COOPERATIVE GAMES

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COOPERATIVE GAMES
2
Non-Cooperative games
I want the maximum payoff to Player II
I want the maximum payoff to Player I
Player I
Player II
3
1.9 Cooperative games
Good morning mate!
Hi buddy!, how should we play today ?
Player I
Player II
4
Players are allowed to
1.9 Cooperative games

• 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

• 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
  pij1
• where A (aij), B (bij).
• We refer to C as The Cooperative Payoff Set.

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1.9.1 Example

• Suppose we denote a general cooperative
• payoff pair as (c1, c2)

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1.9.1 Example
C2
6
4
2
0
C1
-2
-4
-6
11
1.9.1 Example
C2
6
4
2
0
C1
-2
-4
-6
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1.9.1 Example
C2
6
4
2
0
C1
-2
-4
-6
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1.9.1 Example
C2
Cooperative Payoff Set
6
4
2
0
C1
-2
-4
-6
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Observe

• both players want non-negative values
• what if (5,2) is treated as a shared payoff
  that both players can share the 527 units of
  payoff?

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1.9.1 Example
C2
Cooperative Payoff Set with side-payments, first
quadrant.
6
4
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 gt c'2
• and/or c1 gt 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)

• Security levels (both have saddles)
• v1 1, v2 3

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V11, v2 3
C2
6
4
2
0
C1
-2
(1, 3)
-4
-6
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V11, v2 3
C2
Non-dominated points (Pareto optimal boundary)
6
4
2
0
C1
-2
-4
-6
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V11, v2 3
C2
Non-dominated points
6
Security level bounded
4
2
0
C1
-2
-4
-6
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V11, v2 3
C2
6
Negotiation Set
4
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?