Network Games: On the Extension of Nashs Theory to Networks Lacra - - PowerPoint PPT Presentation

Network Games: On the Extension of Nash’s Theory to Networks

Lacra Pavel*

Systems Control Group University of Toronto

∗based on joint work with Dian Gadjov, Peng Yi

Farzad Salehisadaghiani NecSys 2018, Groningen, 27-28 August, 2018

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Motivation

Social Networks Wireless Ad-hoc Networks

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What is a Game?

Setup: a number of players /agents (2 or more), or decision-makers. Each player has a finite number of choices or has a continuum of choices (actions/decisions). At the end of the game there is some payoff to be gained or cost to be paid by each player. In standard optimization: one decision-maker who aims to optimize some

• bjective function (goal).

In a game: multiple decision-makers (self-interested players /agents) with own individual goals (cost/payoff).

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What is a Game?

Setup: a set of N players /agents, V = {1, . . . , N}. Each player i ∈ V has an action xi selected from his action set Ωi (finite or continuous). has an individual payoff (utility) Ui or cost function Ji takes an action xi to maximize its own payoff Ui, which is equivalent to minimizing its own cost (loss) function, Ji. His success in making decisions depends on the others’ decisions. Ji(x), where x = (x1, · · · , xN) is the action profile of all players, or, denoted Ji(xi, x−i), where x = (xi, x−i) and where x−i denotes the actions of others except player i. Denote such a game by G(V, Ωi, Ji). Game theory = the mathematical framework that studies the strategic interaction among multiple decision-makers.

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Solution Concepts

A player is rational if he makes choices that optimize his expected utility (minimize expected cost). A strategy can be regarded as a rule for choosing an action, e.g. Security strategy: minimize your own maximum (worst) expected cost. Minimax Solution: when each player uses a security strategy. Prisoners’ Dilemma Game example: Action choices: "Confess" (defect) or "Not Confess", Cost matrix: years in prison M =

• (5, 5)

(0, 15) (15, 0) (1, 1)

• "Confess" is the security strategy for either players.

Minimax solution is ("Confess", "Confess") ⇒ (5, 5) years in prison.

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Solution Concepts

Best-response (BR) strategy: Play the action that gives you the lowest cost given your opponents’ actions/strategies. Given his opponents’ play x−i, a BR strategy for player i is x∗

i such that

Ji(x∗

i , x−i) ≤ Ji(xi, x−i),

∀xi ∈ Ωi (Nash) Equilibrium Solution x∗: when each player uses a best-response (BR) strategy Ji(x∗

i , x∗ −i) ≤ Ji(xi, x∗ −i),

∀xi ∈ Ωi, ∀i ∈ I denoted as x∗ = (x∗

i , x∗ −i) (action N-tuple or profile)

An equilibrium solution is at the intersection of all BR strategies. Key: No player has an incentive to unilaterally change its action => No Regret. We might expect a set of rational agents to choose an equilibrium solution.

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Solution Concepts

Prisoners’ Dilemma Game: ("Confess", "Confess") is in fact the unique equilibrium ⇒ cost (5, 5) years in prison. M =

• (5, 5)

(0, 15) (15, 0) (1, 1)

• Matching Pennies Game: DOES NOT have an equilibrium solution!

M = (−1, 1) (1, −1) (1, −1) (−1, 1)

• Randomize choices: if each player chooses Head (H) with 50% probability and

Tail (T) with 50% probability, the expected payoff for both is (0, 0), and no regret ⇒ is the equilibrium solution in randomized (mixed) strategies. von Neuman (1928): Every 2-player zero-sum game has an equilibrium in mixed strategies.

• J. Nash (1949): Every N-player game G(V, Ωi, Ji) with finite action sets has an

equilibrium in mixed strategies (called Nash equilibrium (NE)). Debreu, Glicksberg, Fan (1952): Every N-player infinite game G(V, Ω, Ji) with non-empty, compact, convex Ω, Ji continuous in x = (xi, x−i) and convex in xi, has a pure Nash equilibrium (NE).

