[PPT] - Bargaining and Coalition Formation Dr James Tremewan PowerPoint Presentation

SLIDE 1

Bargaining and Coalition Formation

Dr James Tremewan (james.tremewan@univie.ac.at) Cooperative Game Theory1

1These slides are based largely on Chapter 18, Appendix A of

”Microeconomic Theory” by Mas-Colell, Whinston, and Green.

SLIDE 2

Cooperative Game Theory

Cooperative Games

A cooperative game is a game in which the players have

complete freedom of preplay communication and can make binding agreements.

In contrast, in a non-cooperative game no communication is

admitted outside the formal structure of the game, and commitments must be self-enforcing, i.e. part of a subgame perfect equilibrium.

There are many solution concepts in cooperative game theory,

e.g:

The Nash bargaining solution was an example of a solution

concept for a class of cooperative games.

The core includes outcomes that may result from coalitional

competition (descriptive).

The Shapley value is motivated as a ”fair” division of surplus

(normative).

2/23

SLIDE 3

Cooperative Game Theory

Games in Characteristic Form

Cooperative games are described in characteristic form. The

characteristic form summarizes the payoffs available to different coalitions.

The set of players is denoted I = 1, ..., I. Nonempty subsets

S, T ⊂ I are coalitions.

An outcome is a list of utilities u = (u1, ..., uI) ∈ RI. The

relevant coordinates for a coalition S are uS = (ui)i∈S.

A utility possibility set is a nonempty, closed set US ⊂ RS where

uS ∈ US and u′S ≤ uS implies u′S ∈ US i.e. utility freely disposable.

A game in characteristic form (I, V ) is a set of players I and a

rule V (·) that associates to evry coalition S ⊂ I a utility possibility set V (S) ⊂ RS .

3/23

SLIDE 4

Cooperative Game Theory

Characteristic Form: Examples

Nash bargaining game (using notation from earlier in course):

I = {1, 2}; V ({1}) = d1, V ({2}) = d2, and V ({1, 2}) = S.

Three-player example (I = {1, 2, 3}):

4/23

SLIDE 5

Cooperative Game Theory

Superadditive Games

A game is superadditive if two (non-overlapping) coalitions can

do at least as well together as alone.

Formally: A game in characteristic form (I, V ) is superadditive if

for any coalitions S, T ⊂ I such that S ∩ T = ∅ we have: if uS ∈ V (S) and uT ∈ V (T), then (uS, uT) ∈ V (S ∪ T).

We will look only at superadditive games: as in the bilateral

section, we are primarily interested in situations where there are gains from cooperation.

5/23

SLIDE 6

Cooperative Game Theory

Transferable Utility (TU)

Much of the literature focuses on TU games (as will we), i.e.

where utility can be transfered costlessly between coalition

members. Sufficient conditions for TU are:
Players can make side-payments.
Players utility is linear in money.
Excludes, for example, situations where bribes are illegal.
In TU games, V (S) = {uS ∈ RS :

i∈S

uS

i ≤ v(S)} for some v(s).

i.e. coalition S chooses a joint action to maximise their total

utility, denoted v(S) which can be allocated amongst S in any way.

v(S) is called the worth of coalition S.

6/23

SLIDE 7

Cooperative Game Theory

TU Games: Example

Boundaries in TU games are hyperplanes in RS:

7/23

SLIDE 8

Cooperative Game Theory

TU Games: Simplex Representation

Normalising utilities such that V ({i}) = 0∀i allows us to represent an n-player TU game on an (n-1) dimension simplex:

8/23

SLIDE 9

Cooperative Game Theory

Two Games in Characteristic Form

A three-player game is defined by:
v({1, 2, 3}) = 1,
v({1, 3}) = v({2, 3}) = 1, v({1, 2}) = 0,
v({1}) = v({2}) = v({3}) = 0.
What do you think will/should happen in this game?
A three-player game is defined by:
v({1, 2, 3}) = 10,
v({1, 2}) = 10, v({1, 3}) = 3, v({2, 3}) = 2,
v({1}) = v({2}) = v({3}) = 0.
What do you think will/should happen in this game?

