Multiagent Systems
Bayesian Games
© Manfred Huber 2012
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Bayesian Games
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Games so far can model some uncertainty
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Uncertainty about actions taken by other players
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Uncertainty about resulting next game
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(normal form, imperfect information game)
(stochastic games)
But agents share knowledge about the game
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Number of players
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Actions/strategies available to all players
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Payoff associated with each strategy
© Manfred Huber 2012
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Bayesian Games
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Bayesian games include uncertainty about the
game that is played in terms of the payoff.
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Agents play a game from a set of possible games
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All games have the same number of agents and the
same strategy space for all agents
Games occur according to a prior probability distribution
Only limited, agent-specific information about which
game is available, giving each agent a different posterior
probability about which game is being played
Uncertainty about actions taken by other players
© Manfred Huber 2012
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Bayesian Game
Information Set Definition
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Bayesian games can be defined using the
concept of Information Sets
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A Bayesian Game in Information Set notation is a
tuple (N, G, P, I)
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© Manfred Huber 2012
N is a set of agents
G is a set of games, all defined for the same strategy
space for each agent
P is a prior over games
I=(I1, …, IN) is the set of vectors of information sets for
the agents, each defining a partition over the set of
games for the corresponding agent.
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Bayesian Game
Information Set Definition
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Bayesian games with Information Sets
I2,1
I1,1
I2,1
3, 3
1, 7
1, 1
1, 0
0, 0
1, 2
7, 1
5, 5
0, 1
1, 1
2, 1
0, 0
P=0.1
I1,2
P=0.2
0, 3
5, 1
0, 1
1, 0
3, 1
2, 2
1, 1
0, 0
P=0.3
© Manfred Huber 2012
P=0.3
P=0.1
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Extensive Form with Chance Nodes
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Bayesian games with information sets can be
translated into extensive form games of
imperfect information
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A “Nature” agent performs the game selection in
the first node according to the prior distribution
Information sets serve the same function as in
imperfect information games
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Thus only set information is available to the agents
Utilities at leaf nodes are expected values given
the prior probabilities and information sets
© Manfred Huber 2012
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Extensive Form with Chance Nodes
Nature
I1,1, I2,1
U
L
R
5 5
,
3 3
D
L R
I1,2, I2,2
I1,1, I2,2
U
L
D
R
3 7 7 3 7 7 0, 0
,
,
,
3 3 3 3 3 3
© Manfred Huber 2012
I1,2, I2,1
L R
U
L R
1, 2 2,1 0, 0 0, 3
L
D
U
R
L R
D
L
R
5,1 3,1 2, 2 0,1 1, 0 1,1 0, 0
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Bayesian Game
Epistemic Type Definition
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Bayesian games can be defined using the
concept of Epistemic Type
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A Bayesian Game in Epistemic Type notation is a
tuple (N, A, Θ, p, u)
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N is a set of agents
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A=(A1,..,An) is the joint action space of the players
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Θ=(Θ1,…, ΘN) is the set of vectors of type spaces
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p: Θ
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© Manfred Huber 2012
[0..1] is the prior over types
u=(u1,..,uN) is the vector of utility functions for the
agents, where ui: AxΘ
R is the utility function of
agent i
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Bayesian Game
Epistemic Type Definition
© Manfred Huber 2012
θ
p
θ1,1 , θ2,1
0.3
θ1,1 , θ2,2
0.3
θ1,2 , θ2,1
0.3
θ1,2 , θ2,2
0.1
a1
U
U
D
D
U
U
D
D
U
U
D
D
U
U
D
D
a2
L
R
L
R
L
R
L
R
L
R
L
R
L
R
L
R
u
5/3 , 5/3
1 , 7/3
7/3 , 1
7/3 , 7/3
0,0
1,2
2,1
0,0
0,3
5,1
3,1
2,2
0,1
1,0
1, 1
0,0
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Strategies in Bayesian Games
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Pure strategies are mappings from the
agent’s epistemic type to an action
si :!!i " Ai
Mixed strategies are probability distributions
over pure strategies
si :!!i " #(Ai )
© Manfred Huber 2012
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Utility in Bayesian Games
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In Bayesian games utility can be defined in
different ways depending on the amount of
information about the type of the game
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Ex-ante: Before any information about the type is
available
Ex-interim: When only the agent’s type
information is available
Ex-post: When all agents’ type information is
available
© Manfred Huber 2012
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Utility in Bayesian Games
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Ex-ante expected utility
#
&
#
&
EUi (s) = ) %% p(! ) ) %% " s j (a j | ! j )(( ui (a, ! )((
'
! !* $
a!A $ j!N
'
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Ex-interim expected utility
$
'
$
'
EUi (s | ! i ) = * && p(! !i | ! i ) * && # s j (a j | ! j ))) ui (a, ! i , ! !i )))
(
!!i "+!i %
a"A % j"N
(
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Ex-post expected utility
#
&
EUi (s, ! ) = ) %% " s j (a j | ! j )(( ui (a, ! )
'
a!A $ j!N
© Manfred Huber 2012
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Nash Equilibrium
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As in other games we can define an agent’s
best response to other agents’ strategies
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Best Response is an action that leads to the
highest ex-ante utility when used with the
strategies of the other agents
BRi (s!i ) = arg max s'i ( EUi (s'i , s!i ))
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© Manfred Huber 2012
Using ex-interim expected utility would lead to the same
result since strategies are defined as mappings from the
agent’s type to actions
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Nash Equilibrium
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A Bayes-Nash equilibrium is a mixed strategy
that is a simultaneous best response for all
agents
!i!!!si " BRi (s#i )
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Can compute induced normal form over pure
strategies
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Either with ex-ante or ex-interim values
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© Manfred Huber 2012
Both will yield the same equilibria
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Ex-post Equilibrium
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An ex-post Bayes-Nash equilibrium is a mixed
strategy that is for all complete types a
simultaneous best response for all agents in
terms of ex-post expected utility (i.e.
assuming complete type information)
!! , i!!!si " arg max s'i "Si ( EUi (s'i , s#i , ! ))
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Ex-post Bayes-Nash is a strategy in which agents
do not need to have accurate beliefs about the
type distribution
© Manfred Huber 2012
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Bayesian Games
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Bayesian games allow to model uncertainty
about the game that is being played
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Applies to situations where the agents do not
know what game they are participating in
Each agent can know a different amount about
the game
Bayes-Nash equilibrium can be solved using
solution approaches from normal form games
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Induced normal form of Bayesian games
© Manfred Huber 2012
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