A Three-Valued Semantics for Negotiated Situation of Multi-Agent

System Based on BATNA and WATNA

Yang Song

and Ken Satoh

National Institute of Informatics, Japan

Keywords:

Multi-Agent System, Negotiation, BATNA and WATNA, G

odel Logic, Modal Logic.

Abstract:

Negotiation plays a crucial role in the dispute resolution systems. In the negotiation, the agents usually need

to compromise with each other because their preferences are different. To provide the best or acceptable

suggestion in the negotiation, BATNA (Best Alternative To a Negotiated Agreement) and WATNA (Worst

Alternative To a Negotiated Agreement) can be a method to express the preference of each agent. In this

paper, our aim is to formalize the negotiated situations of multi-agent system in a logical method based on

BATNA and WATNA. We consider each given suggestion as a possible world in modal logic, and provide a 3-

valued valuation based on G

odel logic to judge whether a suggestion is over the BATNA, below the WATNA,

or between the BATNA and WATNA of each agent, which is to show whether the suggestion is acceptable,

rejectable or undecided to the agent. Moreover, by using the modal operator we can check whether there exists

a best or acceptable suggestion for all agents in a negotiated situation.

1 INTRODUCTION

Today, Online Dispute Resolution (ODR) became

more important because it could reduce the costs to

resolve the dispute and due to demand for non face-to-

face negotiation. In such systems, negotiation plays

a crucial role since the ultimate goal of the system

is to provide a satisfactory solution among disputing

agents.

Negotiation is also an important topic in multi-

agent systems (Kraus, 1997), and therefore many

studies are aimed to formalize the negotiated situ-

ation. Dunne (Dunne et al., 2005; Dunne, 2005)

considered the negotiation as resource allocation be-

tween agents, and therefore provided the deﬁnition

of resource allocation setting and the model of re-

source allocation in which several agents exchange

resources. Ragone (Ragone et al., 2006) gave a logic-

based framework to automate the one-shot bilateral

negotiation considering the demand and preference

of agents. Endriss (Endriss and Pacuit, 2006) devel-

oped a dynamic modal logic that can be used to model

scenarios where agents negotiate over the allocation.

Yang (Yang et al., 2018) forced on the system of per-

sonalized product supply chain and provided a multi-

https://orcid.org/0009-0006-1020-130X

https://orcid.org/0000-0002-9309-4602

agent negotiation mechanism based on personalized

index.

In this research, we want to provide logical se-

mantics to express the negotiated situations of multi-

agent systems in ODR. Here, we consider the negoti-

ation as a selection of several suggestions among the

agents. Normally, a lot of suggestions by deputing

agents and neutral third party are suggested in the ne-

gotiation. One agent may be glad to accept a sug-

gestion while the other agent rejects it because they

have different preference, therefore sometimes agents

need to compromise with each other. In this paper, we

consider that not each suggestion is either accepted or

rejected, but some of them are not decided at ﬁrst. As

a famous example of negotiation, the case Two sis-

ters arguing over an orange (Follett, 2011) can be

considered as obtaining four suggestions as follows:

• The elder sister has the whole orange while the

younger sister has none (suggested by elder sis-

ter).

• The younger sister has the whole orange while the

elder sister has none (suggested by younger sis-

ter).

• The orange is cut into half and each sister has half

of the orange (they compromise with each other).

• The elder sister has the peel (for cooking) and the

younger has the juice (for breakfast), as the best

Song, Y. and Satoh, K.

A Three-Valued Semantics for Negotiated Situation of Multi-Agent System Based on BATNA and WATNA.

DOI: 10.5220/0012394200003636

Paper published under CC license (CC BY-NC-ND 4.0)

In Proceedings of the 16th International Conference on Agents and Artiﬁcial Intelligence (ICAART 2024) - Volume 1, pages 333-340

ISBN: 978-989-758-680-4; ISSN: 2184-433X

333

solution.

In this example, we can see that each sister wants to

have the whole orange at ﬁrst, which is their desire. If

the fourth suggestion is not mentioned, they can ac-

cept the third suggestion as the result of compromise,

otherwise they prefer the forth suggestion to solve this

negotiation. Using BATNA and WATNA, we can ex-

plain this situation. The BATNA and WATNA are im-

portant concept in negotiation and the meanings are

as follows (Notini, 2005):

• BATNA: Best Alternative To a Negotiated Agree-

ment.

• WATNA: Worst Alternative To a Negotiated

Agreement.

