An order-centric treatment of the Bayesian supermodular game

Abstract

We take an order-centric approach to an incomplete-information version of the supermodular game (SG). In particular, we first introduce concepts related to ordered normal form games and the stochastic dominance order. Then, we work on a Bayesian SG, for which we show the existence of a monotone equilibrium and its monotonic trend as the player type distribution varies. Our results complement those that appeared in the Bayesian-SG literature.

Appendices

Appendix A: Proof of Proposition 1

By hypotheses and Fact 4, we see that $\bar{B}$ is a monotone map from Q to Q. By Fact 3, we obtain the existence of Γ’s largest pure Nash equilibrium in Q.

Appendix B: Proof of Lemma 1

By hypotheses and Fact 4, we know that $\bar{\mathcal{B}}$ is a monotone map from ${\mathcal{Q}}\cap Q^{T}$ to itself. We can then obtain the desired result by applying Fact 3.

Appendix C: Proof of Lemma 2

Given $x\in {\mathcal{I}}(T,S)\cap Q^{T}$ and θ∈T, consider the following subset of S _n :

$$M_n(x(\theta),\theta)=\operatorname {argmax}_{x'_n\in S_n}f_n(x'_n,x_{-n}(\theta ),\theta),$$

(9)

which is a non-empty complete sublattice of S _n due to the fact that S _n is a complete lattice, f _n ’s order upper semi-continuity and supermodularity in $x'_{n}$ under the given x _−n (θ)∈Q _−n , and Fact 2. Note that $x\in{ \mathcal{I}}(T,S)$. So under the above x and given θ ¹,θ ²∈T with θ ¹≤θ ², we have x _−n (θ ¹)≤x _−n (θ ²) as well. By this, the fact that f _n has increasing differences in $x'_{n}$ and $x'_{-n}\in Q_{-n}$ as well as in $x'_{n}$ and θ for fixed $x'_{-n}\in Q_{-n}$, we know that $f_{n}(x'_{n},x_{-n}(\theta),\theta)$ has increasing differences in $x'_{n}$ and θ. Thus, Fact 1 will lead us to

$$ M_n(x(\theta^1),\theta^1)\leq_s M_n(x(\theta^2),\theta^2).$$

(10)

By definition, we have $\bar{\mathcal{B}}_{n}(x)=(\sup M_{n}(x(\theta),\theta )\mid\theta\in T)$. From (10), we know that $\bar{\mathcal{B}}_{n}(x)$ is monotone in θ. For every θ∈T, since Γ(θ) is Q-preserving, we know that $\bar{B}(x(\theta),\theta)\in Q$ just because x(θ)∈Q. Thus, we have $\bar{\mathcal{B}}(x)\in {\mathcal{I}}(T,S)\cap Q^{T}$.

Appendix D: That weak affiliation along with the ≤ _iib partial order is weaker than affiliation

Suppose probability $p\in {\mathcal{P}}(R^{\mid N\mid})$ possesses a joint probability density function q[p]≡(q[p](θ)∣θ≡(θ ₁,…,θ _n )∈R ^∣N∣) with respect to the Lebesgue measure on R ^∣N∣. According to Milgrom and Weber (Milgrom and Weber 1982), we may call p affiliated when q[p] is log-supermodular in θ. It was shown that the affiliation of p will lead to (4) for each n∈N, where p _−n|n is the conditional probability kernel between θ _−n ≡(θ _n′∣n′≠n) and θ _n that is induced by p.

Let $p^{1},p^{2}\in{ \mathcal{P}}(R^{\mid N\mid})$ be two affiliated probabilities that possess densities q[p ⁱ]_−n|n , for i=1,2, for their conditional probability kernels with respect to the Lebesgue measure on R ^∣N∣−1. When p ² is greater than p ¹ in the affiliated ranking, i.e., when p ¹≤ _ar p ², it will follow that q[p ⁱ]_−n|n (θ _−n ∣θ _n ) is log-supermodular in (i,θ _−n )∈{1,2}×R ^∣N∣−1 for every n∈N and θ _n ∈R. But this will lead to (5) for every n∈N.

