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)

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 ! !

“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.

“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”

Preliminaries (game) Notations

Preliminaries (core)

Preliminaries (core) Note

Example (core) 1 2 3 4

Example (core) 1 2 3 4

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

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

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

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

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.

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

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)

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.

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.

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.

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.

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!

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)

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!!

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.

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.

(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.

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)

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

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,

< + - 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

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.

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..

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..

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.

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.

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

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

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

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

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

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

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.

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.

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.

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

Example 1 2 4 super additive, but not convex 3

Example 1 2 4 super additive, but not convex 3

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

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

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)