Stochastic Game with Interactive Information Acquisition:
Pipelined Perfect Markov Bayesian Equilibrium
Version 05 October, 2023

A finite-player infinite-horizon stochastic game can be characterized by a tuple $\mathtt{B}=\left<N,S,\left\{A_{i}\right\},\left\{R_{i}\right\},T,\mathring{T},\delta\right>$ in which

• •

  There is a finite number of players, denoted by $N=\{1,2,\dots,n\}$.

• •

  The finite set of states that players can encounter at each period is denoted by $S$.

• •

  The finite set of actions that player $i$ can take is denoted by $A_{i}$.

• •

  The payoff function of player $i$ is denoted by: $R_{i}:S\times A\mapsto\mathbb{R}$, where $A=\prod_{i\in N}A_{i}$.

• •

  The state evolves over time according to $T:S\times A\mapsto\Delta\left(S\right)$; i.e., the probability of $s_{t+1}$ is given by $T(s_{t+1}|s_{t},a_{t})$ when period-$t$ state is $s_{t}\in S$ and action profile is $a_{t}\in A$. $\mathring{T}(\cdot)\in\Delta(S)$ is the initial distribution of the state.

• •

  $\delta\in(0,1)$ is the common discount factor.

For notational compactness, we use history, denoted by $h_{t}\in H$, to capture the pair of state and action profile of the last period; i.e., $h_{t}\equiv(s_{t-1},a_{t-1})$ and $H\equiv S\times A$. We assume that $h_{t}$ is common knowledge; i.e., $s_{t}$ and $a_{t}$ become publicly observable at the end of each period $t$.

We consider that the agents do not observe the realizations of the state at the beginning of each period. Instead, in each period $t$, the agents engage in interactive information acquisition to obtain additional information about the unobserved state $s_{i,t}\in S$.

Based on history $h_{t}\in H$, agents’ interactive information acquisition leads to an information profile, denote by $\mathcal{I}\left[h_{t}\middle|g_{t}\right]\equiv\left<p(\cdot|h_{t},g_{t}),\tau(\cdot,g_{t}),\Theta\right>$ at the beginning of each period $t$, where $g_{t}\equiv(g_{i,t})_{i\in N}\in G\equiv\prod_{i\in N}G_{i}$ represents the profile of the agents’ choices of information acquisition. Here, $\Theta\equiv\prod_{i\in N}\Theta_{i}$ is the profile of finite type spaces for the agents. $p(\cdot|h_{t},g_{t})\in\Delta(S\times\Theta)$ is the joint probability of the state and the type profile. $\tau(\cdot|s_{t},g_{t})\in\Delta(\Theta)$ is the signaling rule profile that generates a type profile $\theta_{t}=(\theta_{i,t})_{i\in N}\in\Theta$ for the agents. The signaling rule profile is independent if $\tau(\theta_{t}|s_{t},g_{t})=\prod_{i\in N}\tau_{i}(\theta_{i,t}|s_{t},g_{i,t})$, or correlated if the marginal concerning each agent $i$, $\tau_{i}(\theta_{i,t}|s_{t},g_{i,t},g_{-i,t})$, depends on $g_{-i,t}$. In this work, we focus on independent signaling rules. In addition, we refer to $g_{t}=(g_{i,t})_{i\in N}\in G$ as the cognition choice profile where each $g_{i,t}\in G_{i}$ is agent $i$’s cognition choice that indexes the agent’s choice of $\tau_{i}(\cdot|s_{t},g_{i,t})$. We assume that the cardinalities of $\mathcal{G}_{i}$ and $\Theta_{i}$ are the same for all agents; i.e., $\left|\mathcal{G}_{i}\right|=\left|\mathcal{G}_{j}\right|$ and $\left|\Theta_{i}\right|=\left|\Theta_{j}\right|$, for all $i\neq j$.

