Introduction to Game Theory Applied to
Spectrum Sharing
Luiz DaSilva
Professor of Telecommunications
Summer School on Spectrum Aggregation and Sharing for 5G
Networks
EURECOM, Sophia Antipolis, France, 18 October 2016
Acknowledgements
This presentation was made possible with funding from:
 The European Commission through Horizon 2020 project
WiSHFUL (grant agreement 645274)
 The Science Foundation Ireland
Trinity College Dublin, The University of Dublin
Trinity College Dublin
Founded 1592
Trinity College Dublin, The University of Dublin
Objectives
 Introduce basic concepts of cooperative and non-cooperative games
 Illustrate the modeling of wireless communications/networks
problems using game theory
 Provide a brief overview of more advanced concepts in game theory,
such as coalitional games and Bayesian games
 Present some examples from the recent wireless comms literature
that use game theory as a tool for analysis and design
 Introduce the WiSHFUL project, which provides tools for
experimentation in wireless networks, including spectrum sharing
What is game theory?
A set of analytical tools from economics and mathematics to
predict the outcome of complex interactions among rational
entities
Interactions among adaptations
performed by autonomous
elements in a network
Economic models of spectrum
markets, including licensed
shared access
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in the context of resource /
spectrum sharing…
Incentives for OTT service
providers to rely on own
infrastructure versus contracting
with MNOs
(…)
Components of a game
A set of 2 or more players
A set of actions for each player
A set of preference relationships for each player for each
possible action tuple
 usually expressed as a utility function
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Cognitive adaptations as a game
Cognitive Radio
Cognitive radios in network
Available waveforms
(modulation, coding,
operating frequency)
Game
Player set
Action set
Objective function
(e.g., increasing, concave
function of SINR)
Utility function
Distributed channel selection as a game
Spectrum-agile Radio
Autonomous radios in a
network
Available channels
Game
Player set
Objective function
(e.g., derived from
network connectivity
graph and conflict graph)
Utility function
Action set
Opportunistic spectrum access as a game
Cognitive Radio
Secondary users in a network
Potentially available channels
Objective function
(e.g., 0 if any conflicts
with primary user,
increasing w/ # of
channels used
otherwise)
Game
Player set
Action set
Utility function
Pricing of spectrum as a game
Radios
Secondary users in a network
(bandwidth, bid)
Game
Player set
Action set
Objective function
(e.g., consumer surplus)
Utility function
Formal model
Normal form games
G  N , A, ui 
N – Set of players
Ai – Set of actions available to player i
A – Action space
{ui} – Set of individual payoff (utility) functions
Nash equilibrium
John Nash (1928-2015)
A point from which no user can benefit by unilaterally deviating
An action tuple a is a Nash equilibrium if, for every player i in N and
every action bi in Ai,
ui (a)  ui (bi , a i )
Existence and uniqueness depend on the structure of the game
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Mixed strategies
Randomizations over actions
A player can randomize over the set of actions available to her
Denote by σi a mixed strategy available to player i
– A probability distribution
– For discrete action sets, σi(ai) is the probability assigned to
action ai
Each player now tries to randomize her expected utility
N
ui ( )   ( j (a j ))ui (a)
aA
j 1
Pareto optimality
Vilfredo Pareto (1848-1923)
A resource allocation solution is Pareto optimal if no player can be
made happier without sacrificing the welfare of at least one other
player
– A measure of efficiency in resource allocation
– In multi-objective optimization, the Pareto frontier is often
sought
Nash equilibria are not necessarily Pareto optimal
– The prisoner’s dilemma is a famous example of unique Nash
equilibrium that is not Pareto optimal
Predictive power of Nash equilibrium
A consistent prediction of the outcome of the game
– If all players predict the NE, it is reasonable to assume that they will play it
– Once reached, there is no reason to believe any player will deviate, and
the system will remain in equilibrium until conditions change
But not without its issues…
– If players start from an action profile that is not an NE, are we sure they
eventually reach the NE? (Convergence)
– What if there are multiple NEs? Is one more likely than the others?