In this example, the BATNA of the elder sister is to

have the peel for cooking, therefore she is glad to ac-

cept the ﬁrst and forth suggestion. Also, the BATNA

of the younger sister is to have the juice for breakfast,

therefore she is glad to accept the second and forth

suggestion. In this example, both of their WATNAs

are to have half of the orange (peel or juice), therefore

they can accept the third suggestion as a compromise,

while rejecting the ﬁrst or second one because it is

below one sister’s WATNA.

There are several studies that employ BATNA and

WATNA to the ODR systems. Lodder (Lodder and

Zelznikow, 2005) considered calculating BATNAs as

the ﬁrst step of their three-step model of the negotia-

tion support systems. Andrade (Andrade et al., 2010)

developed an architecture supported by a JADE plat-

form based on BATNA and WATNA, while consid-

ering that they are useful to take into account when

making or accepting a proposal. In this paper, we con-

sider a logical method based on BATNA and WATNA,

that provide a 3-valued valuation as follows:

• The value 1 means that the suggestion is over the

BATNA, which will be accepted by the agent.

• The value 0.5 means that the suggestion is be-

tween the BATNA and the WATNA, which may

be accepted or rejected by the agent.

• The value 0 means that the suggestion is below

the WATNA, which must be rejected by the agent.

In this paper, our aim in this paper is not how to

reach the solution, but to deﬁne what the negotiated

point is under the assumption that every information

is provided about each agent’s desire, for example.

We then try to consider how to reach such a negoti-

ated point. We give a suggestion model like modal

logic and show that we can use the modal operators

to compare the suggestions. Moreover, we provide

some formulas which can express the best suggestion

and best acceptable suggestion of one agent and of a

group. Also, we can express some features of negoti-

ation by our logic.

The structure of the rest of the paper is as fol-

lows. In Section 2, we introduce the previous stud-

ies, G

odel propositional and modal logic as the tech-

nical background. In Section 3, we propose the syntax

and semantics of our logic. We regard the symbols of

agents as the atomic propositions informally, and then

provide a 3-valued suggestion model. Moreover, we

provide an explanation of the concrete example “two

sisters arguing over an orange” by our semantics In

Section 4, we consider other normal and dynamic op-

erators to extend our semantics and give the axioms.

Finally, in Section 5, we conclude and give some di-

rections for future works.

2 PREVIOUS STUDY

2.1 G

odel 3-Valued Logic

In classical logic, the valuation is 2-valued, i.e., each

formula is either true or false. t is natural for humans,

however, that sometimes two values seem to be not

enough, e.g., when considering paradoxes like the liar

sentence.

odel provided a many-valued propositional logic

with ﬁnite or inﬁnite values between 0 and 1, where

0 and 1 are considered as false and true, respec-

tively. G

odel 3-valued logic is the simplest G

odel

logic whose valuation is {1, 0.5, 0}. The language

is built over a countable set of propositional variables

with binary connectives ∧, ∨, → and constant ⊥. >

is deﬁned as ⊥ → ⊥ and the negation ¬ϕ is deﬁned

as ϕ → ⊥. In G

odel 3-valued logic, we have the fol-

lowing truth tables for the operators ¬, ∧, ∨ and →:

(Robles, 2014)

A ¬A

1 0

0.5 0

0 1

A∧B 1 0.5 0

1 1 0.5 0

0.5 0.5 0.5 0

0 0 0 0

A∨B 1 0.5 0

1 1 1 1

0.5 1 0.5 0.5

0 1 0.5 0

A→B 1 0.5 0

1 1 0.5 0

0.5 1 1 0

0 1 1 1

odel 3-valued logic is axiomatized by the fol-

lowing axioms and rules (Robles, 2014):

• A1. A → A

• A2. (A ∧ B) → A and (A ∧ B) → B

• A3. A → (A ∨ B) and B → (A ∨ B)

• A4. [A ∧ (B ∨C)] → [(A ∧ B) ∨ (A ∧C)]

• A5. [(A → B) ∧ A] → B

• A6. A → (B → A)

ICAART 2024 - 16th International Conference on Agents and Artiﬁcial Intelligence

334

• A7. (A → ¬B) → (B → ¬A)

• A8. ¬(A ∧ B) → (¬A ∨ ¬B)

• A9. ¬A → (A → B)

• A10. (A ∨ ¬B) ∨ (A → B)

• (Adj) From A and B, infer A ∧ B

• (MP) From A and A → B, infer B

• (Trans) From A → B and B → C, infer A → C

• (CI∧) From A → B and A → C, infer A → B ∧C

• (E∨) From A → C and B → C, infer A ∨ B → C

2.2 G

odel Modal Logic

In modal logic, we use the modal operators  and ♦

to express the necessity and possibility. ϕ stands for

“ϕ is necessary” while ♦ϕ stands for “ϕ is possible”.

