Job rotation or specialization? A dynamic matching model analysis

Abstract

Which is better for a firm, job rotation or specialization, can be considered as an endogenously formed worker-indivisible job matching problem. We model this problem as a firm’s profit maximization problem under uncertainty, with and without overlapping generations. In both models, we show that among all possible job allocations, the rotation and specialization schemes are the only variations that can be optimal in terms of profits. Moreover, the rotation scheme is better when the productivity difference between post- and under-training workers is smaller, the uncertainty about job continuation in the future is more significant, or the slope of seniority wages is larger. The results indicate that firms in different environments prefer different worker-job matchings.

Appendices

A Proof of Proposition 1

We complete the proof of Proposition 1 by checking the profit comparison in the following three cases.

Case 1: $p>\frac{(1-\lambda )-2(w-w^{*})}{2\delta (2^{\alpha }-1-w)}$. We show that for each $k\in \{1,\ldots ,8\}$, $\pi (x^{R1})>\pi (x^{k})$. First, under Claim 1, $\pi (x^{R1})>\pi (x^{S1})$. Second, from Table 2, for each $k\in \{1,\ldots ,8\}\setminus \{2,7\}$, $\pi (x^{R1})>\pi (x^{k})$. It remains to show the inequality for $k\in \{2,7\}$.

$$\begin{aligned} \pi (x^{R1})-\pi (x^{2})= & {} \pi (x^{R1})-\pi (x^{7})=\lambda +\delta \{\lambda -\lambda ^{1/2}+w-w^{*}\}+\delta ^{2}p(2^{\alpha }-1-w)\\> & {} \lambda +\delta \{\lambda -\lambda ^{1/2}+w-w^{*}\}\\{} & {} +\frac{\delta }{2}\{(1-\lambda )-2(w-w^{*})\}=\lambda +\delta \left( \frac{1}{2}-\lambda ^{1/2}+\frac{\lambda }{2}\right) , \end{aligned}$$

where the inequality follows from $p>\frac{(1-\lambda )-2(w-w^{*})}{2\delta (2^{\alpha }-1-w)}$. Let us define the function $f:[0,1]\rightarrow \mathbb {R}$ by $f(\lambda )=\lambda +\delta \left( \frac{1}{2}-\lambda ^{1/2}+\frac{\lambda }{2}\right) .$ Then, $f(0)=\frac{\delta }{2}<1=f(1)$, $f^{\prime }(\lambda )=1+\frac{\delta }{2}\left( 1-\frac{1}{\lambda ^{1/2}}\right)$, and $f^{\prime \prime }(\lambda )=\frac{\delta }{4}\lambda ^{-3/2}>0$. Thus, function f is concave and there is a unique $\lambda ^{*}$ such that $f^{\prime }(\lambda ^{*})=0$, i.e., $\lambda ^{*}=\left( \frac{\delta }{\delta +2}\right) ^{2}\in (0,1)$. Because our calculation gives us $f(\lambda ^{*})=\frac{2\delta ^{2}+4\delta }{2(2+\delta )^{2}}>0,$ we can conclude that for any $\lambda \in (0,1)$, $f(\lambda )>0$. Thus, $\pi (x^{R1})>\pi (x^{2})$ and $\pi (x^{R1})>\pi (x^{7})$.

Case 2: $p<\frac{(1-\lambda )-2(w-w^{*})}{2\delta (2^{\alpha }-1-w)}$. We show that for each $k\in \{1,\ldots ,8\}$, $\pi (x^{S1})>\pi (x^{k})$. First, under Claim 1, $\pi (x^{S1})>\pi (x^{R1})$. Second, from Table 2, for each $k\in \{1,\ldots ,8\}\setminus \{2,7\}$, $\pi (x^{R1})>\pi (x^{k})$, and thus $\pi (x^{S1})>\pi (x^{k})$. It remains to show the inequality for $k\in \{2,7\}$.

$$\begin{aligned} \pi (x^{S1})-\pi (x^{2})= & {} \pi (x^{S1})-\pi (x^{7})=\lambda +\delta \{1-\lambda ^{1/2}-(w-w^{*})\}-\delta ^{2}p(2^{\alpha }-1-w)\nonumber \\> & {} \lambda +\delta \{1-\lambda ^{1/2}-(w-w^{*})\}-\frac{\delta }{2}\{(1-\lambda )-2(w-w^{*})\}=f(\lambda ), \end{aligned}$$

(5)

where the inequality follows from $p<\frac{(1-\lambda )-2(w-w^{*})}{2\delta (2^{\alpha }-1-w)}$. As shown in Case 1, for each $\lambda \in (0,1)$, $f(\lambda )>0$. Thus, under inequality (5), $\pi (x^{S1})>\pi (x^{2})$ and $\pi (x^{S1})>\pi (x^{7})$.

Case 3: $p=\frac{(1-\lambda )-2(w-w^{*})}{2\delta (2^{\alpha }-1-w)}$. The last part of the proposition is now straightforward from the arguments in cases 1 and 2.$\square$

B Proof of Proposition 2

To prove Proposition 2, it remains to show lemmas 1 and 2. We start with some preliminaries.

1.1 B.1 Preliminaries

We need to calculate the value function depending on the states $\omega ^{O}$ and $\omega ^{A}$. Note that $\omega ^{O}$ is the set of assignable jobs at some period, and $\omega ^{A}$ is the set of available jobs at some period. For the simplicity of notation, when we write the state $(\omega ^{O},\omega ^{A})$, for example, we denote $(\omega ^{O},\omega ^{A})=(\emptyset ,A;A,B)$ for $(\{\emptyset ,A\};\{A,B\})$ where we omit the braces for sets. Thus, for value functions, we denote $V(\omega ^{O},\omega ^{A})=V(\emptyset ,A;A,B)$ for $V(\{\emptyset ,A\},\{A,B\})$. Because of the symmetric roles of jobs A and B, we only need to calculate the value function for the following five cases. (1) $V(\emptyset ;A)=V(\emptyset ;B)=V(\emptyset ,B;A)=V(\emptyset ,A;B)$, (2) $V(\emptyset ;A,B)$, (3) $V(\emptyset ,A;A)=V(\emptyset ,B;B)=V(\emptyset ,A,B;B)=V(\emptyset ,A,B;A)$, (4) $V(\emptyset ,A;A,B)=V(\emptyset ,B;A,B)$, and (5) $V(\emptyset ,A,B;A,B)$.

