Comparison of information structures in stochastic games with imperfect public monitoring

Abstract

This paper studies the impact of an improvement of information structure upon the perfect public equilibrium payoff set in discounted stochastic games with imperfect public monitoring. We first suggest three partial orders on information structures in stochastic games. Although each of them reduces to the notion of garbling in repeated games (Kandori in Rev Econ Stud 59:581–593, 1992), we find that an improvement of information in terms of our two garbling notions does not imply an expansion of the equilibrium payoff set for some games. We also show that more informativeness in terms of our third notion of garbling is sufficient for the expansion, thereby extending the well-known monotonicity result in Kandori (1992) to stochastic games.

Appendix

1.1 A Omitted proofs

1.1.1 A.1 Proof of Lemma 2

Proof

Fix $\delta \in [0,1)$. Let $W \subseteq \mathbb {R}^{ |S| \times N}$ be self-generating. Then, for each $v = (v_1, \dots , v_{|S|}) \in W$ and $s \in S$, we can find $\alpha ^v_s \in \prod _{ i \in I} \Delta A_i$ and $w^v_s : Y \rightarrow W$ which decomposes $v_s$ for each s. If there are more than one such pairs, choose one among them arbitrarily.

Pick $v = (v_1, \dots , v_{|S|}) \in B(W)$. We want to show that for each state s, $v_s \in E (s; \delta )$. Let $\tilde{\mathcal {H}}_P^0 \equiv \{ \varnothing \}$ and $\tilde{\mathcal {H}}_P^k \equiv \{ \varnothing \} \times (S \times Y)^k$ for $k \ge 1$, i.e., the set of ex-ante public histories (that is, history before$s^k$ is realized). For each $k \ge 0$, define $\nu ^k: \tilde{\mathcal {H}}_P^k \rightarrow W$ recursively as follows: let $\nu ^0 ( \varnothing ) := v$ and, for each $k \ge 1$ and $\tilde{h}^k = (\tilde{h}^{k-1}, s, y)$ where $\tilde{h}^{k-1} \in \tilde{H}_P^{k-1}$, $s \in S$ and $y \in Y$,

$$\begin{aligned} \nu ^{k} (\tilde{h}^k) := w^{ \nu ^{k-1} (\tilde{h}^{k-1}) }_s (y) \in W. \end{aligned}$$

That is, $\nu ^k (\tilde{h}^k) \in \mathbb {R}^{|S| \times N}$ is the continuation payoff vector corresponding to ex-ante history $\tilde{h}^k$.

Define a public strategy as follows: for any $k \ge 0$ and (ex-post) public history $h^k \in \mathcal {H}_P^k$ (see Sect. 2)

$$\begin{aligned} \sigma ( h^k ) := \alpha ^{ \nu ^k (\tilde{h}^{k})}_{s(h^k)} \end{aligned}$$

where $\tilde{h^k} \in \tilde{\mathcal {H}}_P^k$ is such that $(\varnothing , h^k) = (\tilde{h}^k, s(h^k) )$. Recall that $s(h^k)$ is the most recent state in $h^k$. Choose a state s. Then for each $k \ge 0$,

$$\begin{aligned} \begin{aligned} v_s&= (1- \delta ) g ( s, \sigma ( s) ) + \delta \sum _{ s^1 , y^0 } p(s^1, y^0 | s, \sigma (s)) w^{v}_{s} (s^1, y^0)\\&= (1- \delta ) g(s, \sigma (s )) + \delta \sum _{ s^1 , y^0 } p(s^1, y^0 | s, \sigma (s))\\&\quad \times \left( (1- \delta ) g (s^1, \sigma (s, y^0, s^1)) + \delta \sum _{ s^2 , y^1} p(s^2, y^1 | s^1, \sigma (s, y^0, s^1)) w_{s^1}^{ \nu ^1 (\varnothing , s,y^0) }(s^2, y^1) \right) \\&= (1- \delta ) \sum _{ \tau =0}^{k} \delta ^\tau \sum _{h^\tau \in \mathcal {H}_P^\tau } g (s (h^\tau ), \sigma (h^\tau )) \mathbf {P}_\tau ^\sigma (h^\tau )\\&\quad + \delta ^{k+1} \sum _{h^{k+1} \in \mathcal {H}_P^{k+1}} \mathbf {P}_{k+1}^\sigma (h^{k+1}) \sum _{ s^{k+1}, y^{k}} p ( s^{k+1}, y^k | s (h^k), \sigma (h^k))w^{\nu ^k (\tilde{h}^{k}) }_{s (h^k) } ( s^{k+1}, y^{k}), \end{aligned} \end{aligned}$$

where for each $k \ge 0$, $\mathbf {P}_k^\sigma (\cdot )$ is the probability measure on $\mathcal {H}_P^k$ induced by $\sigma$ and the initial state s. As W is bounded, as $k \rightarrow \infty$,

$$\begin{aligned} \begin{aligned} v_s&= (1- \delta ) \sum _{ k=0}^{\infty } \delta ^k \sum _{h^k \in \mathcal {H}_P^k} g(s (h^k), \sigma (h^k )) \mathbf {P}_k^\sigma (h^k) \\&=(1- \delta ) \mathbb {E}_{ \mathbf {P}^\sigma } \left[ \sum _{k=0}^\infty \delta ^k g ( s (h^k), a (h^k) | s_0 = s \right] . \end{aligned} \end{aligned}$$

The incentive compatibility of $\sigma$ follows from the one-shot deviation principle. Therefore $v_{s} \in E (s;\delta )$. $\square$

1.1.2 A.2 Proof of Proposition 2

Proof

In order to show $E(\delta )$ is a fixed point of $B(\cdot ; \delta )$, it suffices to prove $E(\delta ) \subseteq B(E(\delta );\delta )$ since this will also imply $B(E(\delta ); \delta ) \subseteq E(\delta )$ by Lemma 2. Let $v \in E(\delta )$ and $\sigma \in \Sigma _P$ be the corresponding public strategy profile. Then for each state s,

$$\begin{aligned} v_s = U (\sigma ; s) = (1- \delta ) g (s, \sigma (s)) + \delta \sum _{ t \in S, y \in Y} U ( \sigma |_{ (s, y)};t) p(t,y |s, \sigma (s)). \end{aligned}$$

As $\sigma$ is a PPE, for each y and t, $U (\sigma |_{(s,y)};t ) \in E (t;\delta )$, and any player i does not have an incentive to deviate from $\sigma _i (s)$. Thus, $v_s \in B_s (E(\delta );\delta )$; and so $v \in B (E(\delta );\delta )$.

Suppose $W \subseteq \mathbb {R}^{ |S| \times N}$ is a bounded fixed point of $B(\cdot ;\delta )$. Then, $W = B(W;\delta ) \subseteq E(\delta )$ by Lemma 2. This implies that $E(\delta )$ is the largest bounded fixed point of B. $\square$