A Kripke modal M = (S, R,V ) is usually used for the

semantics of modal logic, where S is a set of possible

worlds (states), R is an arbitrary function, and V is the

valuation function.

In G

odel modal logic, the semantics is similar

with normal modal logic, where the frame may be

a fuzzy or a crisp Kripke frame and the valuation is

a G

odel many-valued valuation. Here, since our se-

mantics is similar to the crisp model, we introduce

the semantics of crisp G

odel modal logic as follows

(Rodriguez and Vidal, 2021):

Deﬁnition 1 (Semantics). A crisp G

odel Kripke

model is a tuple (S, R,V ), where S is a set of possi-

ble worlds, R : S × S → {0, 1} is an arbitrary func-

tion, and V : Prop × S → [0, 1] (closed interval) is a

odel valuation function where Prop is a non-empty

set of propositions. We can extend the valuation V

to interpretations I by the following conditions where

p ∈ Prop:

I(⊥, s)=0

I(p, s)=V (p, s)

I(¬ϕ , s)=

(

1 I(ϕ , s) = 0

0 I(ϕ , s) > 0

I(ϕ ∧ ψ , s)=min(I(ϕ, s), I(ψ, s)),

I(ϕ ∨ ψ , s)=max(I(ϕ , s), I(ψ, s)),

I(ϕ → ψ , s)=

(

1 I(ϕ , s) ≤ I(ψ, s)

I(ψ , s) Otherwise

I(ϕ , s)=in f ((I(ϕ,t) : sRt and t ∈ S), 1),

I(♦ϕ , s)=sup((I(ϕ , t) : sRt and t ∈ S), 0),

The axioms for modal operators are shown as fol-

lows: (Rodriguez and Vidal, 2021)

• K



: (ϕ → ψ ) → (ϕ → ψ )

• K

♦

: ♦(ϕ ∨ ψ) → (♦ϕ ∨♦ψ)

• T



: ϕ → ϕ

• T

♦

: ϕ → ♦ϕ

• 4



: ϕ → ϕ

• 4

♦

: ♦♦ϕ → ♦ϕ

• B

: ϕ → ♦ϕ

• B

: ♦ϕ → ϕ

• 5

: ♦ϕ → ♦ϕ

• 5

: ♦ϕ → ϕ

• D : ♦>

3 LANGUAGE AND SEMANTICS

3.1 Language

Usually, the symbol of agent is sometimes used as a

subscript to combine the modal operator to the for-

mula, however, the propositions are usually indepen-

dent from agents. In this paper, we combine the

agents and the atomic propositions, i.e., we use the

symbol of agents as the propositions and then provide

the language of our logic.

Deﬁnition 2 (Language). The language L is written

as follows in BNF:

L 3 ϕ ::= i|⊥|ϕ ∧ ϕ|ϕ ∨ ϕ|ϕ → ϕ |ϕ |♦ϕ

where i ∈ Ag and Ag is a non-empty set.

We consider the meaning of the elements in syntax as

follows where i, j ∈ Ag:

• i means that agent i accepts the suggestion.

• Ag is a set of propositions and each element i ∈ Ag

means that “agent i accepts the suggestion”.

• ⊥ can be considered as an agent that rejects all

suggestions.

• i ∧ j means that agent i and j accept the sugges-

tion.

• i ∨ j means that either agent i or j accepts the sug-

gestion.

• i → j means that if agent i accepts the suggestion,

then agent j will also accept the suggestion.

• i means that agent i accepts all given sugges-

tions.

• ♦i means that agent i accepts some given sugges-

tion.

Remark 1. In our research, the idea that using the

symbol of agents as the propositions came from

Agotnes (

Agotnes et al., 2011). However, here we

provide different readings.

A Three-Valued Semantics for Negotiated Situation of Multi-Agent System Based on BATNA and WATNA

335

Here, we deﬁne the negation as ¬ϕ ≡ ϕ → ⊥ and

> ≡ ⊥ → ⊥ as normal logic. Therefore, we can read

the formula ¬i and > as follows:

• ¬i means that agent i rejects the suggestion.

• > can be considered as an agent that always ac-

cepts all suggestions.

Remark 2. It may seem strange that we don’t deﬁne

suggestions in the language. Actually, we consider

the suggestions as possible worlds, therefore “agent i

accepts the suggestion” is the same as “proposition i

is valid in the possible world” in modal logic.

