NonMonotonicOffers Bargaining Protocol

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Non-Monotonic-Offers Bargaining Protocol

• Presenter Sean
• Authors Pinata Winoto, Gordon McCalla,
• Julita Vassileva
• Dept. of Computer Science
• University of Saskatchewan
• Saskatoon, SK S7N5A9, Canada
• Best Paper of AAMAS04

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Problem and approach

• Bargain
• Which bargaining protocol is good?
• Monotonic-offer protocol (M-protocol)
• Non-monotonic-offer protocol (N-protocol)
• Approach to get this conclusion
• Basic model
• Formal analysis based on basic model
• Use simulation to answer the part which analysis
  can not answer
• Evaluation criteria
• Total surplus
• Number of breakdown/success

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Some notions

• Valuation reservation price, the bottom line
• Offer bid
• Belief The possibility that the offer will be
  accept
• Monotonic offer (weak monotonic offer) the offer
  is either non-increasing or non-decreasing all
  the time
• Non-monotonic offer offer is not monotonic.

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An example of the bargaining under M-Protocol
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An example of the bargaining under N-Protocol
From the point of successful rate, N-protocol is
better
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Basic model

• Assumptions
• All agents are rational, self-interested, and
  utility maximizer
• Rational As a rational agent, a buyer will never
  offer a value that generates negative
  profit/surplus and it always prefers an offer
  that generates higher expected gain
• Only two agents in the bargain over a
  single-attribute item
• Buyer show a decreasing valuation over time till
  deadline Td
• Both parties do not know any information about
  their opponents
• Time deadline
• Valuation
• Bargaining strategy
• Buyer move first
• Only bargain the price

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Proposition 1 (belief)

• Xt buyers offer at time t
• P(Xt) buyers belief that its offer Xt is going
  to be accepted by a seller
• Proposition 1
• P(Xt) is an increasing function of Xt

A buyer believes that offering a higher price may
have a higher chance of being accepted by the
seller
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Expected gain

• EGt expected gain at time t
• qt likelihood of failure (breakdown) caused by
  the seller at time t
• Bt buyer valuation at time t
• weight put for the expected gain in the next
  period
• EGt1 prediction of the expected gain made in
  next period
• buyer valuation if the negotiation breaks
  down

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Expected gain
The gain when the negotiation does not break down
The gain when the negotiation breaks down
Surplus when bargain succeeds in time t
Surplus when bargain doesnt succeeds in time t
Suppose no surplus is generated from a breakdown
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Some intuitions

• qt decrease as time goes by (both parties
  approach their time deadlines), qt1gtqt
• Bt1ltBt
• EGt1ltEGt
• lt 1
• Buyer will reduce p(xt) if its offer is not
  accepted by the seller, or increase it if the
  sellers conteroffer yt is very close to xt

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Example shifting of a buyers belief towards
sellers acceptance rate
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Proposition 2 (More opportunities)

• Proposition 2. A series of consecutive offers
  ltx1, x2 , ,xt, fgt is preferred than ltx1, x2 , ,
  xt-1, fgt for 0 xt lt f
• Proposition 2 states that an additional
  opportunity to submit an offer is always
  preferred than ending up with a breakdown, no
  matter what the value of additional offer is,
  i.e. xt could be higher, lower or equal to xt-1
• Prove EGt gt 0 EGf.

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Type-I agent

• A type-I agent uses a typical evaluation function
  in accepting an offer by its opponent, i.e.,
  accept an offer if it is at least as good as the
  counteroffer that would be sent by the agent in
  the next period.
• Example if a buyer plans to offer 5 in the next
  period, while the seller is currently asking for
  4, then the buyer will accept the sellers
  current offer instead of offering 5 in the next
  period.
• Formal definition

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Proposition 3 (offer increase or decrease)

• Proposition 3 Let all agents under N-protocol be
  type-I agents and let this be common knowledge.
  In order to maximize its expected gain, a buyer
  will monotonically increase its offers when Bt -
  xt gtgt ?EGt1, and decrease its offers when Bt -
  xt ? ?EGt1.
• Prove
• Let xt1 xt2 ?x, p(xt2) p(xt1) - ?p
• EG1t1EG2t1, since the buyer can offer any
  price at time t1, which maximizes its expected
  gain
• A rational buyer will choose a higher offer only
  if EG1t1gtEG2t1
• 
  Equation (2)

