A Proof of the relations
We give a proof for player 1’s first criterion (i.e., find a relationship between \(V_1^{1c}(\tau ,x)\) and \(V_1^{1c}(\tau +1,x)\)). In the case of player 2 and both players’ second criteria, the line of reasoning is the same. Using (29), construct the Bellman function of player 1 as the game reaches step \(\tau \):
$$\begin{aligned}&V_1^{1c}(\tau ,x)=\max _{u_{1\tau }^c,\ldots ,u_{1n}^c} \left\{ \frac{\theta _{\tau }}{\sum \nolimits _{l=\tau }^n \theta _l} \sum _{n_2=\tau }^{n} \frac{\omega _{n_2}}{\sum \nolimits _{l=\tau }^n \omega _l} \delta ^{\tau } p_1 u_{1\tau }^c\right. \\&\left. \qquad +\,\sum _{n_1=\tau +1}^{n} \frac{\theta _{n_1}}{\sum \nolimits _{l=\tau }^n \theta _l} \left[ \sum _{n_2=n_1}^{n} \frac{\omega _{n_2}}{\sum \nolimits _{l=\tau }^n \omega _l} \sum _{t=\tau }^{n_1}\delta ^t p_1 u_{1t}^c +\sum _{n_2=\tau }^{n_1-1} \frac{\omega _{n_2}}{\sum \nolimits _{l=\tau }^n \omega _l} \sum _{t=\tau }^{n_2}\delta ^t p_1 u_{1t}^c+V_1^a(\tau ,n_1)\right] \right\} \\&\quad =\frac{\theta _{\tau }}{\sum \nolimits _{l=\tau }^n \theta _l} \delta ^{\tau } p_1 u_{1\tau }^c + \sum _{n_1=\tau +1}^{n} \frac{\theta _{n_1}}{\sum \nolimits _{l=\tau }^n \theta _l} \left[ \sum _{n_2=n_1}^{n} \frac{\omega _{n_2}}{\sum \nolimits _{l=\tau }^n \omega _l} \left( \sum _{t=\tau +1}^{n_1}\delta ^t p_1 u_{1t}^c+\delta ^{\tau } p_1u_{1\tau }^c\right) \right. \\&\left. \qquad +\sum _{n_2=\tau }^{n_1-1} \frac{\omega _{n_2}}{\sum \nolimits _{l=\tau }^n \omega _l} \left( \sum _{t=\tau +1}^{n_2}\delta ^t p_1 u_{1t}^c+\delta ^{\tau } p_1 u_{1\tau }^c\right) +V_1^a(\tau ,n_1)\right] =\delta ^{\tau } p_1 u_{1\tau }^c +\sum _{n_1=\tau +1}^{n} \frac{\theta _{n_1}}{\sum \nolimits _{l=\tau }^n \theta _l} \\&\qquad \cdot \left[ \sum _{n_2=n_1}^{n} \frac{\omega _{n_2}}{\sum \nolimits _{l=\tau }^n \omega _l} \sum _{t=\tau +1}^{n_1}\delta ^t p_1 u_{1t}^c +\sum _{n_2=\tau +1}^{n_1-1} \frac{\omega _{n_2}}{\sum \nolimits _{l=\tau }^n \omega _l} \sum _{t=\tau +1}^{n_2}\delta ^t p_1 u_{1t}^c+V_1^a(\tau ,n_1)\right] \\&\quad =\delta ^{\tau } p_1 u_{1\tau }^c+ \sum _{n_1=\tau +1}^{n} \frac{\theta _{n_1}}{\sum \nolimits _{l=\tau +1}^n \theta _l} \frac{\sum \nolimits _{l=\tau +1}^n\theta _l}{\sum \nolimits _{l=\tau }^n\theta _l} \left[ \sum _{n_2=n_1}^{n} \frac{\omega _{n_2}}{\sum \nolimits _{l=\tau +1}^n \omega _l} \frac{\sum \nolimits _{l=\tau +1}^n\omega _l}{\sum \nolimits _{l=\tau }^n\omega _l}\sum _{t=\tau +1}^{n_1}\delta ^t p_1 u_{1t}^c\right. \\&\left. \qquad +\sum _{n_2=\tau +1}^{n_1-1} \frac{\omega _{n_2}}{\sum \nolimits _{l=\tau +1}^n \omega _l} \frac{\sum \nolimits _{l=\tau +1}^n\omega _l}{\sum \nolimits _{l=\tau }^n\omega _l} \sum _{t=\tau +1}^{n_2}\delta ^t p_1 u_{1t}^c + \frac{\omega _{\tau }}{\sum \nolimits _{l=\tau }^{n}\omega _l}\sum _{t=\tau }^{n_1}\delta ^t p_1 u_{1t}^a +\frac{\sum \nolimits _{l=\tau +1}^{n}\omega _l}{\sum \nolimits _{l=\tau }^{n}\omega _l} V_1^a(\tau +1,n_1)\right] \\&\quad =\delta ^{\tau } p_1 u_{1\tau }^c+ P_{\tau }^{\tau +1} V_1^{1c}(\tau +1,x) +C_{1\tau }\sum _{n_1=\tau +1}^n \theta _{n_1}\sum _{t=\tau }^{n_1}\delta ^t p_1 u_{1t}^a, \end{aligned}$$