Given the state dynamics specified by the base game $\mathtt{B}$, we assume that the information profile $\mathcal{I}\left[h_{t}\middle|g_{t}\right]$ satisfy

	$\displaystyle\sum\nolimits_{\theta_{t}\in\Theta}p\left(s_{t},\theta_{t}\middle|h_{t},g_{t}\right)=T\left(s_{t}\middle|h_{t}\right),$		(1)
	$\displaystyle\tau\left(\theta_{t}\middle|s_{t},g_{t}\right)=\frac{p(s_{t},\theta_{t}|h_{t},g_{t})}{T(s_{t}|h_{t})}.$		(2)

Given the base game $\mathtt{B}$, define the set of available signaling rule profiles by

$$\mathcal{T}\left[G,\Theta\right]\equiv\left\{\tau\middle|\begin{aligned} &\tau_{i}\left(\cdot,g_{i,t}\right):S\mapsto\Delta\left(\Theta_{i}\right),\forall i\in N,\\ &g_{i,t}\in G_{i},\textup{s.t. }\exists p(\cdot|h_{t},g_{t})\in\Delta(S\times\Theta)\\ &\textup{ satisfying }(\ref{eq:common_prior_cond_1})\textup{ and }(\ref{eq:common_prior_cond_2}),\forall h_{t}\in H\end{aligned}\right\}.$$

(3)

After all agents made their cognition choices, each agent $i$ privately receives a type $\theta_{i,t}\in\Theta_{i}$ with probability $\tau_{i}(\theta_{i,t}|s_{t},g_{i,t})$. Based on his type, agent $i$ chooses an action $a_{i,t}\in A_{i}$.

Each agent $i$’s information acquisition induces a cognition cost. In this work, we restrict attention to the cognition cost that depends on the true state and the action taken by the agent. That is, after choosing $g_{i,t}$, agent $i$ suffers a cost $C_{i}(s_{t},a_{i,t})$ that is realized at the end of each period when the true state is $s_{i,t}$ and agent $i$ takes action $a_{i,t}$. This cost scheme prices agent $i$’s information acquisition based on the consequences of the information acquisition (i.e., the agent’s local action $a_{i,t}$) when the true state is $s_{i,t}$.

Let $\Gamma\equiv\left\{\mathcal{T}^{\dagger}\left[G,\Theta\right],C\right\}$ denote the cognition scheme for some $\mathcal{T}^{\dagger}\left[G,\Theta\right]\subseteq\mathcal{T}\left[G,\Theta\right]$, where $C=\{C_{i}\}_{i\in N}$. The base game $\mathtt{B}$ and the cognition scheme $\Gamma$ induces a stochastic game with interactive information acquisition (SGIA), denoted by $\mathtt{B}^{\Gamma}$. Each agent $i$’s decision making in each period $t$ is described as follows.

• •

  Contingent on the history $h_{t}$, agent $i$ uses a pure-strategy selection policy $\beta_{i,t}(h_{t})\in G_{i}$ to select $g_{i,t}\in G_{i}$.

• •

  Contingent on the type $\theta_{i,t}$, agent $i$ uses a mixed-strategy policy $\pi_{i}$ to choose mixed action $\pi_{i,t}(\theta_{i,t},h_{t})=\left(\pi_{i}(a|\theta_{i,t},h_{t})\right)_{a\in\mathcal{A}_{i}}\in\Delta(A_{i})$.

We say that a policy profile $\pi=(\pi_{i,t},\pi_{-i,t})_{t\geq 1}$ is feasible if it satisfies the following constraints:

	$\displaystyle\pi_{i,t}(a_{i,t}|\theta_{i,t})\geq 0,\forall a_{i,t}\in A_{i},\theta_{i,t}\in\Theta_{i},i\in N,t\geq 1,$			(FE1)
	$\displaystyle\sum\nolimits_{a_{i,t}\in A_{i}}\pi_{i,t}(a_{i,t}|\theta_{i,t})=1,\forall\theta_{i,t}\in\Theta_{i},i\in N,t\geq 1.$			(FE2)

With reference to Fig. 1, the following events occur in each period $t$ of $\mathtt{B}^{\Gamma}$.