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Game Theory: from Classical Setting to Learning

Classical Setting: Originated and used in economics and social sciences. Relies on equilibrium analysis based on NE or its refinements. Offers traditional explanation for when and why a NE arises: in one-shot games (players interact for only a single round), from analysis and introspection by sophisticated players when the game and the rationality of the players are all common knowledge, complete information. Learning Game Theory: Alternative justification - more relevant for engineering or social systems. in repeated games (players interact with each other for multiple rounds of the same game), a NE arises as the limiting point of repeated play in which (less than fully) rational players update their behaviour. information available is critical.

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Learning Game Theory

Myopic Learning: Simple and rule-of-thumb rules, no forecasting. Examples: Best-response (BR) play. Fictitious play (play optimally/BR against the empirical distribution of past plays of opponents). Gradient-play ("better-response" play). Reinforcement-learning (payoff-based play). Evolutionary Games: Select strategies according to performance against the aggregate and random mutations.

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Network Games

Multi-agent (MA) systems or networks Complex problems Multiple entities with individual goals and capacity to act Limited distributed resources Central controller issues Applications Transportation Communication Economics Social Networks . . . Major challenge Efficient coordination between agents (centralized?)

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Game Theory in Network Games

Congestion control games in networks (internet, transportation): [Altman, Basar &

Srikant’02, Alpcan & Basar’05, Yin, Shanbhag & Mehta’11].

Power allocation in communication networks (wireless, optical): [Alpcan & Basar’03, Pan

& Pavel’09, Menache & Ozdaglar’11].

Multi-agent (mobile sensors/robot) network formation: [Cortes, Martinez et al.’02’13, Arslan,

Marden & Shamma’07].

Resource allocation game in cloud computation: [Ardagna et al.’13]. Demand response management in smart grids: [Zhu & Frazzoli’16, van der Schaar et al.’16, Ye &

Hu’17].

fase

Players / Agents: network nodes, routers, channels, robots or network users. share network resources (bandwidth, power, capacity). Approaches: infinite (continuous-kernel) games, finite action games, evolutionary games.

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Network Games

Centralized coordination is often impractical in networks, or consumes excessive bandwidth and energy => want distributed learning algorithms/dynamics. Challenges: complexity of agents’ networked interaction local/partial information delayed/asynchronous communication curse of dimensionality non-stationary environment (multiple agents learning at the same time). Differences: MA consensus/agreement: dynamic but decoupled agents, with individual properties for each agent (e.g. incremental passivity, convexity); only consensus/agreement type optimality (translates in quadratic objective). Distributed optimization: optimality in terms of an aggregate objective, each agent optimizes a part of it (cooperate) and has complete authority

• ver full argument (decoupled problems).

Game setting: each agent optimizes own objective (does not cooperate) and has authority only over his own actions, but his objective depends on the others (coupled problems and incomplete authority over the full argument/profile).

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Network Games: Setting

3 1 N 2 3 1 2 GI GC N Interference Graph Communication Graph

Each player/agent i has an individual cost Ji or payoff Ui (his goal). Ji may be affected by the decision of any subset of players => strategic interaction modelled by interference graph GI (conflict, in graphical games). Information obtained over a network => communication graph GC. GI and GC can be identical or not, complete or sparsely connected, directed or undirected, static or time-varying.

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Network Games: Problem

3 1 N 2 3 1 2 GI GC N Interference Graph Communication Graph

Design agent learning rules (algorithms/dynamics) that achieve a (G)NE collective configuration, while considering agents’ networked interaction and communication (rely on local information), minimize superfluous communication or processing overhead, and are provably convergent in a large class of games. Setting: non-cooperative in the way actions are taken (each agent minimizes its

• wn individual cost), but collaborative (agents share some information with

neighbours to compensate for the lack of global information on others). Tools: convex optimization, graph theory, monotone operator theory, multi-agent dynamical systems & control.