(Refinements to the concept of NE)
– Vulnerable to deviations by a coalition of players
(…)
Predictive power of Nash equilibrium
Games model cooperation and competition between “intelligent
decision makers”
 Game theory is multi-agent decision theory
Must model preference relations for these decision makers, and
from those derive utility functions
 The utility function can be the “weakest link” in the model
 There is no fixed recipe for how to determine the appropriate
utility function… but for particular applications there may be
certain properties that the function is expected to have
(…)
Non-cooperative vs. cooperative game theory
• e.g., power control and
interference games
• e.g., spectrum sharing among
equals
• establish a Nash equilibrium
• establish a bargaining solution
• establish a path to the Nash
equilibrium (e.g., best response
for potential games)
• establish a path to the bargaining
solution
Cooperative games
Intuition
To understand the outcome of a bargaining process, we should
not focus on trying to model the process itself, but
instead we should list the properties, or axioms, that we expect
the outcome of the bargaining process to exhibit. (Nash)
Agreement Point
A possible outcome of
the bargaining game
Bargaining Solution
A map from a bargaining
game to a solution
Disagreement Point
The expected outcome of the
game if players do not
come to an agreement
Bargaining solutions
A bargaining solution φ is a mapping that associates with each
problem (U, uo) a unique point in U
Note: uo is the utility achieved at disagreement point a0
Many such mappings have been proposed in the game theory
literature (including by Nash)
Nash’s axioms
A prelude to the Nash bargaining solution
 Individual rationality
 Pareto optimality
 Invariance to affine transformations
 Independence of irrelevant alternatives
 Symmetry
We call φ(U, uo) the Nash bargaining solution iff it satisfies the five
axioms above
Example (non-convex utility space)
[Juris, DaSilva, Han, MacKenzie, Komali, IEEE TWC 2009]
 A – NBS (unique
maximizer of NP for
both U and its convex
hull)
 B – KSS (WPO)
 A – PO extension of
KSS
 C – ES
 D – PO extension of
ES
Suppose: u0 = 0
Application to spectrum sharing and capacity
[Juris, DaSilva, Han, MacKenzie, Komali, IEEE TWC 2009]
Incentives for infrastructure deployment by OTT
service providers
[Kibiłda, Malandrino, DaSilva, IEEE ICC, 2016]
• Games between operators
 How much infra-structure to deploy
individually and how much to deploy
collectively?
 Spectrum versus infra-structure sharing
• Games between operators and over-the-top
service providers
 Should the OTT deploy its own infrastructure?
• Model as a cooperative game
Other (more complex) game theoretic models
• Hierarchy of decision makers
• Stackelberg games
• Uncertainty of player types
• Bayesian games
• Sub-sets of players cooperating
• Coalitional games
• Setting the rules of the game
• Mechanism design
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Coalitional games
Motivation
 The basic formulation of non-cooperative games is that players act
alone
 There are many practical cases in which subsets of players can
benefit from entering into coalitions
 E.g., spectrum sharing agreements among neighbors
 The grand coalition is the coalition of all players
 It does not necessarily maximize social welfare
 E.g., spectrum reuse may be more advantageous
 We are interested in finding the coalition structure that does
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Coalitional games
Definitions
 An N-player coalition game is defined by the pair (N, v), where v is a
real-value function defined on the subsets of N (the coalitions),
called the value of the game, with v() =0.
 A coalition structure (CS) is a partition of N into exhaustive and
disjoint subsets. We will denote the set of all CSs as C.
 The cardinality of C is given by the Bell number
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Coalitional games
Welfare maximization
 Let us denote by V(Cj) the value of coalition structure Cj, the sum of
the values of all coalitions in this CS
 The welfare-maximizing CS C* is given by
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Coalitional games
Stability and equilibrium concepts
 Internal stability: no player has an incentive to leave its current
coalition and join one of the existing coalitions
 External stability: no coalition has an incentive to merge with one of
the other existing coalitions
 A coalition structure is a multicoalition equilibrium if its internally
and externally stable
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Coalitional games
Imputations and the core
 An imputation vector x is a vector of payoffs to the N players such
that
1.
xi ≥ v({i}) for all i in N
2.