where
$$\begin{aligned} P_{\tau }^{\tau +1}=\frac{\sum \nolimits _{l=\tau +1}^n\omega _l}{\sum \nolimits _{l=\tau }^n\omega _l} \frac{\sum \nolimits _{l=\tau +1}^n\theta _l}{\sum \nolimits _{l=\tau }^n\theta _l}, C_{1\tau } =\frac{\omega _{\tau }}{\sum \nolimits _{l=\tau }^{n}\omega _l} \frac{1}{\sum \nolimits _{l=\tau }^{n}\theta _l} . \end{aligned}$$
Similarly, we establish a relationship between \(V_2^{1c}(\tau ,x)\) and \(V_2^{1c}(\tau +1,x)\) in the form
$$\begin{aligned} V_2^{1c}(\tau ,x)=\delta ^{\tau } p_2 u_{2\tau }^c + P_{\tau }^{\tau +1} V_2^{1c}(\tau +1,x)+ C_{2\tau }\sum _{n_2=\tau +1}^n\omega _{n_2}\sum _{t=\tau }^{n_2}\delta ^t p_2 u_{2t}^a, \end{aligned}$$
where
$$\begin{aligned} C_{2\tau } =\frac{\theta _{\tau }}{\sum \nolimits _{l=\tau }^{n}\theta _l} \frac{1}{\sum \nolimits _{l=\tau }^{n}\omega _l}. \end{aligned}$$
Hence,
$$\begin{aligned} V_1^{c}(\tau ,x)= & {} V_1^{1c}(\tau ,x)+V_2^{1c}(\tau ,x)=\delta ^{\tau }(p_1 u_{1\tau }^c+p_2 u_{2\tau }^c)+ P_{\tau }^{\tau +1} V_1^{c}(\tau +1,x)\\&+C_{1\tau }\sum _{n_1=\tau +1}^n\theta _{n_1}\sum _{t=\tau }^{n_1}\delta ^t p_1 u_{1t}^a +C_{2\tau }\sum _{n_2=\tau +1}^n\omega _{n_2}\sum _{t=\tau }^{n_2}\delta ^t p_2 u_{2t}^a. \end{aligned}$$
According to the above, by analogy we get the relations for the second cooperative payoffs of both players.
B Proof of Theorem 2
Our analysis begins with the occurrence of step n. Both players have zero payoffs at the next step \(n+1\). Hence, the optimal cooperative strategies coincide with the Nash equilibrium ones, and the payoffs have the form
$$\begin{aligned} V_1^{c}(n,x)=A_1 x, V_2^{c}(n,x)=A_2 x^2 . \end{aligned}$$
Now, suppose that the game reaches step \(n-1\). In this case, the problem (35) reduces to
$$\begin{aligned} (V_1^{c}(n-1,x)-A_1 x)(V_2^{c}(n-1,x)-A_2 x^2) \rightarrow \max _{u_{1n-1}^c,u_{2n-1}^c}, \end{aligned}$$
(40)
where
$$\begin{aligned} V_1^{c}(n-1,x)= & {} \delta ^{n-1} (p_1 u_{1 n-1}^c+p_2 u_{2 n-1}^c)+P_{n-1}^{n} V_1^{c}(n,\varepsilon x-u_{1 n-1}^c-u_{2 n-1}^c)\\&+C_{1 n-1}\theta _n\delta ^n p_1 u_{1n}^a +C_{2 n-1}\omega _n\delta ^n p_2 u_{2n}^a, \\ V_2^{c}(n-1,x)= & {} -2\delta ^{n-1}m u_{1 n-1}^c u_{2 n-1}^c + P_{n-1}^{n} V_2^{c}(n,\varepsilon x-u_{1 n-1}^c-u_{2 n-1}^c). \end{aligned}$$
As usual, we seek for the linear strategies of the form \(u_{i n-1}^c=\gamma _{i n-1}^c x\). From the first-order conditions we get the relation