• 1.

  Nature draws a state $s_{t}\in S$ according to $T(\cdot|h_{t})\in\Delta\left(S\right)$.

• 2.

  Each agent $i$ selects a cognition choice $g_{i,t}\in\mathcal{G}$, based on the common history, which determines $\tau_{i}$. These selections are simultaneous.

• 3.

  Based on the cognition choice profile $g_{t}$, a type profile $\theta_{t}=\left(\theta_{i,t}\right)_{i\in N}$ is drawn with probability $\tau(\theta_{t}|s_{t},g_{t})$.

• 4.

  Each agent $i$ privately observes his type $\theta_{i,t}$ and then chooses an action $a_{i,t}$ with probability $\pi_{i,t}(a_{i,t}|\theta_{i,t})$.

• 5.

  The state $s_{t}$ and the action profile $a_{t}=(a_{i,t},a_{-i,t})$ (or, $h_{t+1}=(s_{t},a_{t})$) become public information, and the state $s_{t}$ is transitioned to a new state $s_{t+1}$ according to $T(\cdot|h_{t+1})\in\Delta(S)$.

According to the Ionescu Tulcea theorem [16], $\{\mathring{T},T,\tau\}$, the agents’ policy profile $<\beta,\pi>$, and a cost profile (for a certain cost scheme) uniquely define a probability measure, denoted by $P[\tau,\beta,\pi]$, over $(S\times G\times\Theta\times A)^{\infty}$. Let $\mathbb{E}^{\tau}_{\beta,\pi}[\cdot]$ denote the expectation operator with respect to $P[\tau,\beta,\pi]$. In addition, given any $h_{t}$ and $(h_{t},\theta_{i,t})$, we obtain unique probability measures (perceived by agent $i$) $P[\tau,\beta,\pi|h_{t}]$ and $P[\tau,\beta,\pi|h_{t},\theta_{i,t}]$ over $S\times G_{-i}\times\Theta\times A\times(S\times G\times\Theta\times A)^{\infty}$ and $S\times\Theta_{-i}\times A\times(S\times G\times\Theta\times A)^{\infty}$, respectively, for all $i\in N$. In particular, $P[\tau,\beta,\pi|h_{t}]$ models the uncertainty at period $t$ for each agent $i$ at the beginning of the selection stage while $P[\tau,\beta,\pi|h_{t},\theta_{i,t}]$ models the uncertainty for each agent $i$ at the beginning of the primitive stage. Let $\mathbb{E}^{\tau}_{\beta,\pi}[\cdot|h_{t}]$ and $\mathbb{E}^{\tau}_{\beta,\pi}[\cdot|h_{t},\theta_{i,t}]$, respectively, denote the expectation operators with respect to $P[\tau,\beta,\pi|h_{t}]$ and $P[\tau,\beta,\pi|h_{t},\theta_{i,t}]$.

Given $P[\tau,\beta,\pi|h_{t}]$, agent $i$’s period-$t$ history value function (H value function) is defined by

$\displaystyle J_{i,t}(h_{t}|\tau,\beta,\pi)\equiv\mathbb{E}^{\tau}_{\beta,\pi}\left[\sum_{k=t}^{\infty}\delta^{k-t}\left(R_{i}(\tilde{s}_{k},\tilde{a}_{k})+C_{i}\left(\tilde{s}_{k},\tilde{a}_{i,k}\right)\right)\middle|h_{t}\right].$

(H)

