Bargaining in Markets with One-Sided
Competition: Model and Analysis
Nicola Gatti
ngatti@elet.polimi.it
DEI, Politecnico di Milano, Piazza Leonardo da Vinci 32, Milano, 20133, Italy
Results Overview
2
The paper aims at studying models for bilateral negotiations with
rational agents when they are in competition
• Modeling result: extension of the alternating-offers protocol to the
situation wherein:
• There are more buyers and one seller, and
• The seller can exploit the outside-option
• Algorithmic result: development of an efficient solving algorithm
for computing agents’ equilibrium strategies (this result is not
discussed in the presentation)
Nicola Gatti
Bilateral Negotiation (Bargaining)
3
10$
11$
Buyer
Seller
12$
...
Nicola Gatti
Alternating-Offers Protocol
4
• Game theoretical mechanism for bilateral negotiation [Rubinstein,
Econometrica, 1982]
• A buyer and a seller wants to negotiate in an alternate-fashion with
discrete time
• Someone decides the agent that starts to make offers
• Time discounts agents’ utilities (so agents prefer to reach
agreement as soon as possible)
• Agents are supposed to be rational, i.e., they act aiming at
maximizing their expected utility
• The solution is a pair of strategies, a buyer’s one and a seller’s one,
that are in equilibrium (typically, refinement of Nash)
Nicola Gatti
The Formal Model [Di Giunta and Gatti, 2006]
• Players
b (buyer )

 s ( seller )
• Player function
 (0)  i

 (t )   (t  1)
• Actions
offer(x)

accept
exit

• Preferences

5
U b ( NoAgreement)  Us( NoAgreement)  0
(RPb  x)( bi )t t  Tb
U b (x,t)  
1
t  Tb

(x  RPs )( si )t t  Ts
U s(x,t)  
1
t  Ts


0
1
2
3
Nicola Gatti
4
5
6
7
Bargaining in Markets
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• More buyers and more sellers are present in a market
• More buyers are in competition over the purchase of an item from
the same seller
• More sellers are in competition over the sale of an item to the same
buyer
• The alternating-offers protocol does not capture this competition
• We extend the alternating-offers protocol to capture:
• The matching between a buyer and a seller
• The possibility of leaving a negotiation to start a new one with a
different commercial partner
• We consider a situation with a seller and more buyers
Nicola Gatti
Bilateral Negotiation with Outside-Option
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15$
Buyer 1
11$
Buyer 2
Seller
14$
Buyer 3
accept
Buyer 4
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The Formal Model
Devoted to
matching
0
8
Devoted to
negotiation
1
2
3
4
5
6
7
situation
stage
agents
time points
available actions
agent i is not
negotiating
1
buyer
any
matchable, nonmatchable
2
seller
any
match(bj) if bj has made matchable
agent i is
negotiating
2
buyer
alternately
offer(x), accept, exit
2
seller
alternately
offer(x), accept, exit, match(bj) is bj has
made matchable
Nicola Gatti
Closing Remarks
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• The proposed model allows one to effectively capture
competition among agents:
• With one seller and more buyers, the seller gains drastically
more than against the weakest buyer
• The computation of agents’ equilibrium strategies is easy (linear
in the size of the problem)
• In future, we will investigate:
• The comparison between the agents’ expected utilities in our
protocol with respect to the ones in multilateral negotiations
• The analysis of negotiations with uncertainty
Nicola Gatti
Questions
10
Email to: ngatti@elet.polimi.it
Nicola Gatti