User:StellaChoi/sandbox

The infinitely repeated prisoners' dilemma
The infinitely repeated prisoners’ dilemma is a well-known example for the grim-trigger strategy. Let’s say two prisoners play a game as following.

In the prisoners' dilemma, each player has two choices in each stage: If a player defects, they will be punished for the remainder of the game. In fact, both players are better to not to confess(cooperate) than to betray the other and so, playing (C, C) is the cooperative profile while playing (D, D), which is the unique Nash equilibrium in this game, is the punishment profile.
 * 1) Cooperate
 * 2) Defect for an immediate gain

Then, the grim trigger strategy is that the players cooperate in the first round and in the repeated rounds as well as long as no one defects from the agreement. Once the player finds the opponent betrayed in the previous game, the player will play D forever.

In order to evaluate the sub-game perfect equilibrium(SPE) for the following grim-trigger strategy of the game, strategy S* for player i and j is as follows:
 * Play C in every period unless someone has ever played D in the past
 * Play D forever if someone has played D in the past [1]

Then, the strategy is a SPE only if the discount factor is $\delta \geq {\frac{1}{2}}$. In other words, neither Player 1 or Player 2 is incentivized to defect from the cooperation profile if the discount factor is greater than one half. [2]

To prove the that strategy is a SPE, both the cooperation should be the best response to the other player’s cooperation, and the defection should be the best response to the other player’s defection. [1]

Step 1: Suppose that D is never played so far. Then, C is better than D if $$1 \geq 2(1-\delta)$$. This shows that if $$\delta \geq \frac{1}{2}$$, playing C is optimal.
 * Player i’s payoff from C : $$(1-\delta)[1+\delta+\delta^2+ ... ] = (1-\delta) \times \frac{1}{1-\delta} = 1$$
 * Player i’s payoff from D : $$(1-\delta)[2+0+0+ ... ] = 2(1-\delta)$$

Step2: Suppose that someone has played D previously, then Player j will play D no matter what. Since $$ 0 \leq \delta \leq 1$$, player D is definitely optimal.
 * Player i’s payoff from C : $$(1-\delta)[-1+\delta \times 0+\delta^2 \times 0 + ... ] = (1-\delta) \times -1 = \delta - 1$$
 * Player i’s payoff from D : $$(1-\delta)[0+\delta \times 0+\delta^2 \times 0 + ... ] = 0$$

The preceding argument emphasizes that there is no incentive to deviate(no profitable deviation) from the cooperation profile if $$\delta \geq \frac{1}{2}$$, and this is true for every sub-game. Therefore, the grim-trigger strategy for the infinitely repeated prisoners’ dilemma game is a Sub-game Perfect Nash equilibrium.

Grim trigger in International relations
Under the grim trigger in international relations perspective, a nation cooperates only if its partner has never been exploited in the past. Because a nation will refuse to cooperate in all future periods once its partner defects once, the indefinite removal of cooperation becomes the threat that makes such strategy a limiting case.[3] While grim trigger is a limiting case, Folk theorem (game theory) states that a perfect equilibrium can arise if both nations are patient. [4]

Grim trigger in User network interaction game
Game Theory has recently been used in developing future communications system, and the user in the user-network interaction game employing the grim trigger strategy is one of such examples.[5] If the grim trigger is decided to be used in the user-network interaction game, the user will be staying in the network(cooperate) if the network maintains a certain quality, but punishes the network by stopping the interaction and leaving the network as soon as the user found the opponent defect. [6] “Given such a strategy, the network has a stronger incentive to keep the promise given for a certain quality, since it faces the threat of losing its customer forever.” [5]

Comparison with other strategies
Tit for tat and Grim trigger strategies are similar in that players refuse to defect first if they have the ability to punish the other player for defecting. The difference, however, is that Grim Trigger seeks maximal punishment for a single defection while Tit for Tat is more forgiving, offering one punishment for one defection. [7]