Presented by Daniel Choi
Provable Software Lab.
KAIST
• Introduction
• Game Theory
– Classification of Games
– Notations
• Applications to Model Verification
– Bisimulation
– Model Checking
• Conclusion
Game theory is the study of the choice of
strategies by interacting rational agents.
Roger A. McCain, Game Theory: A Nontechnical Introduction to the Analysis of
Strategy (South-western, 2004)
• Nim Game
Take one or more coins at the same row
The player who takes the last coin wins
Example from : Roger A. McCain, Game Theory: A Nontechnical
Introduction to the Analysis of Strategy (South-western, 2004)
• Players
– Shin
– Yun-ho
• Assumption
– Shin starts first
• Strategies for Shin
– Take one coin from the top row
– Take one coin from the second row
– Take both coins from the second row
Yun-ho’s turn
Shin’s turn
Shin
Al wWins
i ns
Shin’s turn
Wins
BoYun-ho
b w i ns
Shin
Al wWins
i ns
Shin
Al wWins
i ns
Yun-ho Wins
Bo b w i ns
• It can formally analyze the game
– E.g. Tree diagram
• Metaphor
– Many interactions can be treated and analyzed
as a game
• Game theoretic analysis is to discover which
strategy is a person’s best response to the
strategies chosen by the others.
• Worst-Case Analysis
– Game between a solver and an adversary
• Network routing
– Game between client and environment
– Game between router and clients
• Load-sharing in distributed systems
– Game between server and client
(or other server)
Requirement
Properties
Target Model
(F
How about different approaches?
Game Theory
W)
Target Model
The model
satisfies the
requirement
properties!!
The model
does not
satisfy the
requirement
properties!!
Does Player 2 always win?
Requirement
Properties
(F
Player 1
W)
Player 2
• Introduction
• Game Theory
– Classification of Games
– Notations
• Applications to Model Verification
– Bisimulation
– Model Checking
• Conclusion
• Non-cooperative game
– The players act individually, each decision of a
player influences the payoff of the other
players
• Cooperative game
– The players are allowed to from coalitions and
combine their decision-making problems
• Normal form
– Decisions of players are simultaneous
– The payoffs are represented by a matrix
– Ex. Prisoner’s Dilemma
• Extensive form
– Decisions of players are sequential
– The payoffs are represented by a tree
– Ex. Nim game
• Al Thinks rationally
– Case 1: Bob confess
• If Al confess then Al will get 10 years and Bob will get 10 years
• If Al does not confess then Al will get 20 years, Bob will get 0 years
– Case 2: Bob does not confess
• If Al confess then Al will get 0 years, Bob will get 20 years
• If Al does not confess then Al will get 1 years, Bob will get 1 years
Al
Confess
Don’t
Confess
10 years, 10 years
0, 20 years
Don’t
20, 0 years
1 year, 1 year
Bob
• Al Thinks rationally
– Case 1: Bob confess
• If Al confess then Al will get 10 years and Bob will get 10 years
• If Al does not confess then Al will get 20 years, Bob will get 0 years
– Case 2: Bob does not confess
• If Al confess then Al will get 0 years, Bob will get 20 years
• If Al does not confess then Al will get 1 years, Bob will get 1 years
Al
Confess
Don’t
Confess
10 years, 10 years
0, 20 years
Don’t
20, 0 years
1 year, 1 year
Bob
• Extensive game
– With perfect information
• Any player knows all the moves made before one’s own
move
– Without perfect information
• A player makes one’s move in the game, one does not
know opponent player’s move
• Computation (run or sequence)
– Interaction between “Player” and “Opponent”
– Represented by a sequence of move
– Opponent always makes the first move
• Definition of Game (MG, λG, PG)
– MG : the set of moves of the
game
– λG : MG → {P, O}
• Labeling function designating
each move as by Player or
Opponent
a1
– PG : the set of alternating
sequences of moves in MG
a2
• Non-empty, prefix-closed
• Example
– MG = {a1, a2, b1, b2, b3}
– λG = {(a1, O), (a2, O),
(b1, P), (b2, P), (b3, P)}
– PG = {ε , a1, a1b1, a2, a2b2, a2b3}
b1
b2
b3
• A game can be seen as specifying the
possible interactions between a System and
its Environment
• Games classify behaviors
– Programs will be modeled by strategies
• Strategies are rules specifying how the System should
actually play
– Deterministic strategy σ on a game G (σ ⊆ PGeven)
• ε∈σ
• sab ∈ σ ⇒ s ∈ σ
• sab, sac ∈ σ ⇒ b = c
• Introduction
• Game Theory
– Classification of Games
– Notations
• Applications to Model Verification
– Bisimulation
– Model Checking
• Conclusion
• Bisimulation relation can be modeled as a
Bisimulation game (Equivalence Game)
– Observer can repeatedly interact with a process by
choosing an available transition from it
– Observer match their selections so that they can
proceed with further corresponding choices
• Equivalence game G(E0, F0)
– Player I and II : Observers who make choices of
transitions
– Player I attempts to show initial processes are different
– Player II attempts to show two processes are
equivalent
• Equivalence game G(E0, F0)
– Player I chooses a transition Ej –a-> Ej+1 and then player II
chooses a transition with the same label Fj –a-> Fj+1
– Player I chooses a transition Fj –a-> Fj+1 and then player II
chooses a transition with the same label Ej –a-> Ej+1
• Player I win
– Player I can choose a transition and player II will be unable
to match it
• Player II win
– If the play is infinite
– If the play reaches the position (En, Fn) and both processes
have no available transitions
G(Clock, Clock2)
Clock
Clock2
Clock and
Clock and
Clock2 is
Clock2 is
different!
