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# 📢 ｜ Developed joint transportation matching technology to list combinations of transportation routes with high cooperation effect at high speed

### Developed joint transportation matching technology to list combinations of transportation routes with high cooperation effect at high speed

If you write the contents roughly
The fair cost burden is calculated based on the theory of cooperative games (* 6).

October 2021, 10 Japan Pallet Rental Co., Ltd. National University Corporation Gunma University Press release materials Transport with high cooperation effect ...　→ Continue reading

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# Cooperative game

Cooperative game(Kyoryoku game,British: cooperative game) IsGame theoryIn multipleplayerCoalition is a game when it is considered possible to act.Alliance actions in cooperative games are said to take place when the gains of each affiliated player are increased.

In order to carry out the alliance action, in advanceNegotiationAnd each other bindingagreementIs believed to be necessary.According to this idea, a research plan to explain cooperative games from non-cooperative games that negotiateNashIt's called a program.

## Mathematical definition

Cooperative games are a subset of every N S (Partnership) Is given by specifying the value.Mathematically, this game (Affiliated games) Is a finite set of playersAnd functions Defined by.this function isCharacteristic function Also called (characteristic function).A cooperative game is a set of players N and a set of characteristic functions v.Represented by.Characteristic functions are often used to express and analyze cooperative games, and v is sometimes called a game.

functionIt is,It is interpreted that the reward is associated with each of the alliances in.The value v (S) of the characteristic function for a certain affiliated S represents the best value that the player of S can obtain.TheAlliance valueCalled.NormallySuppose (do not reward partnerships where no one participates).

Also, contrary to the rewards in affiliated games,Cost function that associates the costs of each alliance in There is also a method of describing using.This is called a cost game.The value obtained by the cost function indicates the cost paid by the affiliated players.The concept in affiliated games can be easily rewritten to the concept in cost games.

### Duality

Is a function of the reward game. OfDual game(dual game) Cost game function The value of is defined as follows.

Intuitively, dual games are affiliated by not participating in the overall partnership N Ofopportunity costIt is thought to represent (opportunity cost).

Reward gameIs a cost game as wellIt is decided as a dual reward game.Cooperative games and their dual games are equivalent in some ways, and they share many qualities.For example, in a game and its dual gameコ アAre equal. (For more information about the duality of cooperative games, see (Bilbao 2000) checking. )

### Subgame

A partner game In Is a set of non-empty players. AtSubgameNaturally

Is determined.

In other words, focus on the alliances included in S.Subgames are defined for the overall alliance N Solution concept Is useful because it allows us to apply to partnerships smaller than N.

## Characteristic Function properties

Two A and BDisjoint() In the case of a tie-up, the value of the large tie-up between A and B is greater than or equal to the sum of the values ​​alone.That is,

if .

Superadditivity is a characteristic of characteristic functions, and (Owen 1995, P. 213) is assumed to be satisfied.

### Monotonic

The bigger the alliance, the bigger the reward: .

### The nature of simple games

"Simple game''(Simple game; Voting game) Is a cooperative game that takes only a gain (value) of 1 or 0. A tie-up with a gain of 1 is a "winning tie-up (winning tie-up)", and a tie-up with a gain of 0 is a "defeat tie-up (losing tie-up)". Called.Simple games are usually affiliated Defined as, in this case A tie-up that belongs to is regarded as a winning tie-up, and a tie-up that does not belong to is regarded as a defeat tie-up.It is often assumed that the simple game is non-empty and does not include the empty set.

• Simple game However, "monotonous" means that an alliance including a winning alliance always becomes a winning alliance.That is, And If It means that
• Simple game However, "proper" means that the complement of the winning tie-up (supplementary tie-up) is always a defeat tie-up.That is, If It means that
• Simple game However, "strong" means that the complement of the defeat tie-up (supplementary tie-up) is always a victory tie-up.That is, If It means that
• Simple game When is strong in proper, one tie-up wins and its supplementary tie-up loses.That is, Are equivalent. (Proper and strong simple game affiliated game Expressed in, any alliance about, Will be. )
• A "rejection player" in a simple game is a player who is an element (member) of any winning alliance.In other words, in a simple game in which there are denial players, alliances without denial players are always lost.Simple game "Weak" means that there is a denial player.That is, all winning alliancesIntersection Is to be non-empty.
• A "dictator" in a simple game is a denial player whose alliance, including that player, is a winning alliance.The dictator does not belong to the defeat alliance.
• Simple game "Career" is a set And any tie-up about, Is the same value.Players who do not belong to the career are ignored.When a simple game has a finite career (even Sometimes the simple game is "finite" (even if it is infinite).
• Simple game "Nakamura number"Is the smallest number of winning alliances where the intersection is an empty set.According to Nakamura's theorem, this number is an index to measure the degree of rationality, and it is known that even less than this number of options can be handled well.

