Question

Assume that the stag-hare game is now played repeatedly. What should the discount factor be for such a game to enforce cooperation? Assume the Grim-Trigger strategy. Show your work to justify your conclusions.

Answer 1

The Repeated Game- In a repeated game, a phase game is played over and over by similar players. A phase game is characterized by a limited arrangement of players N={1,… ,n}, a limited arrangement of unadulterated activities for every player Ai, i∈N, and the players' utilities for each activity profile u:A↦Rn, where A=×i∈NAi is the arrangement of unadulterated activity profiles. Additionally, an unadulterated activity of player I is meant ai∈Ai and an unadulterated activity profile is a∈A.

Every player i∈N may randomize over his unadulterated activities ai∈Ai. This characterizes a blended activity qi with the end goal that qi(ai)≥0 for each ai∈Ai and ∑ai∈Aiqi(ai)=1. The arrangement of likelihood circulations over Ai is meant Qi and Q=×i∈NQi. A blended activity profile is indicated by q=(q1,… ,qn)∈Q. A help of a blended activity is the arrangement of unadulterated activities that is played with a carefully certain likelihood: Supp(qi)={ai∈Ai|qi(ai)>0}. We likewise characterize Supp(q)=×i∈NSupp(qi), and for each a∈Supp(q), we let πq(a) be the likelihood that the activity profile an is acknowledged whether the blended activity profile q is played: πq(a)=∏j∈Nqj(aj). In unadulterated methodologies, we make the limitation that qi(ai)=1 for one activity ai∈Ai.

In a model with public connection, the players watch an acknowledgment ω∈[0,1] of a public lottery and they can condition their activity dependent on the sign ω. For instance, two players may consent to make a move a1 if ω≤1/2 and a2 in any case. This way the players can facilitate their activities with the end goal that they randomize between the results (a1,a1) and (a2,a2), and keep away from the results (a1,a2) and (a2,a1). The last results would be acknowledged whether standard blended procedures were utilized and no open  relationship gadget was accessible.

Using Grim Trigger Strategy

Think about the rehashed detainee's situation.

The technique endorses that the player at first participates, and keeps on doing as such if the two players collaborated at all past occasions.

si (a1, . . . , aT) = D if at = (C,C) for some t = 1, . . . , T.

si (a1, . . . , aT) = C in any case.

Note that a player deserts if possibly she or her rival absconded before.

An automaton for player i is (X, x0 , f, g ).
X is a set of states.
x0 is the initial state of the automaton.
f : X x A X is the transition across states, as a function of the play.
g: X Ai is the play output at each state.

The machine of the Grim Trigger Strategy is as per the following:

There are two states: C in which C is picked, and D, in which D is picked.

The underlying state is C.

On the off chance that the play isn't (C,C) in any period then the state changes to D.

In the event that the robot is in state D, there it remains until the end of time.

In the event that I can expand her result by going amiss, at that point she can do as such by straying to D in the primary time frame.

She acquires the surge of settlements (3, 1, 1, . . .) with limited normal

(1 − d)[3 + d + d 2 + d 3 + · ·] = 3(1 − d) + d.

Along these lines player I can't expand her result by going amiss if and just if

2 ≥ 3(1 − d) + d, or d ≥ 1/2.

Thus, in the event that d ≥ 1/2, at that point playing the troubling trigger methodology by the two players is a Nash harmony of the interminably rehashed Prisoner's Dilemma.