This document contains a number of formal dynamic properties that are relevant for multi-issue negotiation

Bosse, T.

Jonker, C.

Treur, J.

VU Research Portal

VU Research PortalFormalisation of Dynamic Properties of Multi-Issue NegotiationBosse, T.; Jonker, C.; Treur, J.2006document versionPublisher's PDF, also known as Version of recordLink to publication in VU Research Portalcitation for published version (APA)Bosse, T., Jonker, C., & Treur, J. (2006). 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May. 2021Formalisation of Dynamic Properties of Multi-Issue Negotiation Tibor Bosse1, Catholijn M. Jonker1, and Jan Treur1,2 1 Vrije Universiteit Amsterdam, Department of Artificial Intelligence, De Boelelaan 1081a, 1081 HV Amsterdam, The Netherlands {tbosse, jonker, treur}@cs.vu.nl  http://www.cs.vu.nl/~{tbosse, jonker, treur} 2 Utrecht University, Department of Philosophy, Heidelberglaan 8, 3584 CS Utrecht, The Netherlands bid_alternation(γ:trace) Over time the bids of A and B alternate: thus for all two different moments in time t1, t3, that A generated a bid, there is a moment in time t2, with t1 < t2 < t3, such that A received a bid generated by B. ∀ A, B: AGENT, ∀ b1, b3: BID, ∀ t1, t3: t1 < t3 & state(γ, t1, output(A)) |== to_be_communicated_to_by(b1, B, A) & state(γ, t3, output(A)) |== to_be_communicated_to_by(b3, B, A)  ∃b2, ∃t2: t1 < t2 < t3 & state(γ, t2, input(A)) |== communicated_to_by(b2, A, B)   is_followed_by(γ:trace, A:AGENT, t1:time, b1:BID, B:AGENT, t2:time, b2:BID) In a negotiation process γ bid b1 at time t1 is followed by a bid b2 at time t2 iff bids b1 and b2 are subse-quent bids in γ. state(γ, t1, output(A)) |== to_be_communicated_to_by(b1, A, B) & state(γ, t2, output(B)) |== to_be_communicated_to_by(b2, B, A) & t1 < t2 & [ ∀t3, ∀C, D: AGENT, ∀b3: BID: t1 < t3 < t2  state(γ, t3, output(C)) |=/= to_be_communicated_to_by(b3, C, D) ]   agent_consecutively_bids_to(γ:trace, A:AGENT, t1:time, b1:BID, t2:time, b2:BID, B:AGENT) In a negotiation process γ agent A consecutively bids b1 at time t1 and then b2 at time t2 to agent B. state(γ, t1, output(A)) |== to_be_communicated_to_by(b1, A, B) & state(γ, t2, output(A)) |== to_be_communicated_to_by(b2, A, B) & t1 < t2 & [ ∀t3, ∀b3: BID: t1 < t3 < t2  state(γ, t3, output(A)) |=/= to_be_communicated_to_by(b3, A, B) ]   stop_criterion(γ:trace, A:AGENT, t2:time) The stop criterion holds for agent A at time t, if at time t agent A receives a bid by negotiation partner B that is at least as good as the last bid made by A. ∃t1, ∃B: AGENT, ∃b1, b2: BID: state(γ, t2, input(A)) |== communicated_to_by(b2, A, B) & state(γ, t1, output(A)) |== to_be_communicated_to_by(b1, B, A) & is_followed_by(γ, t1, b1, t2, b2) & util(γ, A, b1) ≤ util(γ, A, b2)  negotiation_continuation(γ:trace) For both A and B, unless the stop criterion holds, a new proposal is generated by A upon receival of a proposal by B. ∀t, ∀A, B: AGENT, ∀b1: BID: ¬stop_criterion(γ, A, t) & state(γ, t, input(A)) |== communicated_to_by(b1, A, B)   [ ∃b2: BID ∃t2: t2 > t & state(γ, t2, output(A)) |== to_be_communicated_to_by(b2, B, A) ]    strictly_dominates(b1:BID, b2:BID, A:AGENT, B:AGENT) A bid b1 dominates a bid b2 with respect to agents A and B iff both agents prefer bid b1 over bid b2.  ∀vA1, vA2, vB1, vB2 : real : util(A, b1, vA1) & util(A, b2, vA2) & util(B, b1, vB1) & util(B, b2, vB2)   vA1 > vA2  &  vB1 > vB2   weakly_dominates(b1:BID, b2:BID, A:AGENT, B:AGENT) A bid b1 dominates a bid b2 with respect to agents A and B iff both agents prefer bid b1 over bid b2.  ∀vA1, vA2, vB1, vB2 : real : util(A, b1, vA1) & util(A, b2, vA2) & util(B, b1, vB1) & util(B, b2, vB2)   vA1 ≥ vA2  &  vB1 ≥ vB2   strictly_better_social_welfare(b1:BID, b2:BID, A:AGENT, B:AGENT) The social welfare of bid b1 is better than that of bid b2 with respect to agents A and B iff the sum of the utility values of bid b1 is bigger than the sum of the utility values of bid b2. See also [6,10]. ∀vA1, vA2, vB1, vB2 : real : util(A, b1, vA1) & util(A, b2, vA2) & util(B, b1, vB1) & util(B, b2, vB2)   vA1 + vB1 > vA2 + vB2  strictly_better_equitability(b1:BID, b2:BID, A:AGENT, B:AGENT) A bid b1 has a better equitability than bid b2 with respect to agents A and B iff the difference in the utility values of bid b1 is less than the difference in utility values of bid b2. ∀vA1, vA2, vB1, vB2 : real : util(A, b1, vA1) & util(A, b2, vA2) & util(B, b1, vB1) & util(B, b2, vB2)   | vA1 - vB1 | < | vA2 - vB2 |  ε-equitability(b:BID, A:AGENT, B:AGENT, ε:real) A bid b has ε-equitability with respect to agents A and B iff the difference in the utility values of bid b is less than ε. Thus, a bid that has an equitability of 0 has a maximum equitability. This definition corre-sponds to the idea of Raiffa to maximize the minimum utility [10]. ∀vA, vB : real : util(A, b, vA) & util(B, b, vB)   | vA - vB | ≤ ε  pareto_inefficiency(b:BID, A:AGENT, B:AGENT, ε:real) With respect to agents A and B, the Pareto inefficiency of a bid b is the number ε that indicates the dis-tance to the Pareto Efficient Frontier according to some distance measure d in utilities. Here d(b1, b2) is the distance between the bids b1 and b2 when viewed as points in the plane of utilities.  ∀vA, vB : real : util(A, b, vA) & util(B, b, vB)   pareto_distance(vA, vB) = ε    making_global_concession(γ:trace, A:AGENT, t1:time, b1:BID, t2:time, b2:BID, B:AGENT) In a negotiation process γ agent B makes a global concession to agent B with respect to bid b1 at time t1 and bid b2 at time t2 iff both bids are consecutive, and b2 has a lower utility than b1, from A’s perspec-tive. A similar property could be defined stating that an agent receives a global concession from another agent. agent_consecutively_bids_to(γ, A, t1, b1, t2, b2, B)  & ∀vA1, vA2 : real : util(A, b1, vA1) & util(A, b2, vA2)   vA1 > vA2   configuration_differs(b1:BID, b2:BID) Two bids b1 and b2 differ in configuration iff there is an issue that has a different value in both bids. Similar properties could be defined stating that two bids differ in configuration in at least x issues. ∃a: ISSUE, ∃v1, v2: VALUE:  value_of(b1, a, v1) & value_of(b2, a, v2) & v1 ≠ v2   agent_views_agent_makes_config_variation(γ:trace, A:AGENT, B:AGENT, t1:time, b1:BID, t2:time, b2:BID) In the view of agent A, agent B varies the configuration, but not the utility. Note that one agent can both be agent A and B, or A and B can refer to different agents. agent_consecutively_bids_to(γ, A, t1, b1, t2, b2, B)  &  configuration_differs(b1, b2) & ∀vA1, vA2 : real : util(A, b1, vA1) & util(A, b2, vA2)   vA1 = vA2   agent_views_agent_makes_strict_ε-progression(γ:trace, A:AGENT, B:AGENT, t1:time, b1:BID, t2:time, b2:BID, ε:real) In the view of agent A, the two consecutive bids b1 and b2 made at times t1 and t2 by agent B show mini-mum ε-progression in utility iff the second bid is at least ε higher than the first bid. Note that one agent can both be agent A and B, or A and B can refer to different agents. agent_consecutively_bids_to(γ, A, t1, b1, t2, b2, B)  &  ∀vA1, vA2 : real : util(A, b1, vA1) & util(A, b2, vA2)   vA2 - vA1 > ε   strict_pareto_monotony(γ:trace, tb:time, te:time) A negotiation process γ is Strictly Pareto-monotonous for the interval [t1, t2] iff for all subsequent bids b1, b2 in the interval b2 dominates b1: ∀t1, t2, ∀A, B: AGENT, ∀b1, b2: BID [ tb ≤ t1 < t2 ≤ te & is_followed_by(γ, A, t1, b1, B, t2, b2) ]  strictly_dominates(γ, b2, b1, A, B)   weak_pareto_monotony(γ:trace, tb:time, te:time) A negotiation process γ is Weakly Pareto-monotonous for the interval [t1, t2] iff for all subsequent bids b1, b2 in the interval b2 weakly dominates b1: ∀t1, t2, ∀A, B: AGENT, ∀b1, b2: BID [ tb ≤ t1 < t2 ≤ te & is_followed_by(γ, A, t1, b1, B, t2, b2) ]  weakly_dominates(γ, b2, b1, A, B) 

