**4. Generation of carbenes**

The synthesis of carbene molecules is generally based on elimination and fragmentation reactions. Carbenes are formed as intermediate products when groups attached to the carbon atom are broken as a result of photolysis, thermolysis or reaction with metals.

#### **4.1 Fragmentation reactions**

Diazo compounds, tosylhydrazones, ketenes, three-membered carbon rings, strained alkenes and heterocyclic compounds are generally used as carbene precursors in fragmentation reactions.

#### *4.1.1 Diazo compounds*

Diazo compounds (RR'C=N2), constitute a principal class of carbene precursors, known since the first preparation of ethyl diazoacetate by Curtius in 1883 [24]. Diazo compounds, which have 1,3-dipolar structures (**Figure 5a**), are generally converted to related carbenes by easily removing nitrogen gas when heated or photolyzed in aprotic solvents as shown in **Figure 5b**. This conversion has been proven by spectroscopic methods. The electronic spectrum of methylene produced in the gas phase by flash photolysis of diazomethane was recorded by Herzberg and Shoosmith [25]. Various metal complexes are used as catalyst for carbene formation to occur at low temperatures. Since diazo compounds are generally unstable and easily degradable compounds, they must be synthesized before each reaction.

#### **Figure 5.**

*(a) Resonance structure of 1,3-dipolar diazo compound. (b) Generation of carbene from diazo compounds.*

Diazo compounds, which have a unique reactivity due to their 1,3-dipole and ylide structures, are useful synthetic products in organic synthesis. However, their large-scale use has been avoided due to their toxicity and unpredictable explosive behavior [26].

#### *4.1.2 Tosylhydrazones*

Where the diazo compound is somehow unstable and dangerous to use, it is usually better to use a diazo precursor. The simplest and most common compounds used for this are hydrazones. Ketones and aldehydes easily react with hydrazine to form hydrazone compounds. The oxidation of hydrazones with metal salts such as Ag2O, HgO, MnO2, Pb(OAc)4 gives the diazo compound (**Figure 6**).

The most widely known and used carbene precursors are tosylhydrazones which are prepared from the reaction of aldehydes and ketones with p-toluenesulphonyl hydrazide.

*Carbene*

**Figure 6.** *Generation of carbenes from hydrazones compound.*

p-Toluenesulpnonylhydrazones (tosylhydrazones) of aldehydes and ketones undergo base-catalyzed thermal decomposition with loss of p-toluenesulfinate to give intermediate diazo compounds [27]. This method is called the Bamford-Stevens reaction (**Figure 7**).

**Figure 7.** *Bamford-Stevens reaction.*

When the N-H proton in the tosylhydrazone molecule is removed with a base such as NaH or NaOCH3, the formed anion is stabilized by the tosyl group. Therefore, the resulting salt can be isolated and stored for a long time. This salt can be converted to carbene by heating at any time in situ. After the nitrogen gas is separated, alkene formation is observed as a result of 1,2-hydrogen shift. When the alkene formation mechanism was examined comprehensively, it was found that the products formed may vary depending on the reaction conditions [28]. In the Bamford-Stevens reaction, if the decomposition of the tosylhydrazone salt is done in aprotic solvents, carbene is formed by the removal of nitrogen gas. The resulting carben usually forms an alkene as a result of 1,2-hydrogen shift. However, if the salt is decomposed in protic solvents, the diazonium salt is formed first. Diazonium salts, especially the aliphatic ones, are not stable, they turn into carbenium ions by removing nitrogen gas.

*Basic Information about Carbenes DOI: http://dx.doi.org/10.5772/intechopen.100425*

#### *4.1.3 Ketenes*

Ketenes can eliminate CO molecule on thermolysis or photolysis to generate carben. Since ketenes are not readily available precursor and polymerize under the reaction conditions, they are not widely used. Ketene has been used extensively to generate CH2.

$$\mathbf{c}\mathbf{^{\mathcal{R}}\_{\mathsf{B}}} \xrightarrow[\mathsf{GC}\mathbf{^{\mathcal{R}}}]{} \mathbf{^{\bullet}\_{\mathsf{F}}} \mathbf{^{\bullet}\_{\mathsf{F}}} \mathbf{^{\bullet}\_{\mathsf{F}}} \mathbf{^{\bullet}\_{\mathsf{B}}}$$

#### *4.1.4 Three-membered carbon ring*

Three-membered rings that have a high ground state energy due to steric strain often decompose to give carbene intermediates by heating or irradiation. For example; photolysis of 1,1-dichloro-2-phenylcyclopropane apparently gives CCl2 and photolytic decomposition of oxiranes yield arylcarbenes and related species as shown in **Figure 8**.

Substituted phenyloxiranes provide convenient precursors for substituted arylcarbenes, such as diphenylcarbene [29], phenylmethylcarbene [30], cyanophenylcarbene [31, 32], and methoxycarbonylphenylcarbene [33]. Selective cleavage was observed with unsymmetrical precursors, the thermodynamically more stable isomer appears to be favored.

#### **Figure 8.**

*Generation of carbene from cyclopropanes and epoxides.*

The most useful carbene precursors of the three-membered ring are diazirines. Diazirines, the cyclic isomers of diazoalkanes, also decompose under the influence of heat and light to give carbenes [34–37]. As shown in **Figure 9**, they are prepared from ketones by reaction with ammonia and chloramine followed by oxidation of the resulting diaziridine. This method is quite widely used especially for halocarbenes.

#### **Figure 9.** *Preparation and decomposition of diazirines.*

#### *4.1.5 Strained alkenes*

If the alkene is extremely sterically hindered then the π-bond is weakend due to the substantially reduced p–p overlap and distortion from planarity. As a result, the ground state energy is raised and then dissociation to carbenes become possible by heating. The well-known example of this process is the reversible dissociation of tetranaphth-1-ylethene into bis(naphth-1-yl) carbene at 250°C [38] (**Figure 10**).

**Figure 10.** *Thermal dissociation of tetranaphth-1-ylethene.*

#### *4.1.6 Heterocycles*

Various five-membered heterocyclic compounds decompose to give carbenes by heating or irradiation (**Figure 11**). The decomposition of 1,5-dihydro-1,3,4 oxadiazoles give carbenes at about 80°C with loss of nitrogen followed by the carbonyl fragment.