After a type $\theta_{i,t}$ is realized, each agent $i$ forms a posterior belief, denoted by $\mu_{i}(\cdot|\theta_{i,t},h_{t})\in\Delta(S\times\Theta_{-i})$, over the state $s_{t}$ and other agents’ contemporaneous types $\theta_{-i,t}$. Due to (3), there is a (period-$t$) common prior $p$ such that each posterior belief satisfies

$$\mu_{i}\left(s_{t},\theta_{-i,t}\middle|\theta_{i,t},h_{t},g_{t}\right)=\frac{p\left(s_{t},\theta_{i,t},\theta_{-i,t}\middle|h_{t},g_{t}\right)}{\sum\nolimits_{s_{t},\theta_{-i,t}}p\left(s_{t},\theta_{i,t},\theta_{-i,t}\middle|h_{t},g_{t}\right)}.$$

With abuse of notation, we use $\mu_{i}(s_{t}|\theta_{i,t},h_{t})$ and $\mu_{i}(\theta_{-i,t}|\theta_{i,t},h_{t})$ for the marginals of $\mu_{i}(s_{t},\theta_{-i,t}|\theta_{i,t},h_{t})$. Given $h_{t}$ and $\theta_{i,t}$, we define each agent $i$’s expected immediate reward (due to agent $i$’s uncertainty about $s_{t}$) by

$$\overline{R}_{i}\left(h_{t},\theta_{i,t},a_{t}\right)\equiv\sum\limits_{s_{t}\in S}\left(R_{i}(s_{t},a_{t})+C_{i,t}\left(s_{t},a_{i,t}\right)\right)\mu_{i}(s_{t}|\theta_{i,t},h_{t}).$$

Given $P[\tau,\beta,\phi|h_{t}]$, agent $i$’s period-$t$ history-type value function (HT value function) is defined by

		$\displaystyle V_{i,t}(h_{t},\theta_{t}|\tau,\beta,\pi)\equiv\mathbb{E}^{\tau}_{\beta,\pi(\cdot|\theta_{t})}\Bigg{[}\overline{R}_{i}(h_{t},\theta_{i,t},\tilde{a}_{t})$		(HT)
		$\displaystyle+\sum_{k=t+1}^{\infty}\delta^{k-t+1}\left(R_{i}(\tilde{s}_{i,k},\tilde{a}_{k})+C_{i,k}\left(\tilde{s}_{i,k},\tilde{a}_{i,k}\right)\right)\Bigg{|}h_{t},\theta_{i,t}\Bigg{]}.$

Here, $V_{i,t}(\cdot)$ depends on $\theta_{-i,t}$ through the policy profile $\pi(\cdot|\theta_{t})$ to take expectation over the current-period action profile. Finally, agent $i$’s period-$t$ history-type-action value function (HTA value function) is defined by

		$\displaystyle Q_{i,t}(h_{t},\theta_{i,t},a_{t}|\tau,\beta,\pi)\equiv\overline{R}_{i}(h_{t},\theta_{i,t},a_{t})$		(HTA)
		$\displaystyle+\mathbb{E}^{\tau}_{\beta,\pi}\left[\sum_{k=t+1}^{\infty}\delta^{k-t+1}\left(R_{i}(\tilde{s}_{i,k},\tilde{a}_{k})+C_{i}\left(\tilde{s}_{i,k},\tilde{a}_{k}\right)\right)\middle|h_{t},\theta_{i,t}\right].$

Here, $Q_{i,t}(\cdot)$ is independent of $\theta_{-i,t}$ given the action profile $a_{t}$.