3.2 Semantics

In the negotiation, each agent will accept it, reject it

or hesitate for a given suggestion, therefore we can

provide a 3-valued valuation as follows

• The value 1 means that the suggestion is over the

BATNA, which will be accepted by the agent.

• The value 0.5 means that the suggestion is be-

tween the BATNA and the WATNA, which may

be accepted or rejected by the agent.

• The value 0 means that the suggestion is below

the WATNA, which must be rejected by the agent.

Then, we provide our suggestion model based on

odel 3-valued as follows:

Deﬁnition 3 (Suggestion model). A suggestion

model M is a pair (S,V ), where S is a non-empty ﬁnite

set of possible worlds, and V : Ag × S → {1, 0.5, 0} is

a 3-valued valuation.

Semantically speaking, we read the model as follows:

• the set of possible worlds S is the set of given sug-

gestions in the system;

• the 3-valued valuation expresses whether each

suggestion is over BATNA, between BATNA and

WATNA or below WATNA of each agent as we

showed before.

Deﬁnition 4 (Interpretation). Given a suggestion

model M = (S,V ), we can extend the valuation V to

interpretations I by the following conditions:

I(⊥, s)=0

I(i, s)=V (i, s)

I(¬ϕ , s)=

(

1 I(ϕ , s) = 0

0 I(ϕ , s) ≥ 0.5

I(ϕ ∧ ψ , s)=min(I(ϕ, s), I(ψ, s)),

I(ϕ ∨ ψ , s)=max(I(ϕ , s), I(ψ, s)),

I(ϕ → ψ , s)=

(

1 I(ϕ , s) ≤ I(ψ, s)

I(ψ , s) Otherwise

I(ϕ , s)=min(I(ϕ ,t) : t ∈ S),

I(♦ϕ , s)=max(I(ϕ , t) : t ∈ S),

Remark 3. In this research, we deﬁne the sugges-

tion model considering the Kripke model, however,

we don’t deﬁne the relation R because we consider

that all suggestions are shown to all agents, therefore

they can judge every suggestion is over BATNA, be-

tween BATNA and WATNA, or below WATNA, and

moreover they can compare them.

As we use a G

odel 3-valued valuation, there are

some important differences from classical (modal)

logic:

• Double negation cannot be removed. Actually,

¬¬i here means that agent i may accept the sug-

gestion (suggestion s is over i’s WATNA) be-

cause from semantics we can see I(¬¬i, s) = 1 iff

I(¬i, s) = 0 iff I(i, s) ≥ 0.5.

• ϕ ∧ ψ is not the same as ¬(¬ϕ ∨ ¬ψ ). One coun-

terexample is that, assume I(ϕ, s) = I(ψ , s) = 0.5,

then we have I(ϕ ∧ ψ, s) = 0.5 while I(¬(¬ϕ ∨

¬ψ), s) = 1. Also, ϕ ∨ ψ and ¬(¬ϕ ∧ ¬ψ) are

different and the counterexample is similar.

• ϕ is not the same as ¬♦¬ϕ. one counterexam-

ple is that, assume S = {s} therefore I(ϕ , s) =

I(♦ϕ , s) = I(ϕ , s). Let I(ϕ , s) = 0.5, then we have

I(ϕ , s) = 0.5 while I(¬♦¬ϕ, s) = 1. Also, ♦ϕ

and ¬¬ϕ are different and the counterexample

is similar.

Remark 4. Actually, there exist other 3-valued log-

ics, e.g., strong Kleene 3-valued logic, that can also

be used as a 3-valued valuation. In our research, the

main reason we choose G

odel three-valued logic in

our semantics is that we consider that the formula

i → i should always hold (it is not a tautology in strong

Kleene logic).

From the interpretation, we can deﬁne the satis-

faction relation |= as model logic as follows:

M, s |= ϕ iff I(ϕ, s) = 1

Also, we can give a weaker relation as:

M, s |=

ϕ iff I(ϕ, s) 6= 0

As we remarked, we can see that M, s |=

ϕ iff

M, s |= ¬¬ϕ.

By this semantics, we can formalize some statements

as follows:

• agent i will accept suggestion s (suggestion s in M

is over i’s BATNA): M, s |= i.

• agent i may accept suggestion s (suggestion s in

M is over i’s WATNA): M, s |=

• agent i may accept and may reject suggestion s

(suggestion s in M is between i’s BATNA and

WATNA): M, s 6|= i and M, s 6|= ¬i.

• group G will accept suggestion s (suggestion s is

over the group G’s BATNA): M, s |=

i∈G

ICAART 2024 - 16th International Conference on Agents and Artiﬁcial Intelligence

336

Moreover, we can deﬁne M |= ϕ and |= ϕ as follows:

M |= ϕ iff for all s ∈ S that M, s |= ϕ

|= ϕ iff for all M that M |= ϕ .