**Figure 11.** *Fragmentation of five-membered heterocycles to carbenes.*

## **4.2 α-Elimination**

The hydrolysis of chloroform in basic medium was probably the first reaction in which intermediate carbene formation was suggested by Geuther [39]. The investigation the mechanism of this reaction by Hine and co-workers initiated the modern era of carben chemistry in the early 1950s [40–42]. In the basic environment, the acidic proton of chloroform separates and trichloromethyl anion is formed.

*Basic Information about Carbenes DOI: http://dx.doi.org/10.5772/intechopen.100425*

Dichlorocarbene is formed as a result of the removal of the chloride anion from the carbanion. By hydrolysis of dichlorocarbene in aqueous media, carbon monoxide is formed (**Figure 12**). These types of reactions are called α-elimination reactions because the hydrogen and chlorine released in the formation of carbene are attached to the same carbon atom.

$$\begin{array}{ccccccccc} \text{Cl}^{-} & \text{Cl}^{-} & & & \text{Cl}^{2} \\ & & & \text{Cl}^{2} & & \\ \text{Cl}^{2} & & & \text{Cl}^{2} & & \\ & & & \text{Cl}^{2} & & \\ & & & & \text{Cl}^{2} & \\ \end{array} \quad \begin{array}{ccccccccc} \text{Cl}^{-} & & & \text{Cl}^{2} & & \\ & & & \text{Cl}^{2} & & \\ & & & \text{Cl}^{2} & & \\ & & & \text{Cl}^{2} & & \\ & & & \text{Cl}^{2} & & \\ \end{array} \quad \begin{array}{ccccccccc} \text{Cl}^{-} & & & \text{Cl}^{2} & & \\ & & & \text{Cl}^{2} & & \\ & & & \text{Cl}^{2} & & \\ & & & \text{Cl}^{2} & & \\ \end{array}$$

**Figure 12.** *α-Elimination reactions.*

In experiments with aprotic solvents and strong bases, it was determined that carbene was formed and added to the double bond electrons to form cyclopropane derivatives as shown in **Figure 13**. This reaction is the best method for cyclopropane synthesis in organic chemistry.

$$\text{3.3.} \xrightarrow[]{\text{C}} \text{ \downarrow\text{C}}^{\text{\downarrow\text{C}}} \text{ \downarrow\text{C}}^{\text{\downarrow\text{C}}} \xrightarrow[]{\text{A}} \text{8.08}^{\text{\downarrow\text{C}}} \text{ + } \text{H}\_{\text{C}}^{\text{\downarrow\text{C}}}$$

**Figure 13.** *Cyclopropane synthesis.*

Before the reaction, the solvent used must be dried very well. Because the water in the environment can easily react with the carbene formed, and it also reacts with the base in the environment and completely prevents the formation of carbenes. However, in Makosza, a study carried out in a two-phase system, showed that carbenes were formed in the presence of phase transfer catalysts in aqueous medium and added to double bonds (**Figure 14**) [43].

**Figure 14.** *Carbene formation in a two-phase system.*

Since the reaction is between two phases, effective mixing is very important for this reaction. Although the reaction is carried out in the presence of water, the carbene is generated and reacts in the organic phase.

#### *4.2.1 Simmon-Smith reaction*

It is not possible to synthesize methylene carbene by the α-elimination method. However, there are methods that form methylene carbene adducts. The most commonly used method is the reaction of alkenes with diiodomethane in the presence of zinc. As a result of the reaction, cyclopropane compounds are formed (**Figure 15**) [44, 45].

**Figure 15.** *Simmon-Smith reaction.*

During this reaction, methylene carbene is not formed in the free form. First, zinc and diiodomethane react to form a carbenoid intermediate, which acts as a carbene. Later, since this product is unstable, it transforms into zinc iodide by transferring the methylene group to the double bond as shown in **Figure 16** [46].

**Figure 16.** *Mechanism of Simmon-Smith reaction.*

#### **5. Carbene reactions**

Singlet and triplet carbene exhibit different reactivity. Singlet carbenes generally participate in reactions as either electrophiles or nucleophiles. Singlet carbenes which have unfilled p-orbitals should be electrophilic. Triplet carbenes can be considered as diradicals and participate in stepwise radical additions. Triplet carbenes must pass through an intermediate with two unpaired electrons, while single carbenes can react in a single concerted step. Because carbenes have two modes of reactivity, singlet methylene gives stereospecific reactions where as triplet methylene gives stereoselective reactions [47].

Carbenes are highly reactive intermediate due to electron deficiency. Carbenes react instantly in various ways in the environment where they are formed. Carbene reactions are classified under four main groups.

#### **5.1 Cycloaddition reaction of carbenes**

Since carbenes generally react electrophilically, they give a [2 + 2] cycloaddition reaction with double bonds to form cyclopropane compounds (**Figure 17**). This

*Basic Information about Carbenes DOI: http://dx.doi.org/10.5772/intechopen.100425*

**Figure 17.** *Cycloaddition reaction of carbenes.*

method, the most characteristic reaction of carbene intermediates, has now been widely used as a synthetic route to cyclopropane since 1954 [6].

According to the theory proposed by Skell and Woodworth [47], singlet carbenes are added the double bonds simultaneously in a single step. The opening of double bond electrons and the formation of new carbon–carbon bonds occur simultaneously. On the other hand, triplet carbenes are not added in double bonds in a single step because this addition is forbidden by orbital symmetry. Triplet carbenes form cyclopropane compounds by adding double bonds as a result of a multi-step reaction.

Since carbenes are generally electrophilic compounds, they prefer double bonds where the electron density is high when adding systems that contain more than one double bond. In particular, singlet carbenes (dihalocarbenes) show more regioselective properties as they are more stable. For example, in the isotetralin compound, which has two different double bonds, dihalocarbene selectively adds the central double bond, although it has a steric hindrance. However, the more reactive ethoxycarbonylcarbene cannot act selectively due to its shorter lifetime and adds the double bonds located on the outer part of the molecule as shown below.

