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Bayesian Games

© Petteri Nurmi 2003

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

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

- 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

- 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

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

- 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

- 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

- 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

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.

- 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

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

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 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 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

- 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

- 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

- 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∈Mθ∈Θρ(θ) σ(m| θ)u(m,a,θ)

© Petteri Nurmi 2003

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

- 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

- 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

- 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

- Ad Hoc Networks
- Auctions
- Social learning
- The web search game
- Voting

© Petteri Nurmi 2003

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

- 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

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

- 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

- 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.

- 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

- 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.

- 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

- 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

Summary

© Petteri Nurmi 2003

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

- 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.

- 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

- 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

- 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

Bayesian Games

by Petteri Nurmi

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

© Petteri Nurmi 2003

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