Appendix E: Proof of Lemma 3

Let $(n,\theta_{n})\in{ \mathcal{N}}$, ordered chain $C_{n,\theta_{n}}\subset {\mathcal{S}}_{n,\theta_{n}}= S_{n}$, and ϵ>0 be given. By (S1′), we know the existence of some $y_{n,\theta_{n}}\in C_{n,\theta_{n}}$, so that, for any $x_{-n}\in\Pi_{n'\neq n}{\mathcal{M}}(T_{n'},S_{n'})$, any θ _−n ∈T _−n , and any $x_{n,\theta_{n}}\in C_{n,\theta_{n}}$ with $x_{n,\theta_{n}}\geq y_{n,\theta_{n}}$, it will follow that

$$f_n(x_{n,\theta_n},x_{-n}(\theta_{-n}),\theta_n,\theta_{-n})<f_n(\sup C_{n,\theta_n},x_{-n}(\theta_{-n}),\theta_n,\theta_{-n})+\epsilon.$$

(11)

Taking expectation on θ _−n with respect to probability p _−n|n (θ _n ), we obtain

$$f^B_{n,\theta_n}(x_{n,\theta_n},x_{-n})<f^B_{n,\theta_n}(\sup C_{n,\theta_n},x_{-n})+\epsilon.$$

(12)

After symmetrically tackling the case involving $\inf C_{n,\theta_{n}}$, we can conclude that $f^{B}_{n,\theta_{n}}$ is order upper semi-continuous in $x_{n,\theta_{n}}$ for fixed x _−n . So Γ^B will satisfy (S1).

When each Γ(θ) satisfies (S2), we know that

(13)

for any n∈N, θ _n ∈T _n , $x^{1}_{n,\theta_{n}},x^{2}_{n,\theta _{n}}\in{ \mathcal{S}}_{n,\theta_{n}}$, $x_{-n}\in\Pi_{n'\neq n}{\mathcal{M}}(T_{n'},S_{n'})$, and θ _−n ∈T _−n . Taking expectation on θ _−n with respect to probability p _−n|n (θ _n ), we obtain

(14)

So Γ^B will satisfy (S2).

When each Γ(θ) satisfies (S3), we know that

(15)

for any $x^{1},x^{2}\in\Pi_{n'\in N}{\mathcal{M}}(T_{n'},S_{n'})$ with x ¹≤x ², n∈N, and θ∈T. Taking expectation on θ _−n with respect to probability p _−n|n (θ _n ), we obtain

(16)

So Γ^B will satisfy (S3). Therefore, it is supermodular as an ordered normal form game.

Appendix F: Proof of Lemma 4

For fixed n∈N, $x^{1}_{n},x^{2}_{n}\in S_{n}$ with $x^{1}_{n}\leq x^{2}_{n}$, and $x_{-n}\in\Pi_{n'\neq n}{\mathcal{I}}(T_{n'},S_{n'})$, let g:T→R be such that

$$ g(\theta_n,\theta_{-n})=f_n(x^2_n,x_{-n}(\theta_{-n}),\theta_n,\theta _{-n})-f_n(x^1_n,x_{-n}(\theta_{-n}),\theta_n,\theta_{-n}).$$

(17)

As the game family is supermodular and parametrically monotone, f _n (y _n ,y _−n ,ζ _n ,ζ_−n ) has increasing differences in y _n and y _−n as well as in y _n and (ζ _n ,ζ_−n ). Hence, as x _−n is monotone in θ _−n , we have that g is monotone in both θ _n and θ _−n . Define G(θ _n ), so that

$$ G(\theta_n)=\int_{T_{-n}}g(\theta_n,\theta_{-n})\cdot p_{-n|n}(d\theta _{-n}|\theta_n).$$

(18)

By the membership of p in ${\mathcal{W}}$, we know that p _−n|n (θ _n ) is stochastically monotone in θ _n . Hence, G(θ _n ) is monotone in θ _n . But from (6), (17), and (18), we have

$$G(\theta_n)=f^B_{n,\theta_n}(x^2_n,x_{-n})-f^B_{n,\theta_n}(x^1_n,x_{-n}).$$

(19)

Thus, $f^{B}_{n,\cdot}(\cdot,x_{-n})$, when viewed as a function of θ _n and x _n , has increasing differences in the two variables.