#### **5.2 Dimerization reaction of carbenes**

When carbenes cannot find a reagent that can react in the environment in which they are formed, they dimerize to fill the electron gap in their outer orbitals and turn into olefins (**Figure 18**).

**Figure 18.** *Dimerization reaction of carbenes.*

#### **5.3 Insertion reaction of carbenes**

Another characteristic reaction that carbenes give to become stable is that carbenes insert between C-H or C-C bonds (**Figure 19**).

**Figure 19.** *Examples of carbene insertion reaction.*

Methylene, when there is no double bond to react it is not selective and insert into a random C-H bond in the liquid phase. Methylene acts selectively in the gas phase and preferentially inserts into tertiary C-H bonds as shown in **Figure 20**.

**Figure 20.** *Insertion of methylene.*

The electronic structure of carbene plays an important role in the insertion reactions (**Figure 21**). Singlet carbenes generally insert into bonds in a single step with retention of configuration. The situation is different with triplet carbenes. Since triplet carbenes act as radicals, they first abstract hydrogen from the C-H bond and form two new radicals. With the combination of these radicals, a new C-C bond is formed. Meanwhile, as the radical configuration undergoes isomerization, a racemic mixture is formed as a result of the reaction. In the insertion of triplet carbenes, tertiary carbon-hydrogen bonds are primarily preferred because they form stable radicals.

**Figure 21.** *Mechanism of singlet and triplet carbene in the insertion reaction.*


Z t bðÞ¼ P1 C1 þ F~ 5 � � þ P1 BΤ 3 þ D2 � �P1 þ P1 BΤ 3 þ D2 � �P2 h iX t b ð Þ þ P1 ~~ D2 þ D2 � �Z t bð Þþ P1 D3 þ D~ 2 � �Z t ~ ð Þ, Z t b~ ðÞ¼ P1 ~ C1 þ ~ C2 � � þ P1 B~ Τ 1 þ D3 � �P1; þP2 ~ C1 þ ~ C2 � �; þP2 B~ Τ 1 þ D3 � �P1 � þ P1 B~ Τ 1 þ D3 � �P2 þ P2 B~ Τ 1 þ D3 � �P2 � X t b ð Þ þ ð Þ P1 þ P2 DΤ 3 þ D3 � �Z t bð Þþ ð Þ P1 þ P2 F~ 3 þ D~ 3 � �Z t ~ ð Þ: (75)

Supposing that (I2 denotes the 2 � 2 unit matrix)

$$\begin{aligned} \text{(A2.2)} \quad \bar{\hat{N}}\_2^{-1} &:= \left[ I\_2 - \mathcal{P}\_1 \left( \bar{\tilde{\mathcal{D}}}\_2 + \overline{\overline{\mathcal{D}}}\_2 \right) \right]^{-1} \text{ and} \\ \bar{\hat{\mathcal{N}}}\_2^{-1} &:= \left[ I\_2 - (\mathcal{P}\_1 + \mathcal{P}\_2) \left( \mathcal{D}\_3^{\top} + \overline{\overline{\mathcal{D}}}\_3 \right) \bar{\hat{N}}\_2^{-1} \mathcal{P}\_1 \left( \mathcal{D}\_3 + \overline{\overline{\mathcal{D}}}\_2 \right) - (\mathcal{P}\_1 + \mathcal{P}\_2) \left( \overline{\overline{\mathcal{F}}}\_3 + \overline{\overline{\mathcal{D}}}\_3 \right) \right]^{-1} \text{ exist}, \end{aligned} \tag{76}$$

we get

$$
\hat{Z}(t) = \Sigma\_0(\mathcal{P}\_1, \mathcal{P}\_2) \hat{X}(t), \tilde{Z}(t) = \tilde{\Sigma}\_0(\mathcal{P}\_1, \mathcal{P}\_2) \hat{X}(t), \tag{77}
$$

we get

where.

8

>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>><

>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>:

8

>>>>>>>>>>>>>>>>>>>><

>>>>>>>>>>>>>>>>>>>>:

Σ1ð Þ P1;P2 ≔N �1

Σ~1ð Þ P1;P2 ≔N �1

Σ2ð Þ P1;P2 ≔N �1

þ D3N �1

þ P1~

þP1DΤ 3 N �1 2 P1 h F~ 5 þ BΤ

Σ~2ð Þ P1;P2 ≔N �1

þ P2 B~ Τ

þP1DΤ 3 N �1 2 P1 h F~ 5 þ BΤ

2 P2 ~

�

C2 þ P1D

2 P1 C1 þ BΤ

2 P1 C1 þ B~ Τ

2 P1 F~ 5 þ BΤ

n

C1 þ ~ C2 � � þ P2 B~ Τ

> Τ 3 þ P2 DΤ

2 P2 ~

�

1 þ D3 � �P2 þ P1D

0 ¼ P\_ 1 þ P1A1 þ AΤ

0 ¼ P\_ 2 þ P2 A1 þ A2

þ ~ C Τ 1 þ P1B~ 1

þ ~ C Τ

þ P2 F~ 1 þ B2

C1 þ ~ C2 � � þ P2 B~ Τ

3P1 þ D3P1 C1 þ B~ Τ

3 N �1

1 þ D3 � �P1 þ P1 B~ Τ

� � � �P1D2Σ0ð Þþ P1;P2 P1D~ Τ

� � � �,

1 P1 þ DΤ

3 þ D3

Τ 3 þ P2 DΤ

� � þ A1 þ A2

� �P2 þ P1A2 þ A

� �Σ~2ð Þþ P1;P2 F~ Τ

2 þ P1B~ 2 þ P2 B~ 1 þ B~ 2

� �Τ

Τ 2

� � � �Σ~0ð Þ P1; P2 , P1ð Þ¼ T G2, P2ð Þ¼ T 0:

Z tðÞ¼ Σ1ð Þ P1;P2 X tð Þþ Σ2ð Þ P1;P2 Xb ð Þt , Z t ~ ðÞ¼ Σ~1ð Þ P1;P2 X tð Þþ Σ~2ð Þ P1;P2 X t bð Þ, (81)

