Prisoner's Dilemma

The police had been watching two suspects of bank robbery for some time but they didn't have sufficient evidence for a conviction. One day, these two guys were caught stealing packs of bubble gum in a supermarket. The police put them into separate rooms, visited each of them and offered the same deal: If one testifies (defects from the other) for the prosecution against the other and the other remains silent, the betrayer goes free and the silent suspect receives the full 10-year sentence for bank robbery. If each betrays the other, each receives a 5-year sentence for bank robbery. If both remain silent, both suspects will be sentenced to only a 1-month jail for theft in supermarket. Each suspect was assured that the other would not know about the betrayal before the end of the investigation and each was told that the other was also offered the same deal. Thus, each suspect was facing a dilemma of choosing either to co-operate with the police (betray the other) or not to co-operate with the police (remain silent).
How would the suspect act?
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The possible outcomes (pay-off) can be summarized diagrammatically as follows:

	B remains silent	B betrays
A remains silent	A: 1 month B: 1 month	A: 10 years B: goes free
A betrays	A: goes free B: 10 years	A: 5 years B: 5 years

In 'win-lose' terminology, the table looks like this:

	B remains silent	B betrays
A remains silent	win-win	lose much-win much
A betrays	win much-lose much	lose-lose

Clearly from the diagrams above, the best result (for BOTH suspects) could be achieved IF both co-operated with each other and not with the police by remaining silent. Then each would only be sentenced to a 1-month jail for theft. However, since they were shut up separately, they would not be able to negotiate with each other for a co-operation, and they could only guess what the outcomes would be for different scenarios. Without loss of generality, if suspect A remained silent, he had 50% chance for 1-month jail and 50% chance for 10-year jail. His average pay-off was 5 years and a half month. If A betrayed, he had 50% chance to go free and 50% chance for 5-year jail. His average pay-off was 2 and a half years. A might think that, regardless what B chose, he would always receive a higher pay-off (lesser sentence) by betraying. A could actually say, "No matter what B does, I personally am better off betraying than remaining silent. Therefore, for my own sake, I should betray." All things being equal, suspect B would act similarly. Then they both betrayed each other, and both actually got a lesser pay-off (5 years) than they would get by remaining silent (1 month).

This classical prisoner's dilemma illustrates the different strategies that the player (suspect) would take in the game with different perspectives. If the player could look at the broad picture (which only the police could but the suspects could not), he would opt for remaining silent to achieve the best result. If the player looked at the game only from his own perspective (which the suspects were constrained to do), the strategy to remain silent is clearly dominated by that to betray. So the only possible equilibrium for the game is for all players to betray. No matter what the other player does, one player will always gain a greater benefit by playing betrayal. Since in any situation, playing betrayal is more beneficial than playing remaining silent, all rational players will betray. It also shows how a win-win situation (for the suspects) can be turned into a lose-lose situation (as exactly what the police wanted).