Advanced Microeconomics (ES30025)
Topic Three: Bargaining (ii)
Advanced Microeconomics (ES30025)
Topic Three: Bargaining (ii) - Finite Alternating Offers Bargaining
Outline: 1.
2.
3.
1.
Alternating Offers Bargaining
Bargaining Without Impatience
Bargaining with Impatience
Alternating Offers Bargaining
This model, which is widely used by theorists, is based on non-cooperative game theory.
Take the following example. There are 2 players: a buyer, Bill (B), and a seller, Sally (S).
They are negotiating a sales contract. The maximum price Bill is willing to pay is $300. The
minimum price Sally is willing to accept is $200. The price P must therefore lay in the range
$200 < P < $300. To simplify the discussion, assume that price will be at least $200, and so
bargaining can be phrased in terms of the division of the surplus $100. Thus, if P = $245, this
is an allocation of $45 of the surplus to the seller Sally and $55 of the surplus to the buyer
Bill. I.e., the surplus $100 is the ‘cake’ being divided. We shall consider a dynamic game and
look for the subgame perfect equilibrium. We write lower case p to represent price in excess
of $200: that is, p = P - $200.
2.
Bargaining Without Impatience
Suppose Bill moves first, proposing price p1 . Sally accepts or rejects. If she accepts, sale
takes place at price p1 . If she rejects, she proposes an alternative price p2 . Bill accepts or
rejects her proposal. If he accepts, sale takes place at price p2 . If he rejects, bargaining ends,
there is no sale, and they have both lost an opportunity to gain.
The game is shown in Figure 1 in extensive form, with payoffs written in the order (Bill’s
payoff, Sally’s payoff). Throughout, we assume that if a player is indifferent between
accepting and rejecting, they accept. This is a convenient simplification that does not affect
the solution significantly.
We solve by backward induction. Consider Bill’s choice in the final round. He will accept
any price proposal no higher than $100. Go back to Sally’s proposal. She knows that Bill will
accept any price proposal no higher than $100. She wants the highest price possible. So she
proposes p2 = $100 (i.e. p2 = P2 − $200 = $100 ⇒ P2 = $300 ). Go back to Bill’s offer at the
beginning of the game
( p1 ) . If he proposes less than $100 ( p1 < $100) Sally will reject it.
She knows that Bill will then accept her counterproposal of p2 = $100 . If he proposes
p1 = $100 , Sally accepts it (she will get $100 later anyway). Therefore Sally receives the
entire surplus. However, it is easily checked that if Sally had moved first, the entire surplus
would have gone to Bill, with price at zero. There is a second-mover advantage. Furthermore,
the result depends on the number of rounds in the game, i.e., the number of times an offer is
made (here 2). If we had a third round, or if we deleted a round (leaving only one), it would
be found that there is a first-mover advantage. Thus, with an even number of rounds there is
a second-mover advantage. With an odd number of rounds there is a first-mover advantage.
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Advanced Microeconomics (ES30025)
Topic Three: Bargaining (ii)
Bill
Offer
Sally
Accept
Reject
Sally
(100-p1, p1)
Offer
Bill
Accept
(100-p2, p2)
Reject
(0, 0)
Figure 1
These are unsatisfactory results. We chose the first mover arbitrarily. There is nothing in the
set-up of the game to require on or the other player to move first. Similarly, there may be no
particular reason why the number of rounds should be odd or even.
3.
Bargaining with Impatience
Now take into account further that bargaining takes time. Suppose that players are impatient
to come to an agreement as soon as possible. For example, by waiting longer a player may
miss a profitable opportunity in another activity. Also, suppose the number of rounds is large,
so that impatience can have a significant effect. Suppose that they each apply a discount rate
of 3 percent. That is, a player is indifferent between having $1 in one round, or having $1.03
one round later. Assume that there are 100 rounds (N.B. we will not write out the extensive
form!). Bill moves first.
We apply backward induction.1 For the same reasons as in the previous game, she proposes
$100, i.e., p100 = $100. In the 99th round Bill knows that in the 100th round Sally will
propose that Bill pays $100. But she is indifferent between receiving [1/(1 + 0.03)]]*$100 =
$97.09 in the 99th round and $100 in the 100th round. If Bill offers less than $97.09, then
Sally will reject his offer and will wait until the final round, when the price will be $100,
which will be accepted. If Bill offers $97.09, then Sally will accept. This is just as valuable to
her as $100 one round later. And Bill gets $2.91, which is preferable to $0 in the next round
(Bill has taken advantage of Sally’s impatience). Thus Bill therefore offers p99 = $97.09,
which Sally accepts.
But the above only occurs if the 99th round is reached. Consider round 98, when Sally makes
the proposal. This is a chance for her to take advantage of Bill’s impatience. Sally knows that
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In the 100th round Sally makes a proposal.
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Advanced Microeconomics (ES30025)
Topic Three: Bargaining (ii)
if the 99th round is reached Bill will get $2.91 of the surplus. But he would be just as happy
if he got [1/(1 + 0.03)] percent of $2.91 in the 98th round. Sally therefore proposes that he
pay p98 = $97.17, leaving him a surplus of [[1/(1 + 0.03)] *$2.91 = $2.83. Bill is willing to
accept this. Sally is happy if it is accepted since she gets more than she would in the 99th
round, and does not have to wait the extra round to be paid. Therefore, Sally proposes the
price p98 = $97.17, which Bill accepts.
But the above only occurs if the 98th round is reached. Round 97 is a chance for Bill
to take advantage of Sally’s impatience. Since Sally would accept $97.17 in round 98, she
would be willing to accept [1/(1 + 0.03)]*$97.17 = $94.34 in round 97. Bill proposes p97 =
$94.34, which Sally accepts. Following this pattern backwards, Sally proposes a price in
round 96, Bill in 95, Sally in 94, etc.
Solution
It can be shown that, following this pattern, Bill will propose price p1 = $51.90 in round 1,
which Sally accepts. So Sally gets $51.90 of the surplus and Bill gets $48.10.
Comments
Again, the solution is sensitive to who moves first. Sally gets a bigger share because she
moves last. But we have (a) a large number of rounds; and (b) each player is impatient - so
the other player can take advantage of this. As a result, the division of the surplus is close to
50:50. It is possible, however, that impatience is asymmetric. For example, Sally may be
very impatient, using a discount rate of 6 percent, while Bill’s discount rate remains at 3
percent. Each player is still assumed to know the other’s discount rate. Now Bill can take
advantage of Sally’s greater impatience. With all other assumptions unchanged, the solution
is that Bill offers $33.51 in the first round, and Sally accepts the offer. Note that the division
of the surplus is now $33.51 for Sally and $66.49 for Bill. Thus, approximately, Bill (whose
rate of discount is half that of Sally’s) gets twice as much of the surplus as she does. As the
number of rounds is made larger, this result holds more exactly.
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