1.1.1 B.1.1 Calculation of $V(\emptyset ;A)=V(\emptyset ;B)=V(\emptyset ,B;A)=V(\emptyset ,A;B)$

We calculate $V(\emptyset ;A)$, which is obviously equal to $V(\emptyset ;B)$, $V(\emptyset ,B;A)$, and $V(\emptyset ,A;B)$. We have the following three candidates for $V(\emptyset ;A)$, depending on period-t allocations.

$V^{11}$				$V^{12}$				$V^{13}$
	Alloc. at t		States at $t+1$		Alloc. at t		States at $t+1$		Alloc. at t		States at $t+1$
	$s=1$	$s=2$			$s=1$	$s=2$			$s=1$	$s=2$
$a^{t-1}$	$\emptyset$	$\emptyset$	$\omega _{t+1}^{A}=\{A\}$	$a^{t-1}$	$\emptyset$	$\emptyset$	$\omega _{t+1}^{A}=\{A\}$	$a^{t-1}$	$\emptyset$	$\emptyset$	$\omega _{t+1}^{A}=\{A\}$
$a^{t}$	$A^{*}$	A	$\omega _{t+1}^{O}=\{\emptyset ,A\}$	$a^{t}$	$\emptyset$ ($A^{*}$)	$A^{*}$ ($\emptyset$)	$\omega _{t+1}^{O}=\{\emptyset ,A\}$	$a^{t}$	$\emptyset$	$\emptyset$	$\omega _{t+1}^{O}=\{\emptyset \}$

Denote

$$\begin{aligned} V^{11}= & {} \pi (\emptyset ,A^{*};A)+\pi (\emptyset ,A;A)+\delta V(\emptyset ,A;A),\\ V^{12}= & {} \pi (\emptyset ,\emptyset ;A)+\pi (\emptyset ,A^{*};A)+\delta V(\emptyset ,A;A)\text { or }\pi (\emptyset ,A^{*};A)+\pi (\emptyset ,\emptyset ;A)+\delta V(\emptyset ,A;A),\\ V^{13}= & {} \pi (\emptyset ,\emptyset ;A)+\pi (\emptyset ,\emptyset ;A)+\delta V(\emptyset ;A). \end{aligned}$$

Claim 2

$V^{11}>V^{12}$ and $V^{11}>V^{13}$. Thus, $V(\emptyset ;A)=V^{11}$.

Proof

First, $V^{11}-V^{12}=\pi (\emptyset ,A;A)-\pi (\emptyset ,\emptyset ;A)$. Thus, since $\pi (\emptyset ,A;A)>\pi (\emptyset ,\emptyset ;A)$ according to Assumption 8, we have $V^{11}>V^{12}$. We next show $V^{11}>V^{13}$. Note that $V^{11}-V^{13}=\{\pi (\emptyset ,A^{*};A)-\pi (\emptyset ,\emptyset ;A)\}+\{\pi (\emptyset ,A;A)-\pi (\emptyset ,\emptyset ;A)\}+\delta \{V(\emptyset ,A;A)-V(\emptyset ;A)\}.$ By Assumption 8, $\pi (\emptyset ,A^{*};A)>\pi (\emptyset ,\emptyset ;A)$ and $\pi (\emptyset ,A;A)>\pi (\emptyset ,\emptyset ;A)$. Also, as any allocation at state $(\emptyset ;A)$ is possible at state $(\emptyset ,A;A)$, we have $V(\emptyset ,A;A)\ge V(\emptyset ;A)$. Thus, $V^{11}>V^{13}$. $\square$

Thus, we have

$$\begin{aligned} V(\emptyset ;A)=V^{11}=\pi (\emptyset ,A^{*};A)+\pi (\emptyset ,A;A)+\delta V(\emptyset ,A;A). \end{aligned}$$

(6)

Since $V(\emptyset ,A;A)=\frac{\pi (A,A^{*};A)+\pi (A,A;A)}{1-\delta }$ (this will be shown in Section B.1.3),

$$\begin{aligned} V(\emptyset ;A)=\pi (\emptyset ,A^{*};A)+\pi (\emptyset ,A;A)+\frac{\delta }{1-\delta }\left( \pi (A,A^{*};A)+\pi (A,A;A)\right) . \end{aligned}$$

1.1.2 B.1.2 Calculation of $V(\emptyset ;A,B)$

We have the following three candidates for $V(\emptyset ;A,B)$, depending on period-t allocations.

$V^{21}$
	Alloc. at t		States at $t+1$	States at $t+1$	States at $t+1$
	$s=1$	$s=2$
$a^{t-1}$	$\emptyset$	$\emptyset$	$\omega _{t+1}^{A}=\{A,B\}$	$\omega _{t+1}^{A}=\{A\}$	$\omega _{t+1}^{A}=\{B\}$
$a^{t}$	$B^{*}$	B	$\omega _{t+1}^{O}=\{\emptyset ,B\}$	$\omega _{t+1}^{O}=\{\emptyset ,B\}$	$\omega _{t+1}^{O}=\{\emptyset ,B\}$

$$\begin{aligned}{} & {} V^{21}=\pi (\emptyset ,B^{*};A,B)+\pi (\emptyset ,B;A,B)+(1-2p)\delta V(\emptyset ,B;A,B)\\{} & {} +p\delta V(\emptyset ,B;A)+p\delta V(\emptyset ,B;B). \end{aligned}$$

$V^{22}$
	Alloc. at t		States at $t+1$	States at $t+1$	States at $t+1$
	$s=1$	$s=2$
$a^{t-1}$	$\emptyset$	$\emptyset$	$\omega _{t+1}^{A}=\{A,B\}$	$\omega _{t+1}^{A}=\{A\}$	$\omega _{t+1}^{A}=\{B\}$
$a^{t}$	$A^{*}$	A	$\omega _{t+1}^{O}=\{\emptyset ,A\}$	$\omega _{t+1}^{O}=\{\emptyset ,A\}$	$\omega _{t+1}^{O}=\{\emptyset ,A\}$

$$\begin{aligned} V^{22}= & {} \pi (\emptyset ,A^{*};A,B)+\pi (\emptyset ,A;A,B)+(1-2p)\delta V(\emptyset ,A;A,B)\\{} & {} +p\delta V(\emptyset ,A;A)+p\delta V(\emptyset ,A;B). \end{aligned}$$

