Imagine you
are a criminal whose life of crime has finally caught up to you. You and your partner, Al, have been caught by
the police for a minor crime and both of you are being interrogated separately.
The police
correctly suspect that you and Al have committed larger crimes but have no
evidence for a conviction, so they set up a bargain. If both you and Al confess to your other
crimes you will both serve 8 years in prison.
If one of you confesses and the other does not, the confessor will go
free and the silent one will serve 20 years in prison. If both of you are silent, then there is only
enough evidence to convict you of the minor crime, and both of you will only
serve 5 years in prison. What do you do?
|
Al Confesses
|
Al is Silent
|
You Confess
|
You serve 8 years and
Al serves 8 years
|
You go free and
Al serves 20 years
|
You Are Silent
|
You serve 20 years and
Al goes free
|
You serve 5 years and
Al serves 5 years
|
This
Prisoner’s Dilemma was first conceived by Flood and Dresher in 1950 to
illustrate the relationship between competitive and cooperative behavior between
two parties in game theory.
A good
illustration of the risks and rewards presented in the Prisoner’s Dilemma is a
card game described in the dystopian young adult book Matched by Ally Condie.
As one of
the characters explains the game, “They each put down a card at the same
time. If they both have an even card,
they each get two points. If they’re
both odd, then they each get one point … If one is even and one is odd, the
person who puts down the odd card gets three points. The person who puts down the even one gets
zero.”
|
Player 2 Odd
|
Player 2 Even
|
Player 1 Odd
|
Player 1 gets 1 pt and
Player 2 gets 1 pt
|
Player 1 gets 3 pts and
Player 2 gets 0 pts
|
Player 1 Even
|
Player 1 gets 0 pts and
Player 2 gets 3 pts
|
Player 1 gets 2 pts and
Player 2 gets 2 pts
|
Note that in
this version of the Prisoner’s Dilemma, the goal is to gain something positive,
like receiving points, instead of avoiding something negative, like a prison
sentence. But the dilemma is
similar. On your turn, do you lay down
an odd card for 1 or 3 points, or do you lay down an even card for 0 or 2 points?
To help
solve these dilemmas, you can use the Nash Equilibrium, which is the solution set
of best decisions for each party given that no party has anything to gain by
changing his or her decision. In the card game, you should lay down an odd
card, because you would have nothing to gain by changing your mind even if you
know what the other player will play.
(If the other player lays down an even card, you will get 3 points, so
changing your mind to play an even card to get 2 points would not make
sense. If the other player lays down an
odd card, you will get 1 point, so changing your mind to play an even card and
get 0 points would also not make sense.)
Similarly, as a prisoner you should confess, because you would have
nothing to gain by changing your decision even if you know Al’s decision. (If Al remains silent, you will be let free,
so changing your decision to also remain silent and get a 5 year prison
sentence would not make sense. If Al
confesses, you will receive an 8 year sentence, so changing your decision to
remain silent and get a 20 year prison sentence would also not make sense.)
Of course, by
symmetry the Nash Equilibrium would be for the other player to also lay down an
odd card, and for Al to confess as well.
However, this solution which would not give the best overall result for
both parties. In the card game, the best
overall result would be for both players to lay down even cards for 2 points
each, for the maximum total reward of 4 pts.
|
Player 2 Odd
|
Player 2 Even
|
Player 1 Odd
|
Player 1 gets 1 pt and
Player 2 gets 1 pt
1 + 1 = 2 pts
total
|
Player 1 gets 3 pts and
Player 2 gets 0 pts
3 + 0 = 3 pts
total
|
Player 1 Even
|
Player 1 gets 0 pts and
Player 2 gets 3 pts
0 + 3 = 3 pts
total
|
Player 1 gets 2 pts and
Player 2 gets 2 pts
2 + 2 = 4 pts
total
|
With the interrogated
prisoner situation, the best overall result would be for both you and Al to
remain silent, which will result in the two of you receiving only a 5 year
prison sentence, for the minimum total penalty of 10 years.
|
Al Confesses
|
Al is Silent
|
You Confess
|
You serve 8 years and
Al serves 8 years
8 + 8 = 16 years
total
|
You go free and
Al serves 20 years
0 + 20 = 20 years
total
|
You Are Silent
|
You serve 20 years and
Al goes free
20 + 0 = 20 years
total
|
You serve 5 years and
Al serves 5 years
5 + 5 = 10 years
total
|
In both
cases, the only way to achieve the best overall result would be for both parties
to cooperate and trust each other. (In
the case of the prisoners which are interrogated separately, a deal between the
two prisoners would have to be made beforehand.) In contrast, the Nash Equilibrium assumes no
cooperation, just competition.