English

Formalisation of Dynamic Properties of Multi-Issue Negotiation

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DSpace at VU

Formalisation of Dynamic Properties 
of Multi-Issue Negotiation 
Tibor Bosse1, Catholijn M. Jonker1, and Jan Treur1,2 
1
 Vrije Universiteit Amsterdam, Department of Artificial Intelligence, De Boelelaan 1081a, 
1081 HV Amsterdam, The Netherlands 
{tbosse, jonker, treur}@cs.vu.nl  
http://www.cs.vu.nl/~{tbosse, jonker, treur} 
2
 Utrecht University, Department of Philosophy, Heidelberglaan 8, 
3584 CS Utrecht, The Netherlands 
bid_alternation(γ:trace) 
Over time the bids of A and B alternate: thus for all two different moments in time t1, t3, that A generated 
a bid, there is a moment in time t2, with t1 < t2 < t3, such that A received a bid generated by B. 
∀ A, B: AGENT, ∀ b1, b3: BID, ∀ t1, t3: 
t1 < t3 & 
state(γ, t1, output(A)) |== to_be_communicated_to_by(b1, B, A) & 
state(γ, t3, output(A)) |== to_be_communicated_to_by(b3, B, A)  
∃b2, ∃t2: t1 < t2 < t3 & 
state(γ, t2, input(A)) |== communicated_to_by(b2, A, B) 
 
 
is_followed_by(γ:trace, A:AGENT, t1:time, b1:BID, B:AGENT, t2:time, b2:BID) 
In a negotiation process γ bid b1 at time t1 is followed by a bid b2 at time t2 iff bids b1 and b2 are subse-
quent bids in γ. 
state(γ, t1, output(A)) |== to_be_communicated_to_by(b1, A, B) & 
state(γ, t2, output(B)) |== to_be_communicated_to_by(b2, B, A) & 
t1 < t2 & 
[ ∀t3, ∀C, D: AGENT, ∀b3: BID: 
t1 < t3 < t2  state(γ, t3, output(C)) |=/= to_be_communicated_to_by(b3, C, D) ] 
 