By using such deﬁnition we can express the fea-

ture of the all given suggestions in model M as fol-

lows:

• agent i will accept each given suggestion (all sug-

gestions in M are over i’s BATNA): M |= i.

• agent i may accept each given suggestion (all sug-

gestions in M are over i’s WATNA): M |=

• agent i will accept at least one suggestion s (there

exists some suggestion that over i’s BATNA):

M |= ♦i.

• the features for a group of agents are similar as

above.

3.3 Best Suggestion and Best

Acceptable Suggestion

One of the advantage of our logic is that, using our

semantics, we can deﬁne the best suggestion and best

acceptable suggestion by formulas as follows:

Deﬁnition 5 (Best suggestion). In a suggestion

model, if M, s |= ♦i → i, then suggestion s is (one of)

the best suggestion for agent i in model M. And for a

group G ⊆ Ag, if M, s |=

i∈G

(♦i → i), then sugges-

tion s is (one of) the best suggestion for group G in

model M.

We explain why we deﬁne the above semantically.

M, s |= ♦i → i means that I(♦i → i, s) = 1, which

holds if and only if I(♦i, s) ≤ I(i, s). I(♦i, s) stands

for the biggest value of agent i among the set of sug-

gestions S, therefore I(♦i, s) ≤ I(i, s) means that there

doesn’t exist a better suggestion than s, thus s is con-

sidered as (one of) the best suggestion. The consider-

ation of the best suggestion for group G is similar.

However, sometimes even the best suggestion will

also be rejected by the agent since it may be below

the WATNA of this agent. This case occurs when

I(♦i, s) = 0, in other words, all of the suggestions are

below the WATNA therefore every suggestion is the

best suggestion and would be rejected. To avoid this

case, we can deﬁne the best acceptable suggestion as

follows:

Deﬁnition 6 (Best acceptable suggestion). In a sug-

gestion model, if M, s |= ♦i → i ∧ ¬¬i, then sugges-

tion s is the best acceptable suggestion for agent i

in model M. And for a group G ⊆ Ag, if M, s |=

i∈G

(♦i → i ∧ ¬¬i), then suggestion s is the accept-

able best suggestion for group G i in model M.

Here, we add the condition ¬¬i to express the accept-

able. The reason is that we read M, s |= ¬¬i as “agent

i may accept the suggestion s” in our semantics.

There are some properties for the best suggestion

and best acceptable suggestion.

• A best acceptable suggestion is also a best sugges-

tion for an agent or a group. It is easy to see from

the two deﬁnitions.

• In every model, there always exists at least one

best suggestion for each agent, while there may

not exist one for a group. The reason is that by

the semantics there always exist s ∈ S that for one

agent i: I(i, s) = I(♦i, s), while a group of agents

may not have the same best suggestion.

• In every model, there may exist more than one

best acceptable suggestion and may not exist one

for an agent or a group. The counter case is that

we showed above if I(♦i, s) = 0.

• If I(i, s) = 1 (M, s |= i), then suggestion s must be

one of the best (acceptable) suggestions for agent

• If I(i, s) = 0 (M, s |= ¬i), then suggestion s cannot

be the best acceptable suggestion for agent i (but

may be the best suggestion).

Similar to the deﬁnition of best suggestion, we can

also deﬁne the worst suggestion as follows:

Deﬁnition 7 (Worst suggestion). In a suggestion

model, if M, s |= i → i, then suggestion s is (one of)

the worst suggestion for agent i in model M. And for

a group G ⊆ Ag, if M, s |=

i∈G

(i → i), then sug-

gestion s is (one of) the worst suggestion for group G

i in model M.

The reason for this deﬁnition is similar since I(i, s)

stands for the smallest value of agent i among the set

of suggestions S.

We can give some meaningful formula by syntax

to express the property of the negotiation as follows:

• 6|= i ∨ ¬i: a suggestion is not either accepted or

rejected, because it may be not decided(0.5).

• ¬(i ∧ ¬i): a suggestion can not be accepted and

rejected at the same time.

• (♦(i ∧ ¬ j) ∧ ♦(¬i ∧ j) ∧ ♦((♦i → i) ∧ (♦ j →

j))) → ♦(i ∧ j): we can read it as “if there exists

a suggestion that over i’s BATNA while below j’s

WATNA, and there exists a suggestion that below

i’s WATNA while over j’s BATNA, and there ex-

ists a best suggestion for group i and j, then there

exists a suggestion over both i and j’s BATNA”.

