Slides - Polimi

Towards Automated Bargaining in Electronic
Markets: a Partially Two-Sided Competition Model
N. Gatti, A. Lazaric, M. Restelli
{ngatti, lazaric, restelli}@elet.polimi.it
DEI, Politecnico di Milano, Piazza Leonardo da Vinci 32, Milano, 20133, Italy
Aim and Outline
2
• Aim
We aim at providing a satisfactory extension of the alternating-offers
protocol to electronic markets and at game theoretically analyzing it
• Outline
1. We discuss the bargaining problem in electronic markets
2. We propose a protocol that extends the alternating-offers
protocol to electronic markets
3. We study agents’ equilibrium strategies with complete
information
4. We provide a solving algorithm and we experimentally
evaluate it
AMEC IX 2008
Gatti, Lazaric, Restelli
Introduction to the Bargaining Problem
3
• A bargaining situation involves two parties, which can cooperate
towards the creation of a commonly desirable surplus, over whose
distribution both parties are in conflict [Serrano 2008]
• Bargaining is the most common form of negotiation and plays a
crucial role in automated negotiations
• Bargaining is studied in depth both as cooperative problem [Nash
1953] and non-cooperative problem [Rubinstein 1982]
• The alternating-offers protocol [Rubinstein 1982] is considered the
principal protocol for bilateral negotiations and it has received a lot
of attention
• In economics, to analyze human transactions [Osborne and
Rubinstein 1990]
• In computer science, to automate electronic transactions [Kraus
2001]
AMEC IX 2008
Gatti, Lazaric, Restelli
Alternarting-Offers with Deadlines (informal)
4
• It is an extensive-form game in which agents alternately act, e.g. s acts at
t=0, b acts at t=1, s acts at t=2, and so on
• The model studied in computer science [Fatima 2002] is an extension of
[Stahl 1972] and [Rubinstein 1982]
• Game mechanism:
• The agent that acts at t=0 is a parameter of the protocol
• Agents’ allowed actions are:
• Offer a value x
• Accept the last opponent’s offer
• Exit the negotiation
• Agents’ preferences:
• Agents have opposite preferences and temporal discounting factors
• Agents have reservation values, e.g.
• Buyer’s reservation value expresses the maximum price at which she would
buy the item
• Seller’s reservation value expresses the minimum price at which she would
sell the item
• Agents have deadlines and after these they strictly prefer not to reach
any agreement rather than to reach
AMEC IX 2008
Gatti, Lazaric, Restelli
Alternating-Offers with Deadlines (formal)
• Players
b (buyer )

 s ( seller )
• Player function
 (0)  i

 (t )   (t  1)
• Actions
offer( x )

accept
exit

• Preferences
U b ( NoAgreement )  U s ( NoAgreement )  0
( RVb  x)  ( bi ) t
U b ( x, t )  
1