In this section, we define a new equilibrium concept for the game $\mathtt{B}^{\Gamma}$. Our focus lies on the stationary equilibrium, and as such, we omit the time indexes of the value functions and variables, unless explicitly mentioned otherwise. First, we construct

	$\displaystyle\mathtt{EV}_{i}\left(h,\theta_{i}\middle|\tau,\beta,\pi;V_{i}\right)\equiv\mathbb{E}^{\tau}_{\beta}\left[V_{i}(h,\theta_{i},\tilde{\theta}_{-i}|\beta,\tau,\pi)\middle|h,\theta_{i}\right].$		(4)
	$\displaystyle\Pi\left[\beta;\mathtt{B}^{\Gamma}\right]\equiv\left\{\pi\middle|\begin{aligned} &\mathtt{EV}_{i}\left(h,\theta_{i}\middle|\tau,\beta,\left(\pi_{i},\pi_{-i}\right);V_{i}\right)\\ &\geq\mathtt{EV}_{i}\left(h,\theta_{i}\middle|\tau,\beta,\left(\pi^{\prime}_{i},\pi_{-i}\right);V_{i}\right),\\ &\forall i\in N,\theta_{i}\in\Theta_{i},\pi^{\prime}_{i},(\ref{eq:FE1}),(\ref{eq:FE2})\end{aligned}\right\}.$		(5)

The equilibrium concept of PPME builds upon the concepts of Markov perfect equilibrium [8] and perfect Bayesian equilibrium [9].

Hence, in a PPME problem, each agent $i\in N$ tries to solve the optimization problem given $h\in H$:

$\displaystyle\max\limits_{\beta_{i},\pi_{i}}$

$\displaystyle J_{i}\left(h\middle|\tau,\left(\beta_{i},\beta_{-i}\right),\left(\pi_{i},\pi_{-i}\right)\right),\textup{ s.t. }\left(\pi_{i},\pi_{-i}\right)\in\Pi\left[\beta_{i},\beta_{-i};\mathtt{B}^{\Gamma}\right].$

(9)

First, we extend the agents’ strategy profile $\left<\beta,\pi\right>$ to $\left<\beta,\pi,V\right>$. Given $V_{i}$ as a variable, we define the following term based on (4):

$\displaystyle\mathtt{MV}_{i,t}\left(h_{t},\theta_{i,t}\middle|\tau,\beta;V_{i,t}\right)\equiv\mathbb{E}^{\tau}_{\beta}\left[V_{i,t}(h_{t},\theta_{i,t},\tilde{\theta}_{-i,t})\middle|h,\theta_{i}\right].$

(14)

If we fix a $\beta$, Proposition 1 implies that $\left<\pi,V\right>$ of a PPME profile $\left<\beta,\pi,V\right>$ needs to be a global minimum of (OPT) with $Z(\pi,V|\tau,\beta)=0$. Equivalently, given $\pi_{-i}$, each $V_{i}$ needs to be a fixed point of the following equation, for all $i\in N$, $h\in H$, $\theta_{i}\in\Theta_{i}$,

$$\begin{cases}&\begin{aligned} &\mathtt{MV}_{i}(h,\theta_{i}|\tau,\beta;V_{i})\\ &\geq\mathbb{E}^{\tau}_{\beta_{-i},\pi_{-i}}\Big{[}\mathbf{Q}_{i}(h,\theta_{i},a_{i},\tilde{a}_{-i}|\tau,\beta;V_{i})\Big{|}\theta_{i},h\Big{]},\end{aligned}\\ &\forall a_{i}\in A_{i},i\in N,h\in H,\theta_{i}\in\Theta_{i}.\end{cases}$$

(EQ4)

where dependence of the RHS of (EQ4) on $V_{i}$ is due to (12). At the cognition stage, the optimality of PPME requires that the agents’ choice $g$ among all possible options is optimal. Given any $J_{i}(\cdot)$, define, for all $i\in N$, $\theta_{i}\in\Theta_{i}$, $h\in H$,