Given $x\in\Pi_{n'\in N}{\mathcal{I}}(T_{n'},S_{n'})$, now consider the following non-empty complete sublattice of ${\mathcal{S}}_{n,\theta_{n}}$ for some arbitrary n∈N and θ _n ∈T _n :

$$ M^B_{n,\theta_n}(x)=\operatorname {argmax}_{x'_n\in S_n}\;f^B_{n,\theta_n}(x'_n,x_{-n}).$$

(20)

By the above increasing-difference result on $f^{B}_{n,\theta_{n}}$ and Fact 1, we know that $(M^{B}_{n,\theta _{n}}(x)\mid\allowbreak \theta_{n}\in \nobreak T_{n})$ is monotone in θ _n in the strong set order sense. Consider $\bar{B}^{B}_{n,\theta_{n}}(x)=\sup M^{B}_{n,\theta _{n}}(x)$. In view of (20), we know that $(\bar{B}^{B}_{n,\theta _{n}}(x)\mid\theta_{n}\in T_{n})$ is monotone in θ _n , or equivalently, inside ${\mathcal{I}}(T_{n},S_{n})$, for the given $x\in\Pi_{n'\in N}{\mathcal{I}}(T_{n'},S_{n'})$.

Appendix G: Proof of Theorem 1

Identify the current Γ^B and $\Pi_{n\in N}{\mathcal{I}}(T_{n},S_{n})$ with, respectively, Γ and Q in Proposition 1, and then apply the proposition.

Appendix H: Proof of Lemma 5

For fixed $(n,\theta_{n})\in{ \mathcal{N}}$, $x^{1}_{n},x^{2}_{n}\in{ \mathcal{S}}_{n,\theta _{n}}$ with $x^{1}_{n}\leq x^{2}_{n}$, and $x_{-n}\in\Pi_{n'\neq n}{\mathcal{I}}(T_{n'},S_{n'})$, we introduce g:T _−n →R so that

$$ g(\theta_{-n})=f_n(x^2_n,x_{-n}(\theta_{-n}),\theta_n,\theta _{-n})-f_n(x^1_n,x_{-n}(\theta_{-n}),\theta_n,\theta_{-n}).$$

(21)

Since the original game family is supermodular and parametrically monotone, we know that f _n (y _n ,y _−n ,ζ _n ,ζ_−n ) has increasing differences in y _n and y _−n as well as in y _n and ζ_−n . Hence, as x _−n is monotone in θ _−n , we know that g is monotone in θ _−n . Define G(p) so that

$$ G(p)=\int_{T_{-n}}g(\theta_{-n})\cdot p_{-n|n}(d\theta_{-n}\mid\theta_n).$$

(22)

The above monotonicity of g in θ _−n dictates that G(p) is monotone in p. But by (7), (21), and (22), we have

$$G(p)=f^B_{n,\theta_n}(x^2_n,x_{-n},p)-f^B_{n,\theta_n}(x^1_n,x_{-n},p).$$

(23)

So $f^{B}_{n,\theta_{n}}(\cdot,x_{-n},\cdot)$, as a function of x _n and p, has increasing differences in its two variables.

Appendix I: Proof of Theorem 2

Identify the current $(\Gamma^{B}(p)\mid p\in{ \mathcal{W}})$ and $\Pi_{n\in N}{\mathcal{I}}(T_{n},S_{n})$ with, respectively, (Γ(θ)∣θ∈T) and Q in Proposition 2, and then apply the proposition.