( x  RV s )  ( si ) t
U s ( x, t )  
1

AMEC IX 2008
Gatti, Lazaric, Restelli
t  Tb
t  Tb
t  Ts
t  Ts
5
Known Results
6
• One-issue complete information settings
• Agents’ equilibrium strategies can be simply inferred by
backward induction similarly to [Stahl 1972]
• Multiple-issue complete information settings
• Several procedures can be followed to negotiate different issues
(e.g., price and quality), the most efficient is the in-bundle: all
the issues are negotiated together
• The problem of finding agents’ equilibrium strategies with
multiple-issues can be reduced (in linear time in the number of
the issues) to the problem of negotiating one issue [Di Giunta
and Gatti 2006; Fatima et al. 2006]
• Uncertain information settings
• Several partial results have been provided in narrow
uncertainty settings, e.g. [Rubinstein 1985; Cramton et al. 2004;
Gatti et al. 2008]
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Gatti, Lazaric, Restelli
Bargaining and Markets
7
• Within a market of bargaining agents two aspects coexist:
• The matching of two opponents (a buyer and a seller)
• The negotiation between two matched opponents
• Classic models from economic literature do not effectively capture
the negotiation between autonomous agents in electronic markets
[Rubinstein and Wolinsky 1985; Binmore et al. 1989]:
• They assume the matching between two agents to be random
• They assume all the buyers (sellers) to be the same (agents have
the same parameter values) and agents have no deadline
• In electronic markets, we expect that:
• Agents can choose the opponent with which to negotiate
• Agents can be different, having different values for the utility
parameters
AMEC IX 2008
Gatti, Lazaric, Restelli
Our Original Contributions
8
1. We provide a satisfactory model for capturing bargaining in
markets
• Our model rules both the matching between agents and the
negotiation
• Our model extends the alternating-offers protocol, i.e.
in presence of one buyer and one seller, agents’ equilibrium strategies
are those in the original protocol
2. Given the negotiation model, we study agents’ equilibrium
strategies when information is complete
AMEC IX 2008
Gatti, Lazaric, Restelli
The Proposed Protocol (1)
•
•
•
•
9
We consider the following situation:
• The items sold by the sellers are equal
• All the sellers have exactly one item to sell
• All the buyers are interested in buying exactly one item
Agent characterization
• We denote by bi the i-th buyer agent – her parameters will be RPbi, Tbi, and bi –
and by sj the j-th seller agent – her parameters will be RPsj, Tsj, and sj
• Each agent, both bi and sj, will be characterized by a time point denoted by Abi
and Asj , respectively, where she enters the market
Matching mechanism
(At each time point)
• At first, each bi announces the seller with which she wants to be matched
• Then, each sj chooses the buyer to match among the ones that have announced sj
Negotiation mechanism
• Once two opponents matched at time t, they start to negotiate at time t+d (the
value of d is set by the negotiation platform)
• The agent that starts the negotiation is selected by the negotiation platform at
random with probability 0.5
• During the negotiation agents can make the classic actions available in the
alternating-offers protocol
AMEC IX 2008
Gatti, Lazaric, Restelli
The Proposed Protocol (2)
10
• Protocol extension:
• Each time point is divided in two stages
• In the first stage, each non-matched buyer announces the seller
with which she wants to be matched (matchable(sk)) or
announce that she wants not to be matched (nonmatchable)
• In the second stage:
• Each non-matched buyer can wait for a time point (wait) or leave
the market (exit)
• Each non-matched seller can wait for a time point (wait), match a
buyer that has announced her in the first stage (match(bi)), or leave
the market (exit)
• Each matched buyer and each matched seller negotiate alternately
as prescribed by the classic protocol
• Action redefinition:
• Action exit imposes agents to leave the market (in addition to
the negotiation)
AMEC IX 2008
Gatti, Lazaric, Restelli
The Proposed Protocol (3)
state
11
stage
agent
time points
available actions
1
bi
any
nonmatchable, matchable(sk)
if sk is not matched
bi
any
wait, exit if bi has
made nonmatchable
sj
any
wait, match(bi) if bi has
made matchable(sj), exit
bi
alternately
offer, accept, exit
sj
alternately
offer, accept, exit
bi (sj) is not
matched
2
bi and sj are
negotiating
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Gatti, Lazaric, Restelli
An Example
12
• A simple setting
• Three buyers: b1, b2, b3
• Two sellers: s1, s2
• All the agents are present in the market from time t = 0
• The value of d is set to d = 1
b1
b2
b3
At
Att t==01
stage 1
nonmatchable matchable(s1)
• •bb
andss1 2match
match
2 3and
stage
2 start
wait
exit
offer(…)
wait
• •they
they
starttotonegotiate
negotiate
atatt t==12
t=0
t=1
t=2
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t=3
…
Gatti, Lazaric, Restelli
time
s2
match(b2)
match(b
wait 3)
matchable(s21)
wait
• •the
theplatform
platformselects
selectsbs2 2to
toopen
openthe
thenegotiation
negotiation
• b1 leaves the market
s1
Main Results (1)
13
• In presence of one buyer and one seller
• Agents prefer to match themselves immediately rather than to
wait for one or more time points and subsequently match
themselves
• Once two agents were matched, their equilibrium strategies are
exactly those in the classic alternating-offers protocol
• If action exit does not impose agents to leave the market
• Once two agents were matched, their equilibrium strategies are
different from those in the classic alternating-offers protocol
• The buyer or the seller can exploit the action exit to leave the
negotiation and subsequently start a new negotiation with the
same opponent
AMEC IX 2008
Gatti, Lazaric, Restelli
Main Results (2)
14
• In presence of more buyers and more sellers
• The problem is essentially a matching problem, since, once two
opponents matched, they negotiate as is prescribed by classic
alternating-offers protocol
• We provide an algorithm that produces the equilibrium
matching for a large range of the parameters and we
experimentally evaluate it
AMEC IX 2008
Gatti, Lazaric, Restelli
The Solving Algorithm
15
The algorithm develops in three steps
1. The outcomes of all the possible negotiations are calculated: they
are 2·m·n, where m is the number of buyer and n is the number of
sellers
2. The utility expected by each agent from matching each possible
opponent is computed and agents’ preferences over the opponents
to match are found:
bi: s1  s3  s4  s5  s8  …  sn
it requires m linear searches among n elements and n linear
searches among m elements and
3. Iteratively, each pair (bi, sj) such that sj is the first choice for bi and
bi is the first choice for sj is removed from the problem
• The proof can be easily produced
• The asymptotically computational complexity is O(m·n)
AMEC IX 2008
Gatti, Lazaric, Restelli
Experimental Evaluation
16
• We evaluate the success of the proposed algorithm
• Experimental setting
• For each value of min{m, n}{1, . . . , 25} we have considered 105
different settings
• In each settings agents’ parameters are chosen with uniform
probability distribution from the following ranges: i  (0, 1), Ti
 {2, 100}, RPbi = 1, RPsi = 0
• Experimental results
min{m,n}
success
min{m,n}
success
min{m,n}
success
2
~99.7%
6
~90.1%
10
~74.2%
3
~98.2%
7
~86.1%
15
~55.5%
4
~96.2%
8
~82.4%
20
~37.0%
5
~93.4%
9
~78.1%
25
~24.7%
AMEC IX 2008
Gatti, Lazaric, Restelli
Conclusions and Future Works
17
• Conclusions
• The problem of bargaining in electronic markets is of extraordinary
importance
• Literature lacks of satisfactory model
• We provide a satisfactory bargaining model that extends the classic
alternating-offers protocol in electronic markets
• We analyze agents’ equilibrium strategies with complete information
• The computational complexity of the proposed algorithm is O(m·n)
• Future works
• We will complete our solving algorithm by resorting to Gale-Shapley
stable marriage algorithm
• We will enrich the bargaining model by introducing the outside option
(i.e., the possibility of leaving a negotiation to start a new negotiation)
• We will study agents’ equilibrium strategies in presence of uncertainty
AMEC IX 2008
Gatti, Lazaric, Restelli
18
Thank for your attention
AMEC IX 2008
Gatti, Lazaric, Restelli

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