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Network Games: Some Results

3 1 N 2 3 1 2 GI GC N Interference Graph Communication Graph

NEP in general or in aggregative games with central node authority: GI Complete, GC star, [Gilpin ’07, Facchinei & Pang’07, Parise et al’15] NEP in aggregative games or 2-network zero-sum games, center-free: GI Complete, any GC ⊂ GI [Koshal, Nedic & Shanbhag’12], [Maojiao ’17], [Gharesifard & Cortes’13]. NEP in general monotone games: GI Partial, GC = GI, [Zhu & Frazzoli’12], [Li et al.’13] NEP in general monotone games: GI Complete or Partial, any GC ⊂ GI, [Salehi &

Pavel’16,’18, Gadjov Pavel’18, Yi & Pavel’18] 15 / 56

Problem Statement

A game G(V, Ωi, Ji) is played over a communication graph GC ⊂ GI: Set of players: V = {1, . . . , N}, Player i’s action set and cost function: Ωi, Ji(x). Interference Graph: GI complete or partial, undirected or directed

3 1 4 2 J1(x1) := J1(x1, . . . , x4) 3 1 4 2 J1(x1) := J1(x1, x2) 3 1 4 2 J1(x1) := J1(x1, . . . , x4) 3 1 4 2 J1(x1) := J1(x1, x3) GI GI GI GI Complete Interference Graph Partial Interference Graph Partial Interference Digraph Complete Interference Digraph

Communication Graph: GC ⊂ GI partial, undirected or directed

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Assumptions

Communication Graph

GC is connected and undirected.

Existence of NE

Ωi ⊂ Rni is non-empty, compact and convex. Ji(xi, x−i) is C1, convex in xi, and jointly continuous in x = (xi, x−i) ∈ Ω ⊂ Rn.

Pseudo-gradient F(x) := (∇xiJi(xi, x−i))i∈V , F : Ω → Rn

F is strictly monotone: (F(x) − F(y))T (x − y) > 0 ∀x, y ∈ Ω, x = y. F is strongly monotone: (F(x) − F(y))T (x − y) ≥ µx − y20 ∀x, y ∈ Ω, µ > 0. Note: F monotone ≡ incrementally passive ≡ EIP . Objective: Design a distributed algorithm to find the NE of G(V, Ωi, Ji) using only partial information over GC.

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NE Characterization

Given G(V, Ωi, Ji)

NE Problem: Each player i aims to minimize his own cost function selfishly:    minimize

xi

Ji(xi, x−i) subject to xi ∈ Ωi ∀i ∈ V. coupled

2-Player Game.

NE x∗ = (x∗

i , x∗ −i) lies at the intersection of all best-response maps,

x∗

i ∈ arg min xi∈Ωi Ji(xi, x∗ −i),

∀i ∈ N (fixed-point of overall best-response map).

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NE Characterization

Given G(V, Ωi, Ji)

NE Problem: Each player i aims to minimize his own cost function selfishly:    minimize

xi

Ji(xi, x−i) subject to xi ∈ Ωi ∀i ∈ V. coupled NE characterized via a variational-inequality (VI) for pseudo-gradient F, (V I) (x − x∗)T F(x∗) ≥ 0, ∀x ∈ Ω i.e., 0 ∈ F(x∗) + NΩ(x∗), where NΩ(x) is normal cone of Ω at x,

• r, in terms of partial gradient w.r.t. own action, ∇xiJi(xi, x−i),

(FP) x∗

i = PΩi(x∗ i − αi∇xiJi(x∗ i , x∗ −i)), ∀i ∈ V.

where αi > 0, Ω = Πi∈V Ωi. Knowledge required: own cost Ji and all others’ actions x−i. ⇒ Perfect-decision information.

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NEP in Perfect Information Case: Projected-Gradient

Perfect information of x−i, i.e., agents observe all or instantaneously exchange information over a complete GC graph.