The following are widely known about the relationship between the properties (axioms) of simple games (eg, Peleg, 2002, Section 2.1).[1]):

• A weak simple game is proper.
• Having a dictator in a simple game is equivalent to having it strong and weak.

More generally, in addition to the four traditional qualities of a simple game (monotonous, proper, strong, no veto player), finite or "computable"[2] The relationship between the six properties, including Kumabe and Mihara, 2011 has been completely elucidated.[3]), The results can be summarized in the table below "Existence of Simple Games".For example, when a "type" defined by a combination of four traditional properties includes an infinite game, it can be seen that the type includes both computable and non-computable ones.

Existence of simple games[4]
TypeUnlimited calculationLimited computableInfinite calculation impossibleInfinite computable
1111No.yesyesyes
1110No.yesNo.No.
1101No.yesyesyes
1100No.yesyesyes
1011No.yesyesyes
1010No.No.No.No.
1001No.yesyesyes
1000No.No.No.No.
0111No.yesyesyes
0110No.No.No.No.
0101No.yesyesyes
0100No.yesyesyes
0011No.yesyesyes
0010No.No.No.No.
0001No.yesyesyes
0000No.No.No.No.

The restrictions on the typical characteristics of simple games (monotonous, proper, strong, veto player-free, finite) have been fully elucidated.[5]..Especially when a simple game that can be calculated by an algorithm and has no veto player has a Nakamura number greater than 3, the simple game is known to be proper and not strong.

## Solution concept

Cooperative games describe rewards for partnerships.Players only participate in the alliance if they are more profitable than if they did not.Therefore, in order to find out what kind of alliance is actually formed, it is necessary to evaluate the relative power relationship between different alliances and the strength of different players within each alliance.It is an important purpose of cooperative games to think about how to distribute rewards to each player, and various solution concepts are presented for this purpose.

A central assumption in cooperative games is that a global alliance N is formed.Here we must work in a fair manner to distribute the v (N) obtained from the overall alliance to the players. (Even if an individual alliance or a small alliance is formed, this assumption is not limited because the concept of solution can be applied to the subgame if the subgame is obtained from the formed alliance.)

The solution concept shows the distribution obtained for each player Given by the vector.Various fairness standards have proposed multiple solution concepts.

### The nature of the concept of the solution

The concept of a solution may have several properties.Here are some properties that may appear in the concept of solutions.

Also, of the gain vectors, the overall rationality is satisfied.Semi-allocation (pre-imputation), overall rationalityIndividual rationalityWhat meetsAllocation Called (imputation).Most solution concepts give a distribution as a game solution.

• Efficiency ・ Overall rationality

The property that the gain vector of the solution distributes the alliance value of the whole alliance.That is,

Says that holds true.

The property that all players can get more gain than they can get by themselves.

Say that.

• Symmetry

Gain vectorIs a symmetrical player, Give equal gain to) Nature.hereSymmetrical playerIsA player who holds, That is.The symmetric solution concept makes no difference in gain for interchangeable players.

In a game consisting of the sum of two games, the property that the gain to the player is equal to the sum of the gains in each game (summed). Is a gameIs affiliatedSThe sum of the tie-up values ​​of each game is the tie-up value It is a game given as.The concept of an additive solution isFor all players Allocate the total value of the gains obtained in (XNUMX) as the gain.

• Nature of Null Player

The property that the gain given to Null Players is zero.Null playerIsPlayers who meetThat is.Economically, Null Players make zero contribution to any alliance that does not include themselves.

• Existence

Any game where the solution is based on the concept of solutionvAlso exists.

• Uniqueness

Any game where the solution is based on the concept of solutionvIs also the only one.

• Computational ease

The property that the concept of solution can be calculated efficiently.That is, the number of playersPolynomial time can be calculated with respect to.

### Stable set

Game "stable set" (von Neumann-Morgenstern solution) (von Neumann & Morgenstern 1944)) Is the first proposed solution for a game of 3 or more players.

#### Definition of stable set

A stable set is a set of distributions with these two properties.

• "Internal stability": One element of the stable set is the other element支配Not done.
• "External stability": Candidates outside the stable set are governed by at least one element of the stable set.

Von NeumannMorgensternSeen the stable set as a set of socially acceptable behaviors.None are clearly preferred over others, but none of the unacceptable behaviors have better alternatives.

This definition is so general that it is used in a wide variety of game formats.

#### Properties of stable sets

• Stable sets may or may not exist, (Lucas 1969) If present, it is typically not unique. (Lucas 1992). It is usually difficult to find a stable set.

This fact and other difficulties have led to the development of many other solution concepts.

