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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

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

- 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

A game of head-on collision

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-iS-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. siSii(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/nn/
- Hence, = n=1Nn/n
- Per user connection blocking probability (Erlang’s form)
- K() (K/K!) / (k=0Kk/k!)

Formulation leading to equilibrium

- Net arrival rate of nth user:
- n(1 - K())
- Mean number of occupied circuits for the nth user:
- n() 1/nn(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.

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