The Nash
Equilibrium is probably best known by the movie A Beautiful Mind (2001), a biography of the mathematician John Nash.
In this
movie, a new dilemma is presented to introduce the viewers to game theory. A young Nash and his college friends are sitting
in a bar, and some girls walk in, including one beautiful blonde girl.
While Nash’s
friends are arguing over who will get the blonde girl, Nash realizes that if
they all go for her, they will all cancel each other out and no one will get
her, and then no one will get any other girl, either, because none of the other
girls will like being the second choice.
But if everyone goes for the other girls, everyone will get a girl,
which will result in the best overall result.
For simplicity, the following table shows the dilemma for just two of
the guys, Nash and Hansen:
|
Hansen Goes for the Blonde
|
Hansen Doesn’t Go for the
Blonde
|
Nash Goes for the Blonde
|
Nash gets no girl and
Hansen gets no girl
|
Nash gets the blonde and
Hansen gets a girl.
|
Nash Doesn’t Go for the Blonde
|
Nash gets a girl and
Hansen gets the blonde
|
Nash gets a girl and
Hansen gets a girl
|
Technically,
the movie’s solution for all of them to agree to not go after the beautiful
blonde girl is not a Nash Equilibrium. (If
Hansen goes for the beautiful blonde girl, then Nash should go for one of the
other girls because then at least he will get a girl. But if Hansen doesn’t go for the beautiful blonde
girl, then Nash should, because then he will get a more beautiful girl.) However, the solution presented is the best
solution if they are allowed to cooperate
and trust each other to make a deal (and if it is assumed that no one will
agree to the unfair deal of only one guy getting the beautiful blonde girl),
just like how in the card game the best cooperative solution would be for both
players to lay down an even card, or just like how with the interrogated
prisoners the best cooperative solution would be for both prisoners to keep
silent.
A real life
Prisoner’s Dilemma occurred during the 2014 FIFA World Cup for Germany and the
United States. Both countries were in
Group G along with Portugal and Ghana.
By June 22,
2014, each team had played 2 games. Both
Germany and USA were leading the pool with 4 points each, having gained 3
points for a win and 1 point for a tie.
Portugal and Ghana both had 1 point each, 1 point for a tie and 0 points
for a loss.
Teams
|
MP
|
W
|
D
|
L
|
GF
|
GA
|
+/-
|
Pts
|
Germany
|
2
|
1
|
1
|
0
|
6
|
2
|
4
|
4
|
USA
|
2
|
1
|
1
|
0
|
4
|
3
|
1
|
4
|
Portugal
|
2
|
0
|
1
|
1
|
2
|
6
|
-4
|
1
|
Ghana
|
2
|
0
|
1
|
1
|
3
|
4
|
-1
|
1
|
There were
two games left on June 26, 2014 to finish pool play: Germany vs USA and
Portugal vs Ghana. After these games the
top two teams would advance, determined first by points and secondly by goal
difference. The following table shows
the possible outcomes for the two games (the outcomes with “or” would depend on
the goal difference):
|
Portugal Wins and
Ghana Loses
|
Portugal and
Ghana Tie
|
Portugal Loses and
Ghana Wins
|
USA Wins and
Germany Loses
|
USA and (Germany or Portugal) advance
|
USA and Germany advance
|
USA and (Germany or Ghana) advance
|
USA and
Germany Tie
|
USA and Germany advance
|
USA and Germany advance
|
USA and Germany advance
|
USA Loses and
Germany Wins
|
Germany and (USA or Portugal) advance
|
USA and Germany advance
|
Germany and (USA or Ghana) advance
|
As you can
see, as long as game the game between Germany and USA was close in score, both
Germany and USA would advance. Some
sport commentators jokingly suggested that Germany and USA should agree to tie,
because then both teams would have the certainty of advancing. Setting this situation up in a simplified Prisoner’s
Dilemma, you can give both teams the option of either trying to win or taking it
easy. If both teams take it easy, the
result will be a 0-0 tie, and both teams will advance. If one team tries to win and the other team
takes it easy, the team that is trying to win will win by a wide margin of
goals and advance, and the other team that is taking it easy will lose by a
wide margin of goals and not advance. Finally,
if both teams try to win, the game will likely be close in score (unless one
team is much better than the other), so both teams will probably (but not
certainly) advance.
|
Germany Tries to Win
|
Germany Takes It Easy
|
USA Tries
to Win
|
USA probably advances and
Germany probably advances
|
USA advances and
Germany does not advance
|
USA Takes
It Easy
|
USA does not advance and Germany advances
|
USA advances
Germany advances
|
The Nash
Equilibrium is for both teams to try to win, because each team would have
nothing to gain by changing their strategy even if they know the other team’s
strategy. (If Germany tries to win, USA
also needs to try to win as well so they can probably advance, as opposed to taking
it easy and not advancing. If Germany
takes it easy, USA will advance no matter what, so changing the USA’s strategy
makes no difference.) However, the best
overall result would be for both teams to have the certainty of advancing,
which means cooperating and trusting each other on a deal to purposely tie.