� � � �,

3 P1

3 P2 þ D2ð Þþ P1 þ P2 D2Σ0ðP1;P2Þ þ D~ 2Σ~0ðP1;P2Þ

3 P2 þ D2ð Þþ P1 þ P2 D2Σ0ðP1;P2Þ þ D~ 2Σ~0ðP1;P2Þ

� � � �P1D2Σ0ð Þþ P1;P2 P1D~ Τ

3 P2 þ D2ð Þþ P1 þ P2 D2Σ0ðP1;P2Þ þ D~ 2Σ~0ðP1;P2Þ

Comparing the coefficients of dt in Eq. (73) and putting Eqs.  (77) into Eq. (74), we have

Z tðÞ¼ P1C1 þ P1BΤ 3P1 � �X tð Þþ P1 F~ 5 þ BΤ 3 P2; þD2ð Þ P1 þ P2 ; þD2Σ0ð Þ P1;P2 h þ D~ 2Σ~0ð Þ P1;P2 i X t b ðÞþ P1 ~~ D2Z tð Þþ P1D3Z t ~ ð Þ, Z t ~ ðÞ¼ P1 ~ C1 þ P1B~ Τ 1P1 � �X tð Þþ P2 ~ C1 þ ~ C2 � � þ P1~ C2 þ P2 B~ Τ 1 þ D3 � �P1 þ P1 B~ Τ 1 þ D3 � �P2 � þP1D3P1 þ P2 B~ Τ 1 þ D3 � �P2 þ P1D Τ 3 þ P2 DΤ 3 þ D3 � � � �P1D2Σ0ð Þ P1;P2 þ P1D~ Τ 3 þ P2 F~ 3 þ D~ 3 � � � �P1D~ 2Σ~0ðP1;P2Þ � X t b ð Þþ P1DΤ 3 Z tð Þþ P1F~ 3Z t ~ ð Þ: (79)

Supposing that

$$\begin{aligned} \text{(A2.3) } \overline{\mathcal{N}\_{2}}^{-1} & \coloneqq \left(I\_{2} - \mathcal{P}\_{1}\bar{\tilde{\mathcal{D}}}\_{2}\right)^{-1} \coloneqq \left[I\_{2n} + \mathcal{P}\_{1}\mathcal{D}\_{2}N\_{2}^{-1}\mathcal{D}\_{2}^{T}\right]^{-1} \\\\ \text{and} \quad \overline{\mathcal{N}\_{2}}^{-1} & \coloneqq \left(I\_{2} - \mathcal{P}\_{1}\mathcal{D}\_{3}^{T}\overline{\mathcal{N}\_{2}}^{-1}\mathcal{P}\_{1}\mathcal{D}\_{3} - \mathcal{P}\_{1}\overline{\tilde{\mathcal{F}}}\_{3}\right)^{-1} \coloneqq \left[I\_{2n} - \mathcal{P}\_{1}\left(\bar{\tilde{\mathcal{D}}}\_{1} - \mathcal{D}\_{2}N\_{2}^{-1}\bar{\tilde{\mathcal{D}}}\_{2}\right)^{T}\right]^{T} \\\\ & \qquad \times \left[I\_{2n} + \mathcal{P}\_{1}\mathcal{D}\_{2}N\_{2}^{-1}\mathcal{D}\_{2}^{T}\right]^{-1}\mathcal{P}\_{1}\left(\bar{\tilde{\mathcal{D}}}\_{1} - \mathcal{D}\_{2}N\_{2}^{-1}\bar{\tilde{\mathcal{D}}}\_{2}\right)^{T} - \mathcal{P}\_{1}\left(\bar{\tilde{\mathcal{F}}}\_{3} - \bar{\mathcal{D}}\_{2}N\_{2}^{-1}\bar{\tilde{\mathcal{D}}}\_{2}^{T}\right)\right]^{-1} \quad \text{exist}, \end{aligned} \tag{80}$$

we get

Supposing that (I2 denotes the 2 � 2 unit matrix)

2 ≔ I2 � ð Þ P1 þ P2 DΤ

Z t

2 P1 D3 þ D~ 2 � � ~~

P1 C1 þ F~ 5 þ BΤ

2 ð Þ P1 þ P2 ~

D2 þ D2 h i � � �1

and

N~ �1

2 P1 D3 þ D~ 2 � �

~ ðÞ¼ Σ~0ð Þ P1;P2 X t

1 þ D3 � �

:

3 P2; þD2ð Þ P1 þ P2 ; þD2Σ0ð Þ P1;P2

C2 þ P2 B~ Τ

3 þ D3

D1 � D2N�1

� �Τ "

2 D~ Τ 2

> 2 D~ Τ 2

� ���1

� P1 F~ 3 � D~ 2N�1

1 þ D3 � �

b ð Þ, Z t

C1 þ ~

3 þ D2 � �ð Þ P1 þ P2 h i�

C2 þ B~ Τ

ð Þ P1 þ P2

~ ð Þ,

þ P1~

3 þ P2 DΤ

� X t b ð Þþ P1DΤ

≔ I2n � P1 ~~

2 D~ Τ 2

� � � �

Τ

þ P1 C1 þ F~ 5 þ BΤ

� � � ��1

� ð Þ P1 þ P2 F~ 3 þ D~ 3

ð Þþ P1 þ P2 DΤ

3 þ D2 � �ð Þ P1 þ P2 h i� exist,

N~ �1 2

bð Þ, (77)

3 þ D3 � �

,

P1 þ P1 B~ Τ

P1D2Σ0ð Þ P1;P2

3 Z tð Þþ P1F~ 3Z t

1 þ D3 � �

~ ð Þ: (79)

exist,

(80)

P2

(76)

(78)

