The prisoner's dilemma provides a good starting point to compare the two concepts of Nash equilibrium and Pareto optimal, and to understand l'applicazione in economia. Riprendendo quanto illustrato nella definizione matematica dell'equilibrio di Nash, vediamo la loro applicazione al caso del dilemma del prigioniero. Le possibili scelte per due prigionieri in celle diverse non comunicanti sono parlare (accusando l'altro) o non parlare.
* Se entrambi non parlano avranno una pena leggera;
* Se entrambi parlano, accusandosi a vicenda, avranno una pena un po' più pesante;
* Se faranno scelte diverse, quello che parla avrà la libertà e l'altro avrà una pena molto pesante.
Se entrambi conoscono queste regole e non prendono accordi, la scelta che corrisponde all'equilibrio di Nash è di parlare, per entrambi. Da questo esempio we see that the theory in real cases is not always the best solution (or sometimes not sufficiently realistic).
Both players have the same strategies (two) and the same pay-off (2x2) which is (for brevity denoted confesses confesses he's not there years in prison with the minus sign as they represent losses and gains so negative ): *
Strategies: Si = (c, n)
* Pay-off:
ui (c, c) = -6
ui (c, n) = 0
ui (n, c) = -7
ui (n, n) = -1
It follows immediately that, for both, the dominant strategy is confessing, in fact
ui (c, c)> ui (n,c)
e
ui (c,n) > ui (n,n)
quindi qualunque sia la scelta dell'avversario, scegliere confessa garantisce sempre un guadagno maggiore rispetto a scegliere non confessa. È immediato riconoscere come la combinazione di strategie dominanti confessa-confessa soddisfi la disuguaglianza che definisce l'equilibrio di Nash, infatti per entrambi i giocatori
ui (c,c) > ui (n,c)
(per il secondo giocatore la disuguaglianza è soddisfatta invertendo l'ordine delle strategie). In sostanza, posto che il secondo giocatore confessi, il primo deve scegliere anch'esso confessa, e non può aumentare il proprio guadagno cambiando solo la sua strategia: his pay-off if no-confessed confesses is less than that would balance the game. confesses confesses-is also the only equilibrium of the game, in fact any combination of strategies satisfies the inequality.
The solution of the game is therefore both confess, getting six years in prison each.
The most interesting aspect, however, the prisoner's dilemma is the following: all combinations of strategies, with the exception of Nash equilibrium, is Pareto optimal. In fact, taken any of these combinations, you can not find another that involves at least one player a year reduction in jail without any increase in those of the other. This concept is not applicable to the balance-confessed confesses: the combination does not confess, not confess leads to a reduction of years in prison for both players (one year each instead of 6) and since
ui (n, n) > ui (c, c)
for all i, (c, c) is not a Pareto-optimal solution.
The Pareto optimum is a very important concept in economics: the market objective is to always reach a Pareto optimal, ie to a situation in which, regardless of the actual allocation of resources, can not be found another allocation leading to an increase in wealth of some wealth without taking others. The reason for the importance of excellent Pareto is intuitive: if there is a solution that represents an increase in earnings of someone with no one suffers losses, it means that there are resources that have not been allocated, and therefore would be missing. In the case of the excellent walls, in fact, the further enrichment of one necessarily entails the depletion of another. The prisoner's dilemma highlights a key concept of economics: the rational Pareto optimal from the standpoint of collective, but it is not at all from the individual point of view, in essence, if the N agents in a game (and therefore, by extension, of a market) act in accordance with the individual rationality, ie, with the sole purpose of maximizing their personal gain, does not mean that they reach a Pareto optimal, in which case their actions result in a waste of resources.
The comparison between Nash equilibrium and Pareto optimum then denies the allegations made by Adam Smith, considered up to the first formulation of the equilibrium theory, the "father of modern economics." He maintained that if any member of the group pursues its own interest, can only increase the overall wealth of the group. But today we know that if each member does what is best for themselves, the result obtained is a Nash equilibrium but not necessarily for the better Pareto means you can (and has since proved to be very frequent) that if each agent is only his personal interest, you will come to an inefficient allocation of resources. In the case of prisoner's dilemma, this is obvious: the smallest possible value is 0 years' imprisonment for individuals and for group 2, but if both choose their dominant strategy, we get 6 (12 total for the group).
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