1 2 3 4 5 6

Figure: Complete Graph

Iterative projected-gradient (PG) At iteration k, player i updates his action as (1) xi(k + 1) = PΩi(xi(k) − αi∇xiJi(xi(k), x−i(k))), where αi > 0, or in compact (stacked) form, (2) x(k + 1) = PΩ(x(k) − αF(x(k))), PΩ(x) is projection of x onto the action set Ω, PΩ(x) = argminy∈Ωx − y. Under strict (strong) monotonicity & Lipschitz continuity of F, PG (2) converges to the NE with diminishing (constant) step-size α (Facchinei & Pang).

Ω x

PΩ(x)

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NEP in Perfect Information Case: Projected-Gradient

1 2 3 4 5 6

Figure: Complete Graph

With PΩ = (Id + NΩ)−1, PG in compact form is x(k) − x(k + 1) − αF(x(k)) ∈ NΩ(x(k + 1)), where NΩ is the normal cone of Ω at x. Alternative, continuous-time (differential inclusion) − ˙ x ∈ F(x) + NΩ(x),

• r as projected dynamical system (PGD)

(3) ˙ x = ΠΩ(x, −F(x)), where ΠΩ(x, v) is projection of v onto the tangent cone of Ω at x.

O :

• ΠΩ(x, −F(x))

D :

• ˙

x = u y = x +

y u

feedback of all actions x = (xi)i∈V ∈ Ω

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NEP in Partial-Information Case over Graph GC

Player i knows: his own cost function Ji, Ωi. Player i DOES NOT know: Actions of all other players x−i, but only partially observes them. There is no central node to disseminate info. 1 2 3 4 5 6

• Fig. Connected GC

Idea: For a partial communication graph GC, build an estimate of actions of all

• thers, x−i by communicating with neighbours. Each player i has his own esti-

mate of x−i, denoted xi

−i. 4 2 1 3 4 2 1 3 (x1

1(k), x1 −1(k))

4 2 1 3 4 2 1 3 (x2

2(k), x2 −2(k))

(x3

3(k), x3 −3(k))

(x4

4(k), x4 −4(k))

Ideally all estimates should be the same ⇒ consensus.

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A Typical Consensus Problem

A set of N agents each with a different variable xi (marked as a different colour). Agents communicate locally over a partial graph to achieve consensus on xi e.g. each compares his relative difference versus his neighbours via the Laplacian L,

• j∈Ni

xi − xj := [Lx]i and uses it to iteratively update his own xi ⇒ eventually they all reach consensus

x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10

⇒

x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10

⇒

x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10

iteration k k + 1 . . .

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Consensus of Estimates

A set of N agents communicate with their neighbours. Endow each player i with state xi = (xi

i, xi −i) that estimates all others’

decisions. initially are all N-tuples xi are different. Eventually all xi should be the same N-tuple ⇒ consensus of estimates (learn about all others) k k + 1

x0x1 x2 x3 x4 x5 x6 x7 x8 x9 x10⇒ x0x1 x2 x3 x4 x5 x6 x7 x8 x9 x10⇒ x0x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x0x1 x2 x3 x4 x5 x6 x7 x8 x9 x10⇒ x0x1 x2 x3 x4 x5 x6 x7 x8 x9 x10⇒ x0x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x0x1 x2 x3 x4 x5 x6 x7 x8 x9 x10⇒ x0x1 x2 x3 x4 x5 x6 x7 x8 x9 x10⇒ x0 x0x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x1 x2 x3

Combine consensus of estimates with projected-gradient (PG) so that agents reach consensus and converge to NE of the game.

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Asynchronous Gossip-Based PG Algorithm

At iteration k,

1

Gossiping Step: One player ik (selected at random) communicates a neighbour jk in GC. and they average their estimates xik

−ik = xik

−ik +xjk −ik

2

xjk

−jk = xjk

−jk +xik −jk

2

while the other players i = ik, jk stay "asleep".