$V^{23}$
	Alloc. at t		States at $t+1$	States at $t+1$	States at $t+1$
	$s=1$	$s=2$
$a^{t-1}$	$\emptyset$	$\emptyset$	$\omega _{t+1}^{A}=\{A,B\}$	$\omega _{t+1}^{A}=\{A\}$	$\omega _{t+1}^{A}=\{B\}$
$a^{t}$	$A^{*}$	$B^{*}$	$\omega _{t+1}^{O}=\{\emptyset ,A,B\}$	$\omega _{t+1}^{O}=\{\emptyset ,A,B\}$	$\omega _{t+1}^{O}=\{\emptyset ,A,B\}$

$$\begin{aligned} V^{23}= & {} \pi (\emptyset ,A^{*};A,B)+\pi (\emptyset ,B^{*};A,B)\\{} & {} +(1-2p)\delta V(\emptyset ,A,B;A,B)+p\delta V(\emptyset ,A,B;A)+p\delta V(\emptyset ,A,B;B)\\= & {} \pi (\emptyset ,A^{*};A,B)+\pi (\emptyset ,B^{*};A,B)\\{} & {} +(1-2p)\delta V(\emptyset ,A,B;A,B)+p\delta V(\emptyset ,A;A)+p\delta V(\emptyset ,B;B). \end{aligned}$$

Claim 3

$V^{23}>V^{21}=V^{22}$. Thus, $V(\emptyset ;A,B)=V^{23}$.

Proof

First, $V^{21}=V^{22}$ by symmetry. Next, we have

$$\begin{aligned} V^{23}-V^{22}= & {} \pi (\emptyset ,B^{*};A,B)-\pi (\emptyset ,A;A,B)\\{} & {} +(1-2p)\delta (V(\emptyset ,A,B;A,B)-V(\emptyset ,A;A,B))\\{} & {} +p\delta (V(\emptyset ,B;B)-V(\emptyset ,A;B))\\= & {} -w^{*}+w+(1-2p)\delta (V(\emptyset ,A,B;A,B)-V(\emptyset ,A;A,B))\\{} & {} +p\delta (V(\emptyset ,B;B)-V(\emptyset ;B))> 0. \end{aligned}$$

The last inequality follows from the fact that $w>w^{*}$, $V(\emptyset ,A,B;A,B)\ge V(\emptyset ,A;A,B)$, and $V(\emptyset ,B;B)\ge V(\emptyset ;B)$. $\square$

1.1.3 B.1.3 Calculation of $V(\emptyset ,A;A)=V(\emptyset ,B;B)=V(\emptyset ,A,B;B)=V(\emptyset ,A,B;A)$

We calculate $V(\emptyset ,A;A)$, which is obviously equal to $V(\emptyset ,B;B)$, $V(\emptyset ,A,B;B)$, and $V(\emptyset ,A,B;A)$. We have the following three candidates for $V(\emptyset ,A;A)$, depending on period-t allocations.

$V^{31}$				$V^{32}$				$V^{33}$
	Alloc. at t		States at $t+1$		Alloc. at t		States at $t+1$		Alloc. at t		States at $t+1$
	$s=1$	$s=2$			$s=1$	$s=2$			$s=1$	$s=2$
$a^{t-1}$	A	A	$\omega _{t+1}^{A}=\{A\}$	$a^{t-1}$	A	A	$\omega _{t+1}^{A}=\{A\}$	$a^{t-1}$	A	A	$\omega _{t+1}^{A}=\{A\}$
$a^{t}$	$A^{*}$	A	$\omega _{t+1}^{O}=\{\emptyset ,A\}$	$a^{t}$	$\emptyset$ ($A^{*}$)	$A^{*}$ ($\emptyset$)	$\omega _{t+1}^{O}=\{\emptyset ,A\}$	$a^{t}$	$\emptyset$	$\emptyset$	$\omega _{t+1}^{O}=\{\emptyset \}$

$$\begin{aligned} V^{31}= & {} \pi (A,A^{*};A)+\pi (A,A;A)+\delta V(\emptyset ,A;A),\\ V^{32}= & {} \pi (A,\emptyset ;A)+\pi (A,A^{*};A)+\delta V(\emptyset ,A;A),\\ V^{33}= & {} \pi (A,\emptyset ;A)+\pi (A,\emptyset ;A)+\delta V(\emptyset ;A). \end{aligned}$$

Claim 4

$V^{31}>V^{32}$ and $V^{31}>V^{33}$. Thus, $V(\emptyset ,A;A)=V^{31}$.

Proof

First, $V^{31}-V^{32}=\pi (A,A;A)-\pi (A,\emptyset ;A)$. Then, since $\pi (A,A;A)>\pi (A,\emptyset ;A)$ under Assumption 8, we have $V^{31}>V^{32}$. Next, we show $V^{31}>V^{33}$. Note that $V^{31}-V^{33}=\{\pi (A,A;A)-\pi (A,\emptyset ;A)\}+\{\pi (A,A^{*};A)-\pi (A,\emptyset ;A)\}+\delta \{V(\emptyset ,A;A)-V(\emptyset ;A)\}.$ Under Assumption 8, $\pi (A,A;A)>\pi (A,\emptyset ;A)$ and $\pi (A,A^{*};A)>\pi (A,\emptyset ;A)$. Moreover, $V(\emptyset ,A;A)\ge V(\emptyset ;A)$. Hence, $V^{31}>V^{33}$. $\square$

Thus, we have

$$\begin{aligned}{} & {} V(\emptyset ,A;A)=V^{31}=\pi (A,A^{*};A)+\pi (A,A;A)+\delta V(\emptyset ,A;A)\nonumber \\{} & {} =\frac{\pi (A,A^{*};A)+\pi (A,A;A)}{1-\delta }. \end{aligned}$$

(7)

1.1.4 B.1.4 Calculation of $V(\emptyset ,A;A,B)=V(\emptyset ,B;A,B)$

We calculate $V(\emptyset ,A;A,B)$, which is obviously equal to $V(\emptyset ,B;A,B)$. We have the following three candidates for $V(\emptyset ,A;A,B)$, depending on period-t allocations.

$V^{41}$
	Alloc. at t		States at $t+1$	States at $t+1$	States at $t+1$
	$s=1$	$s=2$
$a^{t-1}$	A	A	$\omega _{t+1}^{A}=\{A,B\}$	$\omega _{t+1}^{A}=\{A\}$	$\omega _{t+1}^{A}=\{B\}$
$a^{t}$	$B^{*}$	B	$\omega _{t+1}^{O}=\{\emptyset ,B\}$	$\omega _{t+1}^{O}=\{\emptyset ,B\}$	$\omega _{t+1}^{O}=\{\emptyset ,B\}$

$V^{41}=\pi (A,B^{*};A,B)+\pi (A,B;A,B)+(1-2p)\delta V(\emptyset ,B;A,B)+p\delta V(\emptyset ,B;A)+p\delta V(\emptyset ,B;B)$. Note that $V^{41}$ is the value of the specialization.