		$\displaystyle\mathtt{IJ}_{i}(h,\theta_{i}|\tau_{-i},\beta_{-i},\pi;J_{i})\equiv\sum\limits_{s,\theta_{-i},a}\Big{(}\overline{R}_{i}(h,\theta_{i},a)+J_{i}\left(s,a\right)\Big{)}$		(15)
		$\displaystyle\times\pi(a|\theta_{i},\theta_{-i})\tau_{-i}\left(\theta_{-i}|s,\beta_{-i}(h)\right)T(s|h),$

where $J_{i}(s,a)$ on the RHS of (15) is a H value function of the next period given current-period $(s,a)$. The optimality of $\tau$ in the cognition stage of a PPME (i.e., constraint (EQ1)) implies that the optimal history value function $J_{i}$ for each agent $i$ needs to be a fixed point while fixing others’ $\tau_{-i}$. That is,

$$\begin{cases}&J_{i}(h)\geq\mathtt{IJ}_{i}(h,\theta_{i}|\tau_{-i},\beta_{-i},\pi;J_{i}),\\ &\forall\theta_{i}\in\Theta_{i},i\in N,h\in H.\end{cases}$$

(EQ3)

Here, (EQ3) is independent of $V$ while (EQ4) is independent of $J$. In order to make $<\beta,\pi>$ as a PPME of $\mathtt{B}^{\Gamma}$, $<\beta,\pi>$ must be chosen such that there exist $\left<J,V\right>$ satisfying

$$\left\{\begin{aligned} &\textup{$J$ is a fixed point of (\ref{eq:OB1}) if and only if}\\ &\textup{$V$ is a fixed point of (\ref{eq:EQ4}).}\end{aligned}\right\}$$

We refer to such a procedure as the Global Fixed-Point Alignment (GFPA).

Since $\beta_{i}$ is a pure strategy, each deterministic choice of $g_{i}=\beta_{i}(h)$ leads to a signaling rule $\tau_{i}(\cdot|\cdot,g_{i})$ that determines a distribution of agent $i$’s period-$t$ types. That is, every $\beta_{i}$ determines a unique $\tau_{i}(\cdot,g_{i}):S\mapsto\Delta(\Theta_{i})$ for every $h$. Hence, for ease of exposition, we use $\tau_{i}$ and $\tau_{i,t}(\cdot|\cdot,g_{i},h)$ (with abuse of notation) to represent $\beta_{i}$ and $\beta_{i}(h)$, respectively; unless otherwise stated. Therefore, in game $\mathtt{B}^{\Gamma}$, each agent $i$ controls $\left<\tau_{i},\pi\right>$.

Given a $\pi$, define the following function of $\tau$, $J$, and $V$:

		$\displaystyle Z^{\mathtt{GFPA}}(\tau,J,V|\pi)$		(OBJ2)
		$\displaystyle\equiv\sum\nolimits_{i,h}\left(J_{i}(h)-\sum\nolimits_{\theta,s}V_{i}(h,\theta)\tau(\theta|s,g,h)T_{s}(s|h)\right).$

Similar to (FE1) and (FE2) for $\pi$, we introduce the following two constraints placed on $\tau$:

	$\displaystyle\tau_{i}(\theta_{i}|s,g_{i},h)\geq 0,\forall i\in N,\theta_{i}\in\Theta_{i},s\in S,g\in G,h\in H,$		(RG1)
	$\displaystyle\sum\nolimits_{\theta_{i}\in\Theta_{i}}\tau(\theta_{i}|s,g_{i},h)=1,\forall i\in N,s\in S,g\in G,h\in H.$		(RG2)

Define the following set:

$$\mathcal{K}^{\mathtt{GFPA}}(\pi)\equiv\left\{\left<\tau,J,V\right>\middle|(\ref{eq:regular_tau}),(\ref{eq:feasible_tau}),(\ref{eq:OB1}),(\ref{eq:EQ4})\right\}.$$

(16)

Let

$\displaystyle\mathcal{E}^{\mathtt{GFPA}}(\pi)=\left\{\begin{aligned} \arg\min\limits_{\tau,J,V}&\;Z^{\mathtt{GFPA}}(\tau,J,V|\pi),\\ \textup{ s.t. }&\left<\tau,J,V\right>\in\mathcal{K}^{\mathtt{GFPA}}(\pi)\end{aligned}\right\}.$

(GFPA)

The proof of Proposition 2 is deferred to Appendix -F. Proposition 2 shows that if $\left<\tau,J,V\right>\in\mathcal{E}^{\mathtt{GFPA}}(\pi)$ with $Z^{\mathtt{GFPA}}\left(\tau,J,V\middle|\pi\right)=0$ for a policy $\pi$, then the profile $\left<\tau,\pi\right>$ is a PPME.