3 1 4 2 3 1 4 2 3 1 4 2 3 1 4 2 ik jk GC : GI : 2

Local Step: If i = ik, jk, player i updates his decision by a projected-gradient (PG) using his estimate xi(k + 1) = TΩi

• xi(k) − αk,i∇iJi
• xi(k), xi

j |j∈NI(i)

• else, if i = ik, jk, he does nothing.

Theorem: Under strict monotone & Lipschitz F, with diminishing steps-size, all estimates converge to consensus and players’ actions x(k) converge almost surely to x∗ the NE of the game. All possible cases for GI (complete/sparse, undirected /directed), GC ⊂ GI, can be handled by appropriately modifying Step 1.

[Salehi & Pavel, Automatica’16, Automatica’18, Lecture Notes LNICST-Springer (GameNets’17)] 25 / 56

NEP via Projected Gradient w. Correction

Gossip-based PGA: converges to NE with diminishing-step sizes (slow); this corresponds to a two-time scale, continuous-time PGD, (fast consensus subsystem, slow PG subsystem). Gossip-based PGA with constant-step sizes converges only to a neighbourhood

• f NE.

Use correction term to overcome this: single-time scale, continuous-time PGD with correction, via passivity and multi-agent agreement constant-step size, discrete-time PGA with correction, via an ADMM approach

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NEP via Projected Gradient Dynamics w. Correction

Given a game G(V, Ωi, Ji) with partial Information, over an undirected communication graph Gc. 1 2 3 4 5 6 Design a new continuous-time PGD (single-time scale) s.t. it converges to the NE.

Approach

1

Endow each player with a state vector that estimates the others’ actions.

2

Add communication to PGD dynamics so the equilibrium is the NE.

3

Exploit passivity (monotonicity) properties of F, L to show that all state vectors reach consensus and converge to the NE (M.A. agreement).

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Augmented PG Dynamics w. Correction

Each individual player dynamics are: ˙ xi

i

˙ xi

−i

• =

    ΠΩi

• xi, −∇xiJi(xi

i, xi −i) − j∈Ni

(xi

i − xj i)

• −

j∈Ni

(xi

−i − xj −i)

    Compactly, with x = (xi)i∈V the stacked dynamics are: (4) ˙ x = RT ΠΩ(Rx, −F(x) − RLx) − ST SLx with F(x) := (∇xiJi(xi))i∈V : ΩN → Rn the extended pseudo-gradient and R, S matrices used to select actions and estimates from x.

Assumption: Extended Pseudo-Gradient

F is θ-Lipschitz continuous, F(x) − F(x′) ≤ θx − x′, ∀x, x′ ∈ ΩN, θ > 0.

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Augmented PG Dynamics w. Correction

O :

• RT ΠΩ(Rx, −F(x))

D : ˙ x = u y = x C :

• yr = Lur

+

yr y ur u

correction (local extra feedback via Laplacian L)

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Convergence: Single-Time Scale

Idea

Decompose RNn into the consensus subspace Cn

N = {1N ⊗ x |x ∈ Rn} and its

• rthogonal complement (Cn

N)⊥.

Extended pseudo-gradient decomposed

F(x) − F(x) = F(x⊥ + x||) − F(x||)

• ff consensus

+ F(x||) − F(x)

• consensus term

where x = x|| + x⊥, x|| ∈ Cn

N, x⊥ ∈ (Cn N)⊥.

Exploit

passivity (monotonicity) of F on Cn

N, due to F(1N ⊗ x) = F(x).

Off Cn

N, balance the shortage of passivity of F by Lipschitz and excess passivity

(strong monotonicity) of L on (Cn

N)⊥. 30 / 56

Convergence: Single-Time Scale

Theorem

Consider a game G(V, Ωi, Ji) over Gc with µ-strongly monotone & Lipschitz F and θ-Lipschitz F. Then, if λ2(L) > θ2

µ + θ, any solution of (4) will converge asymptotically

to 1N ⊗ x∗, and the action components converge to the NE of the game, x∗.