• Also, we can prove that these formulas hold in all

suggestion models.

A Three-Valued Semantics for Negotiated Situation of Multi-Agent System Based on BATNA and WATNA

337

3.4 Axiomatization

As a special case of G

odel modal logic, the axiom-

atization and proof theory can be easily inferred by

the proof of general form (Caicedo and Rodr

ıguez,

2015). Therefore, we don’t show the soundness and

completeness in this paper. Instead, we provide the

new readings of some axioms of S5 G

odel crisp model

as follows, which are to show that our interpretation

is a proper reading for G

odel modal logic.

• The K



axiom (i → j) → (i →  j) can be

considered as “ if agent i accepts a suggestion then

j will also accept it for all suggestion, then if i

accepts all suggestions then j will also accept all

suggestions”.

• The K

♦

axiom ♦(i ∨ j) → (♦i∨♦ j) can be consid-

ered as “ if there exists a suggestion either agent

i or j accepts, then there exists a suggestion that i

will accept or there exists a suggestion that j will

accept”.

• The T



axiom i → i can be considered as “if

agent i accepts all given suggestions, then i will

accept this suggestion”.

• The T

♦

axiom i → ♦i can be considered as “if

agent i accepts this suggestions, then i will accept

some given suggestion”.

3.5 Example

Here, we use a concrete example, two sisters argu-

ing over an orange that we have shown in Section 1,

to explain our semantics. We mention the four given

suggestion again as the possible worlds in the model

as follows:

• s

: The elder sister has the whole orange while the

younger sister has none.

• s

: The younger sister has the whole orange while

the elder sister has none.

• s

: Each sister has half of the orange.

• s

: The elder sister has the peel while the younger

sister has the juice.

We consider that there are three steps in this negotia-

tion.

• Firstly, both sisters want to have the whole orange

and suggestions s

and s

are suggested. We write

the model M

in Figure 1:

Here, e stands for the agent “elder sister” and y

stands for the agent “younger sister”. For sugges-

tion s

is over e’s BATNA and below y’s WATNA,

we have V (e, s

) = 1 and V (y, s

) = 0. Also, since

suggestion s

is over y’s BATNA and below e’s

Figure 1: First model: M

WATNA, we have V (e, s

) = 0 and V (y, s

) = 1.

We can see that M

6|= ♦(e ∧ y) and M

6|=

♦(e ∧

y), which means that there doesn’t exist a sugges-

tion that is over both sisters’ BATNAs or WAT-

NAs, therefore they cannot get a solution in the

ﬁrst step. Also, in model M

, M

6|= ♦((♦e →

e) ∧ (♦y → y)), which means that there is no best

suggestion for both sisters at the ﬁrst step.

• Secondly, sisters notice that they need a compro-

mise thus suggestion s

is provided. The second

model is shown in Figure 2.

Figure 2: Second model: M

Since suggestion s

is between both sisters’ BAT-

NAs and WATNAs, we have V (e, s

) = V (y, s

) =

0.5. In model M

, we have M

6|= ♦(e ∧ y) but

♦(e ∧ y), which mean that there doesn’t

exist a suggestion that over sisters’ BATNAs but

exists one over their WATNAs in the second step.

Also, M

6|= ♦((♦e → e) ∧ (♦y → y)), which

means that there is no best suggestion for two sis-

ters.

• Finally, sisters notice the BATNA of each other

therefore s

is suggested. The third model is

shown in Figure 3.

Since suggestion s

is over(just) both sisters’

BATNA, we have V (e, s

) = V (y, s

) = 1. In

model M

, we have M

♦(e∧ y), which means

that there exists a suggestion over both sisters’

BATNA in the ﬁnal step. Also, M

♦((♦e →

e) ∧ (♦y → y)), which means that there exists a

best suggestion for both sisters.

ICAART 2024 - 16th International Conference on Agents and Artiﬁcial Intelligence

338

Figure 3: Third model: M

4 NEW OPERATORS

In Section 3.2, we formalize the statement “agent i

may accept or reject suggestion s” as M, s 6|= i and

M, s 6|= ¬i by semantics. However, we cannot ex-

press this statement by syntax using the operators

¬, ∧, ∨, →. Therefore, we want to add a new operator

to express such suggestions.

Since the negation ¬ is not symmetrical, a natural

idea is to deﬁne a new negation ∼ as a dual of ∼.

