1. Katsushige FUJIMOTO Fukushima University
Incentives to form the grand coalition
versus
no incentive to split off from the grand coalition
Katsushige FUJIMOTO
Fukushima University
ミクロ経済学ワークショップ
2017/4/24(月)

2. Core ! ! Key words: Allocation rules
Allocations under which no coalition can break away and take a joint action that makes all participants better off.
Core ! !
Allocations under which each player has an incentive to form the grand coalition.
To be proposed ! !

3. “maintaining “ and “forming”
Key words:
Allocation rules
Core ! !
Allocations for maintaining the grand coalition
Allocations to be proposed ! !
Allocations for forming the grand coalition
We will focus on and discuss these two notions,
“maintaining “ and “forming”
the grand coalition.

4. “maintaining “ and “forming”
Key words:
Allocation rules
Is there any relation of strength and weakness between “maintaining” and “forming”
the grand coalition?
do these two notions coincide?
In what kind of situations
Core ! !
Allocations maintaining the grand coalition
Allocations to be proposed ! !
Allocations forming the grand coalition
We will focus on and discuss these two notions,
the grand coalition.
“maintaining “ and “forming”

5. Preliminaries
(game)
Notations

6. Preliminaries
(core)

7. Preliminaries
(core)
Note

8. Example
(core) 1 2 3 4

9. Example
(core) 1 2 3 4

10. Prayers 2,3,& 4
can improve their payoff themselves

11. Players 2,3,&4 would break away from the grand coalition
The grand coalition 1234 would break up!
under (6,0,0,0)
Prayers 2,3,& 4
can improve their payoff themselves

12. So can the players 3 &4.
Prayers 2,& 3
can improve their payoff themselves

13. So can the players 3 &4.
Prayers 2,& 3
can improve their payoff themselves

14. Player 1 wants to improve his payoff.
If he decides to break away from the grand coalition alone,
Then, however, he cannot improve it himself.
Player 1 wants to improve his payoff.
Prayers 1
cannot improve his payoff by himself.
Therefore, he has no incentive to break away from the grand coalition alone.

15. Player 1 wants to improve his payoff.
If he breaks away with player 2,
then, they can earn 2 themselves, therefore, they cannot improve their payoffs.
Player 1 wants to improve his payoff.
Prayers 1 &2
cannot improve both their payoffs by themselves

16. Player 1 has no incentive to break away from the grand coalition!
Neither can the player 1 with 2 & 3.
Player 1 wants to improve his payoff.
In order to improve their payoff, they need at least greater than 4 (v(123) > 4)

17. Example (core) 1 2 3 4 Player 1 want to improve his payoff.
Next, we see the payoff vector (0,2,2,2) which is in the core.
.Indeed, it satisfy the core condition:
“the total amount of payoff in any coalition is greater than or equal to its worth”.
However, player 1 want to improve his payoff.
But he cannot improve it himself.
Indeed,…
Player 1 want to improve his payoff.
He cannot improve it himself.

18. Preliminaries (core) Allocations for maintaining the grand coalition
READ THE SENTENCE
That is, the grand coalition is maintained and stable under core-allocations.
The core is a solution concept for coalitional games that require no coalition (group of players) to break away and take a joint action that makes all participants better off.
That is, no player has the incentive to split from the grand coalition under core allocations.

19. Preliminaries (core) Allocations for maintaining the grand coalition
Here we should note that it does not say that
every player has an incentive to form the grand coalition
under core allocations.
The core is a solution concept for coalitional games that requires that no coalition (group of players) be able to break away and take a joint action that makes all of them better off.
That is, every player has no incentive to split from the grand coalition under core allocations.
Note that this does not imply that
each player has the incentive to form the grand coalition
under core allocations.

20. Note that, there is no rule
Here we should note that it does not say that
no coalition can make others worse off!
Player 1 wants to improve his payoff.
Player 1 wants to improve his payoff.
Note that, there is no rule
that a coalition can make others worse off!
that states
Player 1 wants to improve his payoff.

21. Example
(core) 1 2 3 4
If the player 1 splits from the grand coalition….
If I were the player 1, I would not want to participate in the grand coalition.
If the player 1 splits from the grand coalition,
players 2,3, & 4 should play the 3-person sub-game.
Then, it is easy to see that
The payoff vector (1,1,1) is the unique allocation
where S would not break up, moreover S would be formed.
Here, we consider coalition forming processes!
Player 2,3, & 4 should play
sub-game (S,v ) such as |S|=3, S
Then, (1,1,1) is the unique acceptable allocation
in the subgame!

