Nash Bargaining Solution
Non-c oop e rativ e Bargaining T he ory
Strate gic M ov e s in Bargaining
A rb itration and M od e ration m od e ls
47: Non-cooperative games
Axiomatic ap p roach : if a solu tion comp lie s w ith d e sirab le / se n sib le
p rin cip le s (“ axioms” ), th e n it is u n iq u e (or at le ast d e te rmin e d u p
to on e p arame te r).
Ad v an tag e : allow s to p re d ict ou tcome of n e g otiation s w ith ou t
sp e cifi e d p roce d u re , v e ry simp le solu tion con ce p t.
N on -coop e rativ e ap p roach : g iv e n th e ru le s of th e b arg ain in g
p roce d u re (w h o may d o w h at, w ith w h ich in formation an d
con se q u e n ce s? ), w h at is th e lik e ly ou tcome (e q u ilib riu m)?
Ad v an tag e : cle ar b e h av ioral fou n d ation (in d iv id u al e xp e cte d u tility
maximiz ation ), n o “ ou t of th e b lu e ” axioms.
P D D r. R oland K irste in m d @ roland k irste in.d e
Bargaining, A rb itration, M e d iation
Nash Bargaining Solution
Non-cooperative Bargaining Theory
Strategic Moves in Bargaining
Arbitration and Moderation models
48: 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 sam e” . (D ixit/Nalbeuff )
H owever, non-cooperative games do not only consist of confl icts,
but may also have cooperative aspects (coordination).
PD Dr. Roland Kirstein [email protected]
Bargaining, Arbitration, Mediation
Nash Bargaining Solution
Non-cooperative Bargaining Theory
Strategic Moves in Bargaining
Arbitration and Moderation models
49: Rules of a non-cooperative game
I
players: participants who may influence the outcome
I
their strategies: available options (action combinations)
I
information: perfect/imperfect, complete/incomplete?
I
outcomes: payoffs as a function of strategy combinations.
T wo types of interaction: sequential and simultaneous
(information!)
T wo types of models: normal form/strategic form (“bi-matrix”)
and extensive form (“game tree”).
PD Dr. Roland Kirstein [email protected]
Bargaining, Arbitration, Mediation
Nash Bargaining Solution
Non-cooperative Bargaining Theory
Strategic Moves in Bargaining
Arbitration and Moderation models
50: Strategic form
Normalform or strategic form of a game: [N, S, U], where
I
N = {A; B...} is the set of players, i ∈ N,
I
player i has the strategy set Si ,
I
S is the set of strategy combinations: S = SA × SB × ...
I
Ui is the utility function of i (Ui : S → IR (“payoff”),
I
U is the set of payoff combinations U = UA × UB × ...
PD Dr. Roland Kirstein [email protected]
Bargaining, Arbitration, Mediation
Nash Bargaining Solution
Non-cooperative Bargaining Theory
Strategic Moves in Bargaining
Arbitration and Moderation models
51: Strategic form
Solution concept: Nash eq uilibrium.
1 . Determine P layer 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 . L ook for intersections of the reaction functions; these strategy
combinations are simultaneously best replies to each other.
PD Dr. Roland Kirstein [email protected]
Bargaining, Arbitration, Mediation
Nash Bargaining Solution
Non-cooperative Bargaining Theory
Strategic Moves in Bargaining
Arbitration and Moderation models
52: 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 = {confess; deny }
(hence a 2 × 2 game), and Ui measures the months to be served in
prison according to the following table:
B
deny
confess
-3
0
A
deny
confess
-3
-10
-10
0
-5
-5
with T > R > P > S.
Best replies of A (B) are marked in blue (red)
⇒ (D, D) is the (Pareto-ineffi cient) Nash equilibrium.
PD Dr. Roland Kirstein [email protected]
Bargaining, Arbitration, Mediation
Nash Bargaining Solution
Non-cooperative Bargaining Theory
Strategic Moves in Bargaining
Arbitration and Moderation models
53: 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
C
D
R
R
T
S
S
T
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.
PD Dr. Roland Kirstein [email protected]
Bargaining, Arbitration, Mediation
Nash Bargaining Solution
Non-cooperative Bargaining Theory
Strategic Moves in Bargaining
Arbitration and Moderation models
54: Pure conflict: zero-sum game
2 × 2 games do not always have a unique Nash equilibrium:
B
C
D
A
C
D
0
1
1
0
1
0
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).
PD Dr. Roland Kirstein [email protected]
Bargaining, Arbitration, Mediation
Nash Bargaining Solution
Non-cooperative Bargaining Theory
Strategic Moves in Bargaining
Arbitration and Moderation models
55: Coordination and conflict: “battle of the sexes”
2 × 2 games may have multiple Nash equilibria:
B
Soccer
O pera
0
3
A
O pera
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.
PD Dr. Roland Kirstein [email protected]
Bargaining, Arbitration, Mediation