9/23

SLIDE 10

The Core

The set of feasible utility outcomes with the property that no

coalition could improve the payoffs of all its members.

In TU games the core is the set of utility vectors u = (u1, ..., uI)

such that:

i∈S

ui ≥ v(S)∀S ⊂ I, and

i∈I

ui ≤ v(I)

The core may be:
Empty (strategic instability, no useful prediction, e.g.

divide-the-dollar majority rules).

Large (makes no useful prediction, e.g. divide-the-dollar

unanimity rules).

Non-empty and small (makes sharp prediction).
Note that the core depends only on ordinal utility.

10/23

SLIDE 11

The Core

A TU Game with Non-Empty Core

11/23

SLIDE 12

The Core

A TU Game with Empty Core

12/23

SLIDE 13

The Shapley Value

Describes a ”reasonable” or ”fair” division taking as given the

strategic realities captured by the characteristic form.

Idea of fairness here is egalitarianism: gains from cooperation

should be divided equally.

Two player example (I, v) = ({1, 2}, v):
Gains from cooperation = v(I) − v({1}) − v({2}).
Shi(I, v) = v({i}) + 1

2(v(I) − v({1}) − v({2}))

Can re-write as:
Sh1(I, v) − Sh1({1}, v) = Sh2(I, v) − Sh2({2}, v), and
Sh1(I, v) + Sh2(I, v) = v(I), where Shi({i}, v) = v({i})
The benefit to player one from the presence of player two is the

same as the benefit to player two from the presence of player

ne.
Now generalize this idea to more players...

13/23

SLIDE 14

The Shapley Value

The Shapley value of a game (I, v) is the outcome consistent

with:

Shi(S, v) − Shi(S\{h}, v) = Shh(S, v) − Shh(S\{i}, v), and

i∈S

Shi(S, v) = v(S),

for every subgame (S, v) and all players i, h ∈ S.
In words: the benefit a member of a coalition (player i) gets

from another (player h) joining is equal to the benefit player h would get if already a member and player i was joining.

This outcome is unique.

14/23

SLIDE 15

The Shapley Value

The Shapley Value: Axioms

The Shapley value can also be derived as the unique value

satisfying three axioms (here loosely defined):

Efficiency:

i∈S

Shi(I, v) = v(I), i.e. no utility is wasted.

Symmetry: If (I, v) and (I, v ′) are identical except the roles of

players i and h are swapped, then Shi(I, v) = Shj(S, v ′), i.e. labeling doesn’t matter.

Additivity: If a game is in a particular sense the sum of two
ther games2, then Sh(I, v + w) = Sh(I, v) + Sh(I, w).

2See Shapley (1953) for a precise definition.

15/23

SLIDE 16

The Shapley Value

An ”easy” way to calculate the Shapley value:
Shi(I, v) =

i∈S s!(n−s−1)! n!

(v(S ∪ {i}) − v(S)),

where s = |S| and n = |I|.
Intuition: imagine a coalition of all the players is formed by

including one player at a time in a random order, and each player receives all of the added benefit to the coalition at the time they are included. The Shapley value is the expected value

f this process if all orders are equally likely.
v(S ∪ {i}) − v(S) is the value the new player adds.
s! is number of ways the existing members of S could have

arrived.

(n − s − 1)! is number of ways the remaining players can arrive.
n! is the number of possible ways of the players arriving overall.

16/23

SLIDE 17

Examples

Glove Market: The Shapley Value

A three-player game is defined by:
v({1, 2, 3}) = 1,
v({1, 3}) = v({2, 3}) = 1, v({1, 2}) = 0,
v({1}) = v({2}) = v({3}) = 0.
Player 1:
The value P1 adds to the coalition {3} is 1, and this happens
nly if P3 is added first, then P1 second.
P1 adds 0 to any other coalition.
There are six possible orderings so Shi(I, v) = 1

6.

Player 2: as for P1.

17/23

SLIDE 18

Examples

Glove Market: The Shapley Value

Player 3:
adds 1 to the coalition {1} is, and this happens only if P1 is

added first, then P3 second.

adds 1 to the coalition {2} is, and this happens only if P2 is

added first, then P3 second.

adds 1 to the coalition {1, 2} is, and this happens if P1 is added

first, P2 second, then P3 third, or if P2 is added first, P1 second, then P3 third.

