Strategy Analysis for Competitive Bilateral Multi-Issue Negotiation
Takuma Oishi, Koji Hasebe
Department of Computer Science, University of Tsukuba, Japans2120568@s.tsukuba.ac.jp, hasebe@cs.tsukuba.ac.jp

Keywords:

Negotiation Model, Bilateral Negotiation, Competitive Agent.

Abstract:

In most existing negotiation models, each agent aims only to maximize its own utility, regardless of the utilityof the opponent. However, in reality, there are many negotiations in which the goal is to maximize the relative
difference between one’s own utility and that of the opponent, which can be regarded as a kind of zero-sum
game. The objective of this study is to present a model of competitive bilateral multi-issue negotiation and toanalyze strategies for negotiations of this type. The strategy we propose is that the agent makes predictions
both about the opponent’s preference and how the opponent is currently predicting its own preference. Basedon these predictions, the offer that the opponent is most likely to accept is proposed. To demonstrate theusefulness of this strategy, we conducted experiments in which agents with several strategies, including ours,
negotiated with one another. The results demonstrated that our proposed strategy had the highest averageutility and winning rate regardless of the error rate of the preference prediction.

1

INTRODUCTION

Automated negotiation is a process in which an autonomous agent interacts with another agent (or human) to form an agreement that is desirable for bothparties. Several studies have been conducted on automated negotiations over a period of decades, interacting with fields such as artificial intelligence and ecommerce. (A comprehensive survey of research inthis area is provided by (Baarslag et al., 2016) and(Kiruthika et al., 2020).)
Various negotiation models have been proposed
regarding the number of agents and incomplete information. However, in most of these, each agent’s objective is only to maximize its own utility, regardlessof the utility of its opponent. Therefore, the objectiveof all agents in a negotiation is to achieve a Paretooptimal outcome. However, in the real world, manynegotiations exist in which agents are required not
only to increase their own utilities, but also to maximize the difference between their own utilities andthose of their opponents. Typical examples of this include various tradings such as foreign exchange transactions, and negotiations between parties in competitive relationships, as modeled in some board games
such as CATAN (CATAN GmbH, nd). The negotiation of this type requires more skillful tactics that havenot been considered in existing negotiation models.
The objective of this study is to provide a model ofbilateral multi-issue competitive negotiation, and to
propose a strategy for negotiations of this type. Competitive negotiation is a special type of negotiation inwhich the relative difference between the utilities obtained by an agent and its opponent is evaluated as theactual utility. In this sense, this problem is a kind ofzero-sum game. Another difference between conventional negotiations and competitive ones is that, in theformer, the utility obtained as a result of an agreement
is known in advance, whereas in competitive negotiation, the actual utility is not known in advance because it depends in part on the opponent’s utility.
Generally, in negotiations, agents make decisions
such as the choice of the contents of offers andwhether to agree to the opponent’s offer, while making predictions regarding the opponent’s preferences.
In this study, we focused on decision-making strategies rather than the prediction technique. Specifically,
our proposed strategy comprises two tactics. The firsttactic applies to setting a target value (i.e., the minimum acceptable relative utility) for an agent, basedon the prediction of the opponent’s preference as wellas the prediction of how the opponent will predict theagent’s own preference. The second tactic is used
when choosing the content of the offer to which the
opponent is most likely to agree.
We evaluated the effectiveness of our strategy byconducting bilateral negotiations on three competitive
issues. The experimental results showed that the pro-posed strategy had the highest average utility and winning rate regardless of the error rate of the preference
prediction.
The remainder of this paper is organized as follows. Section 2 presents related work. Section 3 describes the model of competitive negotiation. Section 4 introduces strategies for competitive negotiation. Section 5 presents experimental results. Finally,
Section 6 concludes the paper and presents plans for
future work.

Figure 1: Utility spaces of non-competitive (left) and competitive (right) bilateral negotiation.