(Clock, Clock2), (Clock, tick.Clock
tick 2), (Clock,
tick Clock2) ….
equivalent!
Player II wins
tick
Player I
Player II
G(Clock, Clock2)
Clock
Clock and
Clock’ is
different!
Clock’
tick
(Clock,tick
0)
tick
Clock and
Clock’ is
equivalent!
Player I wins
0
Player I
Player II
• Winning Strategy π
– If the player wins every play in which the player
uses π
• Proposition
For any game G(E,F) either player I or player II
has a history-free winning strategy
• History-free : Rules do not depend on what
happened previously in the play
• If player II has a winning strategy for G(E, F)
then E is game equivalent to process F
• E is game equivalent to F iff E is bisimular to F
– Only if
• Showing that the relation R = {(E, F) | E and F are game
equivalent} is a bisimulation.
– If
– Player I’s move : E –a-> E’ (this is possible move by player I)
Player II can respond with F –a-> F’
(by game equivalent relation)
– Player I’s move : F –a-> F’ (this is possible move by player I)
Player II can respond with E –a-> E’
(by game equivalent relation)
• There is a bisimulation relation R s.t. (E, F) ∈ R,
Construct a winning strategy for player II for the game
G(E, F)
• In any play, whatever move player I makes player II
responds by making sure that the resulting pair of
processes remain in the relation R
U ≡ 1p.(1p.tea.U + 1p.coffee.U)
U and V is
tea
not
bisimulation
relation!
1p
1p
Player I
V ≡ 1p.1p.tea.V + 1p.1pcoffee.V
coffee 1p
1p
1p
G(U, V)
1p tea
U and V is
bisimulation
relation!
1p
Player II
V -1p->1p.tea.V
U -1p->
1p.tea.U + 1p.coffee.U
(U, V) ->
(1p.tea.U + 1p.coffee.U, 1p.tea.V)
It is 1p.tea.V
not bisimulation
1p.tea.U + 1p.coffee.U,
-1p-> coffee.U
relation
1p.tea.V -1p-> tea.V)
(1p.tea.U + 1p.coffee.U, 1p.tea.V)
-> (coffee.U, tea.V)
Player I
Player II
• Introduction
• Game Theory
– Classification of Games
– Notations
• Applications to Model Verification
– Bisimulation
– Model Checking
• Conclusion
• Semantics of Interaction
• Computability Logic: A Formal Theory of
Interaction (In Interactive Computing)
• Model mu-calculus
– Property Checking Game
– Model Checking Game
• Overview of Game Theory and Using to Model the Knowledge of
Multi-Agent System
by Thuy Lien PHAM, Marc BUI, Michel LAMURE In Actes de la Premiè
re Conférence Internationale RIVF'03 Rencontres en Informatique Viet
nam-France, RIVF'03
• Bisimulation, Model Checking and Other Games
by Colin Stirling, In Notes for Mathfit instructional meeting on games
and computation, Edinburgh, June 1997
• Games and Model Mu-Calculus
by Colin Stirling, In TACAS 1996 Lecture Notes in Computer Science 1
055, 298-312, 1996
• Semantics and Logics of Computation
Edited by A. Pitts and P. Dybjer, Cambrige Press, 1996