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Proposition 3 (cont)

• Discussion
• If Bt - xt gtgt ?EGt1, then xt1 gt xt2 is chosen
• As the offer increases, then the slope ?p/?x
  decreases (see Figure 2) and Bt x1t also
  decreases therefore, the buyer has less
  incentive to increase its offer near its
  valuation

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Proposition 4 (limit of offer)

• Proposition 4. Under N-protocol, if all agents
  are type-I agents and this is common knowledge,
  then x8 ? B8 .
• Prove
• (1) q8 ? 1
• EG8 ? 0
• (2) if xt has not been accepted for a certain
  time, then the buyers belief that all offers
  lower than or equal to xt will be accepted also
  decreases to 0.
• ? 0
• (3) forces the buyer to increase its offer
  to be higher than xt. And if this process
  continues, then x8 ? B8.

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Properties of type-I buyers and proposition 5

• From proposition 3 and 4
• If the pressure from the likelihood of breakdown
  increases, then the buyers offer will converge
  quickly to its valuation
• If the buyers valuation decreases sharply, then
  its offer may also decrease (due to the
  decreasing value of the denominator of RHS in
  equation (2))
• If the buyers belief function has a steep curve
  near its valuation, then its offers will also
  converge quickly to its valuation.
• Proposition 5 (success rate)
• If type-I agents are only concerned about the
  success rate, then N-protocol is preferred over
  M-protocol.

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Theorem 1

• N-protocol is at least as good as M-protocol for
  type-I agents.
• Prove
• the expected gain made by agents in N-protocol is
  greater than that in M-protocol.
• agents in N-protocol can always choose offers
  that maximize their expected gain (by Proposition
  3), while agents in M-protocol can only do this
  for increasing offer
• buyers in M-protocol may get stuck in their
  valuation (in case of decreasing valuation),
  which incurs some cost in order to repeat the
  bargaining. In contrast, buyers under N-protocol
  will never get stuck, and by Proposition 2, this
  is preferred. While this is not applied in the
  case of increasing valuation, allowing N-protocol
  causes no cost.
• for agents who are only concerned with success
  rate, N-protocol provides higher opportunity than
  M-protocol (Proposition 5)

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Type-II agent

• A type-II agent uses an additional evaluation
  function in accepting an offer by its opponent,
  accept an offer if it is perceived to generate an
  optimal surplus.
• Example if the buyers current valuation and
  next offer are 100 and 80 respectively, while
  the sellers current offer is 90, then the buyer
  will accept the sellers current offer if it
  predicts that the seller will increase the price
  instead of reducing it in the next period.
• Formal definition

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Proposition 6, 7 and theorem 2

• Proposition 6 Under N-protocol, if all agents
  are type-II agents and this is common knowledge,
  then x8 ? B8 .
• Proposition 7 If type-II agents are only
  concerned about the success rate, then N-protocol
  is preferred than M-protocol.
• Theorem 2. N-protocol is at least as good as
  M-protocol for type-II agents.

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Questions

• Conclusion N-protocol is better than M-protocol
  in terms of success rate and expected gain for
  buyers
• Question this conclusion based on formal
  analysis can be objected since it relies on the
  strict assumption of agents rationality and
  being utility maximizers, which is not realistic
  in the real world.
• Example
• what is the best belief revision for the buyers
  in our analysis above?
• can a buyer really maximize its expected gain
  according to equations (2) or (4)?
• Use simulation to complement the formal analysis

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Simulation design

• Use a simulation to test the N-protocol from the
  sellers perspective, to prove the validity of
  the formal analysis for sellers in more relaxed
  conditions, where agents are myopic and have
  bounded rationality.
• 100 pairs of upward valuation (both buyers and
  sellers have non-increasing valuations over time)
  are generated randomly under a pre-specified
  range.
• Two experiment parts are designed, based on the
  protocol and the strategies used by agents
• Part 1 agents use random strategies in
  bargaining
• Part 2 agents use reactive (behavior-dependent)
  strategies in bargaining

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Simulation design (con)

• The part 1 experiment can be divided into four
  groups according to the strategies used by
  sellers
• Risk-averse seller (R-averse) a seller who
  offers monotonic price and will not offer any
  price below the maximum valuation (in this case
  100)
• Risk-seeking seller (R-seeking) a seller who
  offers monotonic price but may offer any price
  above its valuation
• Nonmonotonic-offer seller (N-seller) a seller
  who offers any price above its valuation and may
  increase it as its valuation increases, but only
  when it is stuck on its valuation
• Nonmonotonic with random change (NR-seller) a
  seller who is similar to N-seller, except that it
  may increase its price randomly (with probability
  equals to0.1) in order to attract type-II buyers.