22. Example (core) 1 2 3 4 Case 1 : 123  1234 (i.e., 1  0 for player 1)
(0,2,2,2)
trivial
In the case
where the coalition 123 has already been formed.
It is trivial that player 1 never want to form the grand coalition and never accept the allocation (0,2,2,2)
where the coalition 234 has already been formed,
if the player 1 does not participate in the grand coalition, then the payoff of each player of 2,3, and 4 remains at 1.
Therefore all players in the coalition 234 would accept any payoff more than 1.
That is,
if the payoff vector (0,2,2,2) is proposed to the player 1,
he would refuse the proposal and propose a re-proposal, for example, (0.3,1.9,1.9,1.9) or (2.7,1.1,1.1,1.1).
(1,1,1)
That is, the player 1 will refuse the proposal (0,2,2,2).
Case 2 : 234  1234
(*,*,*,*)
(1,1,1)

23. Example (core) 1 2 3 4 Under (0,2,2,2),
the grand coalition wouldn ’t break up!
the grand coalition wouldn’t be formed!!
That is, under the allocation (0,2,2,2),
the grand coalition wouldn’t break up!
the grand coalition wouldn’t be formed!!

24. Preliminaries (new allocation)
Each player has an incentive to form a larger coalition
in seeking an increase in its allocation.
It is reasonable to assume
that each player has an incentive to form a larger coalitions
for seeking an increase in its allocation.
Each player cannot earn any on his own.
So, payoff is zero.
Then, these payoff improve both their payoffs.
Therefore, player 1 & 2 agree to cooperate with each other
to improve their payoffs.
Players 1 & 2 agree to cooperate with each other
to improve their payoffs.

25. Let’s see this example.
You can easily see that the allocation (1,1) can be acceptable for each two-person coalitions.
The unique allocations (1,1,1) for three-person coalition improves some acceptable payoffs in any sub-coalitions.

26. (2,2,1,1) can be a potential target to form the grand coalition!.
(2,2,1,1) improves, in a weak sense, all allocations in each sub-games.
(2,2,1,1) can be a potential target
to form the grand coalition!.
Let’s see this example.
You can easily see that the allocation (1,1) can be acceptable for each two-person coalitions.
The unique allocations (1,1,1) for three-person coalition improves some acceptable payoffs in any sub-coalitions.

27. As mentioned before, the unique acceptable allocation for three-person coalition is (1,1,1) in the game.
Therefore all these allocations such payoff vectors can improve some allocation in all its sub-games.
That is these allocation can be interpreted as a potential target to form the grand coalition!
In this case,
the set of all potential target to form the grand coalition
is of the form A(N,v)

28. F.Y.I
Note
That is, these mathematical results can be interpreted as this concluding remarks.

29. 6 2 3 2 3
As mentioned before, the unique acceptable allocation for three-person coalition is (1,1,1) in the game.
Therefore all these allocations such payoff vectors can improve some allocation in all its sub-games.
That is these allocation can be interpreted as a potential target to form the grand coalition!
In this case,
the set of all potential target to form the grand coalition
is of the form A(N,v) 2
super additive,

30. < + - 6 3 2 2 super additive, but not convex
As mentioned before, the unique acceptable allocation for three-person coalition is (1,1,1) in the game.
Therefore all these allocations such payoff vectors can improve some allocation in all its sub-games.
That is these allocation can be interpreted as a potential target to form the grand coalition!
In this case,
the set of all potential target to form the grand coalition
is of the form A(N,v) 3
< 2 + - 2
super additive,
but not convex

31. convex Considering such a game, 4 instead of 3 of previous example.
In 1-person sub-games, every player’s payoff is trivially zero.
In 2-person game, such a game with players 12,
any payoff of the game should improve some payoffs in every sub-game.
In 3-person game, such a game with players 123,
for example, (2,1,1,*) improves some payoffs in every sub-game.

32. So does payoff (1,2,1) of the game with players 234.
Finally, for example, such an allocation (2,1,2,1) improves some allocations in every sub-game.
That is, under an allocation (2,1,2,1) every player has incentive to form the grand coalitions for seeking an increase in its allocation.
In other words, such an allocation can be a potential target to form the grand coalition..

33. So does payoff (1,2,1) of the game with players 234.
Finally, for example, such an allocation (2,1,2,1) improves some allocations in every sub-game.
That is, under an allocation (2,1,2,1) every player has incentive to form the grand coalitions for seeking an increase in its allocation.
In other words, such an allocation can be a potential target to form the grand coalition..

