Summary NBS
Bargaining, Arbitration, Mediation
Dr. Annette Kirstein,
annette.kirstein@ovgu.de
Summer term
Lecture of May 12, 2009
May 13, 2009
Summary NBS
1: A few important insights up to now
• Two approaches to bargaining: axiomatic or non-cooperative.
• Axiomatic approach: bargaining problem, bargaining set,
threat point (outside options).
• If we can assume that bargaining parties follow a solution
concept that satisfies four reasonable properties, and the
bargaining problem is symmetric, then the solution is unique,
• and given by maximization of the Nash product (welfare
function).
• SNBS with two risk-neutral parties = split the surplus.
• SNBS can also be applied to asymmetric bargaining problems
(implicit assumption of equal unexplained bargaining power).
• ANBS is governed by threat point (increasing the own or
decreasing the opponent’s outside option improves own
bargaining position ⇒ strategic moves).
Summary NBS
2: What does the Nash model teach about bargaining?
According to Nash, the predicted outcome of negotiations depends
on three factors:
• the shape of the bargaining set,
• the parties’ threat point,
• the relative bargaining (unexplained) power.
Two concepts from Bazerman/Neale book:
• BATNA = best alternative to a negotiated approach
• bargaining range
Summary NBS
3: What does the Nash model teach about bargaining?
Bargaining zone: the range for the bargaining parameter(s) in
which individual rationality is obeyed.
E.g.: bargaining over π, threat point of player i ∈ {A; B} is di .
The bargaining range (if x denotes A’s share) is [dA , π − dB ].
If the bargaining zone is non-empty, then there is scope for an
agreement which is Pareto-superior to the non-agreement.
However, if the bargaining zone is empty, then no Pareto-superior
agreement exists.
The bargaining range [dA , π − dB ] is empty if, and only if
dA > π − dB or, equivalently,
dA + dB > π
hence, if the collective value of the outside options exceeds the pie
size (compare this to the Nash bargaining problem).
Summary NBS
4: What does the Nash model teach about bargaining?
Bazermann/Neale, ch. 9, prescriptions for rational bargaining:
1. What will you (and your current opponent) do if you don’t
reach an agreement? (Outside options)
2. What is your bargaining zone?
3. What are the true issues in negotiation?
4. How important is each issue to you (the value of an
agreement) and your opponent?
5. Which trade-offs exist? (Shape of bargaining set)
Summary NBS
5: Non-cooperative bargaining theory
Axiomatic approach: if a solution complies with desirable/sensible
principles (“axioms”), then it is unique (or at least determined up
to one parameter).
Advantage: allows to predict outcome of negotiations without
specified procedure, very simple solution concept.
Non-cooperative approach: given the rules of the bargaining
procedure (who may do what, with which information and
consequences?), what is the likely outcome (equilibrium)?
Advantage: clear behavioral foundation (individual expected utility
maximization), no “out of the blue” axioms.
Research goal: under which conditions do the two approaches lead
to identical results? (”Nash program”)
Summary NBS
6: Non-cooperative game theory
Strategic interdependence: in an interactive decision problem
(“game”), each decision-maker (“player”) takes in to account the
consequences of the other players’ decisions on his own situation,
while anticipating the consequences of his own decisions on their
situation.
“The art of outdoing an adversary while the adversary is trying to
do the same”. (Dixit/Nalebeuff)
However, non-cooperative games do not only consist of conflicts,
but may also have cooperative aspects (coordination).
Summary NBS
7: Rules of a non-cooperative game
• players: participants who may influence the outcome
• their strategies: available options (action combinations)
• information: perfect/imperfect, complete/incomplete?
• outcomes: payoffs as a function of strategy combinations.
Two types of interaction, depending on information/observability:
sequential and simultaneous
Two types of models: normal form/strategic form (“bi-matrix”)
and extensive form (“game tree”).
Both models can be used for both types of games!
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8: Strategic form
Normalform or strategic form of a game: [N, S, U], where
• N = {A; B...} is the set of players, i ∈ N,
• player i has the strategy set Si ,
• S is the set of strategy combinations: S = SA × SB × ...
• Ui is the utility function of i (Ui : S → IR (“payoff”),
• U is the set of payoff combinations U = UA × UB × ...
Summary NBS
9: Solution of strategic form game
Solution concept: Nash equilibrium.
1. Determine Player A’s utility maximizing choices against each
possible choice of B ⇒ A’s reaction function.
2. Derive reaction function of the other player(s).
3. Look for intersections of the reaction functions; these strategy
combinations are mutually best replies to each other.
Definition of Nash equilibrium:
If each player has chosen a strategy and no player can benefit by
changing his or her strategy while the other players keep theirs
unchanged, then the current set of strategy choices and the
corresponding payoffs constitute a Nash equilibrium.
Summary NBS
10: Most prominent example: Prisoners’ Dilemma
In the original version, the “PD” was modeled like this: a game
[N, S, U] with N = {A; B}, Si = {blow whistle; shut up}
(hence a 2 × 2 game), and Ui measures the months to be served in
prison according to the following table:
B
shut up
blow whistle
A
-3
shut up
-3
blow whistle
0
0
-10
-10
-5
-5
Best replies of A (B) are marked in blue (red)
⇒ (blow, blow) is the (Pareto-inefficient) Nash equilibrium.
Summary NBS
11: Generalized Prisoners’ Dilemma
Consider the game [N, S, U] with N = {A; B}, Si = {C ; D}
(a 2 × 2 game), and Ui according to the following table:
B
C
D
A
R
C
R
D
T
T
S
S
P
P
with T > R > P > S.
Best replies of A (B) are marked in blue (red)
⇒ (D, D) is the (Pareto-inefficient) Nash equilibrium.
Summary NBS
12: Pure conflict: zero-sum game
2 × 2 games do not always have a unique Nash equilibrium:
B
C
D
A
0
C
1
D
0
1
0
1
0
1
Best replies of A (B) are marked in blue (red)
⇒ no Nash equilibrium in pure strategies.
⇒ unique Nash equilibrium in “mixed” strategies
(not our subject right here).
Summary NBS
13: Coordination and conflict: “battle of the sexes”
2 × 2 games may have multiple Nash equilibria:
B
Soccer
Opera
0
3
A
Opera
Soccer
0
1
1
3
0
0
Best replies of A (B) are marked in blue (red)
⇒ two Nash equilibrium in pure strategies: (C,D) and (D,C).
⇒ third Nash equilibrium in “mixed” strategies.