$V^{42}$
	Alloc. at t		States at $t+1$	States at $t+1$	States at $t+1$
	$s=1$	$s=2$
$a^{t-1}$	A	A	$\omega _{t+1}^{A}=\{A,B\}$	$\omega _{t+1}^{A}=\{A\}$	$\omega _{t+1}^{A}=\{B\}$
$a^{t}$	$A^{*}$	A	$\omega _{t+1}^{O}=\{\emptyset ,A\}$	$\omega _{t+1}^{O}=\{\emptyset ,A\}$	$\omega _{t+1}^{O}=\{\emptyset ,A\}$

$V^{42}=\pi (A,A^{*};A,B)+\pi (A,A;A,B)+(1-2p)\delta V(\emptyset ,A;A,B)+p\delta V(\emptyset ,A;A)+p\delta V(\emptyset ,A;B).$

$V^{43}$
	Alloc. at t		States at $t+1$	States at $t+1$	States at $t+1$
	$s=1$	$s=2$
$a^{t-1}$	A	A	$\omega _{t+1}^{A}=\{A,B\}$	$\omega _{t+1}^{A}=\{A\}$	$\omega _{t+1}^{A}=\{B\}$
$a^{t}$	$A^{*}$	$B^{*}$	$\omega _{t+1}^{O}=\{\emptyset ,A,B\}$	$\omega _{t+1}^{O}=\{\emptyset ,A,B\}$	$\omega _{t+1}^{O}=\{\emptyset ,A,B\}$

$$\begin{aligned} V^{43}= & {} \pi (A,A^{*};A,B)+\pi (A,B^{*};A,B)\\{} & {} \quad +(1-2p)\delta V(\emptyset ,A,B;A,B)+p\delta V(\emptyset ,A,B;A)+p\delta V(\emptyset ,A,B;B)\\= & {} \pi (A,A^{*};A,B)+\pi (A,B^{*};A,B)\\{} & {} \quad +(1-2p)\delta V(\emptyset ,A,B;A,B)+p\delta V(\emptyset ,A;A)+p\delta V(\emptyset ,B;B). \end{aligned}$$

Claim 5

$V^{41}>V^{42}$ and $V^{43}>V^{42}$.

Proof

First,

$$\begin{aligned} V^{41}-V^{42}= & {} \{\pi (A,B^{*};A,B)-\pi (A,A^{*};A,B)\}+\{\pi (A,B;A,B)-\pi (A,A;A,B)\}\\{} & {} +(1-2p)\delta \{V(\emptyset ,B;A,B)-V(\emptyset ,A;A,B)\}\\{} & {} +p\delta \{V(\emptyset ,B;A)-V(\emptyset ,A;B)\}+p\delta \{V(\emptyset ,B;B)-V(\emptyset ,A;A)\}. \end{aligned}$$

By calculation, $\pi (A,B^{*};A,B)>\pi (A,A^{*};A,B)$ and $\pi (A,B;A,B)>\pi (A,A;A,B)$. Moreover, by symmetry, $V(\emptyset ,B;A,B)=V(\emptyset ,A;A,B)$, $V(\emptyset ,B;A)=V(\emptyset ,A;B)$, and $V(\emptyset ,B;B)=V(\emptyset ,A;A)$. Thus $V^{41}>V^{42}$.

We next show $V^{43}>V^{42}$.

$$\begin{aligned} V^{43}-V^{42}= & {} \pi (A,B^{*};A,B)-\pi (A,A;A,B)+p\delta \left( V(\emptyset ,B;B)-V(\emptyset ,A;B)\right) \\{} & {} +(1-2p)\delta \left( V(\emptyset ,A,B;A,B)-V(\emptyset ,A;A,B)\right) \\= & {} \pi (A,B^{*};A,B)-\pi (A,A;A,B)+p\delta \left( V(\emptyset ,B;B)-V(\emptyset ;B)\right) \\{} & {} +(1-2p)\delta \left( V(\emptyset ,A,B;AB)-V(\emptyset ,A;A,B)\right) >0. \end{aligned}$$

The last inequality follows from the fact that $\pi (A,B^{*};A,B)-\pi (A,A;A,B)=(\lambda ^{1/2}-w-w^{*})+2w=\lambda ^{1/2}+w-w^{*}>0$, $V(\emptyset ,B;B)\ge V(\emptyset ;B)$, and $V(\emptyset ,A,B;A,B)\ge V(\emptyset ,A;A,B)$. Thus, $V^{43}>V^{42}$. $\square$

At this stage, we cannot clearly say which is larger, $V^{41}$ or $V^{43}$.

1.1.5 B.1.5 Calculation of $V(\emptyset ,A,B;A,B)$

We have the following four candidates for $V(\emptyset ,A,B;A,B)$, depending on period-t allocations.

• Case 1: $(x_{t,1}^{t-1},x_{t,2}^{t-1})=(A,A)$. Then, the old worker is assigned job A though jobs A and B are assignable. Thus, the value candidate $V^{51}$ in this case is equal to the value when only job A is assignable, i.e., $V^{51}=V(\emptyset ,A;A,B)$.

• Case 2: $(x_{t,1}^{t-1},x_{t,2}^{t-1})=(B,B)$. Then, similarly, the value candidate $V^{52}$ in this case is $V^{52}=V(\emptyset ,B;A,B)$.

• Case 3: $(x_{t,1}^{t-1},x_{t,2}^{t-1})=(A,B)$. Denote the value candidate in this case by $V^{53}$. Then, we have the following four subcandidates for $V^{53}$, depending on the young worker’s assignment $(x_{t,1}^{t},x_{t,2}^{t})$.

  $$\begin{aligned} V^{531}= & {} \pi (A,B^{*};A,B)+\pi (B,A^{*};A,B)\\{} & {} \quad +(1-2p)\delta V(\emptyset ,A,B;A,B)+p\delta V(\emptyset ,A,B;A)+p\delta V(\emptyset ,A,B;B). \end{aligned}$$

  Note that $V^{531}$ is the value of the rotation. $V^{532}=\pi (A,A^{*};A,B)+\pi (B,B^{*};$ $A,B)+(1-2p)\delta V(\emptyset ,A,B;A,B)+p\delta V(\emptyset ,A,B;A)+p\delta V(\emptyset ,A,B;B).$ $V^{533}=\pi (A,A^{*};A,B)+\pi (B,A;A,B)+(1-2p)\delta V(\emptyset ,A;A,B)+p\delta V(\emptyset ,A;A)$ $+p\delta V(\emptyset ,A;B)$. $V^{534}=V^{533}$ by symmetry.