Sh3(I, v) = 1

6 · 1 + 1 6 · 1 + 2 6 · 1 = 2 3

The Shapley value suggests the allocation {1

6, 1 6, 2 3}

Check! The sum of Shi(I, v) is v(I)!

18/23

SLIDE 19

Examples

Glove Market: the Core

What is the core of this game?:
Consider an allocation {u1, u2, u3}.
Suppose u1 > 0. P2 and P3 are better off forming the coalition

{2, 3} and sharing u′

2 = u2 + 1 2u1 and u′ 3 = u3 + 1 2u1.

Therefore in any allocation in the core u1 = 0 (similarly u2 = 0).
No coalition can improve on {0, 0, 1} for all members.
The core is {0, 0, 1}.
The strategic environment means P1 and P2 undercut each
ther leaving zero profits.
The Shapley value ({1

6, 1 6, 2 3}) is more equitable than the core.

However the Shapley value is less equitable than an even split

({1

3, 1 3, 1 3}) because it recognises to some extent the individual

contributions (bargaining power?) of the different players.

19/23

SLIDE 20

Examples

Three Player Game: The Shapley Value

A three-player game is defined by:
v({1, 2, 3}) = 10,
v({1, 2}) = 10, v({1, 3}) = 3, v({2, 3}) = 2,
v({1}) = v({2}) = v({3}) = 0.
Player 1:
adds 10 to the coalition {2} if P2 is added first, then P1 second.
adds 3 to the coalition {3} if P3 is added first, then P1 second.
adds 8 to the coalition {2, 3} if P2 is added first, P3 second and

P1 third; or if P3 is added first, P2 second and P1 third.

P1 adds 0 to any other coalition.
There are six possible orderings so

Sh1(I, v) = 1

6 · 10 + 1 6 · 3 + 2 6 · 8 = 29 6

20/23

SLIDE 21

Examples

Three Player Game: The Shapley Value

Player 2:
adds 10 to the coalition {1} if P1 is added first, then P2 second.
adds 2 to the coalition {3} if P3 is added first, then P2 second.
adds 7 to the coalition {1, 3} if P1 is added first, P3 second and

P2 third; or if P3 is added first, P1 second and P2 third.

P2 adds 0 to any other coalition.
Sh2(I, v) = 1

6 · 10 + 1 6 · 2 + 2 6 · 7 = 26 6

Player 3:
adds 3 to the coalition {1} if P1 is added first, then P3 second.
adds 2 to the coalition {2} if P2 is added first, then P3 second.
adds 3 to the coalition {1, 2} if P1 is added first, P2 second and

P3 third; or if P2 is added first, P1 second and P3 third.

P3 adds 0 to any other coalition.
Sh3(I, v) = 1

6 · 3 + 1 6 · 2 = 5 6

The Shapley value suggests the allocation {29

6 , 26 6 , 5 6}

21/23

SLIDE 22

Examples

Three Player Game: The Core

Consider an allocation {u1, u2, u3}.
If u1 + u2 < 10 P1 and P2 are better off forming the coalition

{1, 2} and sharing u′

1 = u1 + 1 2(10 − u1 − u2) and

u′

2 = u2 + 1 2(10 − u1 − u2).

Therefore in any allocation in the core u1 + u2 ≥ 10.
But v(I) = 10 so u1 + u2 ≤ 10 which means u1 + u2 = 10. Thus

u3 = 0 (from now on we take this as given).

If u1 < 3 P1 is better off forming the coalition {1, 3} and

sharing u′

1 = u1 + ǫ, u′ 3 = 3 − u1 − ǫ.

If u2 < 2 P2 is better off forming the coalition {2, 3} and

sharing u′

2 = u2 + ǫ, u′ 3 = 2 − u2 − ǫ.

The core is the set of allocations {x, 10 − x, 0} where x ∈ [3, 8].

22/23

SLIDE 23

Examples

Three Player Game

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

✉ ✉ ✉

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

(0,10,0) (0,0,10) (10,0,0) (8,2,0) (3,7,0) Shapley value Core

23/23