A Mutual Influence Algorithm for Multiple Concurrent Negotiations
Ka-man Lam and Ho-fung Leung
Department of Computer Science and Engineering, The Chinese University of Hong Kong
Shatin, N. T., Hong Kong, China
{kmlam, lhf}@cse.cuhk.edu.hk
Abstract
Buyers always want to obtain goods at the lowest price. To
do so, a buyer agent can have multiple concurrent
negotiations with all the sellers. It is obvious that if the
buyer obtains a good price from one of the sellers, the buyer
should have more bargaining power in negotiating with
other sellers. Then, other sellers should offer a lower price
in order to make a deal. In this way, the concurrent
negotiations mutually influence one another. In this paper,
we present an algorithm to enable mutual influence among
multiple concurrent negotiations.
Introduction
There are two main components in any negotiation:
negotiation protocol, and negotiation strategies. One of the
most widely used negotiation protocol is the alternating
offers protocol (Osborne and Rubinstein 1994). In this
protocol, a buyer is an agent that wants to obtain a good or
service, while a seller is an agent providing the good or
service. Each agent, buyer or seller, has its own
reservation price. For a buyer, reservation price means the
highest price at which it is willing to buy the good. For a
seller, reservation price means the lowest price at which it
is willing to sell the good. An agent starts the negotiation
by making a proposal. In each round of negotiation, an
agent can quit the negotiation, accept the latest proposal, or
make a counter-proposal.
To enable mutual influence, Nguyen and Jennings (2003,
2004) propose how to coordinate multiple concurrent
negotiations. However, there are several problems with
their model. In this paper, we introduce an algorithm to be
used in the single-issue alternating offers protocol, which
enables mutual influence in multiple concurrent
negotiations. Most importantly, this algorithm ensures the
buyer to obtain the good at a price within the lowest and
the second lowest reservation price among all the sellers.
A Motivating Example
Consider a buyer b, which wants to buy a good. Its
reservation price, RPb is $10, which means it would not
buy the good at a price higher than $10. There are two
sellers, s1 and s2. Their reservation prices, RPs1 and RPs2,
are $7 and $5 respectively, which means the sellers would
not sell the good at a price lower than $7 and $5
respectively.
The buyer starts multiple concurrent negotiations with
the two sellers by proposing, say, $8 for the good. Suppose
sellers s1 and s2 reply with counter proposals of $9 and $8
for the good respectively. To enable mutual influence,
buyer sends a message to seller s1, telling it seller s2 gives
an offer of $8. Here, we assume there is no communication
between the two sellers and time is not critical.
Furthermore, if no agreement can be reached, then no
transaction will occur, and all participating agents will
effectively get zero utility. Since everyone prefers an
agreement to no deal, after knowing that seller s2 offers the
buyer $8, seller s1 rationally makes a counter proposal by
offering a price lower than $8, say $7 to the buyer. To
enable mutual influence again, the buyer sends a message
to seller s2, telling it that seller s1 gives an offer of $7.
Then, by knowing seller s1 offers the buyer $7, seller s2
rationally makes a counter proposal by offering a price
lower than $7, say $6 to the buyer. Similarly, the buyer
again sends a message to seller s1, telling it that seller s2
gives an offer of $6. However, since $6 is lower than s1’s
reservation price, seller s1 cannot make a lower offer
although it wants to make a deal. As seller s1 cannot make
a lower offer, the buyer makes an agreement with seller s2
at a price of $6.
Now suppose before the buyer makes an agreement with
seller s2, there is another seller, s3, whose reservation price
is $4. Then the buyer can send a message to this seller,
telling it that someone offers the good at a price of $6. By
mutual influence, this seller rationally makes a counter
offer of $5. The buyer can again tell seller s2 that seller s3
offers $5. Then seller s2 will rationally replies with a
counter offer of $5, but not a price lower than $5 as this is
its reservation price. By mutual influence again, seller s3
will offer a price lower than $5, at which the buyer will
make an agreement with this seller as this is the lowest
price that the buyer can get from these negotiations.
From this example, we can see that buyer can negotiate
actively by enabling mutual influence, with which buyer
can minimize the price in obtaining the good.