3
2

RELATED WORK

A representative early study on the automated negotiation was conducted by Faratin et al. (Faratin
et al., 1998). This study disseminated a technique
for searching for an agreement point by concessions
and described the basic idea of strategic negotiation.
Fatima et al. (Fatima et al., 2006) analyzed negotiation strategies and equilibrium in bilateral negotiations, where in which there is a negotiation deadline
and the opponent’s preferences are unknown. Jennings et al. (Jennings et al., 2001) classified the existing negotiation strategies by approaches, and studies(Cao et al., 2015; Zheng et al., 2014) proposed negotiation approaches that combine multiple strategies anddynamically change an agent’s behavior according tothat of the opponent.
Our study and those described above focus on bilateral negotiations, but there have also been studies on negotiations involving more than two agents.
For example, (Aydoğan et al., 2014) investigated thestrategies and protocols in such negotiations. There
are also studies, such as (Mansour and Kowalczyk,
2015; An et al., 2011) regarding situations in which
single-issue negotiations between two agents to obtain multiple resources are performed in parallel in
e-commerce.
The negotiation problem targeted by the above
studies aims at maximizing each agent’s own utility
through negotiation. Conversely, our study focuses on
the competitive negotiation problem that aims to maximize the difference between one’s own utility and theopponent’s utility, which has not been thoroughly investigated.
There is a study (Keizer et al., 2017) targeting negotiation in CATAN, which is a specific example ofcompetitive negotiations. Although that study did notpresent a negotiation model, it implemented negotiation strategies by a rule-based technique and machinelearning.

COMPETITIVE BILATERALNEGOTIATION MODEL

In this section, we define competitive bilateral multiissue negotiation model. In addition, we show that
this negotiation with perfect information always leads
to an agreement in which the utility for both agentsare zero.

3.1

Basic Concept

As explained earlier, the objective of a rational agent
in conventional bilateral negotiations is to reach an
agreement that maximizes its own utility regardlessof that of the opponent. Conversely, the objective of
the agent in the competitive bilateral negotiation proposed in this study is to maximize the difference inutility between itself and the opponent. In this sense,
this negotiation can be regarded as a zero-sum game.
Therefore, even if high utility is obtained as a resultof negotiation, it is undesirable to bring high utility
to the opponent as well. Rather, it is desirable that thedifference in utility relative to the other party is larger,
even if one’s own utility is lower.
Figure 1 illustrates the difference between noncompetitive (left) and competitive (right) negotiations. Here, the utility spaces of agents i1 and i2 ineach negotiation are shown, and the green and blue
arrows represent the vector of utility aimed at by i1
and i2 , respectively. In non-competitive negotiations,the combined vector of both agents is indicated by a
red arrow. This means that the goal of the agents is toreach agreement on the Pareto frontier represented by
the curved line. On the contrary, in competitive negotiations, each agent aims both to maximize its own
utility and minimize that of the opponent. Therefore,
the combined vectors of each agent are opposite toeach other as indicated by the red arrows. This implies that each agent aims to reach an agreement on
its own side of the diagonal line that indicates a relative utility of zero for both agents.

3.2

Formal Model

The formal model of negotiation is defined as a tuple of the following components: a set of negotiation
agents, negotiation domain (often called the outcome
space), negotiation protocol, and preference profiles.
(The paper (Baarslag et al., 2016) presents a surveythat provides more details.)
Let I be the set of negotiation agents. Because
this study deals only with bilateral negotiations, we
fix I = {i1 , i2 }. We also use the notation −i to indicate
the opponent of agent i ∈ I. The negotiation domain,or often called the outcome space (denoted by O) represents the set of all possible negotiation outcomes.
The negotiation protocol determines the rules of negotiation, such as the order of offers and the conditions of agreement. The preference profile is a binary
relation over the negotiation domain for each agent,that determines which of any two outcomes is more
preferable. Here, we follow the game-theoretic convention and define the relation by a utility function
U i : O → R (for i ∈ I). However, in competitive negotiations, the goal is to maximize the difference in utilities between the two agents. Therefore, in addition to
the usual utility function, we introduce RU i : O → R,
which represents the relative utility. The formal definitions of these components are given below.
3.2.1

