introduction to game theory application to networks n.
Download
Skip this Video
Loading SlideShow in 5 Seconds..
Introduction to Game Theory application to networks PowerPoint Presentation
Download Presentation
Introduction to Game Theory application to networks

Loading in 2 Seconds...

play fullscreen
1 / 31

Introduction to Game Theory application to networks - PowerPoint PPT Presentation


  • 173 Views
  • Uploaded on

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

loader
I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
capcha
Download Presentation

PowerPoint Slideshow about 'Introduction to Game Theory application to networks' - lieu


An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript
introduction to game theory application to networks

Introduction to Game Theory application to networks

Joy Ghosh

CSE 716, CSE@UB

25th April, 2003

what is game theory
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
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
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 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
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
allais paradox
Allais Paradox
  • 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
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
Some standard games in normal form

Matching Pennies

Tough vs. Chicken

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

A game of head-on collision

iterated deletion of dominated strategies
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
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
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
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
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
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
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 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 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
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
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
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
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
Optimization Problem
  • In a nutshell, the players are solving the following pair of dual linear programming problems
    • Player 1
    • Player 2
application to networks
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
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
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
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!)
formulation leading to equilibrium
Formulation leading to equilibrium
  • 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
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.