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.
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Notes
In general, the algorithm presented in this paper handles multivariate strategies.
A formal characterization of correlated equilibria is mentioned in Theorem 2.1.
Since 〈V,S〉=〈(V T+V)/2,S〉 holds for arbitrary V and symmetric S, the arbitrary V qβ can be treated as symmetric due to Proposition A.1 in Appendix.
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Communicated by David G. Luenberger.
Appendix: Proofs
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))∈ℝl×l is symmetric.
Proof
Let A be partitioned into the block matrix
where A ij ∈ℝm×m with (i,j)∈K l . Let I⊗B∈ℝlm×lm be partitioned into the block matrix
Applying block matrix multiplication, we get
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∈ℝm×n and Q∈ℝm×n, we have 〈P T AQ,B〉=〈A,PBQ T〉.
Proof
It is well known that given arbitrary C∈ℝm×n and D∈ℝn×m, tr(CD)=tr(DC). Since AQ∈ℝm×n, PB∈ℝm×n, PBQ T∈ℝm×m, the following can be derived:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1007%2Fs10957-012-9988-6/MediaObjects/10957_2012_9988_Equu_HTML.gif)
□
Proposition A.3
Given ρ∈ℕ, l∈ℕ, permutation matrix Δ∈ℝlρ×lρ, positive semidefinite \(Z \in \mathbb{S}^{l \rho}\), and constant \(\varLambda \in \mathbb{S}^{\rho}\), there exist column selection matrices P k ∈ℝlρ×ρ and Q k ∈ℝlρ×ρ such that
Proof
We first expand 〈[Δ T ZΔ] k ,Λ〉,
By Proposition (A.2), we have
□
Proposition A.4
Given ρ∈ℕ, l∈ℕ, permutation matrix Δ∈ℝlρ×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 ∈ℝlρ×ρ, Q k ∈ℝlρ×ρ such that
Proof
We first expand (Δ T ZΔ,Λ) l ,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1007%2Fs10957-012-9988-6/MediaObjects/10957_2012_9988_Equ52_HTML.gif)
By Proposition A.3, (52) becomes
Therefore, we have \(\langle V, (\varDelta ^{T} Z \varDelta , \varLambda )_{l} \rangle = \sum_{k \in \mathbf{K}_{l}} [ V ]_{k} \langle \varDelta P_{k} \varLambda Q_{k}^{T} \varDelta ^{T}, Z \rangle = \sum_{k \in \mathbf{K}_{l}} [ V ]_{k} \langle \varDelta E_{k} \otimes \varLambda \varDelta ^{T}, Z \rangle\). □
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Kong, F.W., Kleniati, PM. & Rustem, B. Computation of Correlated Equilibrium with Global-Optimal Expected Social Welfare. J Optim Theory Appl 153, 237–261 (2012). https://doi.org/10.1007/s10957-012-9988-6
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DOI: https://doi.org/10.1007/s10957-012-9988-6