Question

Q.3 Player 1 and player 2 bargain over sharing 600 dollars. The bargaining procedure follows the Rubinstein bargaining model. Player 1's share is 1-e-0.5Ae-0.5A where A is the time interval between subsequent periods. Caleulate player 1's and player 2's share ifA approaches zero.

Answer 1

Solution(s):

In the timed Rubinstein Bargaining Model,the $600 has been bargained and discounted for future time periods in repeated games between player 1 and player 2.

Player 1's share is given as x1=600{(1-e^-0.5 $\Delta$ )/(1-e^-0.5 $\Delta$ e^-.05 $\Delta$ )}

As $\Delta$ approaches 0 we can rewrite the above expression as:-

x1=600{(1-e^-0.5(0))/(1-e^-0.5(0)e^-0.5(0))

x1=600{(1-e^0)/(1-e^0e^0)

x1=600{(1-1)/(1-1))

x1=600(0/0)

x1=600(indeterminate/undefined)

Notice in this case,as the successive gap or interval between future time periods or $\Delta$ approaches 0 the share of player 1 or x1 mathematically increases infinitely or indeterminately.In other words as the interval between successive time periods becomes increasingly shorter until the time gap or lapse completely dissappears,the gain for player 1 increases exponentially or rapidly as the present value of the money remains intact and does not decreases in its worth or value due to no time gap.Hence,any monetary share earned by player 1 in the initial time period would remain intact in value even the interval between time periods completeley dissappears.

Hence,Player 2's share:x2=600-600{(1-e^-0.5 $\Delta$ )/(1-e^-0.5 $\Delta$ e^-.05 $\Delta$ )}

Similarly,we can rewrite the expression for player 2's share as:-

600-600(undefined/indeterminate)

Now,notice that player 2's share of bargaining depends on the periodic gains of player 1 as the $\Delta$ approaches 0 or the time gap between successive period dissappears.In a repeated game,however,an equilibirum outcome of the bargaining indicate that an agreement could be reached based on the derived expressions for the shares of respective players for successive time periods where no player would tend to object.