Computation of Correlated Equilibrium with Global-Optimal Expected Social Welfare

Abstract

In this paper, we propose an algorithm which computes the correlated equilibrium with global-optimal (i.e., maximum) expected social welfare for single stage polynomial games. We first derive tractable primal/dual semidefinite programming (SDP) relaxations for an infinite-dimensional formulation of correlated equilibria. We give an asymptotic convergence proof, which ensures solving the sequence of relaxations leads to solutions that converge to the correlated equilibrium with the highest expected social welfare. Finally, we give a dedicated sequential SDP algorithm and demonstrate it in a wireless application with numerical results.

Appendix: Proofs

Proposition A.1

Given l∈ℕ, m∈ℕ, $A \in \mathbb{S}^{l m}$, and $B \in \mathbb{S}^{m}$, (A,B) _l :=tr _l (A ^T(I⊗B))∈ℝ^l×l is symmetric.

Proof

Let A be partitioned into the block matrix

$$\left[\begin{array}{c@{\quad}c@{\quad}c}A_{11} & A_{ij} & \ldots \\A_{ji} & A_{22} & \ldots \\\vdots & \vdots & \ddots \end{array}\right]$$

(48)

where A _ij ∈ℝ^m×m with (i,j)∈K _l . Let I⊗B∈ℝ^lm×lm be partitioned into the block matrix

$$\left[\begin{array}{c@{\quad}c@{\quad}c}B & 0 & \ldots \\0 & B & \ldots \\\vdots & \vdots & \ddots \end{array}\right].$$

(49)

Applying block matrix multiplication, we get

$$\mathrm{tr}_{l}\bigl(A^T(I \otimes B)\bigr) =\left[\begin{array}{c@{\quad}c@{\quad}c}\mathrm{tr}(A_{11}B) & \mathrm{tr}(A_{ij}B) & \ldots \\\mathrm{tr}(A_{ji}B) & \mathrm{tr}(A_{22}B) & \ldots \\\vdots & \vdots & \ddots \end{array}\right].$$

(50)

Since A and B are symmetric as given, $A_{ij} = A_{ji}^{T}$ implies tr(A _ij B)=tr(A _ji B). □

Proposition A.2

Given m∈ℕ, n∈ℕ, $A \in \mathbb{S}^{m}$, $B\in \mathbb{S}^{n}$, and column selection matrices namely P∈ℝ^m×n and Q∈ℝ^m×n, we have 〈P ^T AQ,B〉=〈A,PBQ ^T〉.

Proof

It is well known that given arbitrary C∈ℝ^m×n and D∈ℝ^n×m, tr(CD)=tr(DC). Since AQ∈ℝ^m×n, PB∈ℝ^m×n, PBQ ^T∈ℝ^m×m, the following can be derived:

 □

Proposition A.3

Given ρ∈ℕ, l∈ℕ, permutation matrix Δ∈ℝ^lρ×lρ, positive semidefinite $Z \in \mathbb{S}^{l \rho}$, and constant $\varLambda \in \mathbb{S}^{\rho}$, there exist column selection matrices P _k ∈ℝ^lρ×ρ and Q _k ∈ℝ^lρ×ρ such that

$$\big\langle \bigl[ \varDelta ^T Z \varDelta \bigr]_{k},\varLambda \big\rangle =\big\langle Z, \varDelta P_{k} \varLambda Q_{k}^T \varDelta ^T \big\rangle, \quad k \in \mathbf{K}_{l}.$$

Proof

We first expand 〈[Δ ^T ZΔ] _k ,Λ〉,

$$\big\langle \bigl[ \varDelta ^T Z \varDelta \bigr]_{k},\varLambda \big\rangle = \big\langle P_{k}^T \varDelta ^T Z \varDelta Q_{k}, \varLambda \big\rangle = \big\langle (\varDelta P_{k})^T Z \varDelta Q_{k} , \varLambda \big\rangle.$$

By Proposition (A.2), we have

$$\big\langle (\varDelta P_{k})^T Z \varDelta Q_{k}, \varLambda \big\rangle = \big\langle Z , \varDelta P_{k} \varLambda (\varDelta Q_{k})^T \big\rangle = \big\langle Z , \varDelta P_{k} \varLambda Q_{k}^T \varDelta ^T \big\rangle.$$

 □

Proposition A.4

Given ρ∈ℕ, l∈ℕ, permutation matrix Δ∈ℝ^lρ×lρ, positive semidefinite $Z \in \mathbb{S}^{l \rho}$, constant $\varLambda \in \mathbb{S}^{\rho}$, and an arbitrary $V \in \mathbb{S}^{l}$, there exist column selection matrices P _k ∈ℝ^lρ×ρ, Q _k ∈ℝ^lρ×ρ such that

$$\big\langle V, \bigl(\varDelta ^T Z \varDelta , \varLambda \bigr)_{l}\big\rangle = \sum_{k \in \mathbf{K}_l} [ V ]_k \big\langle \varDelta E_k \otimes \varLambda \varDelta ^T, Z \big\rangle.$$

(51)

Proof

We first expand (Δ ^T ZΔ,Λ) _l ,

(52)

By Proposition A.3, (52) becomes

$$\left[\begin{array}{c@{\quad}c@{\quad}c}\langle Z, \varDelta P_{1,1} \varLambda Q_{1,1}^T \varDelta ^T\rangle & \cdots & \cdots \\\cdots & \langle Z, \varDelta P_{k} \varLambda Q_{k}^T \varDelta ^T \rangle & \cdots \\\cdots & \cdots & \cdots \end{array}\right].$$

(53)

Therefore, we have $\langle V, (\varDelta ^{T} Z \varDelta , \varLambda )_{l} \rangle = \sum_{k \in \mathbf{K}_{l}} [ V ]_{k} \langle \varDelta P_{k} \varLambda Q_{k}^{T} \varDelta ^{T}, Z \rangle = \sum_{k \in \mathbf{K}_{l}} [ V ]_{k} \langle \varDelta E_{k} \otimes \varLambda \varDelta ^{T}, Z \rangle$. □