$V^{531}$
	Alloc. at t		States at $t+1$	States at $t+1$	States at $t+1$
	$s=1$	$s=2$
$a^{t-1}$	A	B	$\omega _{t+1}^{A}=\{A,B\}$	$\omega _{t+1}^{A}=\{A\}$	$\omega _{t+1}^{A}=\{B\}$
$a^{t}$	$B^{*}$	$A^{*}$	$\omega _{t+1}^{O}=\{\emptyset ,A,B\}$	$\omega _{t+1}^{O}=\{\emptyset ,A,B\}$	$\omega _{t+1}^{O}=\{\emptyset ,A,B\}$

$V^{532}$
	Alloc. at t		States at $t+1$	States at $t+1$	States at $t+1$
	$s=1$	$s=2$
$a^{t-1}$	A	B	$\omega _{t+1}^{A}=\{A,B\}$	$\omega _{t+1}^{A}=\{A\}$	$\omega _{t+1}^{A}=\{B\}$
$a^{t}$	$A^{*}$	$B^{*}$	$\omega _{t+1}^{O}=\{\emptyset ,A,B\}$	$\omega _{t+1}^{O}=\{\emptyset ,A,B\}$	$\omega _{t+1}^{O}=\{\emptyset ,A,B\}$

$V^{533}$
	Alloc. at t		States at $t+1$	States at $t+1$	States at $t+1$
	$s=1$	$s=2$
$a^{t-1}$	A	B	$\omega _{t+1}^{A}=\{A,B\}$	$\omega _{t+1}^{A}=\{A\}$	$\omega _{t+1}^{A}=\{B\}$
$a^{t}$	$A^{*}$	A	$\omega _{t+1}^{O}=\{\emptyset ,A\}$	$\omega _{t+1}^{O}=\{\emptyset ,A\}$	$\omega _{t+1}^{O}=\{\emptyset ,A\}$

$V^{534}$
	Alloc. at t		States at $t+1$	States at $t+1$	States at $t+1$
	$s=1$	$s=2$
$a^{t-1}$	A	B	$\omega _{t+1}^{A}=\{A,B\}$	$\omega _{t+1}^{A}=\{A\}$	$\omega _{t+1}^{A}=\{B\}$
$a^{t}$	$B^{*}$	B	$\omega _{t+1}^{O}=\{\emptyset ,B\}$	$\omega _{t+1}^{O}=\{\emptyset ,B\}$	$\omega _{t+1}^{O}=\{\emptyset ,B\}$

Thus, $V^{53}=\max \{V^{531},V^{532},V^{533},V^{534}\}$.

Claim 6

$V^{531}>V^{532}.$

Proof

This is because $\pi (A,B^{*};A,B)>\pi (A,A^{*};A,B)$ and $\pi (B,A^{*};A,B)>\pi (B,B^{*};A,B)$. $\square$

• Case 4: $(x_{t,1}^{t-1},x_{t,2}^{t-1})=(B,A)$. Let $V^{54}$ be the value in this case. Then, by symmetry, $V^{54}=V^{53}.$

1.2 B.2 Proof of Lemma 1: A necessary and almost sufficient condition for the rotation to be optimal

1.2.1 B.2.1 A necessary condition for the rotation to be optimal

Since our initial condition is $(\emptyset ,A,B;A,B)$, we focus on Section B.1.5 for $V(\emptyset ,A,B;A,B)$. Suppose that the rotation is a profit-maximizing allocation. Then, the value is $V(\emptyset ,A,B;A,B)=V^{531}$. In addition to the inequality of Claim 6, we have the following relations.

$$\begin{aligned} V^{531}\ge & {} V^{533}=V^{534},\end{aligned}$$

(8)

$$\begin{aligned} V^{531}\ge & {} V^{51}=V^{52}=V^{4}, \end{aligned}$$

(9)

where $V^{4}:=V(\emptyset ,A;A,B)=\max \{V^{41},V^{42},V^{43}\}$. Let

$$\begin{aligned} M:=(1-2p)\delta \{V(\emptyset ,A,B;A,B)-V(\emptyset ,B;A,B)\}+p\delta \{V(\emptyset ,A;A)-V(\emptyset ;A)\}. \end{aligned}$$

Then,

$$\begin{aligned} V^{43}-V^{41}= & {} M-(1-w+w^{*}),\end{aligned}$$

(10)

$$\begin{aligned} V^{531}-V^{533}= & {} M-\left( 1-2\lambda ^{\frac{1}{2}}+w^{*}-w\right) . \end{aligned}$$

(11)

Thus, we have

Claim 7

$V^{43}>V^{41}\Rightarrow V^{531}>V^{533}$.

To know the implications from (8) and (9) using (10) and (11), we explore the value of M. Since $V(\emptyset ,A,B;A,B)=V^{531}$ and $V(\emptyset ,A;A,B)=V^{4}$, we have

$$\begin{aligned} M= & {} (1-2p)\delta (V^{531}-V^{4})+p\delta (V(\emptyset ,A;A)-V(\emptyset ;A)). \end{aligned}$$

(12)

In the above M, we have the three unknowns: The first unknown is $V^{531}$. It follows from Section B.1.5 that $V^{531}=\pi (A,B^{*};A,B)+\pi (B,A^{*};A,B)+(1-2p)\delta V^{531}+p\delta V(\emptyset ,A;A)+p\delta V(\emptyset ,B;B)$. Thus, solving this equation for $V^{531}$, we have

$$\begin{aligned} V^{531}= & {} \frac{1}{1-(1-2p)\delta }\nonumber \\{} & {} \Big (\pi (A,B^{*};A,B)+\pi (B,A^{*};A,B)+p\delta V(\emptyset ,A;A)+p\delta V(\emptyset ,B;B)\Big ). \end{aligned}$$

(13)

The second unknown is

$$\begin{aligned} V(\emptyset ,A;A)-V(\emptyset ;A)= & {} V(\emptyset ,A;A)-\pi (\emptyset ,A^{*};A)-\pi (\emptyset ,A;A)-\delta V(\emptyset ,A;A)\quad (\because ((6)))\nonumber \\= & {} \pi (A,A^{*};A)+\pi (A,A;A)-\pi (\emptyset ,A^{*};A)-\pi (\emptyset ,A;A)\quad (\because ((7)))\nonumber \\= & {} (1+\lambda )^{\alpha }-w-w^{*}+2^{\alpha }-2w-(\lambda ^{\alpha }-w^{*})-(1-w)\nonumber \\= & {} (1+\lambda )^{\alpha }+2^{\alpha }-\lambda ^{\alpha }-1-2w. \end{aligned}$$

(14)

The third unknown is $V^{4}$ for which we have two cases from (10): $M<1+w^{*}-w$ and $M\ge 1+w^{*}-w$.

