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Bayesian Games. Bayesian Games. Definitions and Static Bayesian Games Example of solving a static Bayesian game Sender-Receiver Bayesian Games Equilibrium concepts An example of Bayesian games in Computer Science: Ad Hoc Networks: Modelling cooperation. What is a Bayesian game?.

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slide1
Bayesian Games

© Petteri Nurmi 2003

bayesian games
Bayesian Games
  • Definitions and Static Bayesian Games
  • Example of solving a static Bayesian game
  • Sender-Receiver Bayesian Games
  • Equilibrium concepts
  • An example of Bayesian games in Computer Science: Ad Hoc Networks: Modelling cooperation

© Petteri Nurmi 2003

slide3
What is a Bayesian game?
  • A strategic form game with incomplete information
    • What is incomplete information?
      • Some players don’t know the payoff of the others
      • Incomplete information is not imperfect information
      • Imperfect information = Players don’t observe the actions of others correctly
  • Classic reference:
  • Harsanyi J., 1967-1968,
  • Games with incomplete information played by Bayesian players
  • Management Science 14: 159-182; 320-334; 486-502

© Petteri Nurmi 2003

harsanyi s model
Harsanyi’s Model
  • The game is transformed into a game of imperfect information
    • A prior move by nature
    • Nature’s move determines player’s ”type”
    • The equilibrium of this game is the Bayes-Nash equilibrium
  • John C. Harsanyi (29.5. 1920 – 2000)
  • Nobel prize in 1994 together with
  • John F. Nash Jr.
  • Reinhard Selten
  • "for their pioneering analysis of equilibria in the theory of non-cooperative games"

© Petteri Nurmi 2003

mathematical intermezzo
Mathematical Intermezzo
  • A Bayesian Game consists of the following components
    • A (finite) set of players N = {1,…, n}
    • An action set for each player Ai ; A = χi ∈ N Ai
    • A type set Θi ; Θi = χi ∈ N Θ i
    • A probability function pi: Θ i Δ (Θ -i)
    • A payoff function: ui: A x Θ  ℝ

© Petteri Nurmi 2003

slide6
Type?
  • In Harsanyi model nature selected player’s type  What is a type?
    • Any private information (or is not common knowledge) that is relevant to the players’ decision making
    • Such as?
      • Player’s payoff function
      • Player’s beliefs about other players’ payoff functions
      • Beliefs about what players believe his beliefs are
      • … and so on

© Petteri Nurmi 2003

type continued
Type continued
  • Let us denote Θi player i’s type
  • Type Θi is observed by player i only
  • p(Θi | Θ-i ) denotes player i’s conditional probability about his opponents types given his type

Θ-i = (Θ1 , …, Θi-1, Θi+1 , …, ΘI )

  • We assume that the marginal pi (Θi) is strictly positive.

© Petteri Nurmi 2003

strategy
Strategy
  • A pure-strategy space Si represents choices of actions
  • σ(Θi) is the strategy player i chooses when his type is Θi
  • Mathematically defined:
    • Strategy is a mapping from set of types to the set of Actions
      • σ(Θi) = Ai ; σi : Θi Ai∀ i ∈ N (pure)
      • σ(Ai | Θi) = α ; σi : Θi Δ(Ai) ∀ i ∈ N (mixed)
        • α represents the (conditional) probability that player i chooses action i when the type is Θi

© Petteri Nurmi 2003

equilibrium
Equilibrium
  • Bayesian equilibrium = Bayes-Nash equilibrium
    • Player i maximizes her expected utility conditional on type Θi

si (Θi) ∈ arg max  p(Θi | Θ-i) ui (s i’, s-i (Θ-i), (Θi, Θ-i))

si’ ∈ SiΘ-i

© Petteri Nurmi 2003

example
Enter

Don’t

0, -1

2, 0

1.5, -1

3.5, 0

B

2, 1

3, 0

2, 1

3, 0

DB

LOW

Example
  • Consider the following game
    • Player I decides whether to build a new factory
    • Simultaneously player II decides whether to enter or not
    • Player I’s decision depends on her building cost that is unknown to player II