First, we decompose each type space $\Theta_{i}$ into $\Theta_{i}=\Theta^{\natural}_{i}\cup\{\hat{\theta}_{i}\}$ such that the constraint (RG2) can reformulated as, for all $i\in N$, $s\in S$, $g\in G$, $h\in H$,

		$\displaystyle\sum\nolimits_{\theta_{i}\in\Theta^{\natural}_{i}}\tau_{i}\left(\theta_{i}\middle|s,g_{i},h\right)\leq 1,$		(17)
		$\displaystyle\tau_{i}\left(\hat{\theta}_{i}\middle|s,g_{i},h\right)+\sum\nolimits_{\theta_{i}\in\Theta^{\natural}_{i}}\tau_{i}\left(\theta_{i}\middle|s,g_{i},h\right)=1.$

Given $\tau_{-i}(\theta_{-i}|s,g_{-i},h)=\prod_{j\neq i}\tau_{j}(\theta_{i}|s,g_{j},h)$, define for all $i\in N$,

$\displaystyle\mathtt{IV}_{i}(h,\theta_{i}|\tau_{-i};V_{i})\equiv\sum\limits_{\theta_{-i},s}V_{i}(h,\theta_{i},\theta_{-i})\tau_{-i}(\theta_{-i}|s,g_{-i},h)T_{s}(s|h).$

Construct the vector $X^{s}_{i}\equiv\left(J_{i}(h),V_{i}(h,\cdot),\tau_{-i}\left(\cdot|s,g_{-i},h\right)\right)$. Define

$$\lambda_{i}\left(X^{s}_{i};\theta_{i},h\right)\equiv J_{i}(h)-\mathtt{IV}_{i}(h,\theta_{i}|\tau_{-i};V_{i}).$$

Here, $\lambda_{i}\left(X^{s}_{i};\theta_{i},h\right)$ is a function of $J_{i}$, $V_{i}$, and $\tau_{-i}(\cdot|s,g_{-i},h)$ and is independent of $\pi$ and $\tau_{i}$.

For any $s\in S$, $h\in H$, $\theta_{i}\in\Theta_{i}$, define the following function

	$\displaystyle Z^{\mathtt{LFPA}}_{i}\left(X^{s}_{i},\tau_{i};s,h\right)$	$\displaystyle\equiv\sum\nolimits_{\theta^{\prime}_{i}\in\Theta^{\natural}_{i}}\lambda_{i}(X^{s}_{i};\theta^{\prime}_{i},h)\tau_{i}(\theta^{\prime}_{i}|s,g_{i},h)$		(18)
		$\displaystyle+\lambda_{i}(X^{s}_{i};\hat{\theta}_{i},h)\tau_{i}(\hat{\theta}_{i}|s,g_{i},h).$

Define the following set

$\displaystyle\overline{\mathcal{K}}\left[s,g_{-i},h\right]\equiv\left\{\left<X^{s}_{i},\tau_{i},\right>\middle|\begin{aligned} &\tau_{i}(\hat{\theta}_{i}|s,g_{i},h)\geq 0\\ &\tau_{i}(\theta_{i}|s,g_{i},h)\geq 0,\forall\theta_{i}\in\Theta^{\natural}_{i}\\ &\lambda_{i}(X^{s}_{i};\theta^{\prime}_{i},h)\geq 0,\forall\theta^{\prime}_{i}\in\Theta_{i}\end{aligned}\right\}.$

