Prisoner’s dilemma
Prisoner’s dilemma was studied by Merrill Flood and Melvil Dresher in 1950 and after, formalized
by Albert W. Tucker by the matrix payoff and to whom we have to also the name of the dilemma.
The prisoner’s dilemma is a classical example of the theory of games, namely, a situation where the
players take strategic decisions, or the situation where each players consider the action and the
reaction of others players.
This game is “not cooperative” becouse it is not possible for the players to make arrangements to
take the most advantageous strategy for both.
“Two suspects, A and B, are arrested by the police. The police does not have sufficient evidence to
find the culprit and, having locked up the two prisoners in two different cells, questioned both by
offering them the following prospects: if one confesses (C) and the other does not confess (NC) he
has not confessed will serve 10 years in prison while the other will be free; if both do not confess,
then the police will condemn them to one year in prison; if, however, confess both worth to be
served will be equal to 5 years in prison. each prisoner can reflect on the strategy to choose from,
in fact, confess or not to confess. in any case, neither of the two prisoners may know the choice
made on the other prisoner”.
This is the popular form in which we know the Prisoner’s dilemma.
The prospective:
● If both accept the proposal, both they will be sentenced, but the judgment will mitigated by
collaboration.
● If neither accepts, it will be very mild for both.
● If only one of the two agree, the other will undergo a condemnation, being recognized as the only
culprit.
Now I will represent the matrix payoff:
prisoner A/
prisoner B
confesses
Not
confesses
confesses
-5, -5
0, 10
Not
confesses
-10, 0
-1, -1
The choises :
● If the two prisoners could interact and choose a common strategy, probably would opt not to
mention.
● Having to choose without knowing the intention of the companion, the strategy that minimizes
the risk is that of betraying.
● Since the individual is brought to betray, the situation has reached inevitably a solution suboptimal (the total punishment imposed is not the minimum possible).
The first to be questioned is the prisoner A ( whose payoff are listed first and in red).
for each prisoner is "better" confess because, irrespective of the choice player, his payoff is higher
confessing. In fact, regardless of the play by A to B will be cheaper confess.
Let's examine why: if A had confessed (C), would agree to confess to B (C) because in this way
sconterebbero both five years in prison; if A had not confessed (NC), B would agree more so
confess (C) because doing so he would be free, while A would discount 10 years in prison.
The balance of the game, therefore, is placed in C-C: the two prisoners will discount both five years
in prison. This balance is called "Pareto-inefficient" because despite being the rational one, it is not
the best of situations.
According to Game Theory, the choice of confessing (C) operated by the two prisoners in the light
of the above reasoning is called "dominant strategy": It is in fact defined as the optimal strategy
regardless of what the opponent does.
Is useful to point out how the balance in dominant strategy is a special case of Nash Equilibrium:
It can say that,
● I do the best I can given what you do.
● You do the best you can given what I do.
The paradox
The prisoner's dilemma has caused interest as an example of game in which the rationality seems
apparently fail axiom, so to create an action that procures more damage to both sides of the NC-NC
alternative choice. The game theory scholars point out that those who think in this way you
probably imagine a different game, in which the victory is assessed on the sum of years in prison.
At this point we could ask ourselves:
● It is possible that there is no logical conclusion to allow the prisoner to hope to remain in prison
one year or no?
● It is possible that the logic does not reach any other solution besides the agreement to be
sentenced to five years with no hope?
We can reach a solution but we need some clarifications..
A) You should assume that all the characters have an almost perfect logical capacity. This does
not mean you have to be good, altruistic or other, only that everyone understands the game
the same way, and make no mistake.
B) Since the point A) it is easy to see that all take the same decision. There can not be one that
is clever at the expense of the other, because this automatically would mean that others will
like him. Only the "inattentive" reader may think to be clever to a single character.
At this point it seems clear that if one of the prisoners understand that the conclusions he reaches
are the same who gets the other, choose NC is the only possible action.
References :
http://www.dei.unipd.it/
http://utenti.quipo.it
http://www.prisoners-dilemma.com/
http://www.beyondintractability.org
Simona Ippoliti
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