• The positive fraction of cooperative games isコ アIt has a unique stable set consisting of. (Owen 1995, p. 240.).
• The positive fraction of cooperative games is It has a stable set that distinguishes between human players.At least such a stable set Eliminate discriminated players. (Owen 1995, p. 240.)

#### Domination of allocation

As a game,EachIt will be the distribution of. Alliance to meet (However,) Exists The ThedominateThat.

That is, at this time, the S playersThan the gain gained byPrefer the gain gained by If is used, it would threaten to leave the overall alliance.

### コ ア

#### Core definition

The "core" is a set of vectors that distribute rewards to players in a game, and satisfies the following conditions.

• "Efficiency": Players should make a "major alliance" (a partnership consisting of all players) and the total reward for each person should be equal to the value of the major alliance.
• "Strategic stability" or "equilibrium": No collaboration can betray a major collaboration.
(For example, no partnership can be greater than the total amount of compensation for each member. (Questionable))

here,If is a game,CoreIs a set of gain vectors as follows.

In other words, the core is a set of allocations that sets the total gains of the members of the Alliance S to be greater than or equal to the Alliance value v (S).In other words, if the gain is obtained by the gain vector of the core, there is no incentive to get out of the overall alliance N and obtain a large amount of gain in any alliance S.

Note that the core can be an empty set.

#### Core of simple games in preference profile

For simple games, a set of options When each player's preferences are defined above, there is a different concept of "core" than above. What is a "preference profile"? PreferenceList of (columns) That is.here Is "Individual Is the profile In, choices Choice "Prefer more".Simple game And preference profile When given"Dominant relationship" above Is defined as: Is a winning alliance (ie, ) Exists and all Is to be. "Preference profileSimple game about Ofコ ア' Is a relationshipA set of choices that are not governed by Set about The set of maximal elements above):

It is,BecomeIs equivalent to the absence of.

The "Nakamura number" of the simple game is the smallest number of winning alliances that have an empty set in common (although it may be possible to empty the intersection by collecting this number of winning alliances, less than this. You can never empty the intersection by collecting as many as you want).According to Nakamura's theorem, profiles of all non-circular preferences (as well as transitive preferences) Core Is non-empty is a set of choices Is finite and its concentration (number of elements) is It is equivalent to being smaller than the Nakamura number. According to a variant of that theorem by Kumabe and Mihara, any profile consisting of preferences with maximal elements Core Is non-empty because the cardinality of the choice set is It is equivalent to being smaller than the Nakamura number.Detail is"Nakamura number"reference.

### kernel

The kernel is one of the vectors that allocates rewards.

• Efficiency
• Individual rationality

Is satisfied.

## Cooperative game example

Multiple companiesLet's consider a consortium.eachCompany OfProfit,

And

Here, for exampleIndicates the profit when companies A and B cooperate.In this example, it can be said that "superadditivity" is always established.for example,( Is. ) If it is additive, it is better to partner with the wholegainBecomes larger.However, for individual companies, whether or not to partner depends on the distribution of gains.

Company when three companies jointlyGain eachAnd

As an example, the gain isConsider the case of.in this case,Therefore, companies A and B are affiliated with only two companies, and gainIt is advantageous to receive.Therefore, under this condition, companies A and B will refuse the alliance of the three companies including C.Affiliation with this situation AboutAllocationIs allocatedThe支配To do.

On the other hand, allocationIn the case of, neither of the two companies' alliance can increase the gains of all the companies that participated in the alliance.Only such allocationコ アBelong to.

## footnote

1. ^ Peleg, Bezalel (2002). Chapter 8 Game-theoretic analysis of voting in committees. 1.pp. 395–423. two:10.1016 / S1574-0110 (02) 80012-1. ISSN 15740110.
2. ^ The definition that a simple game is "computable" isResults similar to Rice's theoremSee.In particular, any finite game is computable.
3. ^ Kumabe, Masahiro; Mihara, H. Reiju (2011). “Computability of simple games: A complete investigation of the sixty-four possibilities”. Journal of Mathematical Economics 47 (2): 150–158. two:10.1016 / j.jmateco.2010.12.003. ISSN 03044068.
4. ^ Fixed Table 2011 in Kumabe and Mihara (1). There are 16 Type Is determined by four traditional qualities: monotonous, proper, strong, and no veto player.For example, type 1110 refers to simple games that are monotonous (1), proper (1), strong (1) with veto players (0).That line is the absence of finite and uncomputable things in the type 1110 game, the existence of finite and computable things, the absence of infinite and uncomputable things, and the absence of infinite and computable things. Show that it is.
5. ^ Kumabe, Masahiro; Mihara, H. Reiju (2008). “The Nakamura numbers for computable simple games”. Social Choice and Welfare 31 (4): 621–640. two:10.1007 / s00355-008-0300-5. ISSN 0176-1714 .