This slide was used at the 54th IEEE Conference on Decision and Control, Osaka, Japan.

1. December 18 Fri., 2015, 09:10-09:30, Regular Session: Modeling 1, Frb04.3 @ 802
Transition Models of Equilibrium
Assessment in Bayesian Game
Kiminao Kogiso
University of Electro-Communications
Tokyo, Japan
The 54 Conference on Decision and Control
Osaka International Convention Center, Osaka, Japan
December 15 to 18, 2015
Supported by
JSPS Grant-in-Aid for Challenging Exploratory Research
2014 to 2016

2. Outline
2
Introduction
Static Bayesian Game
Novel Form in Bayesian Nash Equilibrium
Dynamics in Equilibrium Assessment
Simulation
Conclusion

3. Introduction
3
Strategic game enabling to consider uncertainties in player’s decisions.
player: a reasonable decision maker
action: what a player chooses
utility: a player’s preference over the actions
type: a label of player’s private valuation (what the player really feels)
belief: a probability distribution over the types
(degree of feeling, tendency, proclivity,…)
Static Bayesian Game[1]
[1] Harsanyi, 1967. [2] Alpcan and Basar, et al., 2011, 2013. [3] Roy, et al., 2010. [4] Liu, et al., 2006. [5] Akkarajitsakul, et al., 2011.
A Bayesian game used in engineering problems to analyze a Bayesian
Nash equilibrium or to design a game mechanism.
network security[2,3], intrusion detection[4,5,6], belief learning[7]
electricity pricing[8,9], mechanism design[10]
[6] Sedjelmachi, et al., 2014, 2015. [7] Nachbar, 2008. [8] Li, et al., 2011, 2014. [9] Yang, et al., 2013. [10] Tao, et al., 2015.

4. Introduction
4
Insufficient tools and concepts[11]
Bayesian Nash equilibrium plays key roles in game analysis & design.
equilibrium analysis: for given belief, find a Bayesian Nash Equilibrium(BNE).
belief learning: for given BNE, find a corresponding belief.
mechanism design: for given utility, find rules to achieve a desired BNE.
Objective of this talk
Derive a dynamical state-space model whose state involves a BNE.
derive a novel condition related to the BNE,
discover a map (discrete-time system) defined by the novel condition,
confirm a time response of the map.
[11] Powell, 2011.
Challenge: prepare tools & concepts to apply our model-based fashion
to analysis and design of the game.

5. Bayesian Game
Player set
Action set
Type set
Utility
Strategy (mixed)
Belief
Static Bayesian Game: General
5
Two-player two-action Bayesian game w/ two types
G(N, A, ⇥, u, µ, S)
N := {1, 2}
A := A1 ⇥ A2
⇥ := ⇥1 ⇥ ⇥2
u := (u1, u2)
µ := (µ1, µ2)
S := (S1, S2)
ai 2 Ai := {a, ¯a} 8i 2 N
✓i 2 ⇥i := {✓, ¯✓} 8i 2 N
µi 2 ⇧(⇥i) 8i 2 N
Si : ⇥i ! ⇧(Ai) 8i 2 N
si 2 Si(⇥i) 8i 2 N
⇧(X) : a probability distribution over a finite set X
Ui(✓i, ✓ i) :=

ui(a, a, ✓i, ✓ i) ui(a, ¯a, ✓i, ✓ i)
ui(¯a, a, ✓i, ✓ i) ui(¯a, ¯a, ✓i, ✓ i)
: utility matrix8i 2 N, 8✓ 2 ⇥
ui : A ⇥ ⇥ ! < 8i 2 N
i 2 N

6. Static Bayesian Game: Example
6
Service of tennis
2, 2 0, 1
1, 21, 1
flat
spin
flat spin
0, 1 1, 2
0, 11, 2
flat
spin
flat spin
sideline
1, 0 1, 1
2, 00, 1
flat
spin
flat spin
1, 3 1, 2
0, 32, 2
flat
spin
flat spin
centerline
s1(a|✓)
s1(¯a|✓)
s1(¯a|¯✓)
s1(a|¯✓)
s2(a|¯✓) s2(¯a|¯✓)s2(¯a|✓)s2(a|✓)
center line ✓ side line ¯✓
✓¯✓
¯a
¯a ¯a
¯a
¯a ¯a
¯a¯aa
a a
a
µ1(✓)
µ1(¯✓)
µ2(¯✓)µ2(✓)
a
a a
a
type
belief

7. Bayesian Nash Equilibrium
7
Equilibrium assessment
definitions of Bayesian Nash Equilibrium(BNE)
using an ex-ante expected utility:
using a best response to opponent strategy:
EUi(si, s i) EUi(s0
, s i) 8s0
i 2 Si, s0
i 6= si
is denoted as the Bayesian Nash equilibrium.
A strategy profile , satisfying , is also a BNE.s = (si, s i)
Given a prior common probability , for any , the strategy satisfyingi 2 N sp(µ)
si 2 BRi(s i, µ) 8i 2 N
the pair of considered as key variables of the Bayesian game.
Equilibrium Assessment : a pair of a belief and the corresponding BNE.(ˆµ, ˆs)
(µ, s)
equilibrium analysis[10]: find a BNE .ˆs9 ˆµ,
[10] Y. Shoham and K. Leyton-Brown, Multiagent Systems, Cambridge University Press, 2009.

