Question

6. A parent P and a young child C are at home together during quarantine. At 10 am one day, P asks C to play quietly so P can take a Zoom meeting. C can either play quietly, which gives him 10 jollies, or make a lot of noise and mess, which gives him 20 jollies. P obtains 0 jollies from taking the Zoom meeting while C is quiet, and −10 jollies from the Zoom meeting if C makes a lot of noise. In addition, at 12 pm, P may or may not let C watch a TV show. Suppose that P and C both gain 100 jollies when C watches the show: C likes the show, and P likes 30 minutes of quiet.

   1. (a) At 9:30 am, P threatens to withhold TV from C if C makes noise during the Zoom meeting. Is this threat credible? Draw an extensive form and find the SPNE to answer this question.

   2. (b) Suppose that P gains only 1 jolly when C watches the TV show, instead of 100. Now is it credible to threaten withholding it?

Answer 1

a.)

The extensive form game can be as follows -

In this, the first payoff is for the Child C and the second payoff will be for the Parent P. C has two choices, whether to make noise or to make no noise. Given each of his two options, P also has two corresponding options, P can either allow C to watch TV or not allow C to watch TV. Given that watching TV gives both C and P an additional 100 jollies, the payoffs can be as follows -

1. If C is quiet, and gets to watch TV, then C gets 10+100=110, and P gets a quiet zoom meeting plus a quiet afternoon so 0+100=100 => (110,100)

2. If C is quiet, and doesn't gets to watch TV, then C gets 10+0=10, and P gets a quiet zoom meeting but no quiet afternoon so 0+0=0 => (10,0)

3. If C is not quiet, and gets to watch TV, then C gets 20+100=120, and P does not get a quiet zoom meeting but a quiet afternoon so -10+100=90 => (120,90)

4.If C is not quiet, and does not get to watch TV, then C gets 20+0=20, and P does not get a quiet zoom meeting and also not a quiet afternoon so -10+0=-10 => (20,-10)

Tracing backwards, we can find the sub-game perfect equilibrium for this game.

We start with player P's strategy. It is clear from the payoffs, that allowing C to watch TV is the best strategy for him. Given that he chooses to allow C to watch TV, it is best for C to make noise, as that gives C the maximum payoff. Hence, the Nash Equilibrium will be (Make noise, Give TV) = (120, 90).

b.)

If the payoff from watching TV is 1 for both instead of 100, the payoffs will be as follows -

1. If C is quiet, and gets to watch TV, then C gets 10+1=11, and P gets a quiet zoom meeting plus a quiet afternoon so 0+1=1 => (11,1)

2. If C is quiet, and doesn't gets to watch TV, then C gets 10+0=10, and P gets a quiet zoom meeting but no quiet afternoon so 0+0=0 => (10,0)

3. If C is not quiet, and gets to watch TV, then C gets 20+1=21, and P does not get a quiet zoom meeting but a quiet afternoon so -10+1=9 => (21,9)

4.If C is not quiet, and does not get to watch TV, then C gets 20+0=20, and P does not get a quiet zoom meeting and also not a quiet afternoon so -10+0=-10 => (20,-10)

We start with player P's strategy. It is clear from the payoffs, that allowing C to watch TV is still the best strategy for him. Given that he chooses to allow C to watch TV, it is best for C to make noise, as that gives C the maximum payoff. Hence, the Nash Equilibrium will be (Make noise, Give TV) = (21, 9).