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Four Treatments based on agents evaluation
Simulation design (cont)

• Subdivide each group of the part 1 experiments
  into four treatments based on the strategies used
  by sellers and buyers shown in following table
• A total of 16 groups of experiments are conducted
  in part 1, and each group is repeated 300 times
  for each pair of valuations, resulting in 480000
  trials

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Simulation design (cont)

• Realization of type-I and type-II agents
• Type-I
• if my opponents current offer generates higher
  positive surplus than my offer which will be sent
  in the next period, then accept my opponents
  current offer
• Type-II add
• if my opponents current offer generates higher
  positive surplus than the previous one, do
  nothing
• if my opponents current offer generate lower
  positive surplus than the previous one, accept it

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Simulation design (cont)

• The experiments in part 2 are almost the same as
  in part 1, except that most agents consider their
  opponents offers, and react accordingly.
• Four reactive strategies
• Tit-for-tat the proponents move is the same as
  the opponents previous move
• Tit-for-2tat the proponents move is the same as
  the opponents previous two moves
• Tat-for-tit the proponents move is the reverse
  of the opponents previous
• Spread-driven the proponents move tries to
  reduce the spread of negotiation by a constant
  fraction

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Parameters and evaluation criteria

• Criteria
• Total surplus sum of surplus for both buyers and
  sellers
• Number of breakdown/success
• Three types of information are retrieved average
  surplus, average surplus per success transaction,
  and success rate.

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Examples of four pairs (out of 100) of upward
valuation used in the experiments
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Result of experiment in part 1
Agents in N-protocol (the two bottom rows)
generate higher surplus compared to agents in
M-protocol (the two upper rows). The success
rates of N-protocol are also higher compared to
the setting where sellers are risk-taking in
M-protocol (2nd row)
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A comparison between N-seller and NR-seller in
N-protocol
The effect of the strategy used by sellers and
buyers is not significant in N-protocol
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A comparison between R-averse and R-seeking in
M-protocol
In M-protocol, much smaller average surpluses are
observed when the sellers are of type-I, which
implies that type-II sellers outperform type-I
sellers in M-protocol
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Result of experiment in part 2

• This result justifies that N-protocol is better
  than M-protocol
• in terms of average surplus gained by bargainers
  and their
• success rate, when both parties have increasing
  valuations.

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Discussion

• Conclusion maybe not true if the following agents
  exist in the bargaining
• Irrational agents, who increase or decrease their
  offers arbitrarily this can be avoided by the
  restriction of M-protocol.
• Misinformed agents, who cannot accept a
  nonmonotonic offer, therefore retreat from the
  bargaining immediately. These agents may perceive
  a nonmonotonic offer as a sign of a lack of
  seriousness of their opponent, or as a sign of a
  prolonged bargaining, or as a sign of a higher
  likelihood of breakdown. Thus, they will retreat
  from the bargaining, and try to find a new
  opponent.
• Nasty agents, who use non-monotonic offers to
  threaten their opponents, delay the bargaining,
  or mislead their opponents beliefs.

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Related work and future work

• Related work
• The deadline of the bargaining.
• The strategies under incomplete information
• Agents with limited resources
• Learning by buyer and seller
• Most of the bargaining problems are solvable
  using gametheoretic analysis, at the cost of a
  less applicable domain
• Use heuristic approach in the design of
  negotiating agents
• Future work
• Extend experiment by using more sophisticated
  agent strategies
• Other cases, such as decreasing valuation for
  buyer and increasing valuation for seller, or
  both with decreasing valuations but with
  different slope and deadline need to be studied
• Extend the simulation by introducing various
  nasty/irrational agents.

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Thanks and Questions
sean_at_wayne.edu and visit http//mist99.cs.wayne.e
du/sean