[Gadjov & Pavel,TAC’18]

x∗ x Ω ΩN C O x(t)

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Augmented PG Dynamics w. Correction

−R ˙ x = −F(x, z) + u RT

1 s

...

1 s

−S ST + Id Q ⊗ In QT ⊗ In x z ˙ z u x z Block Diagram of Actions x and Estimates z dynamics x = Rx, z = Sx L = L ⊗ In, L = QQT

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NEP via ADMM Projected Gradient w. Correction

Gossip-based PGA: converges to NE with diminishing-step sizes (slow); with constant-step sizes convergence is only to a neighbourhood of NE. ⇒ use correction term to improve this. NE Problem: Each player i aims to minimize his own cost function:    minimize

xi

Ji(xi, x−i) subject to xi ∈ Ωi ∀i ∈ V. (1) Idea: With xi := (xi

i, xi −i) local copy (estimate) for each player i,

NE problem is equivalent to a modified game:        minimize

xi

i∈Ωi

Ji(xi

i, xi −i)

subject to xi = xj ∀j ∈ Ni

• Consensus Constraint

∀i ∈ V, (2) For each player use an Alternating Dual Multipliers Method (ADMM) approach

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NEP via ADMM

NE Problem: For each player i, with xi := (xi

i, xi −i) a local copy of x,

             minimize

xi

i ∈R

Ji(xi

i, xi −i) + IΩi(xi i)

subject to xi = tij; ∀j s.t. (i, j) ∈ E xi = tji; ∀j s.t. (j, i) ∈ E

• Consensus Constraint

∀i ∈ V, (3) Associated augmented Lagrangian but treat separately xi

i (min Ji) and xi −i (0 obj. fcn.):

Lc

i

• xi, tij; {uij, vji}
• := Ji(xi

i, xi −i) + IΩi(xi i)

+

• j∈Ni

uijT (xi − tij) +

• j∈Ni

vjiT (xi − tji) + c 2

• j∈Ni

(xi − tij2 + xi − tji2)

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NEP ADMM Algorithm

For each player i, using projected gradient (PG) with correction for consensus constraints for xi := (xi

i, xi −i), where xi i is his action and xi −i is the estimate of

the others’ actions. xi

i(k + 1) = TΩi

• xi

i(k) − αi∇iJi(xi(k))

• Projected Gradient Method

−αi

• wi

i(k + 1) + c

• j∈Ni

xi

i(k) − xj i (k)

• Correction Term
• wi(k + 1) = wi(k) + c
• xi(k) − xj(k)
• xi

−i(k + 1) =

• 1 − c|Ni|αi
• xi

−i(k) + cαi

• j∈Ni

xj

−i(k)

• Average of Estimates

− αiwi

−i(k + 1)

• Correction Term

1 2 3

35 / 56

ADMM Convergence

Assumption: F is cocoercive, i.e., (F(x) − F(y))T (x − y) ≥ σFF(x) − F(y)2, ∀x ∈ ΩN and y ∈ ΩN.

Theorem

For sufficient choices of step-sizes αi related to game coupling and to connectivity strength of Gc, i.e., if σF > 1 2λmin(c(2D − L) + B), the sequence {xi(k)} ∀i ∈ V generated by the ADMM algorithm converges to 1N ⊗ x∗ (consensus), where x∗ is NE of the game.