Enter

Don’t

B

DB

HIGH

© Petteri Nurmi 2003

example cont
Example cont.
  • This can be seen as a Bayesian game
    • Set of players N = { I, II }
    • Action sets A1 = { B, DB} ; A2 = {Enter, Don’t }
    • Type sets Θ1 = { HIGH, LOW } ; Θ2 = { X }
  • Player II has a singleton set as the type space so we can ignore it.
  • Let cl be low cost and ch high cost types

© Petteri Nurmi 2003

example cont12
Enter

Don’t

Enter

Don’t

0, -1

2, 0

1.5, -1

3.5, 0

B

B

2, 1

3, 0

DB

2, 1

3, 0

DB

HIGH

LOW

Example cont.
  • A strategy for player I is an action for EACH of it types
  • For the high-cost type of player I we have a dominant strategy don’t build

© Petteri Nurmi 2003

example cont13
Enter

Don’t

1.5, -1

3.5, 0

B

2, 1

3, 0

DB

LOW

Example cont.
  • The best-response for player I low-cost type depends on player II’s strategy

u1(B; y; cl) = 1.5y + 3.5(1-y) = 3,5 – 2y

u1(DB; y; ch) = 2y + 3(1 – y) = 3 – y

  • Player I’s low-cost type

prefers building IF y ≤ ½

© Petteri Nurmi 2003

example cont14
Example cont.
  • For player II we must first consider the possibility that the cost is actually low 

u2(E;x) = p + (1-p) [-2x + 2 (1 –x)] = 2 – 4(1 – p)x

 x ≤ 1 / 2(1 – p) (:=w)

  • Now we need to compare the best-response correspondences, for player II the correspondence is

{1} x < w

y*(x) = [0, 1] x = w

{0} x > w

© Petteri Nurmi 2003

example cont15
Example cont.
  • In a similar way we get the correspondence mapping for player I.
  • Now the Bayesian equilibrium is the intersection of the correspondence functions (with a fixed value for p1)

© Petteri Nurmi 2003

sender receiver games
Sender-Receiver Games
  • Game has two players a sender and a receiver
  • Sender sends a signal to receiver who then chooses an appropriate action
  • Player I has private information about his type
  • Player II has only one type, which is considered common knowledge
  • “A move by nature”

© Petteri Nurmi 2003

strategies in s r games
Strategies in S-R games
  • A pure strategy for the Sender is a one-to-one correspondence mapping m: Θ M
  • Let σ(m | Θ) be the probability that a type Θ-sender sends message m  mixed-strategy for sender
  • For receiver let p(a | m) be a mixed-strategy (choose action a if message = m)
  • On-the-path messages

ℳ+(Θ) = {m: ∃ θ ∈ Θσ(m | Θ) > 0 } = supp σ(Θ)

© Petteri Nurmi 2003

payoffs in s r games
Payoffs in S-R games
  • Sender’s (expected) payoff: a∈Aρ(a|m)u(m,a,θ)
  • For the receiver?
    • Must consider every type and every message
    • E(v(a,σ)) = m∈Mθ∈Θρ(θ) σ(m| θ)u(m,a,θ)

© Petteri Nurmi 2003

bayes rule in action
Bayes’ rule in action
  • For any on-the-path message m, the receiver’s posterior belief that player I is of type θ, is pB(θ |m)

pB(θ | m) = p(θ)σ(m|θ)

θ’∈Θp(θ’)σ(m| θ)

  • POSTERIOR rule for updating PRIOR beliefs!