Negotiation Domain

The negotiation domain O is represented as a productof one or more sets (called issues) of possible outcomes. The set of indices for the issues is represented
by J = {1, . . . , j}, where j is the number of issues.
The set of outcomes for each issue is represented by
Ok (k ∈ J), and thus O is defined to be O1 × · · · × O j .
3.2.2

Negotiation Protocol

As with many previous studies on bilateral negotiations, we follow the alternating-offers protocol (Rubinstein, 1982), in which agents take turns making
suggestions while searching for a mutually acceptableoutcome. More specifically, an agent with turn proposes one of the elements of the negotiation domain
as the content of offer. The other agent who received
the proposal selects one of the following three actions.
Accept: Agree to the proposal. Both agents obtain
utility based on the agreed outcome.
Offer (Counter-offer): Reject the proposal andmake a new offer to the other party.
EndNegotiation: Reject the proposal and end the negotiation. Both agents receive a utility of zero.

Times (steps) in the progress of negotiation are represented by discrete values t = 1, 2, . . . ∈ T . Negotiations may have a time limit (denoted by tmax ) and, if
no agreement is reached within it, both agents gain autility value of zero.
3.2.3

Preference Profile

The utility of agent i ∈ I for outcome o ∈ O is definedby the following equation:j

U i (o) = Σk=1 wikVki (ok ),where wik denotes the weight assigned to agent i forjissue k, satisfying Σk=1 wik = 1. Vki : Ok → R is theevaluation function for issue k for agent i. According to convention, in this study, the range of utility isdefined as [0, 1] ⊆ R.
3.2.4

Relative Utility Function

A relative utility function RU i : O → R is introduced
to represent the difference between the utility of thetwo parties in a competitive negotiation. This is defined as
RU i (o) = U i (o) −U −i (o).

3.3Analysis of Perfect Information
Case

In competitive negotiations, if the agents’ preference
profiles are common knowledge, the utility for both
agents will be always zero. We demonstrate this factin game theory.
Here, we assume that the negotiation has a time
limit tmax and that the utility function of the agents is
common knowledge. First, let i be the agent having
a turn at time tmax and consider the optimal action ofi at this time. At tmax , i can choose only Accept orEndNegotiation, and if offer o made by −i at tmax −1 satisfies RU i (o) > RU −i (o), i can obtain a positive
relative utility by choosing Accept. Otherwise (i.e.,
RU i (o) ≤ RU −i (o)), and i obtains relative utility zeroby choosing EndNegotiation. Thus, at tmax − 1, −ishould offer o′ with U i (o′ ) ≤ U −i (o′ ), which results
in a final relative utility of zero for both agents. Using
the same argument in the reverse direction, we findthat at any time t < tmax , both agents make offers withpositive relative utility, leading to the same result.

4

NEGOTIATION STRATEGIESThis section presents some possible strategies forcompetitive negotiations, the effectiveness of which
is analyzed empirically in the next section.

4.1

Tactics

In general, negotiation comprises two decisions:choosing an outcome from the negotiation domain as
one’s own offer and deciding whether to accept the
opponent’s offer. To decide what to offer, a plausible
approach is to set a target value (i.e., the minimum
acceptable relative utility) in advance, and make anoffer that exceeds this target. Furthermore, there are
two possible ways to select an outcome to offer: one
is to select an outcome randomly that exceeds the target value, and the other is to select an outcome that the
opponent is more likely to accept. To decide whether
to accept an offer, a plausible approach is to set a target value in advance, and agree to the opponent’s offer
if it exceeds the target value. Here, these two target
values may generally differ and may vary with time.
Below, we discuss possible tactics for each ofthese decisions.
4.1.1

Tactics to Decide Target Value for Offer

Algorithm 1 Evaluation of T RU with predictiondependent tactic.
−ii
1: Update Uest(i)and Uest(i)2: for all o ∈ O do
3:
if RU i (o) = T RUti then
−i
−i
i
4:
if Uest(i)(o) − Uest(i)(o) > Uest(i)(ot−2 ) −i
Uest(i)(ot−2 ) then5:
Oti .add(o)6:
end if7:
end if
8: end for
9: if Oti ̸= ø then10:
T RUti = T RUti
11: else
12:
T RUti = T RUti − c
13: end if