8. then the pair is an Equilibrium Assessment, where ,✏ :=
⇥
1 1
⇤
(ˆµ, ˆs)
Novel Form Satisfying BNE
8
If the game satisfies the following condition (simultaneous polynomial in ):
Sufficient condition to be BNE
Lemma
8✓i 2 ⇥i, 8i 2 N
G
✏⇣i(ˆs i, ✓i) (✓i)p(ˆµ) = 0
⇣i(ˆs i, ✓i) :=
⇥
Ui(✓i, ✓)ˆs i(✓) Ui(✓i, ¯✓)ˆs i(¯✓)
⇤
,
(✓) :=

1 0 0 0
0 1 0 0 , (¯✓) :=

0 0 1 0
0 0 0 1 .
idea: derived from KKT condition of BNE by cancelation of Lagrangian variables.
point: # of the polynomials: 4, # of the variables: 6; D.O.F. in determining their values.
note: a BNE (mixed strategy) holds the above equation, but some of pure strategy BNEs
do not hold it.
ˆµ
all of EAs

9. Discover Dynamics!
9
Map from EA to EA
Idea to derive dynamics in EA
all of EAs
⇥
EA
(ˆµ, ˆs)
satisfying
the Lemma
all of EAs
⇥
EA
(ˆµ + ˆµ, ˆs + ˆs)
satisfying
the Lemma

10. Given an initial EA, if there exists such that the game satisfies
the following condition w.r.t. utility matrices: ,
Dynamics in Equilibrium Assessment
10
Main result
Theorem
⇥
1 1
⇤
Ui(✓i, ✓)

1
1 1
= 0
⇥
1 1
⇤
Ui(✓i, ¯✓)

2
1 2
= 0
8✓i 2 ⇥i8i 2 N
ˆµ(k + 1) = diag(A1, A2)ˆµ(k)
ˆs(k + 1) = A (ci(k))ˆsi(k)
ci(k) :=
ˆµi(✓i, k + 1)
ˆµi(✓i, k)
, and is a row stochastic matrix.Ai 2 <2⇥2
8i 2 N
= [ 1 2]T
2 <2
then a nonlinear autonomous system in terms of the equilibrium assessment:
transfers from an EA to another EA , where(ˆµ(k), ˆs(k)) (ˆµ(k + 1), ˆs(k + 1))
ci(k) ! 1
A (1) = I
ci(k) :=
ˆµi(✓i, k + 1)
ˆµi(✓i, k)ˆµ(k + 1) = diag(A1, A2)ˆµ(k) ˆs(k + 1) = A (ci(k))ˆsi(k)
ˆµ(k)
ˆµ(k + 1)
stable linear system: time-varying system:
·
ˆs(k)
ˆµ(k)

11. Simulation
11
Trajectory of equilibrium assessment
0.3
0.4
0.5
0.6
0.7
10
0.2
0.4
0.6
0.8
probabilityprobability
step
0 1 2 3 4 5 6 7 8 9
100 1 2 3 4 5 6 7 8 9
s1( )a|θ−− s1( )a|θ−
−
s1( )a|θ
−
− s1( )a|θ
−−
s2( )a|θ−− s2( )a|θ−
−
s2( )a|θ
−
− s2( )a|θ
−−
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
the proposed model
the computation of
the best respose (4)
10
expectedutilityvalue
step
0 1 2 3 4 5 6 7 8 9
EU1
EU2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
10
step
0 1 2 3 4 5 6 7 8 9
valueofci()θi
c ( )1 θ−
c ( )1 θ
−
c ( )2 θ−
c ( )2 θ
−
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
10
probability
step
0 1 2 3 4 5 6 7 8 9
1( )η θ−
1( )η θ
−
2( )η θ−
2( )η θ
−
belief strategy
c expected
utility

12. Conclusion
12
Introduction
Static Bayesian Game
two-players two-actions game with two-types
New Form in Bayesian Nash Equilib.
polynomial conditions in equilibrium assessment
Dynamics in Equilibrium Assessment
discrete-time autonomous time-varying system
convergence of the EA (stability)
Simulation
confirms states updated become EA and converge.
Future works
estimate player’s belief for a given BNE, and
realize a control-theoretic mechanism design method.
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
10
probability
step
0 1 2 3 4 5 6 7 8 9
1( )η θ−
1( )η θ
−
2( )η θ−
2( )η θ
−
belief
0.3
0.4
0.5
0.6
0.7
10
0.2
0.4
0.6
0.8
probabilityprobability
step
0 1 2 3 4 5 6 7 8 9
100 1 2 3 4 5 6 7 8 9
s1( )a|θ−− s1( )a|θ−
−
s1( )a|θ
−
− s1( )a|θ
−−
s2( )a|θ−− s2( )a|θ−
−
s2( )a|θ
−
− s2( )a|θ
−−
strategy
ˆµ(k + 1) = diag(A1, A2)ˆµ(k)
ˆs(k + 1) = A (ci(k))ˆsi(k)
Dynamics in EA: