#### 8. What is "prisoner's dilemma"? How is it related to strictly dominant strategy?

**Prisoner's Dilemma** is a hypothetical situation used in Game Theory to explain Nash Equilibrium and Dominance Rule. The hypothetical situation is:

Two suspects, involved in a crime together, are taken into police custody. The police do not have sufficient evidence to convict them so they can be convicted if only if they confess. The police interrogate them in separate cells so that they couldn't communicate and the police put some conditions before them which can be explained with the following pay-off matrix:

Suspect B | |||
---|---|---|---|

Confess | Not Confess | ||

Suspect A | Confess | (-3, -3) | (0, -6) |

Not Confess | (-6, 0) | (-1, -1) |

The payoffs in the payoff matrix represent years of jail as punishment. Minus sign implies that jail is a loss, not profit. So, suspects would like to minimize year of jail. It exhibits the following conditions:

- Payoffs (-3, -3) implies that both will get three year jail if both confess.
- Payoffs (0, -6) implies that Suspect A will get zero year jail, that is, no punishment and Suspect B will get six years of jail if the former confesses and the latter does not confess.
- Similarly, payoffs (-6, 0) implies that Suspect B will get zero year jail, that is, no punishment and Suspect A will get 6 years of jail if the former confesses and the latter does not confess.
- Lastly, payoffs (-1, -1) implies that both will get only one year jail if both do not confess.

Let us first look at the matrix from A's point of view. Given the conditions above, it is beneficial for A to confess. If he confesses, he will get either three year jail (in case B also confesses) or zero year jail(in case B does not confess). If he does not confess, he will get either six year jail (in case of B confesses) or one year jail (in case B does also not confess). A would choose to confess as zero to three year jail is better than one to six years jail.

Similarly, if you look at the payoffs from B's point of view, you will find the same result, that is, B will also confess. It means that the optimum strategy for both is to confess which represented payoffs (-3, -3).

The situation described above can be found in many type of real life situation where two persons are interdependent for optimization of their respective payoffs. This is what we call **Prisoner's Dilemma**.

**Strictly Dominant Strategy:** It is called a **dominant strategy** if a player in a game can play its optimal strategy regardless of what other player will play. Here, A and B both have the dominant strategy. A's optimal stategy is to confess where he can expect that he will get either 0 or 3 year jail which is better than the the alternative strategy, that is, not to confess where he can get 1 or 6 year jail. He can easily choose his best strategywithout worrying about what B will play. Similar situation is faced by B. The best stategy for B is also to confess without worrying about what A will play.

It is called strictly dominant strategy when the alternative stategy always makes worse off. Here, we can see that alternative strategy (Not Confess) of A makes him better off than the optimal strategy if B does not confess. Same situation is faced by B. Therefore, neither of them has **strictly dominant strategy**.

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