There are two main approaches to determine thetarget relative utility for an agent’s offer. The first
fixes the value through negotiation (called fixed-value
tactic) and the second allows it to vary over time.
The latter can be classified further into two types: thetime-dependent tactic, in which the target value is determined depending only on time, and the predictiondependent tactic, in which the target value is determined depending both on time and on predictions
about preferences.
The definitions of these three tactics are given below, where T RU i is used to denote the target value ofagent i.

Fixed-value tactic: This tactic always sets the target
value to any value greater than zero.
Time-dependent tactic: The target value of agent iin this tactic is defined as follows:
T RUti = mini + (1 − αi (t)) × (maxi − mini ).
This tactic is based on the idea of (Faratin et al.,1998). Here, mini (called the reservation value) andmaxi denote the minimum and maximum relative utilities that agent i desires in negotiation regardless oftime, respectively.
The function αi : T → R is called a timedependent function, and satisfies both αi (0) = κi andi ) = 1 (where κi is a constant). Various timeαi (tmax
dependent tactics can be defined according to the definition of this function. Some well-known ways ofdefining this function include polynomials and exponential functions. In this study, the following polynomial function is used:ii
αi (t) = κi + (1 − κi ) × (min(t,tmax)/tmax)1/β .

This function changes the speed of concession depending on the value of β. When β = 1, yielding progresses at a constant rate (linear); when β > 1, yielding progresses early; and when β < 1, few concessions are made while time t is small, whereas concesi .
sions are suddenly made near tmax
Prediction-dependent tactic: This is introduced inthis study for competitive negotiation. This tactic
makes the following types of predictions:• Prediction of the opponent’s preference.
• Prediction about how the opponent will predict
one’s own preference.
Based on these predictions, the target value is set taking into consideration whether the offer will be accepted by the opponent and the current time.
The detailed algorithm to determine the target−ivalue is as follows. Let Uest(i)be the preference funcition of −i predicted by agent i, and let Uest(i)be thei
prediction of Uest(−i) by agent i.
The algorithm for the target value T RUti at time
t for agent i is described in Algorithm 1. Here, ot−2
denotes the previous offer by agent i, Oti denotes theset of candidates of offers(i.e., the set of possible offers expected to yield a relative utility that matches
the target value TRU for agent i at time t), and c denotes the amount of change in the target value when
agent i concedes.
As shown in Algorithm 1 below, this tactic does
not change the target value if there is an offer thatthe opponent is more likely to agree with that has thesame target value as the previous offer. Otherwise, thetarget value is decreased.

Table 1: Strategies for competitive bilateral negotiation.

Strategy name
PMT
TRT
PMF
TRF
FMF
FRF
4.1.2

Target value for offer
Prediction-dependent
Time-dependent
Prediction-dependent
Time-dependent
Fixed
Fixed

Tactics for Offer Choice

There are two main approaches to choose an offer.
The first, called the random offer tactic, chooses anoffer randomly from the set of offers that might bringa relative utility above the target value. The second
approach, called mislead tactics, is proposed in thisstudy for competitive negotiation. This uses the prediction of the opponent’s preference as well as theprediction of how the opponent will predict one’s own
preference. Based on these predictions, the offer to be
selected is the one that the opponent is most likely to
accept from among the possible outcomes exceeding
the current target value. The formal definition of thistactic is as follows: Let Oti be the set of candidates forthe target value of agent i at time t. Offer ot at time t
by mislead tactics is then expressed as follows:−iiot = arg max(Uest(i)(o) −Uest(i)
(o)).
o∈Oti

4.1.3

Tactics for Determining Agreement

Tactics on agreement can be divided into two categories, depending on whether the target value is fixedor variable. In both cases, agreement is chosen when
the predetermined target value is exceeded. Similar to
the target value used to determine the offer, tactics canbe time-dependent or prediction-dependent. Only thesimple time-dependent tactic is analyzed in this study.