3 þ D3 � �

bðÞ¼ Σ0ð Þ P1;P2 X t

N �1

C2 þ B~ Τ

2 P1 C1 þ F~ 5 þ BΤ

h

X tð Þþ P2 ~

1 þ D3 � �

�

� �

3 þ D2 � �ð Þ P1 þ P2 h i�

C1 þ ~

�

N~ �1

2 ð Þ P1 þ P2 ~

1 þ D3 � �

D2Z tð Þþ P1D3Z t

C1 þ ~ C2 � �

P2 þ P1D

≔ I2n þ P1D2N�1

2 P1D3 � P1F~ 3

� ��1

P1 ~~

P1D~ 2Σ~0ðP1;P2Þ

2 DΤ 2

D1 � D2N�1

� �

2 ≔ I2 � P1 ~~

114 Game Theory - Applications in Logistics and Economy

ð Þ A2:2 N~ �1

we get

where

8

>>>>>>>>>>>><

>>>>>>>>>>>>:

Z t ~ ðÞ¼ P1 ~

~~ N �1

Σ0ð Þ P1;P2 ≔N~ �1

Σ~0ð Þ P1;P2 ≔ ~~

Z tðÞ¼ P1C1 þ P1BΤ

N �1

þ DΤ 3 þ D3 � �

þ D~ 2Σ~0ð Þ P1;P2

C1 þ P1B~ Τ

þP1D3P1 þ P2 B~ Τ

2 ≔ I2 � P1 ~~

2 ≔ I2 � P1DΤ

þ P1D~ Τ

Supposing that

ð Þ A2:3 N �1

and N �1

� �

Inserting Eq.  (73) and putting Eqs. (77) and (81) into them, we get

0 ¼ P\_ 1 þ P1A1 þ AΤ 1 P1 þ P1F~ 1P1 þ Q2 þ ð Þ C1 þ P1B3 Σ1ð Þþ P1;P2 ~ C Τ 1 þ P1B~ 1 � �Σ~1ð Þ P1;P2 , 0 ¼ P\_ 2 þ P2 A1 þ A2 � � þ A1 þ A2 � �Τ P2 þ P2 F~ 1 þ B2 � �P1 þ P1 F~ 1 þ B2 � �P2 þ P2 F~ 1 þ B2 � �P2 þ P1A2 þ A Τ 2 P1 þ P1B2P1 þ F~ 4 þ ð Þ C1 þ P1B3 Σ2ð Þ P1;P2 þ ~ C Τ 1 þ P1B~ 1 � �Σ~2ð Þþ P1;P2 F~ Τ 5 þ P1B2 þ P2 B3 þ B2 � � � �Σ0ð Þ P1;P2 þ ~ C Τ 2 þ P1B~ 2 þ P2 B~ 1 þ B~ 2 � � � �Σ~0ð Þ P1; P2 , P1ð Þ¼ T G2, P2ð Þ¼ T 0: 8 >>>>>>>>>>>>>>>>>>>>< >>>>>>>>>>>>>>>>>>>>: (83)

Note that the system of Riccati equations (83) is not standard, and its solvability is open. Due to some technical reason, we can not obtain the solvability of it now. However, in some special case, P1ð Þt and P2ð Þt are not coupled. Then we can first solve the first equation of P1ð Þt , then that of P2ð Þt by standard Riccati equation theory. We will not discuss for the space limit. And we will consider the general solvability of Eq. (83) in the future.

Instituting Eqs.  Define Y;Z;Z~ � � by Eqs. (71) and (81),

1φb∗

2ð Þþ t 0 BΤ

4 þ B2 þ B~2 � �Τ

Remark 3.3 When we consider the complete information case, that is, W~ ð Þ� disappears and

In this chapter, we have studied a leader-follower stochastic differential game with asymmetric information. This kind of game problem possesses several attractive features. First, the game problem has the Stackelberg feature, which means the two players play as different roles during the game. Thus the usual approach to deal with game problems, such as [6–8, 10], where the two players act as equivalent roles, does not apply. Second, the game problem has the asymmetric information between the two players, which was not considered in [3, 13, 14]. In detail, the information available to the follower is based on some sub-σ-algebra of that available to the leader. Stochastic filtering technique is introduced to compute the optimal filtering estimates for the corresponding adjoint processes, which act as the solution to some FBSDFE. Third, the Stackelberg equilibrium is represented in its state feedback form for the LQ problem under some appropriate assumptions. Some new conditional mean-field FBSDEs and

� �Z t ~ ð Þ h i

ð Þþ t D~ Τ

� �ð Þþ P1 þ P2 0 D~ Τ

1

1 β∗ð Þt

� �Y t bð Þþ 0 D~ Τ

2 given by Eq. (84) is a feedback optimal control of the leader.

Stochastic Leader-Follower Differential Game with Asymmetric Information

1

� �Τ

� X t b ð Þ,

http://dx.doi.org/10.5772/intechopen.75413

Σ0ðP1;P2Þ

(87)

117

ð Þþ P1 þ P2 D2 þ F~ 2

1 � �Σ~0ðP1;P2Þ

1 of the follower can also be represented in a nonanticipating way.

respectively. Then Eq. (69) holds, and u∗

In fact, by Eq. (47), noting Eqs. (68), (71), and (77), we have

ðÞþ t S t ~ð Þub

Σ~0ðP1;P2Þ� þ 0 BΤ

� �X t bð Þþ S t ~ð Þub

∗ 2ðÞþ t BΤ

� �

∗

2 ½F~ Τ

1

G1,t ¼ Ft, Theorems 3.1 and 3.2 coincide with Theorems 2.3 and 3.3 in Yong [13].

system of Riccati equations are introduced to deal with the leader's LQ problem.

Riccati equations. These challenging topics will be considered in our future work.

In principle, Theorems 3.1 and 3.2 provide a useful tool to seek Stackelberg equilibrium. As a first step in this direction, we apply our results to the LQ problem to obtain explicit solutions. We hope to return to the more general case in our future research. It is worthy to study the closed-loop Stackelberg equilibrium for our problem, as well as the solvability of the system of

Finally, the optimal control u∗

1 ð Þt S~Τ

1 ð Þt S~ Τ

1 ð Þt S~ Τ

which is observable for the follower.

�

1 ð Þt bx∗

1 ð Þt 0

1 ð Þt 0 � � � S t ~ð ÞN�1

u∗

1ðÞ¼� t N~ �1

¼ �N~ �1

¼ �N~ �1

þ D~ 2 þ ~~ D2 � �Τ

4. Concluding remarks

We summarize the above analysis in the following theorem.

