Introduction to game theory and negotiation

1
Introduction to game theory and negotiation

• Ville Koskinen
• Raimo P Hämäläinen
• Systems Analysis Laboratory
• Helsinki University of Technology

2
Contents

• Prisoners dilemma and the problem of the
  commons
• Basic concepts in game theory
• Negotiation analysis
• Aid for co-operation
• Example of the methods in negotiation analysis
• Jointly improving direction method

3
Prisoners dilemma

• Two friends, Harold and William, are suspected of
  committing a crime
• They are separated
• They are unable to communicate and act
  co-operatively
• They may take two actions
• to confess or not to confess the crime

4
Consequences

• Neither of them confesses
• Both of them will be convicted of a minor offence
  and sentenced to 1 month in jail
• Both of them confess
• They will be sentenced to jail for 6 months
• Only one of them confesses
• The confessor acts as a witness against the other
• The confessor will be freed
• The other will be sentenced to 9 months in jail

5
Representation as a game

• Players Harold and William
• Strategy for each player, to confess (c) or not
  to confess (nc)
• s1?S1c,nc and s2?S2 c,nc
• 1 refers to Harold and 2 refers to William
• The payoff for each player
• Number of months in prison negative sign,
  denoted by ui(s1,s2), i?1,2

6
Best response strategies

• Williams best response strategy is
• c, if Harold chooses nc
• c, if Harold chooses c
• Likewise, Harolds best response is c
  independently of Williams choice

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Nash equilibrium for the game
is Nash equilibrium solution if each players
strategy is the best response to the other
players strategy
Nash equilibrium
8
Pareto optimal solutions

• A solution is Pareto optimal if any other
  solution gives a worse outcome for at least one
  of the players.

Pareto optimal solutions are referred to as
co-operative solutions
Pareto optimal outcomes
9
The problem of the commons

• Two farmers Harold and William
• They are going to buy goats
• Denote number of goats by gH and gW
• They need to share the village green in which
  they graze their goats during the summer
• In the autumn, they are going to sell their goats

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Payoffs for the players

• Environmental capacity of the field is limited
• The more a goat has grass the better it survives
  and the higher its selling price P(gHgW) is
• Cost of buying and caring for a goat is c

P(gHgW)
gHgW
Gmax
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Representation as a game

• Players Harold and William
• Strategy for each player The number of the goats
  he owns gH and gW
• The payoff for a player Monetary value of owning
  the goats it is the total selling price minus
  the total cost of the goats
• uH(gH,gW)P(gHgW)-cgH
• uW(gH,gW)P(gHgW)-cgW

12
Payoff contours
Williams payoff contours
For simplicity, assume that 1. gi?0, Gmax 2.
P(gH gW) Gmax- gH- gW
Harolds payoff contours
13
Best response curves
Harolds best response curve
Nash equilibrium is at the intersection of the
best response curves
Williams best response curve
14
Nash equilibrium
Williams payoff contours
Harolds payoff contours
15
Pareto optimal solution
Pareto optimal solutions are defined by the
points of tangency of the players payoff contours
16
More Pareto optimal solutions
17
Game in utility set
uW
uH
18
Disagreement point
19
Negotiation analysis

• Provides methods to aid negotiations
• Is defined as technology for co-operation
  (Sebenius, 1992)
• Has its roots in DA and game theory
• Uses sometimes terminology differing from game
  theory
• Players Negotiating parties
• Strategies Issues
• Payoff function Value function

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Third party intervention

• Mediator
• is neutral
• gathers some confidential preference information
  from the parties
• assists the parties to reach an agreement
• Arbitrator
• Is like mediator but instead of assisting the
  parties arbitrator suggests directly a
  reasonable solution for a game, e.g., Nash
  bargaining solution

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Classification of methods

• Are the parties value functions elicited?
• Value function based methods vs. interactive
  methods
• Do the parties take joint problem solving
  attitude?
• Concession based methods vs. joint gains
  searching methods

22
Making concessions
uW
William makes concessions
Williams initial offer
Harold makes concessions
uH
Harolds initial offer
23
Searching joint gains

• Mediated joint gains methods are often referred
  to as SNT-methods (Raiffa, 1982)

24
Jointly improving direction method

• Interactive joint gains searching method (Ehtamo,
  Verkama and Hämäläinen, 1999)
• It is mathematical formalization of SNT-method
• The method guides the parties step-by-step to a
  Pareto optimal point
• It starts from an initial tentative agreement,
  which is referred to as reference point
• First, we present the method in a value function
  based form