As is
usually the case, history shows that the Nash Equilibrium based on competition won
over the solution based on cooperation.
It appeared that Germany and USA both used the strategy to try to win,
and on June 26, 2014 Germany beat USA by a close score of 1-0. Portugal beat Ghana, but only by a score of 2-1,
so both Germany and USA advanced.
The Prisoner’s
Dilemma has application in other areas of life, too, especially in
economics. Let’s say that there is a
shopping plaza on the outskirts of town with a fitness center and a sports
bar. Both businesses would benefit by
having a cable line installed to the shopping plaza; the fitness center would
attract more customers if its customers can watch cable TV while running on the
treadmill, and the sports bar would attract more diners if its diners can watch
their favorite sports teams play while eating.
The cable company offers to run a cable line out to the shopping plaza
for a cost of $1000. Once the cable line
is installed, any business can hook into the cable line at $100. (Also for our example, we’ll say that both
businesses value the cable line at $700, in other words, neither business will
pay more than $700 to have the cable line installed.) Each business can either choose to have the
cable line installed or to not have the cable line installed. If both businesses decide to install the
cable line, they can split the cost of the cable line at $500 each, and sign up
for cable for another $100 each, and receive cable television for a total cost
of $600 each. On the other hand, if both
businesses decide not to install the cable line, neither will pay anything but then
neither will have cable television. Finally,
if one business decides to install the cable line and the other does not, the one
business would pay $1000 for installing the cable line and another $100 for
signing up for cable (for a total of $1100), whereas the other business would
pay nothing for installing the cable line and $100 for signing up for cable.
|
Sports Bar No
|
Sports Bar Yes
|
Fitness Center No
|
FC no cable and
SB no cable
|
FC $100 cable later and
SB $1100 cable
|
Fitness Center Yes
|
FC $1100 cable and
SB $100 cable later
|
FC $600 cable and
SB $600 cable
|
The Nash
Equilibrium would be for both businesses to say no to cable. (If the sports bar says yes to installing the
cable line, the fitness center should say no because it can then receive the
benefit of cable television for $100 instead of $600. If the sports bar says no to installing the
cable line, the fitness center should still say no because the cost of
installing the cable line would be $1100, which is over the value of $700.) However, the best overall result would be for
both businesses to have cable television, which means they should cooperate and
trust each other and make a deal.
The Prisoner’s
Dilemma illustrates how two different economic systems have two different
solutions. In a society based on cooperation
or socialism, a cable deal may be able to be reached (perhaps with government
intervention through tax money).
However, in society based on competition or capitalism, the Nash
Equilibrium in this situation says that there will be no progress concerning cable
television. There are several solutions
to this problem, but the most obvious is that the cable company needs to decrease
its price on installing a cable line and increase its price for signing up, or
it will soon be out of its own business.
(This is one of the reasons why it is important for the government not
to fix costs in capitalistic society.) For
example, if the prices change to $0 for installing the cable line and $600 for
signing up, the Nash Equilibrium changes to both businesses saying yes to
cable, because it will cost less than the value of $700 to receive it, and the
cable company will still make its $1200.
|
Sports Bar No
|
Sports Bar Yes
|
Fitness Center No
|
FC no cable and
SB no cable
|
FC no cable and
SB $600 cable
|
Fitness Center Yes
|
FC $600 cable and
SB no cable
|
FC $600 cable and
SB $600 cable
|
The Prisoner’s
Dilemma can also be applied to similar economic situations, such as building roads
or train tracks, installing lighthouses, launching satellites, and much
more. In each case, a socialistic or a
capitalistic solution can be reached, each with its own pros and cons. However, a capitalistic solution will only
work if there are no fixed costs that restrain private companies from withholding
its services.
In
conclusion, the Prisoner’s Dilemma started out as a fun exercise in game theory
in 1950. But like many mental exercises,
theory turned into practice, and the Prisoner’s Dilemma had applications in the
economic realm as well.
This was interesting and I actually understood it! Yay me!
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