[Salehi, Shi & Pavel, IFAC’17, under review Automatica]

x∗ x Ω ΩN C O x(t)

36 / 56

Economic Game Example

N firms (players) producing a homogeneous commodity, Each one is trying to maximize its own profit. Total quantity produced determines the demand price. Other firms actions influence each one’s profit. North America Asia Europe

37 / 56

Economic Game Example

20-player (quadratic game) xi is the quantity produced by firm i. Overall cost function Ji(xi, x−i) = ci(xi) − xif(x). Production cost ci(xi) = [20 + 10(i − 1)]xi. Demand price f(x) = 2200 −

i∈I xi

North America Asia Europe

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Economic Game Example

Simulations, Random Graph

50 100 150 200 250 300 50 100 150 200 250 time action

Figure: Dynamics on random Gc

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Figure: Random communication graph Gc

39 / 56

Economic Game Example

Simulations, Cyclic Graph

50 100 150 200 250 300 50 100 150 200 250 300 350 400 450 500 time action

Figure: Augmented dynamics on cycle Gc

1 2 3 4 5 20

Figure: Cycle communication graph Gc

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WANET Rate Control Game

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Rate Control Game

Wireless Ad-hoc Network:

Example

Consider a WANET of 15 users. Each user i decides its rate xi, xi ∈ [0, Bi] through path Ri in network. Mobile nodes are connected by 16 links. Each link Lj has maximal capacity Cj Constraint:

r:Lj∈Rr xr ≤ Cj

R4 L15 L13 L12 L14 1 2 16 14 13 15 12 7 6 8 3 4 5 11 10 9 L16 L4 L5 L6 L2 L1 L3 L11 L10 L9 L8 L7 R3 R2 R13 R1 R11 R15 R10 R9 R8 R7 R6 R14 R5 R12

• Adhoc Node

Wireless Link Player’s Path

• 42 / 56

Rate Control Game

Cost function of player i: Ji(xi, x−i) :=

• j:Lj∈Ri

κ Cj −

w:Lj∈Rw xw

• Price function

− χi log(xi + 1)

• Utility function

. Communication graph:

12 3 6 7 8 5 9 10 4 11 1 14 15 13 2 GC 43 / 56

Simulation Results

44 / 56

Social Networks Example

45 / 56

Social Network Game

Consider a social media network of 5 users. Each user i produces xi unit of information that the followers can see in their news feeds. Assume a strongly connected communication digraph GC. i → j : j is a follower of i (j receives xi in his news feed). Assume a strongly connected interference (influence) digraph GI. i → j : j is influenced by i (whom j follows, or from whom gets attention). Idea: GI also includes the attention drawn from the network.

1 3 2 4 5 1 3 2 4 5 GI GC 46 / 56

Social Network Game

The cost function of player i consists of three parts (as in [Goel & Ronaghi, 12]): Ji(x) = Ci(xi) − f 1

i (x) − f 2 i (x)

Ci(xi) [Posting Cost]: A cost that user i pays to produce xi unit of information. Ci(xi) := hixi, hi > 0. f 1

i (x) [Payoff from users whom i follows]: A payoff that user i obtains from

receiving information from his news feed. f 1

i (x) := Li

• j∈Nin

C(i)

qjixj, Li > 0, where qji represents i’s interest in user j’s unit of information. f 2

i (x) [Attention from the network]: An incremental payoff that user i obtains

from receiving attention in his network (via his followers). f 2

i (x) =

• l∈Nout

C (i)

Ll

• j∈Nin

C(l)

qjlxj

• Payoff that l
• btains from his news feed

including from i

−

• j∈Nin

C(l)\{i}

qjlxj

• Payoff that l
• btains from his news feed

except from i

• .

47 / 56

Simulation Results

1000 2000 3000 4000 5000 6000 7000 8000 −0.5 0.5 1 1.5 2 2.5 3 Iteration Unit of information that each user produces x1 x2 x3 x4 x5

1 3 2 4 5 1 3 2 4 5 GI GC

User # of followers Analysis of NE 1 1 — 2 1 — 3 2 Players 3, 4 have more followers ⇒ more attention⇒ 4 3 They produce more information: x∗

4 ≥ x∗ 3 ≥ x∗ 1,2,5.

5 1 Player 5 receives x3, x4 ⇒ x∗

5 ≥ x∗ 1,2. 48 / 56