I(∼ ϕ , s)=

(

1 I(ϕ , s) ≤ 0.5

0 I(ϕ , s) = 1

Remark 5. Actually, Baaz (Baaz, 1996) gave this

deﬁnition of negation operator already in the last of

the paper in G

odel inﬁnite-valued logic. However, the

main he studied was the operator 4 which is deﬁned

as follows:

I(4ϕ , s)=

(

1 I(ϕ , s) = 1

0 Otherwise

It is easy to see that 4ϕ ≡ ¬ ∼ ϕ and ∼ ϕ ≡ ¬4ϕ.

The reason we don’t use operator 4 is, from the se-

mantics we can see that M, s |= ϕ iff M, s |= 4ϕ, thus

we cannot distinguish the readings of two formulas in

syntax since both i and 4i should be read as “agent

i will accept the suggestion”. However, since we can

use 4 and ¬ to express the operator ∼, later we will

provide the axioms of ∼ from the axioms of 4 in that

paper.

By this deﬁnition, we read the ∼ i as “agent i

may reject the suggestion (the suggestion is below the

BATNA of agent i)”. And therefore, we can express

the statement “agent i may accept or reject sugges-

tion” by formula ¬¬i∧ ∼ i.

Also, we have some axioms of ∼ which comes

from the axioms of 4 (Preining, 2010; B

ılkov

a et al.,

2022):

• ¬ ∼ ϕ ∨ ¬¬ ∼ ϕ

• ¬ ∼ (ϕ ∨ ψ) → (¬ ∼ ϕ ∨ ¬ ∼ ψ)

• ¬ ∼ (ϕ → ψ) → (¬ ∼ ϕ → ¬ ∼ ψ )

• ¬ ∼ ϕ → ϕ

• ¬ ∼ ϕ → ¬ ∼ ¬ ∼ ϕ

• From ϕ infer ¬ ∼ ϕ

Also, since we consider the G

odel 3-valued logic,

there exists other rule for example:

From ∼ (ϕ → ψ ) and ∼ (ψ → χ), infer ϕ

5 CONCLUSION AND FUTURE

WORK

In this paper, we provided a 3-valued logical seman-

tics to express the negotiated situation based. We di-

vide the suggestions to three –must be accepted, must

be rejected, may be accepted while may be rejected–

by comparing it with BATNA and WATNA of each

agent. We provided our semantics by using the sug-

gestion model like the Kripke model, where we regard

the possible worlds as suggestions and propositional

variables as names of agents. Then, we formalized

some statements of negotiation in our semantics and

syntax, e.g., the best suggestion, and gave an example

to show how we deﬁne a negotiated situation by our

semantics. Later, we provide some new operators to

express the statement that cannot be shown by normal

operator, and ﬁnally, we showed that our semantics

can express some negotiated states and situations by

considering a concrete example.

Unfortunately, in this research, our semantics is

based on G

odel modal logic and the axioms of the

operators have been already studied. However, in this

paper, we provide an informal reading of modal logic

therefore we show that negotiated states can be shown

by the basic frame which can be considered as an S5

Kripke frame. Therefore, we gave a new perspective

to the study of negotiation and the multi-agent system

of modal logic.

Other future works remained as follows:

• In this paper, we use the 3-valued valuation based

on BATNA and WATNA. One of the future direc-

tions is to consider the valuation as the evaluation

of suggestions from 0 to 1 and to give the bound of

BATNA and WATNA. For example, if we deﬁne

the bound of WATNA as 0.3 and that of BATNA

is 0.8, suggestions with value 0.5 are considered

to be between BATNA and WATNA, while those

with value 0.9 are considered to be over BATNA.

In this paper, we compare the bound of BATNA/

WATNA and the suggestion as the pre-work and

A Three-Valued Semantics for Negotiated Situation of Multi-Agent System Based on BATNA and WATNA

339

then give the 3-valued valuation. Since G

odel

logic can have ﬁnite or inﬁnite values, it is able

to use a many-valued valuation (from 0 to 1) in

the suggestion model.

• In our suggestion model M = (S,V ), we do not

use the accessibility relation as Kripke model be-

cause R doesn’t work since we consider only one

suggestion can be selected and each suggestion

can be compared with each other. If we can se-

lect more than two suggestions and combine them

as a solution, we are able to use the relations to

express whether two suggestions can be selected

together or not. For example, we can use s

∪ s

shows the combine suggestion that includes s

and

, and we can deﬁne the satisﬁed relation as:

– M, s

∪ s

|= ⊥ iff s

– M, s

∪s

|= ϕ iff (M, s

|= ϕ or M, s

|= ϕ ) and

not s

• In this paper, we employed our 3-valued seman-

tics to express the negotiated situations in the

ODR system. Actually, using this semantics we

can show other situations, e.g., the strategy in

game theory. If an agent beneﬁts much from a

strategy, then the agent would be glad to execute

it; If an agent loses much from a strategy, then the

agent would not execute it; and if an agent beneﬁts

or loses little from a strategy, the agent may hes-

itate whether to execute it or not. We can see the

consideration is quite similar with our semantics

based on BATNA and WATNA. In this case, since

not all strategies can be noticed by every agent,

we may need to add different relations for each

agent as epistemic logic.