4.2

Negotiation Strategies

Negotiation strategies are realized by combining thetactics described above. Therefore there are 12 strategies, consisting of three tactics to determine the target
value for the offer, two to choose the content of theoffer, and two to determine the target value as the criterion for agreement. Of particular interest are the six
strategies shown in Table 1.

Offer choice
MisleadRandom offer
MisleadRandom offer
MisleadRandom offer

which the strategies defined in Section 4 were negotiated with each other.

5.1

Parameter Settings

5.1.1

Negotiation Domain

The negotiation domain used in the experiments consisted of three issues, each with ten options. The negotiation domain therefore consisted of a set of 1,000
outcomes.
5.1.2

EXPERIMENTS

In order to evaluate the usefulness of the proposed
negotiation strategy, we conducted experiments in

Negotiation Strategies

For the experiments, we developed negotiation agentsfor each of the six strategies defined in the previous
section, whose parameter settings were presented below.
For the time-dependent tactic, the values of thefour parameters are as follows.
• min = 0.05.
• max = 1.0.
• κ = 0.1.
• β = 1.0.
For the prediction-dependent tactic, the parameterc for concession is set to 0.05. However, if this tactic
makes a concession resulting in T RU ≤ 0, to preventthe agent from making an offer that would hurt itself,
the value of T RU was updated to 0.05 and chose anoffer satisfying RU > 0. In the experiment, the offers
based on the random-offer tactic and mislead tactic
were selected within the range of 0.02 round the valueof T RU. This is because the agent may not find anoffer that yields utility exactly equal to T RU. Thus,
the agent set the value of T RU at time t to RU(ot−2 )(where ot−2 denotes the previous offer), and decided
whether to make concession by Algorithm 1.
5.1.3

5

Target value for agreement
Time-dependent
Time-dependent
Fixed
Fixed
Fixed
Fixed

Preferences

The preferences defined by the weights and utility
functions for i1 and i2 were set as follows, respectively. Here, the same utility functions were used for
all issues.

Figure 2: Average utility for each strategy when the cosine
similarities of predictions were 0.99, 0.95, and 0.9.

Figure 3: Win rate (denoted by red) and draw rate (denotedby orange) for each strategy when the cosine similarities ofpredictions were 0.99, 0.95, and 0.9.

• wi1 = (0.2, 0.3, 0.5).
• V i1 = (0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0).
• wi2 = (0.5, 0.3, 0.2).
• V i2 = (1.0, 0.9, 0.8, 0.7, 0.6, 0.5, 0.4, 0.3, 0.2, 0.1).
Each agent i was not informed about the exact
preferences of its opponent. Instead, it was initiallyinformed of its own and its opponent’s preferenceswith a certain range of error, which mimicked the pre−iidicted value Uest(i)and Uest(i). Specifically, for eachof the weights of the agents’ preferences and the evaluation functions, we gave predictions with errors suchthat the cosine similarity is 0.99, 0.95 and 0.9. Here,
since the difference between the preferences of it’s
own and the opponent can be large or small depending on the predicted values, in the experiment all of
the cases were evaluated.

5.2

Experimental Results

Under the settings described in the previous section,
the results of round-robin matches with six negotiation strategies are shown below.
Figure 2 shows the average utility obtained byeach strategy. Here, the horizontal axis represents