Theorem 3.2 Let (A2.1)�(A2.3) hold, ð Þ P1ð Þt ; P2ð Þt satisfy Eq. (83), X be the b G1,t-adapted solution to Eq. (86), and X be the Ft-adapted solution to Eq. (85). Define Y;Z;Z~ � � by Eqs. (71) and (81), respectively. Then Eq. (69) holds, and u∗ 2 given by Eq. (84) is a feedback optimal control of the leader.

Finally, the optimal control u∗ 1 of the follower can also be represented in a nonanticipating way. In fact, by Eq. (47), noting Eqs. (68), (71), and (77), we have

u∗ 1ðÞ¼� t N~ �1 1 ð Þt S~Τ 1 ð Þt bx∗ ðÞþ t S t ~ð Þub ∗ 2ðÞþ t BΤ 1φb∗ ð Þþ t D~ Τ 1 β∗ð Þt � � ¼ �N~ �1 1 ð Þt S~ Τ 1 ð Þt 0 � �X t bð Þþ S t ~ð Þub ∗ 2ð Þþ t 0 BΤ 1 � �Y t bð Þþ 0 D~ Τ 1 � �Z t ~ ð Þ h i ¼ �N~ �1 1 ð Þt S~ Τ 1 ð Þt 0 � � � S t ~ð ÞN�1 2 ½F~ Τ 4 þ B2 þ B~2 � �Τ ð Þþ P1 þ P2 D2 þ F~ 2 � �Τ Σ0ðP1;P2Þ � þ D~ 2 þ ~~ D2 � �Τ Σ~0ðP1;P2Þ� þ 0 BΤ 1 � �ð Þþ P1 þ P2 0 D~ Τ 1 � �Σ~0ðP1;P2Þ � X t b ð Þ, (87)

which is observable for the follower.

Note that the system of Riccati equations (83) is not standard, and its solvability is open. Due to some technical reason, we can not obtain the solvability of it now. However, in some special case, P1ð Þt and P2ð Þt are not coupled. Then we can first solve the first equation of P1ð Þt , then that of P2ð Þt by standard Riccati equation theory. We will not discuss for the space limit. And

2Σ~1ð Þ P1;P2

Τ of the leader satisfies

X tð Þþ h

3 Σ1ð Þþ P1; P2 F~ 3Σ~1ðP1; P2Þ

X tð Þþ h F~ Τ 4 þ BΤ

DΤ

2Σ~0ð Þ P1;P2

2Σ~2ð Þþ P1;P2 ~~

i X tð Þþ h

F~ 5 þ B~ Τ 3

3Σ0ðP1;P2Þ þ F~ 3Σ~2ðP1;P2Þ þ D~ 3Σ~0ðP1; P2Þ

i X tð Þþ h ~ C2 þ B~ Τ

ð Þþ P1 þ P2 B3Σ1ðP1;P2Þ þ B~ 1Σ~1ðP1;P2Þ

2 P2 þ B~ Τ

i X t bð Þ,

A2 þ F~ 1P2 þ B2ð Þ P1 þ P2

1P2 þ D3ð Þ P1 þ P2

i X t b ð Þ o dt

> i X t b ð Þ � dW t ~ ð Þ,

i X t bð Þdt

> � X t bð ÞdW t ~ ð Þ,

3 Σ1ðP1;P2Þ þ F~ 3Σ~1ðP1;P2Þ

P2 þ D2ð Þ P1 þ P2

2 ð Þ P1 þ P2

(84)

(85)

(86)

we will consider the general solvability of Eq. (83) in the future.

2Σ1ð Þþ P1;P2 D~ Τ

A1 þ F~ 1P1 þ B3Σ1ð Þþ P1; P2 B~ 1Σ~1ðP1;P2Þ

D2Σ1ð Þ P1;P2

� �

1 þ D3 � �

þB3Σ2ð Þþ P1;P2 B2Σ0ðP1; P2Þ þ B~ 1Σ~2ðP1;P2Þ þ B~ 2Σ~0ðP1;P2Þ

ð Þþ P1 þ P2 DΤ

3Σ0ðP1;P2Þ þ F~ 3Σ~2ðP1;P2Þ þ D~ 3Σ~0ðP1; P2Þ

h i

2Σ0ð Þþ P1;P2 D~ Τ

þB3Σ2ð Þþ P1;P2 B2Σ0ðP1;P2Þ þ B~ 1Σ~2ðP1;P2Þ þ B~ 2Σ~0ðP1; P2Þ

i X t b ð Þ � dW tð Þ

Instituting Eqs. (77) and (81) into Eq. (68), we obtain

3 P1 þ ~~

D2Σ2ð Þþ P1;P2 D2Σ~0ðP1;P2Þ

P1 þ DΤ

Τ

3Σ2ð Þþ P1;P2 D

b ðÞ¼ A1 þ A2 þ F~ 1 þ B2

C2 þ B~ Τ

Τ

We summarize the above analysis in the following theorem.

h

3Σ2ð Þþ P1;P2 D

þ ~ C1 þ ~

�

h i

2P1 þ DΤ

2 Σ2ð Þþ P1;P2 F~ Τ

u∗

dX tðÞ¼

8

>>>>>>>>>>>>>>>>>>>>>>>><

>>>>>>>>>>>>>>>>>>>>>>>>:

2ðÞ¼� t N�1

þ DΤ

h

�

þ~~

�

þ h ~ C1 þ B~ Τ 1

Xð Þ¼ 0 X0,

8

>>>>>>>>>>>>><

>>>>>>>>>>>>>:

þDΤ

where Xb is governed by

dX t

þDΤ

Xbð Þ¼ 0 X0:

n

2 BΤ

116 Game Theory - Applications in Logistics and Economy

and the optimal "state" X ¼ x∗ ð Þ ; p

þ C1 þ B~ Τ

Remark 3.3 When we consider the complete information case, that is, W~ ð Þ� disappears and G1,t ¼ Ft, Theorems 3.1 and 3.2 coincide with Theorems 2.3 and 3.3 in Yong [13].