Mutual Influence Algorithm
Suppose there is a buyer, b, with reservation price RPb and
there are n sellers s1 to sn, with reservation prices RPs1 to
RPsn. The mutual influence algorithm can be formalized as
follows.
Step 1. Buyer starts negotiations by proposing a value
v ≤ RPb to sellers s1 to sn, and waits for counter
offers.
Step 2. Suppose a counter proposal COik, from seller si, is
the minimal among the received counter offers in
round k. Buyer sends messages to seller sj, where
j∈[1, n] and j ≠ i, announcing there is a counter
offer COik from one of the sellers, and waits for
counter offers.
Step 3. Repeat step 2 until COjm ≥ COim–1.
Step 4. Accept the counter offer COim–1 from seller si if
COim–1 ≤ RPb, otherwise, announce RPb to sellers
s1 to sn, and repeat step 3 to step 4. If RPb is
already announced and COim–1 > RPb, then no
agreement can be made.
In step 1, buyer starts negotiations rationally by making
an offer smaller than or equal to its reservation price. We
assume, for simplicity, every seller must reply a counter
offer even if the counter offer is the same as the previous
one. In step 2, since everyone prefers an agreement than no
deal, by knowing someone offers COik, those sellers
rationally propose a counter offer greater than their
respective reservation prices, but smaller than COik. Step 2
is repeated until the minimum counter offer the buyer
received in round m, from seller sj, is greater than or equal
to that received, from seller si, in round m – 1. This means
that offer cannot be further lowered by mutual influence.
So, in step 4, buyer accepts the counter offer COim–1, if this
value is smaller than or equal to its reservation price.
However, if this is not the case, buyer tries to lower the
offer further by disclosing its reservation price. But if
reservation price is already disclosed and COim–1 is still
greater than its reservation price, which means that no
seller can offer a price lower than or equal to the buyer’s
reservation price, then no agreement can be made.
This mutual influence algorithm can be applied if there
is no communication between sellers. In this case, since
every seller prefers an agreement to no deal, sellers are
competing with each other and price can be lowered.
However, if sellers are allowed to communicate with each
other, they can collude not to lower the price. If this is the
case, no mutual influence can be made. At the same time,
the mutual influence algorithm can be applied if time is not
critical. Obviously, if time is critical, e.g., buyer loses
utility with time or deadline is short, there may not be
enough time for buyer to benefit from mutual influence.
Theorem 1
By mutual influence, the lowest price that the buyer can
obtain lies within the lowest and the second lowest
reservation prices of all the participating sellers.
Conclusions and Future Work
A good price with one seller should enable buyer to have
more bargaining power in negotiating with other sellers.
This paper presents a mutual influence algorithm in doing
so. By simply sending messages to the sellers, telling them
the minimum offer the buyer receive so far, the buyer can
enable mutual influence among the sellers. On knowing
someone is offering the buyer a lower price, sellers are put
into competition. Since everyone prefers agreement to no
deal, sellers rationally reply the buyer with a lower offer.
One important contribution of this algorithm is that it can
ensure the final price that the buyer can obtain lies within
the lowest and the second lowest reservation prices among
the sellers. Furthermore, if these two values are close, the
final price will be close to the optimal, which is the lowest
reservation price among the sellers.
One important assumption in this algorithm is that
sellers are assumed to believe all the messages from the
buyers and the buyers never lies. In fact, buyer can always
tell lies in messages, telling sellers that another seller is
offering a low price. Then, the sellers may face the
problem of having to choose whether to believe the
message, and deciding what counter-offers to make. The
issues about trust and honesty will be dealt with in future
work. Besides, a mutual influence algorithm for multiissues can also be developed in future work.
References
Nguyen, T. D., and Jennings, N. R. 2003. A heuristic
model for concurrent bi-lateral negotiations in incomplete
information settings. In Proceedings of the Eighteenth
International Joint Conference on Artificial Intelligence,
1467-1469.
Nguyen, T. D., and Jennings, N. R. 2004. Coordinating
multiple concurrent negotiations. In Proceedings of the
Third International Conference on Autonomous Agents
and Multi-Agent Systems, 1064-1071.
Osborne, M. J., and Rubinstein, A. 1994. A Course in
Game Theory. MIT Press.