$$\begin{aligned} \gamma _{2 n-1}^c= & {} \frac{\varepsilon A_2 P_{n-1}^n(p_1-p_2)}{m(\delta ^{n-1}p_1-P_{n-1}^n A_1)+P_{n-1}^n A_2(p_1-p_2)}\nonumber \\&+\gamma _{1 n-1}^c\frac{ m(\delta ^{n-1}p_1-P_{n-1}^n A_1)-P_{n-1}^n A_2(p_1-p_2)}{m(\delta ^{n-1}p_1-P_{n-1}^n A_1)+P_{n-1}^n A_2(p_1-p_2)}. \end{aligned}$$
(41)
Now, we study the situation when step \(n-2\) occurs in the game. Then the problem (35) takes the form
$$\begin{aligned} \left( V_1^{c}(n-2,x)-A_1 x\right) \left( V_2^{c}(n-2,x)-A_2 x^2\right) \rightarrow \max _{u_{1n-2}^c,u_{2n-2}^c}, \end{aligned}$$
(42)
where
$$\begin{aligned} V_1^c(n-2,x)= & {} \delta _1^{n-2} (p_1 u_{1 n-2}^c+p_2 u_{2 n-2}^c) +P_{n-2}^{n-1} V_1^c(n-1,\varepsilon x-u_{1 n-2}^c-u_{2 n-2}^c)\\&+\,C_{1 n-2}\sum \limits _{t=n-1}^n\theta _{t}\delta ^t p_1 u_{1t}^a+ C_{2 n-2}\sum \limits _{t=n-1}^n\omega _{t}\delta ^t p_2 u_{2t}^a, \\ V_2^c(n-2,x)= & {} -2m\delta ^{n-2} u_{1 n-2}^c u_{2 n-2}^c+ P_{n-2}^{n-1} V_2^c(n-1,\varepsilon x-u_{1 n-2}^c-u_{2 n-2}^c). \end{aligned}$$
Searching for the linear strategies \(u_{i n-2}^c=\gamma _{i n-2}^c x\), from the first-order optimality conditions we obtain the following relationship between equilibrium strategies of the players when step \(n-2\) occurs in the game:
$$\begin{aligned}&\gamma _{2 n-2}^c=1/\Bigl (m\Bigl [\delta ^{n-2}p_2-P_{n-2}^{n-1}(\delta ^{n-1}(p_1\gamma ^c_{1n-1} +p_2\gamma ^c_{2n-1})\nonumber \\&\quad +P_{n-2}^{n-1}A_1(\varepsilon -\gamma ^c_{1n-1}-\gamma ^c_{2n-1}))\nonumber \\&\quad +P_{n-2}^{n-1}(p_1-p_2)\Bigl (-2\delta ^{n-1}m\gamma ^c_{1n-1}\gamma ^c_{2n-1}+P_{n-2}^{n-1} A_2 (\varepsilon -\gamma ^c_{1n-1}-\gamma ^c_{2n-1})^2\Bigr )\Bigr ]\Bigr )\nonumber \\&\quad \cdot \varepsilon P_{n-2}^{n-1}(p_1-p_2)(-2\delta ^{n-1}m \gamma ^c_{1n-1}\gamma ^c_{2n-1})+P_{n-2}^{n-1}A_2(\varepsilon -\gamma ^c_{1n-1} -\gamma ^c_{2n-1})^2\nonumber \\&\quad +\gamma ^c_{1n-2}\Bigl [\delta ^{n-2}p_1-P_{n-2}^{n-1}(\delta ^{n-1} (p_1\gamma ^c_{1n-1}+p_2\gamma ^c_{2n-1})\nonumber \\&\quad +P_{n-2}^{n-1}A_1(\varepsilon -\gamma ^c_{1n-1}-\gamma ^c_{2n-1}))\nonumber \\&\quad -P_{n-2}^{n-1}(p_1-p_2)\Bigl (-2\delta ^{n-1}m\gamma ^c_{1n-1}\gamma ^c_{2n-1}+P_{n-2}^{n-1} A_2 (\varepsilon -\gamma ^c_{1n-1}-\gamma ^c_{2n-1})^2\Bigr )\Bigr ] \, \end{aligned}$$
(43)
and the relationship between equilibrium strategies of the players when the game reaches steps \(n-1\) and \(n-2\):
$$\begin{aligned} \frac{\gamma _{2 n-2}^c-\gamma _{1 n-2}^c}{\varepsilon - \gamma _{1 n-2}^c-\gamma _{2 n-2}^c}=\delta \left( \gamma _{2 n-1}^c-\gamma _{1 n-1}^c\right) . \end{aligned}$$
(44)
By continuing the described process until the game reaches step k, we easily obtain the payoffs (36), (37) and the cooperative strategies of the form (38), (39).