## 4. Concluding remarks

In this chapter, we have studied a leader-follower stochastic differential game with asymmetric information. This kind of game problem possesses several attractive features. First, the game problem has the Stackelberg feature, which means the two players play as different roles during the game. Thus the usual approach to deal with game problems, such as [6–8, 10], where the two players act as equivalent roles, does not apply. Second, the game problem has the asymmetric information between the two players, which was not considered in [3, 13, 14]. In detail, the information available to the follower is based on some sub-σ-algebra of that available to the leader. Stochastic filtering technique is introduced to compute the optimal filtering estimates for the corresponding adjoint processes, which act as the solution to some FBSDFE. Third, the Stackelberg equilibrium is represented in its state feedback form for the LQ problem under some appropriate assumptions. Some new conditional mean-field FBSDEs and system of Riccati equations are introduced to deal with the leader's LQ problem.

In principle, Theorems 3.1 and 3.2 provide a useful tool to seek Stackelberg equilibrium. As a first step in this direction, we apply our results to the LQ problem to obtain explicit solutions. We hope to return to the more general case in our future research. It is worthy to study the closed-loop Stackelberg equilibrium for our problem, as well as the solvability of the system of Riccati equations. These challenging topics will be considered in our future work.

## Acknowledgements

Jingtao Shi would like to thank the book editor for his/her comments and suggestions. Jingtao Shi also would like to thank Professor Guangchen Wang from Shandong University and Professor Jie Xiong from Southern University of Science and Technology, for their effort and discussion during the writing of this chapter.

[6] Hamadène S. Nonzero-sum linear-quadratic stochastic differential games and backward-

Stochastic Leader-Follower Differential Game with Asymmetric Information

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119

[7] Wu Z. Forward-backward stochastic differential equations, linear quadratic stochastic optimal control and nonzero sum differential games. Journal of Systems Science and

[8] An TTK, Øksendal B. Maximum principle for stochastic differential games with partial information. Journal of Optimization Theory and Applications. 2008;139(3):463-483

[9] Wang G, Yu Z. A Pontryagin's maximum principle for non-zero sum differential games of BSDEs with applications. IEEE Transactions on Automatic Control. 2010;55(7):1742-1747

[10] Wang G, Yu Z. A partial information non-zero sum differential game of backward stochastic differential equations with applications. Automatica. 2012;48(2):342-352

[11] von Stackelberg H. Marktform und Gleichgewicht (An English translation appeared in the Theory of the Market Economy, Oxford University Press, 1952). Vienna: Springer; 1934

[12] Basar T. Stochastic stagewise Stackelberg strategies for linear quadratic systems. In: Kohlmann M, Vogel W, editors. Stochastic Control Theory and Stochastic Differential

[13] Yong J. A leader-follower stochastic linear quadratic differential games. SIAM Journal on

[14] Bensoussan A, Chen S, Sethi SP. The maximum principle for global solutions of stochastic Stackelberg differential games. SIAM Journal on Control and Optimization. 2015;53(4):

[15] Wang G, Wu Z, Xiong J. Maximum principles for forward-backward stochastic control systems with correlated state and observation noises. SIAM Journal on Control and

[16] Huang J, Wang G, Xiong J. A maximum principle for partial information backward stochastic control problems with applications. SIAM Journal on Control and Optimiza-

[17] Andersson D, Djehiche B. A maximum principle for SDEs of mean-field type. Applied

[18] Li J. Stochastic maximum principle in the mean-field controls. Automatica. 2012;48(2):366-

[19] Yong J. Linear-quadratic optimal control problems for mean-field stochastic differential

equations. SIAM Journal on Control and Optimization. 2013;51(4):2809-2838

forward equations. Stochastic Analysis and Applications. 1999;17(1):117-130

Complexity. 2005;18(2):179-192

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## Notes

The main content of this chapter is from the following two published article papers: (1) Shi, J.T., Wang, G.C., & Xiong, J. (2016). Leader-follower stochastic differential games with asymmetric information and applications. Automatica, Vol. 63, 60–73. (2) Shi, J.T., Wang, G.C., & Xiong, J. (2017). Linear-quadratic stochastic Stackelberg differential game with asymmetric information. Science China Information Sciences, Vol. 60, 092202:1–15.

## Author details

Jingtao Shi

Address all correspondence to<EMAIL_ADDRESS>
School of Mathematics, Shandong University, Jinan, P.R. China

## References

[6] Hamadène S. Nonzero-sum linear-quadratic stochastic differential games and backwardforward equations. Stochastic Analysis and Applications. 1999;17(1):117-130

Acknowledgements

118 Game Theory - Applications in Logistics and Economy

Notes

Author details

Jingtao Shi

References

Theory. 2015;159:989-1015

37(7):1284-1299

1982

University of Wisconsin-Madison; 2008

Nos. RM-1391, RM-1411, RM-1486; 1954-1955

discussion during the writing of this chapter.

Science China Information Sciences, Vol. 60, 092202:1–15.

Address all correspondence to<EMAIL_ADDRESS>
School of Mathematics, Shandong University, Jinan, P.R. China

Jingtao Shi would like to thank the book editor for his/her comments and suggestions. Jingtao Shi also would like to thank Professor Guangchen Wang from Shandong University and Professor Jie Xiong from Southern University of Science and Technology, for their effort and

The main content of this chapter is from the following two published article papers: (1) Shi, J.T., Wang, G.C., & Xiong, J. (2016). Leader-follower stochastic differential games with asymmetric information and applications. Automatica, Vol. 63, 60–73. (2) Shi, J.T., Wang, G.C., & Xiong, J. (2017). Linear-quadratic stochastic Stackelberg differential game with asymmetric information.

[1] Williams N. A solvable continuous time principal agent model. Journal of Economic

[2] Williams N. On Dynamic Principle-agent Models in Continuous Time. Working Paper.