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Steps in the method

• The mediator helps the parties to criticise a
  tentative agreement
• The mediator generates a compromise direction
• The mediator helps the parties to find a new
  jointly preferred point in the compromise
  direction
• If joint gains was reached, go to 1. otherwise
  stop

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Improving directions for Harold
The mediator asks the parties to criticise the
tentative agreement
Tangent of Harolds payoff contour
improving directions
most preferred direction
Direction is improving if by taking a
sufficiently small step along it a preferred
point is reached
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Improving directions for William
Williams most preferred direction
most preferred direction is the gradient of the
value function
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Set of jointly improving directions
Williams improving directions
Jointly improving directions
Harolds improving directions
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Compromise direction
The mediator bisects the angle between the
parties most preferred directions
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Partys most preferred point on the compromise
direction

• Harold prefers points on AC to A
• Harolds most preferred point is B, where the
  direction tangents one of his contours

A
Harolds payoff contours
B
C
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The next tentative agreement

• Each party states his/her most preferred point on
  the compromise direction
• The mediator chooses the point that is closer to
  the tentative agreement as the next tentative
  agreement
• This guarantees that the step is not too long and
  hence the proposal is jointly preferred.

32
The next tentative agreement
Tentative agreement
Next tentative agreement
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Producing joint gains iteratively
The method terminates when the most preferred
directions are opposite
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Developing Pareto frontier
by variation of the reference point
g
W
Pareto optimal
points
g
H
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Interactive form of the method

• We described the method as if the parties value
  functions are explicitly known
• We need only local preference information from
  the parties
• Most preferred direction
• Most preferred point on the compromise direction
• Joint Gains applet implements the interactive
  version of the method
• Available at www.jointgains.hut.fi

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Approximating the most preferred direction

• The mediator draws an ellipse around tentative
  agreement and helps the party to choose the most
  preferred point on that ellipse

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Mediator states series of questions

• Q Which point do you prefer, A or B?
• A I prefer B.

A
B
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Mediator states series of questions

• Q Which point do you prefer, A or B?
• A I prefer A.

B
A
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Most preferred quadrant

• The party prefers to decreasing gW and increasing
  gH

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Mediator states more questions

• Q Which point do you prefer, A or B?
• A I prefer A.
• Partys most preferred point is on arc CB

B
A
C
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Iteration continues

• Mediator chooses two points on arc CB
• Mediator states iteratively more and more
  pairwise comparisons until the party is
  indifferent

B
C
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Approximating partys most preferred direction

• When the party is indifferent
• The mediator chooses a direction going through
  the midpoint of DE
• It approximates partys most preferred direction

E
D
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Eliciting the most preferred point on the
compromise direction

• Mediator helps the party to choose the most
  preferred point on a line segment CD
• Q Which point do you prefer, A or B?
• A I prefer B.
• The most preferred point is on AD

D
B
C
A
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Mediator states more questions

• The mediator chooses two points on AD
• Q Which point do you prefer, E or F?
• A I prefer E.
• The most preferred point is on AF

D
F
E
A
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Approximating the most preferred point on the
compromise direction

• Q Which point do you prefer?
• A I am indifferent.
• The mediator approximates partys most preferred
  point by midpoint of AF

F
A
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Some further readings

• Raiffa H., J. Richardson and D. Metcalfe (2002).
  Negotiation Analysis The Science and Art of
  Collaborative Decision Making. The Belknap Press
  of Harvard University.
• Ehtamo, H. and R.P. Hämäläinen (2001).
  Interactive Multiple-Criteria Methods for
  Reaching Pareto Optimal Agreements in
  Negotiations. Group Decision and Negotiation,
  Vol. 10, 475-491.
• Ehtamo, H., E. Kettunen and R.P. Hämäläinen
  (2001). Searching for Joint Gains in Multi-Party
  Negotiations. European Journal of Operational
  Research, Vol. 130, No. 1, 54-69.
• Ehtamo, H., M. Verkama and R.P. Hämäläinen
  (1999). How to Select Fair Improving Directions
  in a Negotiation Model over Continuous Issues.
  IEEE Transactions on Systems Man and Cybernetics
  Part C Applications and Reviews, Vol. 29,
  26-33.
• Hämäläinen, R.P., E. Kettunen, M. Marttunen and
  H. Ehtamo (2001). Evaluating a Framework for
  Multi-Stakeholder Decision Support in Water
  Resources Management. Group Decision and
  Negotiation, Vol. 10, 331-353.