Question

(Question 2) Ann and Bob are x. Bob may accept or reject. If he accepts, then Ann receives ua = x and Bob receives ub If Bob rejects, then he makes an offer y. Ann may accept of reject this offer. If Ann accepts, then Ann receives u = day and Bob receives ub offer z. Bob may accept of reject this offer. If he accepts, then Ann receives uq = 6,z and Bob engaged in an negotiation over a pie of size 1. In period 1, Ann makes an offer 1 ob(1 -y). If Ann rejects, then she makes a counter (1 2). If Bob rejects, then u = ub receives u 0. Note, both that offers r, y, and z are shares for Ann a) b) equilibrium is ..." c) d) Draw the extensive form game. Find the subgame perfect equilibrium. You should clearly state"The subgame perfect Find the equilibrium payoffs for Ann and Bob How do ob and da influence the equilibrium strategies and equilibrium payoffs?

Answer 1

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given that

Ann and Bob are engagnd M a negotiation over a pie of size 1. In period 1, Ann makes an offer x. Bob may accept or reject. If he accept then Ann receives ua = x and Bob receives ub = 1— x. If Bob rejects, then he makes an offer y. Ann may accent of reject this offer.

If Ann accepts, then Ann receives ua = delta y and Bob receives ub= delta(1 — y). If Ann rejects, then the negotiation goes to arbitration. The arbitrator ants instantaneously, but takes a fee f < 1. He divides the remains of the pie evenly, giving Ann and Bob ua = ub = delta (1— f)/2.

a)extensive form game.:-

Imagine, Ann and Bob, who takes turns making offers about how to divide a pie of size one. Time runs from t = 0, 1, 2, ....n. At time 0,Ann can propose a split (x0, 1 − x0) (with x0 ∈ [0, 1]), which Bob can accept or reject.

If Bob accepts, the game ends and the pie is consumed.

If Bob rejects, the game continues to time t = 1, when Ann gets to propose a split (y1, 1 − y1). Once Bob makes a proposal, Ann can accept or reject, and so on.

We assume that both players want a larger slice, and also that they both dislike delay. Thus, if agreement to split the pie (x, 1−x) is reached at time t, the payoff Ann is δt1x and the payoff for Bob is δt2 (1 − x), for some δ1, δ2 ∈ (0, 1).

b) Subgame Perfect equilibrium:

The subgame perfect equilibrium can be found by using the backward induction, starting from the final offer.

For validity lets assume N = 2. At date 1, Bob will be able to make a final take-it-or-leave-it offer. Given that the game is about to end, Ann will accept any split, so Bob can offer y = 0.Bob thus anticipates that if he rejects Ann’s offer, he can get the whole pie in the next period, for a total payoff of δ2. Thus, to get his offer accepted, Ann must offer Bob at least δ2. It follows that Ann will offer a split (1−δ2, δ2),and Bob will accept.In the N = 2offer sequential bargaining game, the unique SGPE involves an immediate (1 − δ2, δ2) split.It is fairly easy to see how a general N-offer bargaining game can be solved bybackward induction to yield a unique SGPE. But it is is not so obvious.

Suppose player one makes an offer at a given date t. Bobs’s decision about whether to accept will depend on her belief about what he will get if she rejects. This in turn depends on what sort of offer Ann will accept in the next period, and so on.

There is a unique subgame perfect equilibrium in the sequential bargaining game described as follows. Whenever Ann proposes, she suggests a split (x, 1 − x) with x = (1 − δ2) / (1 − δ1δ2). Bob accepts any division giving him at least 1 − x. Whenever Bob proposes, he suggests a split (y, 1 − y) with y = δ1 (1 − δ2) / (1 − δ1δ2). Ann accepts any division giving her at least y. Thus, bargaining ends immediately with a split (x, 1 − x).

In Conclusion it can be verified that no player can make a profitable deviation from her equilibrium strategy in one single period.

c) equilibrium payoffs for ann and bob:

Consider a period when Ann offers. Ann has no profitable deviation. She cannot make an acceptable offer that will get her more than x. And if makes an offer that will be rejected, she will get y = δ1x the next period, or δ2 1x in present terms, which is worse than x.

Bob also has no profitable deviation. If he accepts, he gets 1 − x. If he rejects, he will get 1 − y the next period, or in present terms δ2 (1 − x) = δ2(1 − δ1x).

It is easy to check that 1 − x = δ2 − δ1δ2x. A similar argument applies to periods when Bob offers.

The next step is to show that the equilibrium is unique. For this let v1, v1 denote the lowest and highest payoffs that Ann could conceivably get in any subgame perfect equilibrium starting at a date where she gets to make an offer.

To begin, consider a date where Bob makes an offer. Ann will certainly accept any offer greater than δ1v1 and reject any offer less than δ1v1. Thus, starting from a period in which he offers, Bob can secure at least 1 − δ1v1 by proposing a split (δ1v1, 1 − δ1v1). On the other hand, he can secure at most 1 − δ1v1.

Now, consider a period when Ann makes an offer. To get Bob to accept, she must offer him at least δ2 (1 − δ1v1) to get agreement.

Thus:

v1 ≤ 1 − δ2 (1 − δ1v1) .

At the same time, Bob will certainly accept if offered more than δ2(1 − δ1v1). Thus:

v1 ≥ 1 − δ2 (1 − δ1v1) .

It follows that:

v1 ≥ 1 − δ2 /1 − δ1δ2 ≥ v1.

Since v1 ≥ v1 by definition, we know that in any subgame perfect equilibrium,Ann receives v1 = (1− δ2) / (1 − δ1δ2). Making the same argument for Bob completes the proof.

d) change of equilibrium strategies and equilibrium payoffs:

Maintaining patience is the key. . Note that Ann’s payoff, (1 − δ2) / (1 − δ1δ2), is increasing in δ1 and decreasing in δ2. The reason is that if you are more patient, you can afford to wait until you have the bargaining power.

The first player to make an offer has an advantage. With identical discount factors δ , the model predicts a split between (1/1+ δ, δ/1+ δ)

which is better for Ann. However, as δ → 1, this first mover advantage goes away. The limiting split is (1/2, 1/2).

There is no delay. Bob accepts Ann’s first offer.

The details of the model depend a lot on there being no immediate counter-offers. With immediate counter-offers, it turns out that there are alternative equilibria.