[3] Øksendal B, Sandal L, Ubøe J. Stochastic Stackelberg equilibria with applications to time dependent newsvendor models. The Journal of Economic Dynamics and Control. 2013;

[4] Isaacs R. Differential Games, Parts 1–4. The Rand Corpration, Research Memorandums

[5] Basar T, Olsder GJ. Dynamic Noncooperative Game Theory. London: Academic Press;

[20] Shi J. Sufficient conditions of optimality for mean-field stochastic control problems. In:

[21] Xiong J. An Introduction to Stochastic Filtering Theory. London: Oxford University Press;

[22] Yong J, Zhou X. Stochastic Controls: Hamiltonian Systems and HJB Equations. New York:

Proceedings of 12th ICARCV, Guangzhou, 5–7 December 2012; pp. 747-752

2008

Springer; 1999

120 Game Theory - Applications in Logistics and Economy

*Edited by Danijela Tuljak-Suban*

This book provides a wide range of examples of the uses of game theory, even in situations where such application may seem unsuitable.

This book explores cooperative, competitive, leader-follower games and the free-rider problem—as well as games with the aim of maintaining friendships or team work. The reader will be presented with a wide range of practical applications of game theory.

Published in London, UK © 2018 IntechOpen © wacomka / iStock

Game Theory - Applications in Logistics and Economy

Game Theory

Applications in Logistics and Economy


**1. Introduction**

Salmonellosis or *Salmonella* infection caused by nontyphoid strains is the most common foodborne disease reported from population-based, active laboratory surveillance in the United States (U.S.) (Figure 1). The overall incidence of laboratory confirmed *Salmonella*

Fig. 1. Total number of laboratory-confirmed bacterial and parasitic infection cases and hospitalizations by pathogen in the United States (CDC, 2011).

The Burden of Salmonellosis in the United States 3

years lost (QALYs), and \$3.3 billion in total medical expenditures and lost productivity each

*Salmonella* gastroenteritis is usually a self-limited disease in which the symptom of fever typically resolves within 48 to 72 hours and diarrhea within three to seven days. Complications from the infection may include severe dehydration, shock, collapse, and/or septicemia. Symptoms are usually more severe among infants, young children, elderly, and

Although there are many serotypes of *Salmonella* that are pathogenic to both humans and animals (i.e., approximately 2,500 serotypes have been identified), the vast majority of human *Salmonella* isolates are serotype *S. enterica* subsp. *enterica* (Heymann, 2008)*. S*erovars Typhi and Paratyphi of this serotype, *S. enterica* subsp. *enterica,* are the etiologic agents that cause typhoid and paratyphoid fevers. These types are also common, but are generally found in developing countries, such as those in South America, Africa, and parts of Asia (Heymann, 2008). In developed countries where there is active, coordinated foodborne disease surveillance, other

Sixty to eighty percent of all human salmonellosis cases in the United States occur intermittently and sporadically throughout the population. Clusters of large outbreaks in restaurants, institutions for children, hospitals, and nursing homes have occurred recently and remain major public health threats. These outbreaks are usually the product of contamination from a production source, such as chicken farms, feed blending mills, and slaughterhouses. One of the more well-known *Salmonella* outbreaks in the United States occurred in 2010. This outbreak resulted from contamination in the food production chain, leading to a massive egg recall of over half a billion eggs and more than 2,000 reported cases of *Salmonella*-related illness (Hutchison, 2010). Although less common, outbreaks from food handling by an ill person or carrier have been reported in recent years (Cruickshank et al., 1987; Khuri-Bulos et al., 1994). For instance, in 2000 an ill food handler in a bakery that supplied hamburger buns to restaurants was found responsible for an outbreak among several burger restaurants across Southern California and Arizona. This outbreak was atypical in that it resulted from consumption of commercially distributed bread, which is a

highly unusual vehicle for most foodborne infectious agents (Kimura et al., 2005).

for hand washing likely facilitated the transmission to infants (Boehmer, 2009).

**2.2 Salmonellosis incidence – FoodNet data**

Outbreaks from person-to-person transmission can also be of particular concern, especially among hospital workers who have the potential to spread the bacterium with their hands or through contaminated instruments. Outbreaks of *Salmonella* infection have occurred in places like maternity wards where staff members with contaminated hands and/or the use of contaminated medical instruments result in the transmission of *Salmonella* to babies and mothers (Rowe et al., 1969). In 2008, an outbreak strain of *Salmonella* serotype Tennessee occurred in a neonatal intensive care unit in the United States, where limited access to sinks

Based on FoodNet surveillance data (*see section 4.1*) for nine selected foodborne pathogens from 10 states and three federal agencies (CDC, U.S. Food and Drug Administration, and U.S. Department of Agriculture), a total of 19,089 laboratory-confirmed cases of foodborne infections, 4,247 hospitalizations, and 68 deaths were identified for the year 2010 in the U.S.

year (Batz et al., 2011).

**2.1 Clinical manifestations, serotypes, and outbreaks**

those who are immune-compromised (Scallan et al., 2011).

serovars such as Typhimurium and Enteritidis are frequently reported.

infection was 17.6 cases per 100,000 persons in 2010. This was more than twice the U.S. Healthy People 2010 objective of 6.8 cases per 100,000 persons (Figure 2) (Matyas et al.,2010). Moreover, a recent report released by the Centers for Disease Control and Prevention (CDC) revealed that the incidence of *Salmonella* infections in 2010 was significantly higher than during 2006-2008 representing an increase of about 10% (95% Confidence Interval (CI), 4- 17%). However, other foodborne infections, such as *Campylobacter*, *Listeria*, *Shigella*, STEC O157, *Vibrio*, and *Yersinia*, have all actually decreased during this same period (CDC, 2011). The disease burden of salmonellosis has remained substantial in the United States in spite of ongoing public health and regulatory efforts to prevent and control this infectious disease.

The present chapter discusses the trends in morbidity, mortality, and years of potential life lost attributed to human salmonellosis in the United States. In addition, this chapter provides a snapshot of U.S. public health measures and control policies that are currently in place to protect the public against *Salmonella* infection.


# **New Advances in Soil Pollution and Remediation**

Edited by Fayuan Wang, Liping Li, Lanfang Han and Aiju Liu Printed Edition of the Special Issue Published in *IJERPH*

www.mdpi.com/journal/ijerph

## **New Advances in Soil Pollution and Remediation**