(19)

Then, we define the problem of Local Fixed-Point Alignment (LFPA) by

$\displaystyle\min_{X^{s}_{i},\tau_{i}}Z^{\mathtt{LFPA}}_{i}\left(X^{s}_{i},\tau_{i};s,h\right),\textup{ s.t. }\left<X^{s}_{i},\tau_{i}\right>\in\overline{\mathcal{K}}\left[s,g_{-i},h\right].$

(LFPA)

Let $e_{i}$, $\bm{b}_{i}\equiv(b[\theta_{i}])_{\theta_{i}\in\Theta^{\natural}_{i}}$, $\bm{f}_{i}\equiv(f[\theta_{i}])_{\theta_{i}\in\Theta_{i}}$, respectively, denote the Lagrange multipliers of the constraints $\left\{\tau_{i}(\hat{\theta}_{i}|s,g_{i},h)\geq 0\right\}$, $\left\{\tau_{i}(\theta_{i}|s,g_{i},h)\geq 0,\forall\theta_{i}\in\Theta^{\natural}_{i}\right\}$, and $\left\{\lambda_{i}(X^{s}_{i};\theta^{\prime}_{i},h)\geq 0,\forall\theta^{\prime}_{i}\in\Theta_{i}\right\}$. In addition, the corresponding slack variables are denoted by $w_{i}$, $\bm{q}_{i}\equiv\{q[\theta_{i}]\}_{\theta_{i}\in\Theta^{\natural}_{i}}$, $\bm{z}_{i}\equiv\{z[\theta_{i}]\}_{\theta_{i}\in\Theta_{i}}$, respectively. Then, the Lagrangian of (LFPA) is defined by

		$\displaystyle L_{i}(X^{s}_{i},\tau_{i},e_{i},\bm{b}_{i},\bm{f}_{i},w_{i},\bm{q}_{i},\bm{z}_{i}|s,h)\equiv Z^{\mathtt{LFPA}}_{i}(X^{s}_{i},\tau_{i};s,h)$		(20)
		$\displaystyle+\sum\nolimits_{\theta_{i}\in\Theta^{\natural}_{i}}b[\theta_{i}]\left(q[\theta_{i}]-\tau_{i}(\theta_{i}|s,g_{i},h)\right)$
		$\displaystyle+\sum\nolimits_{\theta_{i}\in\Theta_{i}}f[\theta_{i}]\big{(}z[\theta_{i}]-\lambda_{i}(X^{s}_{i};\theta_{i},h)\big{)}+e_{i}\big{(}w-\tau^{k}_{i}(\hat{\theta}_{i}|s,g_{i},h)\big{)}.$

To simplify the presentation, we omit $s$ and $h$. Taking partial derivatives of $L_{i}$ with respect to $X^{s}_{i}$ and $\tau_{i}$ yields,

$$\begin{cases}&\begin{aligned} &\Delta_{i}\left(X^{s}_{i},\tau_{i},\bm{f}_{i}\right)\\ &\equiv\gradient_{X^{s}_{i}}Z^{\mathtt{LFPA}}_{i}(X^{s}_{i},\tau_{i})-\sum_{\theta_{i}\in\Theta_{i}}f[\theta_{i}]\gradient_{X^{s}_{i}}\lambda_{i}(X^{s}_{i};\theta_{i}),\end{aligned}\\ &\begin{aligned} &D_{i}\left(X^{s}_{i},\tau_{i}(\theta_{i}|\cdot),e_{i},b[\theta_{i}]\right)\\ &\equiv b[\theta_{i}]-e+\frac{\partial}{\partial\tau_{i}(\theta_{i}|\cdot)}Z^{\mathtt{LFPA}}_{i}(X^{s}_{i},\tau_{i}(\theta_{i}|\cdot)),\forall\theta_{i}\in\Theta_{i}.\end{aligned}\end{cases}$$