
in 1944.
The Library of Economics and Liberty (Econlib)
states that “in their book, von Neumann
and Morgenstern asserted that any economic
situation could be defined as the outcome
of a game between two or more players.”
What is a game according to game theory?
Yale economics professor Ben Polak notes a
game has three basic components: players,
strategies, and payoffs.
As we just mentioned, game theory applies
to games involving two or more players.
In a game, players share “common knowledge”
of the rules, available strategies, and possible
payoffs of a game.
However, it is not always the case that players
have “perfect” knowledge of these elements
of a game.
Strategies are the actions that players take
in a game.
Strategy is at the heart of game theory.
Forbes describes the theory presented in A
Theory of Games and Economic Behavior as “the
mathematical modeling of a strategic interaction
between rational adversaries, where each side’s
actions would depend on what the other side
would do.”
The concept of strategic interdependence – the
actions of one player influencing the actions
of the other players -- is one important aspect
of von Neumann’s version of game theory
that is still relevant today.
And then there are payoffs, which one source
describes as the “outcome of the strategy
applied by the player.”
Payoffs could be a wide range of things depending
on the game.
It could be profits, a peace treaty, or getting
a great deal on a car.
One limitation of Von Neumann’s version
of game theory is that it focused on finding
optimal strategies for one type of game called
a zero-sum game.
In a zero-sum game, “one player's loss is
the other player's gain” according to Forbes.
Another source notes that “players can neither
increase nor decrease the available resources”
in zero-sum games.
Critics have noted that life is often not
as simple as a zero-sum game.
More complicated game scenarios are possible
in the real world.
For example, players can do things like find
more resources or form coalitions that increase
the gains of several players.
Game theory has evolved to analyze a wider
range of games such as combinatorial games
and differential games, but we have time to
look at only one.
A classic example of a game often studied
in game theory is called The Prisoner’s
Dilemma.
Different versions of this game are available
on the Internet.
This version is from the Fundamental Finance
website:
“There are two prisoners, Jack and Tom,
who have just been captured for robbing a
bank.
The police don't have enough evidence to convict
them, but know that they committed the crime.
They put Jack and Tom in separate inter[r]ogation
rooms and lay out the consequences:
If both Jack and Tom confess they will each
get 10 years in prison.
If one confesses and the other doesn't, the
one who confessed will go free and the other
will spend 20 years in prison.
If neither person confesses, they will both
get 5 years for a different crime they were
wanted for.”
The Prisoner’s Dilemma contains the basic
elements of a game.
The two players are Jack and Tom.
There are two strategies available to them:
confess or don’t confess.
The payoffs of the game range from going free
to serving 5,10, or 20 years in prison.
As Fundamental Finance explains, “it is
easier to see and compare these outcomes (payoffs)
if they are put into a matrix:
Since Tom's strategies are listed in rows,
or the x-axis, his payoffs are listed first.
Jack's payoffs are listed second because his
strategies are in columns, or on the y-axis.
‘C’ means ‘confess’ and ‘NC’ means
‘not confess.’
This matrix is called ‘Normal Form’ in
game theory.
Moves are simultaneous, which means that neither
player knows the other's decision and decisions
are made at the same time (in this example,
both prisoners are in separate rooms and won't
be let out until they have both made their
decision).”
One common solution to simultaneous games
is known as “dominant strategy.”
Fundamental Finance defines it as the “strategy
that has the best payoff no matter what the
other player chooses.”
Tom does not know if Jack will confess or
not.
He takes a look at his options.
If Jack confesses and Tom does not, Tom will
get 20 years in prison.
If both Jack and Tom confess, Tom will get
only 10 years.
If Jack does not confess and Tom does, Tom
will go free.
The best strategy for Tom is to confess because
it leads to the best payoffs regardless of
Jack’s actions.
Confessing will cause Tom to either go free
or serve less prison time than if he did not
confess.
Jack is in the same situation and has the
same options as Tom.
As a result, the best strategy for Jack is
also to confess because it leads to the same
best payoffs that Tom will get.
One economics website states that a “dominant
strategy equilibrium is reached when each
player chooses their own dominant strategy.”
Why is the strategy of both not confessing
not the best choice?
While this option would give both of them
less prison time than if they confessed, it
would work only if each of them could be sure
the other one would not confess.
It is unknown whether Tom and Jack would be
able to work together with that level of cooperation.
In addition, both are unlikely to choose the
strategy of not confessing because it has
a greater penalty than they would get if they
confessed.
Confessing also gives each of them the possibility
of serving no prison time, which is even less
than 5 years in prison.
The Prisoner’s Dilemma is a good example
of how rationality can be problematic in game
theory.
University of British Columbia - Vancouver
researcher Yamin Htun calls it “one of the
most debatable issues in game theory.”
Htun points out that “almost all of the
theories are based on the assumption that
agents are rational players who strive to
maximize their utilities (payoffs),” yet
studies demonstrate that players do not always
act rationally and that “the conclusions
of rational analysis sometimes fail to conform
to reality.”
As we can see from this game, the most rational
strategy that would give both players less
prison time was not the best choice, while
a choice that involves both players doing
more prison time was.
The Prisoner’s Dilemma also reflects how
other game theorists were able to fix some
of the problems with Von Neumann’s version
of game theory.
One of them was mathematician John Nash.
He found a way to determine optimal strategies
in any finite game.
A New Yorker article describes the Nash equilibrium
as “a particular solution to games—one
marked by the fact that each player is making
out the best he or she (or it) possibly can,
given the strategies being employed by all
of the other players.”
When Nash equilibrium is reached in a game,
none of the players wants to change to another
strategy because doing so will lead to a worse
outcome than the current strategy.
In the Prisoner’s Dilemma, the Nash equilibrium
is the strategy of both players confessing.
There is no other better option for either
player to switch to.
From this game, we can also see another interesting
aspect of the Nash equilibrium.
Mathematician Iztok Hozo points out that “any
dominant strategy equilibrium is also a Nash
equilibrium.”
He explains that this is because “the Nash
equilibrium is an extension of the concepts
of dominant strategy equilibrium.”
However, he notes that the Nash equilibrium
can be used to solve games that do not have
a dominant strategy.
Nash received great praise for the Nash equilibrium
and his other work in game theory – but
not from John von Neumann.
According to Forbes, “Von Neumann, consumed
with envy, dismissed the young Nash's result
as ‘trivial’-- meaning mathematically
simple.”
Others did not share in Von Neumann’s assessment
of Nash’s work.
Nash, Reinhard Selten, and John Harsanyi went
on to share the 1994 Nobel Memorial Prize
in Economic Sciences for their work in game
theory.
When Nash died in 2015, one academic news
website summed up his accomplishments this
way:
“Nash’s most fundamental contribution
to game theory was in opening the field up
to a wider range of applications and different
scenarios to be studied.
[. . .] Without his breakthrough, much of
what followed in game theory might not have
been possible.”
What is your experience with game theory?
Let us know in the comments!
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