 An imputation x is unstable if there exists a coalition S such that
 The set of stable imputations is called the core
 There are cases where the core is non-empty by there is no path
to the formation of the grand coalition
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Spectrum-sharing coalitions
[Khan, Glisic, DaSilva, Lehtomaki, IEEE TCIAIG, 2013]
• N transmitter/receiver pairs [players]
• Channel selection and transmit power
[actions]
• Utility can include network-wide spectrum
efficiency, fairness, network connectivity
• Study the coalition formation process and
the stability of coalitions
Stackelberg games
Heinrich von Stackelberg (1905-1946)
 Hierarchical games
 One player is the leader, others are the followers
 Followers react rationally to the actions of the leader
 Is it better to be the leader or the follower?
 Stackelberg equilibria is the equivalent of Nash equilibria, applied to
hierarchical games
Hierarchical spectrum sharing
[Xiao, Bi, Niyato, DaSilva, IEEE JSAC, 2012]
❶ Primary users (PUs) can charge secondary users
(SUs) for access to spectrum
❷ SUs distributedly select on which sub-bands to
operate
Multiple SUs can occupy the same sub-band and
cooperate in communicating
❸ SUs control their transmit power
Model as inter-related Stackelberg game and coalition
formation game
Derive an algorithm to arrive at the NE for the
individual games and the SE for the hierarchical game
Bayesian game
Thomas Bayes (1702-1761)
Payoffs are not known to all players
 Games of imperfect information
Model as different player types and a set of beliefs
 Beliefs are probability distributions over combinations of rivals’
types
Payoffs depend on actions and on types
(…)
Bayesian games
Formulation (1/2)
Set of players
Ai – set of actions available
to player i
A = A 1 x A 2 x … x An
Qi – set of possible types for
player i
Q = Q1 x Q2 x … x Qn
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pi – probability distribution
over player i’s types
Bayesian games
Formulation (2/2)
Set of outcomes
Mapping from actions to
outcomes
g: A  O
ui} – set of individual utility
functions
ui : Qi x O R
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Supporting D2D communications in cellular bands
[Xiao, Chen, Yuen, Han, DaSilva, IEEE TMC, 2015]
• D2D links [players] compete for subbands occupied by a cellular
subscriber (if interference is
tolerable) or for a sub-band for
exclusive use (otherwise)
• Multiple D2D links can share a subband
• D2D links do not know about others’
preferences, location, link conditions
• Bayesian non-transferable utility
overlapping coalition formation game
• Propose a hierarchical matching
algorithm to achieve a stable, unique
matching structure
Dynamically matching subscribers to operators
[Xiao, Han, Chen, DaSilva, IEEE JSAC, 2015]
• Subscribers [players]
dynamically request channels
of operators
• Bayesian game: subscribers
are unaware of each other’s
preferences
• Belief functions, learning
• Matching market: subscribers
are matched to operators,
then to sub-bands controlled
by the operator
• Design a mechanism that
incentivises truth-telling
The WiSHFUL H2020 project
 Wireless Software and Hardware platforms for Flexible and Unified
radio and network controL
 Offers open, flexible & adaptive software and hardware platforms for
radio control and network protocol development for prototyping
 Support for experimentation with intelligent control of radio and
network settings, enabling intelligent, node-level and network-wide
decisions, driven by higher-level domain-specific application
demands and taking into account external policies (for example
policies for dynamic spectrum access)
 Open call 3 deadline 28 October 2016
www.wishful-project.eu
The WiSHFUL H2020 project
Intelligence framework
WiSHFUL Intelligence Repository
(data aggrega on algorithms, intelligent algorithms
& generic and showcase-specific ac on modules)
Applica on
API
Data Collec on
Component
Data
Collec on
Aggrega on
Intelligence Composi on
Component
Ac on
Component
Data
Analysis
Ac on
WiSHFUL UPIs
Some conclusions and final words
• Game theory is being used to model increasingly complex
interactions among autonomous decision makers
• Models are particularly tailored to autonomous decision making and
reasoning by different network entities - in line with trends in
wireless networks (HetNets, D2D, resource sharing, etc.)
• Models can be applied at different scales: individual transmissions by
nodes, longer-term decisions by transmitters or by users, interactions
among networks, operators, etc.
• Machine learning meets game theory: some learning processes can
be shown to converge to Nash equilibria (e.g., application of learning
automata to dynamic channel selection)
Thank You
luizdasilva.wordpress.com