Question

Consider the following scenario. Carole is an animal-rights activist who owns a sanctuary called Large Cat Rescue, while Joe owns a zoo called Tiger Kingdom. There is an ongoing feud between them with Carole alleging abuse of animals in Tiger Kingdom while Joe counteracts with similar accusations about Large Cat Rescue. In a recent escalation of this feud, Carole sues Joe for making defamatory comments about her personal life on a social media forum called Rumorville. Five weeks from now a judge will decide whether or not Joe is guilty. If found guilty Joe will be ordered to pay $4 million in damages to Carole; if not, there will be no payment. However, Carole and Joe can settle out of court in the four weeks prior to the hearing, in which case they do not go to court in Week 5. The negotiation for settlement proceeds as follows. In each week ?? ∈ {1, 2,3, 4} Carole or Joe can make a settlement offer St and the other party has to decide whether to accept it. Carole and Joe take turns making offers; Carole makes offers in weeks 1 and 3, while Joe gets his turn in weeks 2 and 4. If the offer is accepted in any particular week, the game ends and Joe pays St (the amount decided in week t) to Carole. Carole is risk-averse and her utility from receiving payment x is (x)1/2. She does not discount future payoffs and does not incur any costs of negotiation for going to court. Joe, however, is riskneutral and needs to pay a small fee c > 0 to lawyers for every week the negotiations take place. Use backward induction to analyse the above scenario. For the purpose of your analysis you may assume a probability p=0.7 of Carole winning the court case if the negotiations are not settled.

word limit 1500-2000

Answer 1

Here, there are two players, Carole and Joe, where Carole is the Plaintiff who is suing, and Joe is the Defendant. Court date= week 5, if they go to court, Joe has to pay Carole $4 million. There is the option of an out of court settement which has the following conditions:

At each tt = 1, 3, Carole makes a settlement offer of S₁ or S₃ and if Joe accepts, the game ends there with Joe paying Carole a sum of S₁ or S₃ on week 1 or 3 respectively.

At each tt = 2,4,

Joe makes a settlement offer of S₂ or S₄ and if Carole accepts, the game ends there with Joe paying Carole a sum of S₂ or S₄   on week 2 or 4 respectively.

If they are not able to negotiate till week 4, week 5 they go to court and oe has to pay Carole $4 million with Carole having a probability of 0.7 of winning.

Carol does not incurr any costs of negotiation , however, Joe needs to pay a sum of c>0 to lawyers every week.

Carol is risk-averse and Joe is risk-neutral.

Now, in this case, each party, dpendant or plaintiff will try to maximize his/her expected amount of payoff at the end of the game.

The backward induction analysis of this game is as follows:

The payoff from going to court for Carole is $4 million. However, her utility from it is 1/2, as she is risk averse.

So, as Carol is risk averse:

She will prefer a certain payoff rather than a higher but risky payoff.

So, it will accept a payment of S_T from Joe out of court only if:

U(S_T) > 0.7* U(4) + 0.3 * U(0)

So, in this case for risk averse, the concave utiity function would work: U(x) = x

S_T > 0.7 *  4 million + 0.3* 0

S_T > 0.7 * 2000 = 1400 ,

S_T > 1400 * 1400 = $1,960,000

So, if S_T will be any sum greater than 1,960,000, Carole will prefer to negotiate it out of court, as she is risk averse, and rather than going for a higher sum of $4 million with a probability of 0.3 losing, she prefers a sum of greater than 1,960,000 recieved with certainty.

Now, with every week the amount Joe has to pay increases by c:

So, week 1:

Joe has to pay S_T + c

week 2: S_T + 2c

Week 3: S_T + 3c

Week 4: S_T + 4c

If Carole accepts the settlement offer S₂ of the defendant on week 2, the payoff of Carole wil be S₂

So, if S₂ > $1,960,000, he should accept the offer and if its less, he must reject it. If it is equal to $1,960,000, Carole will be indifferent. So, Carole accepts if and only if S₂ 1,960,000 ,

So, what should Joe offer on second week, Given the risk-averse behaviour of Carole, Joe's payoff from S₂ is:

-S₂ -2c if S₂ 1,960,000,

-$4 million - 2c if S₂ 1,960,000 as if the offer is rejected, they will have to go to court. Note that when S_{2 =} 1,960,000, joe's payoff is -1,960,000 - 2c.

Therefore, the defendant offers S₂ = 1,960,000 on week 2. Now, at week 3, Carole offers a settlement S₃ and Joe could either accept or reject the offer. If Joe rejects the offer, she will get the payoff from settling for S₂ which is 1,960,000 ,

If Joe accepts the offer, he will get the payoff -S₃ - 3c

Hence, Joe will accept the offer only if S₃ + 3c S₂ +2c ,

Therefore, Carole will offer the highest acceptable settlement, to Joe, S₃ = S₂ -c,

at week 4, Joe offers a settlement S₄ and Carole accepts or rejects the offer. If carole rejects it, he will get the payoff from settling S₃ -3c, if he accepts the offer,he will get: S₄ - 4c ,

So, he will accept the offer only if S₄ - 4c S₃ -3c

Then Joe offers the highest acceptable settlement to Carole,

S₄ - 4c S₃ -3c = S4  S₃ +c = S₂ + 2c

S₄ = 1,960,000 + 2c

So, summarizing, at any odd date, weeks 1,3, Joe accepts the offer S₃ or S_`, only if S₃ S₂ + c , and Carole offers

S₃ S₂ + c ,

at an even date , weeks 2, 4, Carole accepts the offer S₂ or S_4, only if S₂ = 1,960,000 , S₄ S₂ + 2c

So, the solution to this problem is: Joe accepts the offer if S₃ S₂ + c , Carole accepts if S₂ = 1,960,000 , S₄ S₂ + 2c