© Petteri Nurmi 2003

bayes equilibrium in s r games
Bayes Equilibrium in S-R games
  • In a Sender-Receiver game the Bayesian equilibrium is a triple (σ, ρ, θ) ∈ℳθxAMx(Δ(θ))M satisfying the following conditions:
      • For all types θ ∈ Θ

supp σ(Θ) ⊂M´(ρ, Θ)

      • For all on-path-messages: ∀ m∈ M+(σ)

supp ρ(m) ⊂A´(ρ´, Θ)

      • The conditional posterior belief system is consistent with Bayes’ rule whenever possible

© Petteri Nurmi 2003

perfect bayesian equilibrium
Perfect Bayesian Equilibrium
  • In a Sender-Receiver game the perfect Bayesian equilibrium is a triple (σ, ρ, θ) ∈ℳθxAMx(Δ(θ))M satisfying the following conditions:
      • For all types θ ∈ Θ

supp σ(Θ) ⊂M´(ρ, Θ)

      • For all messages: ∀ m∈ M(σ)

supp ρ(m) ⊂A´(ρ´, Θ)

      • The conditional posterior belief system is consistent with Bayes’ rule whenever possible

© Petteri Nurmi 2003

other types of equilibrium
Other types of Equilibrium
  • Perfect Bayesian equilibrium in Multi-Stage Games
    • Posterior beliefs are independent, and all types of player i have the same beliefs
    • Bayes rule to update beliefs (history information?)
    • “no signalling what you don’t know”
    • Posterior beliefs need to be consistent with a common joint distribution
  • Extensive-Form games: Sequential Equilibrium
    • See Fudenberg, D., and J.Tirole Game Theory 1991 The MIT Press p. 337-341
  • Trembling-Hand Perfect Equilibrium (p. 351-356)
  • Proper equilibrium (p.356-359)

© Petteri Nurmi 2003

bayesian games in cs
Bayesian games in CS
  • Ad Hoc Networks
  • Auctions
  • Social learning
  • The web search game
  • Voting

© Petteri Nurmi 2003

ad hoc networks
Ad hoc networks
  • MANET = Mobile Ad Hoc Networks
    • A set of mobile hosts, each with a transceiver
    • No base stations; no fixed network infrastructure
    • Multi-hop communication
    • Routing and packet forwarding takes place in a dynamical network topology
  • Game Theory and MANET?
    • Routing mechanisms for “selfish cooperation”

© Petteri Nurmi 2003

cooperation in manet
Cooperation in MANET
  • Reference: Modelling cooperation in Mobile Ad Hoc Networks: A formal description of Selfishness; Urpi A., Bonuccelli M., and Giordano S.
  • Modelling Ad Hoc Networks with Bayesian games
    • The nodes are the players
    • Nodes have to periodically select whether to forward or not
    • Nodes have incomplete information about the total traffic in the network
    • Nodes have local information about their neighbourhood

© Petteri Nurmi 2003

slide27
cont.
  • Important issues for each node
    • Energy consumption
    • The packets are forwarded by someone
  • “A shared medium”
    • Packets are send to every node that is within the transmission range
  • Prior to choosing its next action, a node has an opportunity to analyze its neighbours past behaviour
  • Node most decide to whom to send packets and to whom to discard packets.

© Petteri Nurmi 2003

the model
The Model
  • Time is discrete and divided into timeslots t1,…, tn
  • Node i has the following information in the beginning of frame tk
      • Ni(tk): Set of neighbours, assumed to be fixed during a single frame
      • Bi(tk): The remaining energy units (in the battery)
      • Tij(tk): The traffic node i generated as a source and has to send to node j during frame k. (for each node j in node i’s neighbourhood)
      • Fij(tk-1): The number of packets that j forwarded for i during the previous frame
      • Rij(tk-1): The number of packets i received from j during the previous frame
      • Ȓij(tk-1): The number of packets i received from j during the previous frame as a final destination

© Petteri Nurmi 2003

the forwarding game
The forwarding game
  • Nodes are the players
  • Player i’s type is its energy class e(i) = α, where 0 ≤ α≤ 1
  • Player i as an action sets Sij(tk) i.e. the number of packets she will send to node j, and Fij(tk) the number of packets received from j during the previous frame she will forward to her
  • Player i’s payoff is:

αe(i) Wi(tk) + (1 – αe(i)) Gi(tk)

      • Where
        • Wi(tk) is a measure of the energy spent succesfully
        • Gi(tk) is a the ratio of sent packets over packets that player i wanted to send.