34. Rough Paraphrase (new allocation) The set of potential targets
to form the coalition S!
: Some players have
no incentive to form S.
Here, we will propose a new solution concept A of the game as the set of potential targets to form the grand coalition.
If A of S is empty, that is no goal for some players, that is ,
some players have no incentive to form the coalition S.
Each player has an incentive to form larger coalitions
in seeking an increase in its allocation.

35. Preliminaries (new allocation) S T
Each player has an incentive to form larger coalitions
in seeking an increase in its allocation. S T
Now, we define an new allocation set, that is new solution concept, A(N,v) mathematically
A(N,v) is recursively defined as this way,
That is, A(N,v) is the set of pre-imputation which improve some allocations in all its sub-games.
Then, we have several mathematical results between the core and the allocation set A,
that is, relation between maintaining coalitions and forming coalitions.

36. Relations between Core C(N,v) and A(N,v)
forming
maintaining
Theorem 1
First, theorem 1 shows the allocation set A is a subset of the core of its game.
That is, the notion forming coalitions is stronger notion than maintaining coalitions.
Next, in at most 3-person game or situations,
the allocation set A and the core coincide,
That is, the notions forming and maintaining coincide.
Prop. 4

37. Relations between Core C(N,v) and A(N,v)
F.Y.I.
kernel
nucleolus
First, theorem 1 shows the allocation set A is a subset of the core of its game.
That is, the notion forming coalitions is stronger notion than maintaining coalitions.
Next, in at most 3-person game or situations,
the allocation set A and the core coincide,
That is, the notions forming and maintaining coincide.
Prop. 4

38. Relations between Core C(N,v) and A(N,v)
Prop. 5
Propositions 5 says that
if the number of player is greater than 3,
there are some games or situations
where the grand coalition cannot be formed
and can be maintained.
Propositions 6 says that
there are some games
where the grand coalition can be formed
but some core-allocations are refused.
F.Y.I.
kernel
nucleolus

39. Relations between Core C(N,v) and A(N,v)
Prop. 5
Propositions 5 says that
if the number of player is greater than 3,
there are some games or situations
where the grand coalition cannot be formed
and can be maintained.
Propositions 6 says that
there are some games
where the grand coalition can be formed
but some core-allocations are refused.
Prop. 6

40. Prop. 9 Note A(N,v) of convex games (N,v)
Proposition 9 says that the allocation set is convex. in geometric sense.
Theorem 2 and its corollary say that
A and the core coincide if the game is convex.
Note

41. Th.2 & Cor. 3 Note A(N,v) of convex games (N,v)
Proposition 9 says that the allocation set is convex. in geometric sense.
Theorem 2 and its corollary say that
A and the core coincide if the game is convex.
Note

42. F.Y.I (strong) ε-core : Cε(N,v) least-core : LC(N,v)
That is, these mathematical results can be interpreted as this concluding remarks.
The least-core is the intersection
of all non-empty e-cores.
It can also be viewed
as the e-core for the smallest value
of that makes the set non-empty.

43. F.Y.I K(N,v) : kernel, K*(N,v) : pre-kernel,
N(N,v) : nucleolus, N*(N,v) : pre-nucleolus,
LC(N,v) : least-core
Note
That is, these mathematical results can be interpreted as this concluding remarks.

44. F.Y.I V(N,v) : stable set , M(N,v) : bargaining set
K(N,v) : kernel , N(N,v) : nucleolus
LC(N,v) : least-core
Note
That is, these mathematical results can be interpreted as this concluding remarks.

45. F.Y.I
Recall
Th.2 & Cor. 3
Note
That is, these mathematical results can be interpreted as this concluding remarks.

46. Example 1 2 4
super additive,
but not convex 3

47. Example 1 2 4
super additive,
but not convex 3

48. Example 1 2 4
v1: super additive,
but not convex,
v2: convex

49. Example 1 2 4
v1: super additive,
but not convex
Recall

50. These two notions coincide when (N,v) is convex or |N|<4.
Conclutions!
The existence or non-emptiness of C(N,v) - A(N,v)
suggests to us that
``incentives to form the grand coalition''
is a stronger notion than
``no incentive to split off from the grand coalition''.
That is, these mathematical results can be interpreted as this concluding remarks.
These two notions coincide
when (N,v) is convex or |N|<4.
Conjecture : A(N,v) ≠ ∅　⇒　A(N,v) ∩K(N,v) ≠ ∅
A(N,v) ≠ ∅　⇒　A(N,v) ∩K(N,v) = C(N,v) ∩K(N,v)