Case 1: $M<1+w^{*}-w$. Then, $V^{43}-V^{41}=M-(1-w+w^{*})<0$ and thus, $V^{43}<V^{41}$. Thus, since $V^{41}>V^{42}$ and $V^{43}>V^{42}$ by Claim 5 in Section B.1.4, we have $V^{4}=V^{41}$. Now, it follows from Section B.1.4 that $V^{4}=V^{41}=\pi (A,B^{*};A,B)+\pi (A,B;A,B)+(1-2p)\delta V(\emptyset ,B;A,B)+p\delta V(\emptyset ,B;A)+p\delta V(\emptyset ,B;B).$ Since $V(\emptyset ,B;A,B)=V^{4}$ and $V(\emptyset ,B;A)=V(\emptyset ;A)$, this becomes

$$\begin{aligned}{} & {} V^{4}=\pi (A,B^{*};A,B)+\pi (A,B;A,B)+(1-2p)\delta V^{4}+p\delta V(\emptyset ;A)+p\delta V(\emptyset ,B;B)\\{} & {} \Rightarrow V^{4} =\frac{1}{1-(1-2p)\delta }\left( \pi (A,B^{*};A,B)+\pi (A,B;A,B)+p\delta V(\emptyset ;A)+p\delta V(\emptyset ,B;B)\right) . \end{aligned}$$

Thus, we have obtained the three unknowns. Hence, $(V^{531}-V^{4})$ in M can be calculated as

$$\begin{aligned} V^{531}-V^{4}= & {} \frac{1}{1-(1-2p)\delta }\nonumber \\ {}{} & {} \quad \Big (\pi (B,A^{*};A,B)-\pi (A,B;A,B)+p\delta V(\emptyset ,A;A)-p\delta V(\emptyset ;A)\Big ). \end{aligned}$$

(15)

Thus, we can calculate the value of M as follows.

$$\begin{aligned} M= & {} \frac{(1-2p)\delta }{1-(1-2p)\delta }\Big \{\pi (B,A^{*};A,B)-\pi (A,B;A,B)+p\delta V(\emptyset ,A;A)-p\delta V(\emptyset ;A)\Big \}\\{} & {} +p\delta \left\{ V(\emptyset ,A;A)-V(\emptyset ;A)\right\} \\= & {} \frac{1}{1-(1-2p)\delta }\Big [(1-2p)\delta \left\{ \pi (B,A^{*};A,B)-\pi (A,B;A,B)\right\} +(1-2p)p\delta ^{2}\big \{ V(\emptyset ,A;A)\\{} & {} -V(\emptyset ;A))\big \}+\left\{ 1-(1-2p)\delta \right\} p\delta (V(\emptyset ,A;A)-V(\emptyset ;A))\Big ]\\= & {} \frac{1}{1-(1-2p)\delta }\Big [(1-2p)\delta \big \{\pi (B,A^{*};A,B)-\pi (A,B;A,B)\big \}+p\delta \big \{ V(\emptyset ,A;A)-V(\emptyset ;A)\big \}\Big ]\\= & {} \frac{1}{1-(1-2p)\delta }\Big [(1-2p)\delta \{\lambda ^{\frac{1}{2}}-w-w^{*}-(1-2w)\}+p\delta \{V(\emptyset ,A;A)-V(\emptyset ;A)\}\Big ]\\= & {} \frac{-(1-2p)\delta (1-\lambda ^{\frac{1}{2}}+w^{*}-w)+p\delta (V(\emptyset ,A;A)-V(\emptyset ;A))}{1-(1-2p)\delta }. \end{aligned}$$

Therefore, we substitute this M into (11) to get the following implication from (8).

$$\begin{aligned}{} & {} \frac{-(1-2p)\delta (1-\lambda ^{\frac{1}{2}}+w^{*}-w)+p\delta (V(\emptyset ,A;A)-V(\emptyset ;A))}{1-(1-2p)\delta }\ge 1-2\lambda ^{\frac{1}{2}}+w^{*}-w\nonumber \\\Rightarrow & {} 1-\lambda ^{\frac{1}{2}}+w^{*}-w-\lambda ^{\frac{1}{2}}+(1-2p)\delta \lambda ^{\frac{1}{2}}\le p\delta (V(\emptyset ,A;A)-V(\emptyset ;A)). \end{aligned}$$

(16)

On the other hand, we use (15) to get the following implication from (9).

$$\begin{aligned} V^{531}-V^{41}= & {} \frac{1}{1-(1-2p)\delta }\big \{\pi (B,A^{*};A,B)-\pi (A,B;A,B)+p\delta V(\emptyset ,A;A)-p\delta V(\emptyset ;A)\big \}\ge 0.\nonumber \\\Rightarrow & {} \lambda ^{\frac{1}{2}}-w-w^{*}-(1-2w)+p\delta (V(\emptyset ,A;A)-V(\emptyset ;A))\ge 0\nonumber \\\Rightarrow & {} 1-\lambda ^{\frac{1}{2}}+w^{*}-w\le p\delta (V(\emptyset ,A;A)-V(\emptyset ;A)). \end{aligned}$$

(17)

Therefore, for Case 1, we have the two necessary conditions (16) and (17). In these equations, since $-\lambda ^{\frac{1}{2}}+(1-2p)\delta \lambda ^{\frac{1}{2}}<0$, (17) implies (16). Hence, we have (17) as a necessary condition.

