1
Non-Cooperative games
I want the maximum
payoff to Player I
Player I
I want the maximum
payoff to Player II
Player II
2
1.9 Cooperative games
Good morning mate!
Player I
Hi buddy!, how
should we play
today ?
Player II
3
1.9 Cooperative games
Players are allowed to
•
•
•
•
•
•
discuss strategy before play
make threats
use coercion
strike deals
Thus ....
we have to expand the set of “feasible”
strategies.
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• Players may not be so much out to “beat” each other,
as much as to get as much as they can themselves.
• E.g.
• Trading between two nations
• Negotiations between employer and employee
• Sometimes an action may benefit both competitors
For example one company advertising a new product,
say 30 day contact lenses, makes people aware that
such things are available. So they may ask about their
availability in their usual brand, even though it is not the
one advertising.
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Example
A1
a1 (4,10)
a2 

a3 
A2
A3
A4



(10,6)
• Players may agree to play (a1, A1) 3/4 of the time
and (a3, A4) the rest of the time.
• The expected payoff is then
• (3/4)(4,10) + (1 – 3/4)(10, 6) = (11/2, 9)
–a convex combination of the individual payoffs.
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Expansion of Strategy Space
• Instead of just the payoffs given by the matrix, we can
think of any convex combination of them as a possible
payoff.
• We now focus on the payoffs rather than the strategies
AND this includes all convex combinations of the
possible payoffs given in the matrix.
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• In particular, they may agree to play (ai, Aj) with
some probability say pij. The value of pij will be
agreed upon before the game commences.
• Thus, the set of expected payoffs is now
C := {ij pij(aij,bij): 0 ≤ pij ≤ 1, ij pij=1}
where A = (aij), B = (bij).
We refer to C as The Cooperative Payoff Set.
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1.9.1 Example
È(0, 5)
VÍ
Î (1, - 3)
(5, 2) ×
(1, - 5)Ý
Þ
•Suppose we denote a general cooperative
payoff pair as (c1, c2)
Payoff to II, c2
(c1, c2)
Payoff to I, c1
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1.9.1 Example
È(0, 5)
VÍ
Î (1, - 3)
(5, 2) ×
(1, - 5)Ý
Þ
C2
6
4
2
0
C1
-2
-4
-6
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1.9.1 Example
È(0, 5)
VÍ
Î (1, - 3)
(5, 2) ×
(1, - 5)Ý
Þ
C2
6
4
2
0
-2
-4
C1
-6
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1.9.1 Example
È(0, 5)
VÍ
Î (1, - 3)
(5, 2) ×
(1, - 5)Ý
Þ
C2
6
4
2
0
C1
-2
-4
-6
12
1.9.1 Example
È(0, 5)
VÍ
Î (1, - 3)
(5, 2) ×
(1, - 5)Ý
Þ
C2
6
Cooperative Payoff Set
4
2
0
C1
-2
-4
-6
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Observe…
1. both players want non-negative
values
2. what if (5,2) is treated as a “shared”
payoff that both players can share the
5+2=7 units of payoff?
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1.9.1 Example
C2
6
4
È(0, 5)
VÍ
Î (1, - 3)
(5, 2) ×
(1, - 5)Ý
Þ
Cooperative Payoff Set
with side-payments, first quadrant.
2
0
C1
-2
-4
-6
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Negotiation Set
• An attempt to reduce the size of the solution
space so as to make it easier for the players to
agree on a solution to the game.
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• Definition: A pair of payoffs (c'1, c'2 ) in a
cooperative game is dominated by (c1, c2 ) if
either c1 ≥ c'1 and c2 > c'2
and/or c1 > c'1 and c2 ≥ c'2
• Definition: A pair of payoffs (c1, c2 ) in a cooperative
game is Pareto optimal if it is not dominated.
• Clearly, we should not consider dominated points of
C.
• Furthermore, since (mixed strategy) security levels
can always be attained, each player should get at
least as much as his/her security level.
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1.9.2 Definition
• The negotiation set of a 2-person game with a
cooperative payoff set C is the subset of C
comprising the non-dominated points of C (i.e
the Pareto optimal points) which are not inferior
to the security levels of the two players.
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1.9.1 Example (continued)
È(0, 5)
VÍ
Î (1, - 3)
(5, 2) ×
(1, - 5)Ý
Þ
• Security levels (both have saddles)
• v1 = 1, v2 = –3
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È(0, 5)
VÍ
Î (1, - 3)
(5, 2) ×
(1, - 5)Ý
Þ
V1=1, v2 = –3
C2
6
4
2
0
-2
-4
-6
C1
(1, –3)
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È(0, 5)
VÍ
Î (1, - 3)
(5, 2) ×
(1, - 5)Ý
Þ
C2
6
V1=1, v2 = –3
Non-dominated points
(Pareto optimal boundary)
4
2
0
C1
-2
-4
-6
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È(0, 5)
VÍ
Î (1, - 3)
(5, 2) ×
(1, - 5)Ý
Þ
V1=1, v2 = –3
C2
6
4
Non-dominated points
Security level bounded
2
0
C1
-2
-4
-6
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È(0, 5)
VÍ
Î (1, - 3)
(5, 2) ×
(1, - 5)Ý
Þ
V1=1, v2 = –3
C2
6
4
Negotiation Set
2
0
C1
-2
-4
-6
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Conclusion
• The idea of “negotiation set” gets rid of options that
are clearly not accepted as solutions to the game.
• The reduction can be substantial.
• What do you do if more than one point is left?
In this example
we still have a
whole line of points.
Can we reduce further?
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