© Petteri Nurmi 2003

the forwarding game cont
The forwarding game cont.
  • Player i has prior belief for every player j in its neighbourhood, what its energy class is.
  • A node tries to maximize its payoff function

 SELFISHNESS

  • We need to analyze the game as a repeated (dynamic) game and provide a utility function that makes it profitable to player i to cooperate

© Petteri Nurmi 2003

problems
Problems
  • How to get the forwarding information?
  • Badly defined utility function and/or policy leads to self destruction
  • The usage of time slots
    • There is no synchronization!
  • Too simple decision space?
    • Possible other constaints.

© Petteri Nurmi 2003

problems cont
Problems cont.
  • Malicious and selfish users?
    • Need a stronger policy
      • Punishing vs. Encouraging
        • Punished better suitable because
          • How to reward agents? (better throughput in a network with no authority?)
          • Punishing more suitable to both malicious and selfish users, encouraging/rewarding suitable only for encouraging cooperation
  • Theorem: Cooperation can be enforced in a mobile Ad Hoc network, provided that enough members agree on it and that no node has to forward more traffic than it generates.

© Petteri Nurmi 2003

additional references
Additional References
  • Cooperation in wireless ad hoc networks. Srinivasan V., Nuggehalli P., Chiasserini C-F, and Ramesh R. R., In Proceedings of IEEE Infocom 2003 http://citeseer.nj.nec.com/568937.html
  • Game Theoretic analysis of security in mobile ad hoc networks. Michiardi P., and Molva R. Technical Report RR-02-070, Institut Eurecom 2002.

© Petteri Nurmi 2003

slide34
Summary

© Petteri Nurmi 2003

types of bayesian games
Types of Bayesian games
  • Static Bayesian games
  • Dynamic Bayesian games
    • Sender-Receiver Games
    • Extensive Form Games
    • Multi-Stage Games
  • Equilibrium concepts
    • Bayesian Equilibrium = Bayes-Nash Equilibrium
    • Bayes Equilibrium (in dynamic games)
    • Perfect Bayes Equilibrium

© Petteri Nurmi 2003

applications of static bayesiang
Applications of static BayesianG
  • Packet forwarding in Ad Hoc networks
  • Voting mechanisms
  • Auction mechanisms
  • = MULTI-AGENT SYSTEMS
  • Requires:
    • Simultaneous competition
    • Multiple agents with incomplete information
    • Can also be non-simultaneous competition if the agents/players don’t know each others’ decisions (but have same beliefs that affect their decision-making).

© Petteri Nurmi 2003

applications of dyn bayesiang
Applications of Dyn. BayesianG.
  • Many economic applications
  • Design model for network protocols?
  • Design model for multiprocessor architectures?
  • Bayesian games are a suitable tool for modelling situations where there is interaction between two or more agents and the prior information is incomplete.

© Petteri Nurmi 2003

references
References
  • Fudenberg, D., and J.Tirole Game Theory 1991 The MIT Press
  • Kockesen L., Bayesian Games, http://www.columbia.edu/~lk290/ugbayes.pdf
  • Ratliff J., Static Games of Incomplete Information
  • Myatt D. P., Who Am I Playing? Incomplete Information and Bayesian Games,

http://malroy.econ.ox.ac.uk/dpm/MPhilGameTheory/IncompleteStrategic.pdf

  • Urpi A., Bonuccelli M., Giordano S., Modelling cooperation in mobile ad hoc networks: a formal description of selfishness

© Petteri Nurmi 2003

additional material
Additional Material
  • Eyster E., and M.Rabin Cursed Equilibrium 2000
  • Jackson M., Kalai E., Social Learning in Recurring Games
  • Khoussainov R., and N. Kushmerick Playing the Web Search Game
  • Tenneholtz M., Robust Decision-Making in Multi-Agent Systems

© Petteri Nurmi 2003

the end
The End

Bayesian Games

by Petteri Nurmi

http://www.cs.helsinki.fi/u/ptnurmi/papers.html

© Petteri Nurmi 2003

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