I have the following static game with complete information.
First of all the players play the static game depicted above infinitely repeated. They discount payoffs with the common discount factor $\delta$. And I am interesting in supporting (T,R),(T,R),...) as subgame perfect equilibrium I want to calculate the minimal discount factor needed so that my strategy supports this outcome. And secondly, this static game is assumed to be finite.y related. Now, I am I tested in supporting ((T,L),(D,R),...,(T,L), (D,R)) as a subgame perfect equilibrium. Again I want to implement this outcome as a subgame perfect equilibrium. And I would like to calculate again the minimum discount factor neeeded so that my strategy supports this outcome.
————
For the infinitely repeated case, I guess I should use trigger stategies. I found some examples related to such type questions on the google. But I cannot understand its solutions. And thus, I cannot apply these solutions in my examples. So I cannot write anything about my trails. Sorry for that. Please show me how can I solve thesetypes of questions.
Note that this question is the continue of that question Finding bet response function to the opponent mixed Nash strategy. . I really try to solve these parts but I cannot. I stuck at this point.
Thank you.
You can use the grim trigger strategy for the infinitely repeated games. This method employs the use of a credible threat that would force the player to stick to the outcome agreed upon (given a certain value of delta) and not deviate. Since you cannot deviate, this implies, it holds as a nash equilibrium.
Now, to hold (T,R) as a Nash, both players will say this to each other: We shall play (T,R) till one person defects then play (T,L) or (D,R) forever(depending on who is defecting; if player 1 defects then they play (T,L) forever).
We then compare the payoffs from sticking to this strategy with the payoff that one yields from defecting.
So, from sticking to (T,R), a player gets:
$4+ 4\delta + 4\delta^2 +........$=$\frac{4}{1-\delta}$
By deviating, one gets:
$5 + 1\delta + \delta^2+......=5+ \frac{\delta}{1-\delta}$
We want to support (T,R) as a nash equilibrium and therefore we want:
$\frac{4}{1-\delta} \geq 5+ \frac{\delta}{1-\delta}$
Solving this yields that $\delta \geq 1/4$
Then, given that $\delta$ satisfies the above condition, (T,R) can be supported as a Nash Equilibrium.
EDIT: Finitely repeated case, for periods T=2.
Just realised it right now, you can't hold onto (T,R) as an equilibrium in the finitely repeated games because you can't incentivise the other person to play (T,R) as what will you play in the next stage for that person to benefit from sticking to the outcome?