Case 2: $1+w^{*}-w\le M$. Then, $V^{43}-V^{41}=M-(1-w+w^{*})\ge 0$ and thus, $V^{43}\ge V^{41}$. Thus, since $V^{41}>V^{42}$ and $V^{43}>V^{42}$ under Claim 5 in Section B.1.4, we have $V^{4}=V^{43}$. Now, it follows from Section B.1.4 that

$$\begin{aligned} V^{4}= & {} V^{43}=\pi (A,A^{*};A,B)+\pi (A,B^{*};A,B)+(1-2p)\delta V^{531}+p\delta V(\emptyset ,A;A)+p\delta V(\emptyset ,B;B)\\= & {} 0-w-w^{*}+\lambda ^{\frac{1}{2}}-w-w^{*}+(1-2p)\delta V^{531}+p\delta V(\emptyset ,A;A)+p\delta V(\emptyset ,B;B). \end{aligned}$$

Then, (9) is automatically satisfied as follows.

$$\begin{aligned} V^{531}-V^{4}= & {} V^{531}-V^{43}\\= & {} \left( 1-(1-2p)\delta \right) V^{531}+2w+2w^{*}-\lambda ^{\frac{1}{2}}-p\delta V(\emptyset ,A;A)-p\delta V(\emptyset ,B;B)\\= & {} \pi (A,B^{*};A,B)+\pi (B,A^{*};A,B)+p\delta V(\emptyset ,A;A)+p\delta V(\emptyset ,B;B)\\{} & {} +2w+2w^{*}-\lambda ^{\frac{1}{2}}-p\delta V(\emptyset ,A;A)-p\delta V(\emptyset ,B;B)\quad (\text {by }((13)))\\= & {} \lambda ^{\frac{1}{2}}-w-w^{*}+\lambda ^{\frac{1}{2}}-w-w^{*}+2w+2w^{*}-\lambda ^{\frac{1}{2}}=\lambda ^{\frac{1}{2}}>0. \end{aligned}$$

Thus, we substitute this value $(V^{531}-V^{4})$ into (12) to get $M=(1-2p)\delta \lambda ^{\frac{1}{2}}+p\delta (V(\emptyset ,A;A)-V(\emptyset ;A)).$ We substitute this M into (11) to get the following implication of (8).

$$\begin{aligned}{} & {} (1-2p)\delta \lambda ^{\frac{1}{2}}+p\delta (V(\emptyset ,A;A)-V(\emptyset ;A)\ge 1-2\lambda ^{\frac{1}{2}}+w^{*}-w\nonumber \\\Rightarrow & {} 1-2\lambda ^{\frac{1}{2}}+w^{*}-w-(1-2p)\delta \lambda ^{\frac{1}{2}}\le p\delta (V(\emptyset ,A;A)-V(\emptyset ;A)). \end{aligned}$$

(18)

Therefore, for Case 2, we have the necessary condition (18).

In sum, we have (17) for Case 1 or (18) for Case 2. Since (17) implies (18), we have (17) as a necessary condition for the rotation to be optimal. Hence, $p\ge \frac{1-w+w^{*}-\lambda ^{\frac{1}{2}}}{\delta \{(1+\lambda )^{\alpha }+2^{\alpha }-\lambda ^{\alpha }-1-2w\}}$.

1.2.2 B.2.2 An almost sufficient condition for the rotation to be optimal

Suppose that

$$\begin{aligned} p\delta \left\{ V(\emptyset ,A;A)-V(\emptyset ;A)\right\} >1-w+w^{*}-\lambda ^{\frac{1}{2}}. \end{aligned}$$

(19)

The value of the rotation, $\pi ^{R}$, is

$$\begin{aligned} \pi ^{R}= & {} V^{531} =\pi (A,B^{*};A,B)+\pi (B,A^{*};A,B)\\{} & {} \quad +(1-2p)\delta V(\emptyset ,A,B;A,B)+p\delta V(\emptyset ,A;A)+p\delta V(\emptyset ,B;B). \end{aligned}$$

We need to show

$$\begin{aligned}{} & {} V^{531}\ge V^{532}, \end{aligned}$$

(20)

$$\begin{aligned}{} & {} V^{531}\ge V^{533},\end{aligned}$$

(21)

$$\begin{aligned}{} & {} V^{531}\ge V^{51},\end{aligned}$$

(22)

$$\begin{aligned}{} & {} V^{531}\ge V^{52},\end{aligned}$$

(23)

$$\begin{aligned}{} & {} V^{531}\ge V^{54}=V^{53},\end{aligned}$$

(24)

$$\begin{aligned}{} & {} V^{531}\ge V^{41},\end{aligned}$$

(25)

$$\begin{aligned}{} & {} V^{531}\ge V^{42},\end{aligned}$$

(26)

$$\begin{aligned}{} & {} V^{531}\ge V^{43}. \end{aligned}$$

(27)

Here, the last three inequalities follow from the fact that $V^{531}\ge V^{51}$ and $V^{51}=V^{52}=V(\emptyset ,A;A,B)\equiv V^{4}=\max \{V^{41},V^{42},V^{43}\}$.

Claim 8

The three inequalities (21), (25), and (27) are sufficient for all of the above inequalities.

Proof

Suppose that (21), (25), and (27) hold. First of all, Claim 6 implies (20). Note that when (25) is true, by Claim 5 ($V^{41}>V^{42}$), (26) holds. Hence, it follows from (25), (26), and (27) that $V^{531}\ge V^{4}$. This implies that (22) and (23) are true. On the other hand, since we have (20) and (21), we have $V^{531}=V^{53}$ and thus (24). $\square$

From now on, we will check (21), (25), and (27).

• Check whether $V^{531}\ge V^{533}$, or (21).

$$\begin{aligned} V^{531}-V^{533}= & {} (1-2p)\delta \{V(\emptyset ,A,B;A,B)-V(\emptyset ,B;A,B)\}\\{} & {} +p\delta \{V(\emptyset ,A;A)-V(\emptyset ;A)\}-(1-2\lambda ^{\frac{1}{2}}+w^{*}-w)\\> & {} (1-2p)\delta \{V(\emptyset ,A,B;A,B)-V(\emptyset ,B;A,B)\}\\{} & {} +1-w+w^{*}-\lambda ^{\frac{1}{2}}-(1-2\lambda ^{\frac{1}{2}}+w^{*}-w)\quad (\because ((19)))\\= & {} (1-2p)\delta \{V(\emptyset ,A,B;A,B)-V(\emptyset ,B;A,B)\}+\lambda ^{\frac{1}{2}}\\\ge & {} \lambda ^{\frac{1}{2}}\quad (\because V(\emptyset ,A,B;A,B)\ge V(\emptyset ,B;A,B))\\> & {} 0. \end{aligned}$$

• Check whether $V^{531}\ge V^{41}$, or (25).

  $$\begin{aligned} V^{531}-V^{41}= & {} \pi (B,A^{*};A,B)-\pi (A,B;A,B)\\{} & {} +(1-2p)\delta \{V(\emptyset ,A,B;A,B)-V(\emptyset ,B;A,B)\}+p\delta \{V(\emptyset ,A;A)-V(\emptyset ;A)\}\\\ge & {} \pi (B,A^{*};A,B)-\pi (A,B;A,B)\\{} & {} +p\delta \{V(\emptyset ,A;A)-V(\emptyset ;A)\}\quad (\because V(\emptyset ,A,B;A,B)\ge V(\emptyset ,B;A,B))\\> & {} \pi (B,A^{*};A,B)-\pi (A,B;A,B)+1-w+w^{*}-\lambda ^{\frac{1}{2}}\quad (\because ((19)))\\= & {} \lambda ^{\frac{1}{2}}-w-w^{*}-(1-2w)+1-w+w^{*}-\lambda ^{\frac{1}{2}}=0. \end{aligned}$$

• Check whether $V^{531}\ge V^{43}$, or (27).

  $$\begin{aligned} V^{531}-V^{43}= & {} \pi (B,A^{*};A,B)-\pi (A,A^{*};A,B)\\ {}= & {} \lambda ^{\frac{1}{2}}-w-w^{*}-(0-w-w^{*})=\lambda ^{\frac{1}{2}}>0. \end{aligned}$$

1.3 B.3 Proof of Lemma 2

Suppose that the rotation is not optimal. Then, we will check the two cases: $V^{533}>V^{531}$ and $V^{533}\le V^{531}$.

