Introduction to Game Theory application to networks

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Introduction to Game Theory application to networks. Joy Ghosh CSE 716, CSE@UB 25 th April, 2003. What is Game Theory ?. Study of problems of conflict and cooperation amongst independent decision makers

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### Introduction to Game Theory application to networks

Joy Ghosh

CSE 716, CSE@UB

25th April, 2003

What is Game Theory ?
• Study of problems of conflict and cooperation amongst independent decision makers
• Formal way of analyzing interactions among a group of rational agents who behave strategically
• Games of Strategy rather than Games of Chance!
• Ingredients:
• Players / decision makers
• Choices / feasible actions / pure strategies
• Payoffs / benefits / utilities
• Preferences to payoffs
Some basic concepts
• Group – Any game consisting of more than one player
• with single player the game becomes a decision problem!
• Interaction – Actions of one affects the other
• else it would become simple sequence of independent decisions
• Strategic – Players account for interdependence
• Rationality – Players consistently opt for best choices
• Common Knowledge
• All players know that all players are rational
• Equilibrium – a point of best shared interest for all
Classification of Problems
• Static vs. Dynamic
• In Dynamic problems the sequence of choices are relevant
• Cooperative vs. Non-cooperative
• In non-cooperative games players watch out for their own interests. In cooperative games, players form coalitions with shared objectives.
Decision Theory under Certainty
• Decision problem (A, ≤)
• Finite set of outcomes A = {a1, a2, …. an}
• Preference relation ‘≤’ : a ≤ b => ‘b’ is at least as good as ‘a’
• Completeness – for all a, b in A, either a≤b, or b≤a
• Transitivity – if a≤b and b≤c, then a≤c
• Utility function u: A  R (consist with preference relation)
• For all a,b in A, u(a) <= u(b) iff a ≤ b
• Rational decision maker tries to maximize utility
• Choose outcome a* in A s.t. for all a in A, a ≤ a*
Decision Theory under Uncertainty
• Lottery L = {(a1, p1), (a2, p2), …… (an, pn)}
• Σiεn pi = 1, 0 ≤ pi≤ 1
• Outcome ai occurs with probability pi
• Infinitely many different possible lotteries
• Large number of lottery comparisons
• Preference relation unobservable
• “Under additional restrictions on preferences over lotteries there exists a utility function over outcomes such that the expected utility of a lottery provides a consistent ranking of all lotteries” - John von Neumann and Oscar Morgenstern
• u(L) = Σiεn u(ai).pi
• Lottery A
• A1 (sure win of 3000) vs. A2 (80% chance to win 4000)
• A1 strictly preferred to A2
• Lottery B
• B1 (90% chance to win 3000) vs. B2 (70% chance to win 4000)
• B1 might still be preferred to B2
• Lottery C
• C1 (25% chance to win 3000) vs. C2 (20% chance to win 4000)
• Most people start preferring C2 over C1 even though these two lotteries are variations of the 1st pair in Lottery A
Game Theory – multi agent decision problem
• A normal (strategic) form game G consists of:
• A finite set of agents D = {1, 2, ….. N}
• Strategy sets S1, S2, ... SN = set of feasible actions for agents
• Strategy profile S = S1 x S2 x ... x SN
• Payoff function ui : S  R (i = 1, 2, …. N)
• NOTE: The preference an agent has is to the outcome and not to the individual action
Some standard games in normal form

Matching Pennies

Tough vs. Chicken

Row: gains if pennies matchCol: gains if there is no match

Iterated Deletion of Dominated Strategies
• Common Knowledge
• assumptions about other people’s rational behavior
• Some more definitions:
• S-i = S1 x S2 ....x S(i-1) x S(i+1) ....x SN (strategy sets of others)
• utility function of player i for a given pure strategy:
• ui (s  S) = ui (si, s-i)
• Belief i of agent i = probability distribution over S-i
• for pure strategies the probability distribution is a point distribution
• Player i is rational with beliefs i if:
• si arg max s-iS-i ui (s’i, s-i). i(s-i) for all s’i  Si
• Note: as igets fixed, player i faces a simple decision problem
Dominated Strategies
• Strongly Dominated
• si  Si is strictly dominated if:
•  s’i  Si s.t. ui(s’i, s-i) > ui(si, s-i) for all s-i  S-i
• Weakly Dominated
• if the inequality is weak () for all s-i  S-i, and strong (>) for at least one
• Rational players do not play strongly dominated strategy
Iterated Dominance (deletion)
• M is strictly dominated by L. Rational column player ignores M
• If row player knows column player is rational, he will ignore D
• If column player knows the above, then he will choose L
• With common knowledge about rationality of players U,L is the outcome
Iterated Dominance – Formal Definition

The game is solvable by pure strategy iterated strict dominance only if S contains a single strategy profile

Does the order of elimination matter?
• In games that are solvable by iterated dominance, the speed and order or elimination doesn’t matter.
• This is however not true for weakly dominated strategies.