the results of each strategy, and the three graphs show
the results when the cosine similarities of the preference predictions are 0.99, 0.95, and 0.90, respectively,from the left. The vertical axis shows average utilities
with their maximum and minimum values.
Figure 3 shows the sum of the win and draw rates
for each strategy. Here, the horizontal axis represents
the results of each strategy, and the three graphs show
the results when the preference predictions have cosine similarities of 0.99, 0.95, and 0.90, respectively,from the left. The vertical red and orange graphs represent win and draw rates, respectively.
As shown in Figure 2, PMT and TRT strategies
were found to be superior to the other strategies, witha positive average utility regardless of the error rateof prediction. Specifically, when the cosine similarity of the preference predictions was 0.99, the average utilities of PMT and TRT strategies were about
0.043 and 0.049, respectively. Also, when the cosine
similarity was 0.9, the average utilities of these two
strategies were about 0.168 and 0.187, respectively.
On the other hand, the average utilities of the otherfour strategies always had negative average utility.
From the above results, it can be seen that theutility of both PMT and TRT strategies increases as
the difference in expectations increases. A possible
reason for this is that both the PMT and TRT strategies have time-dependent target values for agreement.
The larger the prediction error, the greater the probability of making a wrong decision about one’s relative gain for a proposal, and the greater the probability of a larger error in the value of that relative gain.
While other strategies choose to agree when their relative gains are large (relative utilities greater than 0),
the PMT and TRT strategies have stricter criteria for
agreement, so they are less likely to agree to an agreement that will actually be to their detriment. As a result, the larger the forecast error, the higher the relative utility of the PMT and TRT strategies, suggestingthat having a time-dependent agreement target value
is important for competitive negotiation.
Figure 3 shows that PMT strategy had a lower win
rate than TRT strategy, but the sum of the draw rate
and the win rate was always the highest for the PMT
strategy. For example, when the cosine similarity ofpreference predictions was 0.99, the win rates of PMT
and TRT strategies were about 0.452 and 0.482, respectively. The draw rates for these strategies wereabout 0.365 and 0.293 respectively. Thus, the sumof the win and draw rates for PMT and TRT wereabout 0.816 and 0.775, respectively, with 4.2 percentage point higher win or draw rate for the PMT strategy.
Examples of the negotiation process for PMT andgotiations.

6

Figure 4: The negotiation process in which PMT won overTRT by the largest utility margin.

Figure 5: The negotiation process in which PMT lost to
TRT by the largest utility margin.

TRT strategies are shown in Figures 4 and 5. Here,the orange and blue plots indicate the history of theoffers by PMT and TRT agents, respectively, and thered star indicates the agreed outcome. Figures 4 and5 both show the results when the Cosine similarity ofthe preference prediction was 0.99.
Figure 4 shows the negotiation process when PMT
wins TRT by the largest margin of utility. In Figure
4, neither agent agreed with the other’s offer and continued to make offers that were profitable for them,but in the end, TRT made a mistake and made an offer that was more profitable for PMT. As a result, theutility of PMT became positive.
Figure 5 shows the negotiation process when PMT
lost by the largest margin of utility to TRT. In Figure
5, both agents made offers that were advantageous tothem until the end. However, in the end, PMT mispredicted the utility of the offer received from TRT andchose to agree, resulting in a negative utility for PMT.
In competitive negotiations, winning is important,but not losing is also important. PMT strategy is also
an excellent strategy in terms of stability. Based on
the above results, the PMT strategy is considered tobe the optimal negotiation strategy in competitive ne-

CONCLUSIONS AND FUTURE
WORKIn this study, we proposed a model of competitive
bilateral multi-issue negotiation, in which an agent’s
utility and that of the opponent are evaluated relative
to each other and the actual utility can be regarded
as a zero-sum game. We also proposed a strategy forthe negotiations of this type, in which the basic idea
is to choose the offer that the opponent is most likely
to accept, based on the prediction of the opponent’s
preference and the prediction of one’s own preference
from the opponent’s perspective.
To demonstrate the effectiveness of the proposed
strategy, we conducted experiments in which agentswith various strategies negotiate competitive threeissue bilateral negotiations. The results show thatthe proposed strategy achieves the highest utility andwinning rate, regardless of the prediction error rate.
We also showed that time-dependent target value foragreement is important for gaining relative utility incompetitive negotiations.
In future work, we will develop a prediction
method required in our negotiation setting, based onsome existing methods such as those using Bayesianestimation (Lin et al., 2006) or heuristics (Jonker andRobu, 2004). We are also interested in applying ourstrategy to the development of agents that play board
games involving competitive negotiation.

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