 
agent_consecutively_bids_to(γ:trace, A:AGENT, t1:time, b1:BID, t2:time, b2:BID, 
B:AGENT) 
In a negotiation process γ agent A consecutively bids b1 at time t1 and then b2 at time t2 to agent B. 
state(γ, t1, output(A)) |== to_be_communicated_to_by(b1, A, B) & 
state(γ, t2, output(A)) |== to_be_communicated_to_by(b2, A, B) & 
t1 < t2 & 
[ ∀t3, ∀b3: BID: 
t1 < t3 < t2  state(γ, t3, output(A)) |=/= to_be_communicated_to_by(b3, A, B) ] 
 
 
stop_criterion(γ:trace, A:AGENT, t2:time) 
The stop criterion holds for agent A at time t, if at time t agent A receives a bid by negotiation partner B 
that is at least as good as the last bid made by A. 
∃t1, ∃B: AGENT, ∃b1, b2: BID: 
state(γ, t2, input(A)) |== communicated_to_by(b2, A, B) & 
state(γ, t1, output(A)) |== to_be_communicated_to_by(b1, B, A) & 
is_followed_by(γ, t1, b1, t2, b2) & 
util(γ, A, b1) ≤ util(γ, A, b2) 
 negotiation_continuation(γ:trace) 
For both A and B, unless the stop criterion holds, a new proposal is generated by A upon receival of a 
proposal by B. 
∀t, ∀A, B: AGENT, ∀b1: BID: 
¬stop_criterion(γ, A, t) & 
state(γ, t, input(A)) |== communicated_to_by(b1, A, B)  
 [ ∃b2: BID ∃t2: t2 > t & state(γ, t2, output(A)) |== to_be_communicated_to_by(b2, B, A) ]  
 
 
strictly_dominates(b1:BID, b2:BID, A:AGENT, B:AGENT) 
A bid b1 dominates a bid b2 with respect to agents A and B iff both agents prefer bid b1 over bid b2.  
∀vA1, vA2, vB1, vB2 : real : 
util(A, b1, vA1) & util(A, b2, vA2) & util(B, b1, vB1) & util(B, b2, vB2)   
vA1 > vA2  &  vB1 > vB2 
 
 
weakly_dominates(b1:BID, b2:BID, A:AGENT, B:AGENT) 
A bid b1 dominates a bid b2 with respect to agents A and B iff both agents prefer bid b1 over bid b2.  
∀vA1, vA2, vB1, vB2 : real : 
util(A, b1, vA1) & util(A, b2, vA2) & util(B, b1, vB1) & util(B, b2, vB2)   
vA1 ≥ vA2  &  vB1 ≥ vB2 
 
 
strictly_better_social_welfare(b1:BID, b2:BID, A:AGENT, B:AGENT) 
The social welfare of bid b1 is better than that of bid b2 with respect to agents A and B iff the sum of the 
utility values of bid b1 is bigger than the sum of the utility values of bid b2. See also [6,10]. 
∀vA1, vA2, vB1, vB2 : real : 
util(A, b1, vA1) & util(A, b2, vA2) & util(B, b1, vB1) & util(B, b2, vB2)   
vA1 + vB1 > vA2 + vB2 
 
strictly_better_equitability(b1:BID, b2:BID, A:AGENT, B:AGENT) 
A bid b1 has a better equitability than bid b2 with respect to agents A and B iff the difference in the utility 
values of bid b1 is less than the difference in utility values of bid b2. 
∀vA1, vA2, vB1, vB2 : real : 
util(A, b1, vA1) & util(A, b2, vA2) & util(B, b1, vB1) & util(B, b2, vB2)   
| vA1 - vB1 | < | vA2 - vB2 | 
 