Deletion Sequence #1: T, L

- (2,1) is the playoff

Deletion Sequence #2: B, R

- (1,1) is the playoff

Nash Equilibrium for pure strategy
• No incentive for a player to deviate from his best response to his/her belief about other player’s strategy
• U,L was the NE in the example of strongly dominated strategies
• Definitions:
• A strategy profile s* is a pure strategy Nash equilibrium of G iff
• ui (si*, s-i*) ≥ ui (si, s-i*) for all players i and for all si Si
• A pure strategy NE is strict if the inequality is strict
• There can be multiple Nash equilibria for a particular G
• Two people trying to meet at one out of 2 places (NY Game!)
Do pure strategies always work?
• Most games are not solvable by dominance
• Coordination game, zero-sum game
• Penny matching Game
• Whatever pure strategy one player chooses, the other can win by choosing a better strategy
• Players have to consider mixed strategies
Mixed strategies - definitions
• Mixed strategy i for player i is a probability distribution over his strategy space Si
• i : Si R+ s.t. siSii(si) = 1
• i is the set of probability distributions on Si
•  = 1 x 2 x … x N
• Player i’s expected payoff with mixed strategies
• ui (i, -i) = si, s-i ui(si, s-i) i(si) -i(s-i)
Mixed strategies – more definitions
• Mixed strategy NE of G is a *   such that:
• ui (i*, -i*) ≥ ui (i, -i*) for all i and for all i i
• In a finite game, support of a mixed strategy i:
• supp (i) = { si Si | i (si) > 0 }
• Proposition
• if i* is a mixed strategy NE and si’, si’’  supp (i*), then ui (si’, -i*) = ui (si’’, -i*)
A mixed strategy example game
• There is no pure strategy NE
• Row plays U with probability 
• Column plays L with probability 
• Players need to be indifferent to their choice of strategies:
• u1 (U, 2*) = u1 (D, 2*)
•  = 2 (1 - )
• u2 (L, 1*) = u2 (R, 1*)
•  + 2 (1 - ) = 4  + (1 -  )
•  = 1/4 ;  = 2/3
• Unique mixed NE
• 1* = 1/4 U + 3/4 D
• 2* = 2/3 L + 1/3 R
Two People Zero Sum Games – Pure Strategy
• One player’s winnings is another player’s loss!
• Each player does the following:
• For each of his/her strategies, compute the maximum of losses that he could incur.
• Choose the strategy with the minimum max loss
Example 2 people 0 sum Game
• Row is player 1; Column is player 2
• If aij > 0, player 1 wins, else player 2
• Player 1:
• i* = arg maxi (minj (aij))
• V(A) = minj a i*j is the gain-floor for the game A
• In this case, V(A) = -2, with i*  {2, 3}
• Player 2:
• j* = arg minj (maxi (aij))
• (A) = maxi a ij* is the loss-ceiling for the game A
• In this case, (A) = 0, with j* = 3
Two People Zero Sum Game – Mixed Strategy
• If (A) = V(A) then A has a point of equilibrium
• Else we need to develop mixed strategy
• Consider the following game:
• For player 1, we have V(A) = 0, with i* = 2
• For player 2, we have(A) = 1, with j* = 2
• No saddle point or equilibrium
• Let players 1, 2 play strategy i with probability xi, yi
Optimization Problem
• In a nutshell, the players are solving the following pair of dual linear programming problems
• Player 1
• Player 2
Application to networks
• Formulation for n users competing for fixed resources
• Generic non-cooperative game
• Each user has access control /parameter n
• Each user receives certain amount n() of network resources
•  (1, 2, …. n)
• n [0, nmax] for some nmax > 0
• n () is a non-decreasing function of n
• n (.) is continuous in n=1N [0, nmax] and is differentiable with respect to n
• If n = 0, n () =0 for all 
• n () maybe interpreted as the QOS received by the nth user
Formulation (contd.)
• Let network charges be fixed at M / unit resources
• Each user tries to maximize his/her net utility
• Un(n()) – M.n()
• Un is non-decreasing and Un(0) = 0;
• U’n is non-increasing, i.e. Un is concave
• Maximum net benefit of nth user
• yn = arg max (Un(n()) – M.n()) = (U’n)-1.(M)
• Action of nth user
• Modify n to make received QOS n() equal to desired yn
User iterations and equilibrium
• After the jth iteration/step, access parameter of user n:
• nj+1 = min (G (yn, n(j), nj), nmax)
• G (y, , )
•  , if  = y
• > , if  < y
• < , if  > y
• Nash equilibria
• A fixed or equilibrium point of this iteration is any *  [0, nmax]
• n* = min (G (yn, n(*), n*), nmax)
• By Brouwer’s fixed point theorem there exists at least one such fixed point.
Non-cooperative game for circuit switched network
• N users compete for K circuits
• nth user’s connection setup request is Poisson with intensity n and arbitrary holding time distribution with mean 1/ n
• Total traffic intensity:   .1/
• Aggregate arrival rate   n=1 N n
• Mean holding time over all connections 1/ = n=1N1/nn/
• Hence,  = n=1Nn/n
• Per user connection blocking probability (Erlang’s form)
• K()  (K/K!) / (k=0Kk/k!)
• Net arrival rate of nth user:
• n(1 - K())
• Mean number of occupied circuits for the nth user:
• n()  1/nn(1 - K(()))
• Thus, n and  depend on all arrival rates 
• Iteration using multiplicative increase and decrease
• nj+1 = min {yn/n . n, nmax}
• or, nj+1 = min {yn.n / (1 - K((j))) , nmax}
• By previous formulation we can find an equilibrium!
References
• Game Theory .NET - college lecture notes (http://www.gametheory.net)
• IE675: Game Theory - Dr. Wayne Bialas, Dept. of IE, SUNY Buffalo, (http://www.acsu.buffalo.edu/~bialas/IE675.html )
• “Computational Finance: Game and Information Theoretic Approach” – Dr. B. Mishra, Dept. of CS, NYU (http://www.cs.nyu.edu/mishra/COURSES/GAME/game.html)
• Introduction to Game Theory – Markus Mobius, Dept. of Economics, Harvard (http://icg.fas.harvard.edu/~ec1052/lecture/index.html)
• Infocom 2003 - “Nash equilibria of a generic networking game with applications to circuit-switched networks” - Youngmi Jin and George Kesidis, Dept. of EE & CS, Pennsylvania State University.