ACKNOWLEDGEMENTS

We would like to thank Takahiro Sawasaki for the

helpful comments. We are also grateful to Wachara

Fungwacharakorn and Kanae Tsushima for the useful

discussions. Finally, but not the least, we would also

like to thank the referees for their careful reading and

helpful comments. This research is supported by AIP

challenge.

REFERENCES

Agotnes, T., van der Hoek, W., and Wooldridge, M. (2011).

On the logic of preference and judgment aggregation.

Autonomous Agents and Multi-Agent Systems, 22:4–

30.

Andrade, F., Novais, P., Carneiro, D., Zeleznikow, J., and

Neves, J. (2010). Using BATNAs and WATNAs in

online dispute resolution. In New Frontiers in Artiﬁ-

cial Intelligence: JSAI-isAI 2009 Workshops, LENLS,

JURISIN, KCSD, LLLL, Tokyo, Japan, November 19-

20, 2009, Revised Selected Papers 1, pages 5–18.

Springer.

Baaz, M. (1996). Inﬁnite-valued G

odel logics with 0-1-

projections and relativizations. In G

odel’96: Logi-

cal foundations of mathematics, computer science and

physics—Kurt G

odel’s legacy, Brno, Czech Republic,

August 1996, proceedings, volume 6, pages 23–34.

Association for Symbolic Logic.

ılkov

a, M., Frittella, S., and Kozhemiachenko, D. (2022).

Paraconsistent G

odel modal logic. In International

Joint Conference on Automated Reasoning, pages

429–448. Springer.

Caicedo, X. and Rodr

ıguez, R. O. (2015). Bi-modal G

odel

logic over [0, 1]-valued kripke frames. Journal of

Logic and Computation, 25(1):37–55.

Dunne, P. E. (2005). Extremal behaviour in multiagent con-

tract negotiation. Journal of Artiﬁcial Intelligence Re-

search, 23:41–78.

Dunne, P. E., Wooldridge, M., and Laurence, M. (2005).

The complexity of contract negotiation. Artiﬁcial In-

telligence, 164(1-2):23–46.

Endriss, U. and Pacuit, E. (2006). Modal logics of negotia-

tion and preference. In European Workshop on Logics

in Artiﬁcial Intelligence, pages 138–150. Springer.

Follett, M. P. (2011). Constructive conﬂict. Sociology of

Organizations: Structures and Relationships, 417.

Kraus, S. (1997). Negotiation and cooperation in multi-

agent environments. Artiﬁcial intelligence, 94(1-

2):79–97.

Lodder, A. R. and Zelznikow, J. (2005). Developing an on-

line dispute resolution environment: Dialogue tools

and negotiation support systems in a three-step model.

Harv. Negot. L. Rev., 10:287.

Notini, J. (2005). Effective alternatives analysis in

mediation:“BATNA/WATNA” analysis demystiﬁed.

URL: https://www. mediate. com/articles/notini1.

cfm.[01/2022].

Preining, N. (2010). G

odel logics–a survey. In International

Conference on Logic for Programming Artiﬁcial Intel-

ligence and Reasoning, pages 30–51. Springer.

Ragone, A., Di Noia, T., Di Sciascio, E., and Donini, F. M.

(2006). A logic-based framework to compute pareto

agreements in one-shot bilateral negotiation. FRON-

TIERS IN ARTIFICIAL INTELLIGENCE AND AP-

PLICATIONS, 141:230.

Robles, G. (2014). A simple Henkin-style completeness

proof for G

odel 3-valued logic G3. Logic and Logical

Philosophy, 23(4):371–390.

Rodriguez, R. O. and Vidal, A. (2021). Axiomatization of

crisp G

odel modal logic. Studia Logica, 109(2):367–

395.

Yang, C., Xu, T., Yang, R., and Li, Y. (2018). Multi-

agent single-objective negotiation mechanism of per-

sonalized product supply chain based on personal-

ized index. Advances in Mechanical Engineering,

10(10):1687814018795785.

ICAART 2024 - 16th International Conference on Agents and Artiﬁcial Intelligence

340