ε-equitability(b:BID, A:AGENT, B:AGENT, ε:real) 
A bid b has ε-equitability with respect to agents A and B iff the difference in the utility values of bid b is 
less than ε. Thus, a bid that has an equitability of 0 has a maximum equitability. This definition corre-
sponds to the idea of Raiffa to maximize the minimum utility [10]. 
∀vA, vB : real : 
util(A, b, vA) & util(B, b, vB)   
| vA - vB | ≤ ε 
 
pareto_inefficiency(b:BID, A:AGENT, B:AGENT, ε:real) 
With respect to agents A and B, the Pareto inefficiency of a bid b is the number ε that indicates the dis-
tance to the Pareto Efficient Frontier according to some distance measure d in utilities. Here d(b1, b2) is 
the distance between the bids b1 and b2 when viewed as points in the plane of utilities.  
∀vA, vB : real : 
util(A, b, vA) & util(B, b, vB)   
pareto_distance(vA, vB) = ε 
 
 
 
making_global_concession(γ:trace, A:AGENT, t1:time, b1:BID, t2:time, b2:BID, B:AGENT) 
In a negotiation process γ agent B makes a global concession to agent B with respect to bid b1 at time t1 
and bid b2 at time t2 iff both bids are consecutive, and b2 has a lower utility than b1, from A’s perspec-
tive. A similar property could be defined stating that an agent receives a global concession from another 
agent. 
agent_consecutively_bids_to(γ, A, t1, b1, t2, b2, B)  & 
∀vA1, vA2 : real : 
util(A, b1, vA1) & util(A, b2, vA2)   
vA1 > vA2 
 
 
configuration_differs(b1:BID, b2:BID) 
Two bids b1 and b2 differ in configuration iff there is an issue that has a different value in both bids. 
Similar properties could be defined stating that two bids differ in configuration in at least x issues. 
∃a: ISSUE, ∃v1, v2: VALUE:  
value_of(b1, a, v1) & 
value_of(b2, a, v2) & 
v1 ≠ v2 
 
 
agent_views_agent_makes_config_variation(γ:trace, A:AGENT, B:AGENT, t1:time, b1:BID, 
t2:time, b2:BID) 
In the view of agent A, agent B varies the configuration, but not the utility. Note that one agent can both be 
agent A and B, or A and B can refer to different agents. 
agent_consecutively_bids_to(γ, A, t1, b1, t2, b2, B)  &  
configuration_differs(b1, b2) & 
∀vA1, vA2 : real : 
util(A, b1, vA1) & util(A, b2, vA2)   
vA1 = vA2 
 
 
agent_views_agent_makes_strict_ε-progression(γ:trace, A:AGENT, B:AGENT, t1:time, 
b1:BID, t2:time, b2:BID, ε:real) 
In the view of agent A, the two consecutive bids b1 and b2 made at times t1 and t2 by agent B show mini-
mum ε-progression in utility iff the second bid is at least ε higher than the first bid. Note that one agent can 
both be agent A and B, or A and B can refer to different agents. 
agent_consecutively_bids_to(γ, A, t1, b1, t2, b2, B)  &  
∀vA1, vA2 : real : 
util(A, b1, vA1) & util(A, b2, vA2)   
vA2 - vA1 > ε 
 
 
strict_pareto_monotony(γ:trace, tb:time, te:time) 
A negotiation process γ is Strictly Pareto-monotonous for the interval [t1, t2] iff for all subsequent bids b1, 
b2 in the interval b2 dominates b1: 
∀t1, t2, ∀A, B: AGENT, ∀b1, b2: BID 
[ tb ≤ t1 < t2 ≤ te & is_followed_by(γ, A, t1, b1, B, t2, b2) ] 
 strictly_dominates(γ, b2, b1, A, B) 
 
 
weak_pareto_monotony(γ:trace, tb:time, te:time) 
A negotiation process γ is Weakly Pareto-monotonous for the interval [t1, t2] iff for all subsequent bids b1, 
b2 in the interval b2 weakly dominates b1: 
∀t1, t2, ∀A, B: AGENT, ∀b1, b2: BID 
[ tb ≤ t1 < t2 ≤ te & is_followed_by(γ, A, t1, b1, B, t2, b2) ] 
 weakly_dominates(γ, b2, b1, A, B) 


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Formalisation of Dynamic Properties of Multi-Issue Negotiation

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