cooperative games

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2

Non-Cooperative games

Player I Player II

I want the maximum payoff to Player I

I want the maximum payoff to Player II

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1.9 Cooperative games

Player I Player II

Good morning mate! Hi buddy!, how should we playtoday ?

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Players are allowed to• discuss strategy before play• make threats• use coercion• strike deals• Thus ....• we have to expand the set of “feasible”

strategies.

1.9 Cooperative games

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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 competitorsFor 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.

A1 A2 A3 A4

a1

a2

a3

(4,10) LM L M

(10,6)

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

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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 := {ij pij(aij,bij): 0 ≤ pij ≤ 1, ij pij=1} where A = (aij), B = (bij).

We refer to C as The Cooperative Payoff Set.

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1.9.1 Example

V =

( 0 , 5 ) ( 5 , 2 )

( 1 , - 3 ) ( 1 , - 5 )

È

Î Í

˘

˚ ˙

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

Payoff to I, c1

Payoff to II, c2

(c1, c2)

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1.9.1 ExampleV =

( 0 , 5 ) ( 5 , 2 )

( 1 , - 3 ) ( 1 , - 5 )

È

Î Í

˘

˚ ˙

0246

-2-4-6

C1

C2

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1.9.1 ExampleV =

( 0 , 5 ) ( 5 , 2 )

( 1 , - 3 ) ( 1 , - 5 )

È

Î Í

˘

˚ ˙

0246

-2-4-6

C1

C2

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1.9.1 ExampleV =

( 0 , 5 ) ( 5 , 2 )

( 1 , - 3 ) ( 1 , - 5 )

È

Î Í

˘

˚ ˙

0246

-2-4-6

C1

C2

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1.9.1 ExampleV =

( 0 , 5 ) ( 5 , 2 )

( 1 , - 3 ) ( 1 , - 5 )

È

Î Í

˘

˚ ˙

0

2

4

6

-2

-4

-6

C1

C2

Cooperative Payoff Set

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Observe…

1. both players want non-negative values2. 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

V =

( 0 , 5 ) ( 5 , 2 )

( 1 , - 3 ) ( 1 , - 5 )

È

Î Í

˘

˚ ˙

0

2

46

-2-4

-6

C1

C2

Cooperative Payoff Setwith side-payments, first quadrant.

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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)

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

V =

( 0 , 5 ) ( 5 , 2 )

( 1 , - 3 ) ( 1 , - 5 )

È

Î Í

˘

˚ ˙

20

V =

( 0 , 5 ) ( 5 , 2 )

( 1 , - 3 ) ( 1 , - 5 )

È

Î Í

˘

˚ ˙

0

2

46

-2-4

-6

C1

C2

V1=1, v2 = –3

(1, –3)

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V =

( 0 , 5 ) ( 5 , 2 )

( 1 , - 3 ) ( 1 , - 5 )

È

Î Í

˘

˚ ˙

0

2

46

-2-4

-6

C1

C2 Non-dominated points(Pareto optimal boundary)

V1=1, v2 = –3

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V =

( 0 , 5 ) ( 5 , 2 )

( 1 , - 3 ) ( 1 , - 5 )

È

Î Í

˘

˚ ˙

0

2

46

-2-4

-6

C1

C2

Non-dominated pointsSecurity level bounded

V1=1, v2 = –3

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V =

( 0 , 5 ) ( 5 , 2 )

( 1 , - 3 ) ( 1 , - 5 )

È

Î Í

˘

˚ ˙

0

2

46

-2-4

-6

C1

C2

Negotiation Set

V1=1, v2 = –3

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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?