Case 1: $V^{533}>V^{531}$. Since $V^{531}>V^{532}$ under Claim 6, we have $V^{53}=V^{533}$.

Claim 9

$V^{41}>V^{43}$ and thus $V^{4}=V^{41}$.

Proof

Since $V^{533}>V^{531}$ holds, we have

$$\begin{aligned} V^{533}-V^{531}= & {} \pi (A,A^{*};A,B)+\pi (B,A;A,B)-\pi (A,B^{*};A,B)-\pi (B,A^{*};A,B)\\{} & {} +(1-2p)\delta \{V(\emptyset ,A;A,B)-V(\emptyset ,A,B;A,B)\}+p\delta \{V(\emptyset ;B)-V(\emptyset ,B;B)\}>0. \end{aligned}$$

Thus,

$$\begin{aligned} V^{41}-V^{43}= & {} \pi (A,B^{*};A,B)+\pi (A,B;A,B)-\pi (A,A^{*};A,B)-\pi (A,B^{*};A,B)\\{} & {} +(1-2p)\{\delta V(\emptyset ,B;A,B)-V(\emptyset ,A,B;A,B)\}+p\delta V\{(\emptyset ;A)-V(\emptyset ,A;A)\}\\> & {} \pi (A,B^{*};A,B)+\pi (A,B;A,B)-\pi (A,A^{*};A,B)-\pi (A,B^{*};A,B)\\{} & {} -\pi (A,A^{*};A,B)-\pi (B,A;A,B)+\pi (A,B^{*};A,B)+\pi (B,A^{*};A,B)\\{} & {} (\because \text {by the inequality derived from }V^{533}>V^{531}\text { in the above})\\= & {} 2\pi (A,B^{*};A,B)-2\pi (A,A^{*};A,B)=2(\lambda ^{\frac{1}{2}}-w-w^{*})-2(0-w-w^{*})=2\lambda ^{\frac{1}{2}}>0. \end{aligned}$$

Thus, $V^{41}>V^{43}$. Moreover, since $V^{41}>V^{42}$ under Claim 5, we have $V^{4}=V^{41}$. $\square$

Claim 10

$V^{51}>V^{533}=V^{53}$ and thus $V^{5}=V^{51}$.

Proof

We have

$$\begin{aligned} V^{51}-V^{533}= & {} V^{4}-V^{533}\\= & {} V^{41}-V^{533}\quad (\because \text { Claim}9)\\= & {} \pi (A,B^{*};A,B)+\pi (A,B;A,B)-\pi (A,A^{*};A,B)-\pi (B,A;A,B)\\{} & {} +(1-2p)\delta \{V(\emptyset ,B;A,B)-V(\emptyset ,A;A,B)\}\\{} & {} +p\delta \{V(\emptyset ,B;A)+V(\emptyset ,B;B)-V(\emptyset ,A;A)-V(\emptyset ,A;B)\}\\= & {} \pi (A,B^{*};A,B)-\pi (A,A^{*};A,B)=\lambda ^{\frac{1}{2}}-w-w^{*}-(0-w-w^{*})=\lambda ^{\frac{1}{2}}>0. \end{aligned}$$

Thus, $V^{51}>V^{533}$. Then, since we know $V^{533}=V^{53}$ and $V^{51}=V^{52}$, we have $V^{5}=V^{51}$. $\square$

Note that we have $V^{51}=V^{4}$, $V^{5}=V^{51}$ ($\because$ Claim 10), and $V^{4}=V^{41}$ ($\because$ Claim 9). Thus, $V^{5}=V^{41}$, which is the specialization value. This means that the specialization is optimal.

Case 2: $V^{533}\le V^{531}$. Then, since $V^{531}>V^{532}$ under Claim 6, we have $V^{53}=V^{531}$, which is the rotation value.

Claim 11

$V^{51}=V^{4}>V^{531}$.

Proof

By definition, $V^{51}=V^{4}$. Suppose to the contrary that $V^{51}=V^{4}\le V^{531}$. Then, $V^{531}\ge V^{4}=V^{51}=V^{52}$, which means that the rotation is optimal. However, this contradicts the initial argument that the rotation is not optimal. $\square$

Claim 12

$V^{41}\ge V^{43}$ and thus $V^{4}=V^{41}$.

Proof

Suppose to the contrary that $V^{41}<V^{43}$. Then, since $V^{42}<V^{41}$ ($\because$ Claim 5), we have $V^{4}=V^{43}$. Thus, by Claim 11, $V^{43}>V^{531}$. However, this inequality contradicts the following.

$$\begin{aligned} V^{43}-V^{531}= & {} \pi (A,A^{*};A,B)+\pi (A,B^{*};A,B)-\pi (A,B^{*};A,B)-\pi (B,A^{*};A,B)\\{} & {} +(1-2p)\delta \{V(\emptyset ,A,B;A,B)-V(\emptyset ,A,B;A,B)\}\\{} & {} +p\delta \{V(\emptyset ,A;A)+V(\emptyset ,B;B)-V(\emptyset ,A;A)-V(\emptyset ,B;B)\}\\= & {} \pi (A,A^{*};A,B)-\pi (B,A^{*};A,B)=-w-w^{*}-(\lambda ^{\frac{1}{2}}-w-w^{*})=-\lambda ^{\frac{1}{2}}<0. \end{aligned}$$

Hence, we have $V^{41}\ge V^{43}$. Moreover, as $V^{41}>V^{42}$ ($\because$ Claim 5), we have $V^{4}=V^{41}$. $\square$

Under claims 11 and 12, we have $V^{51}=V^{4}=V^{41}>V^{531}=V^{53}$. This means that the specialization is optimal.