HMCS Vancouver Pre-Deployment Brief - May 2024 (Web Version).pptx
3 bayesian-games
1.
2. Information in Games
Games with complete information
each player knows the strategy set of all opponents
each player knows the payoff of every opponent for
every possible (joint) outcome of the game
Strong assumption
3. Information in Games
Games with complete information
each player knows the strategy set of all opponents
each player knows the payoff of every opponent for
every possible (joint) outcome of the game
Strong assumption
Does not happen often in real world
e.g. 1: competing sellers don’t know payoffs for each
other
e.g. 2: corporation bargaining with union members
don’t know the (dis)utility of a month’s strike for the
union
5. Incomplete Information
Games in real world scenarios have incomplete
information
How do we model incomplete information?
Consider a player’s beliefs about other players’
preferences, his beliefs about their beliefs about his
preferences, …
6. Modeling Games with
Incomplete Information
Harsanyi’s approach:
Each player’s preferences are determined by a random
variable
The exact value of the random variable is observed by
the player alone
But…
7. Modeling Games with
Incomplete Information
Harsanyi’s approach:
Each player’s preferences are determined by a random
variable
The exact value of the random variable is observed by
the player alone
But…the prior probability distribution of the random
variable is common knowledge among all players
8. Modeling Games with
Incomplete Information
Incomplete information now becomes imperfect
information
Who determines the value of the random variable for
each player?
9. Modeling Games with
Incomplete Information
Incomplete information now becomes imperfect
information
Who determines the value of the random variable for
each player?
Nature
background/culture/external effects on each player
10. Example: Modified DA’s
Brother
Prisoner 2 now can behave in one of two ways
with probability µ he behaves as he did in the original
game
with probability 1-µ he gets “emotional”, i.e., he gets an
additional payoff of –6 if he confesses (rats on his
accomplice)
Who determines µ?
“nature” of player 2
Note: Player 1 still behaves as he did in the
original game
12. Example: Modified DA’s
Brother
Player 2 has 4 pure strategies now
confess if type I, confess if type II
confess if type I, don’t confess if type II
don’t confess if type I, confess if type II
don’t confess if type I, don’t confess if type II
Player’s strategy is now a function of its type
Note:
Player 2’s type is not visible to player 1,
Player 1 still has two pure strategies as before
13. Bayesian Nash Game
Player i’s utility is written as:
ui(si,s-i, θi), where θi Є Θi
θi: random variable chosen by nature
Θi: distribution from which the random variable
is chosen
F(θ1,θ2,θ3,… θI): joint probability distribution of all
players
common knowledge among all players
Θ = Θ1 X Θ2 X Θ3 …X ΘI
Bayesian Nash Game = [I, {Si}, {ui(.
)}, Θ, F(.
)]
14. Example: Bayesian Nash Game
Modified DA’s Brother example
Θ1={1} (some constant element because P1 does not have
types}
Θ2 ={emotional, not emotional}
Θ=Θ1X Θ2= {(emotional), (non-emotional)}
F(Θ) = <0.3, 0.7> (some probability values that give the
probabilities of {(emotional), (non-emotional)}
This is common knowledge
15. Bayesian Nash Game
Pure strategy of a player is now a function of the type
of the player:
si(θi): called decision rule
Player i’s pure strategy set is S iwhich is the set of all
possible si(θi)-s
Player i’s expected payoff is given by:
ûi(s1(.
), s2(.
), s3(.
), ….sI(.
))=Eθ[ui(s1(θ1)...sI(θI), θi)]
16. Bayesian Nash Game as
Normal Form Game
We can rewrite the Bayesian Nash game
[I, {Si}, {ui(.
)}, Θ, F(.
)]
17. Bayesian Nash Game as
Normal Form Game
We can rewrite the Bayesian Nash game
[I, {Si}, {ui(.
)}, Θ, F(.
)]
as
[I, {S i}, ûi(.
)}]
18. Definition: Bayesian Nash Equilibrium
A pure strategy Bayesian Nash equilibrium for the
Bayesian Nash game [I, {Si}, {ui(.
)}, Θ, F(.
)] is a
profile of decision rules (s1(.
), s2(.
), s3(.
), ….sI(.
)) that
constitutes a Nash equilibrium of game [I, {S i},
{ûi(.
)}]. That is, for every i=1…I
ûi(si(.
), s-i(.
)) >= ûi(s’i(.
), s-i(.
))
for all s’i(.
) Є S i where ûi(si(.
), s-i(.
)) is the expected
payoff for player i
19. Proposition
A profile of decision rules (s1(.
), s2(.
), s3(.
), ….sI(.
)) is a
Bayesian Nash equilibrium in a Bayesian Nash
game [I, {Si}, {ui(.
)}, Θ, F(.
)] if and only if, for all i
and all θ’i Є Θi occurring with positive probability
Eθ-i[ui(si(θ’i), s-i(θ-i), θ’i) | θ’i)] >=
Eθ-i[ui(si’, s-i(θ-i), θ’i) | θ’i)]
for all si’ Є Si, where the expectation is taken over
realizations of the other players’ random variables
conditional on player i’s realization of signal θ’i.
20. Outline of Proof (by contradiction)
Necessity: Suppose the inequality (on last
slide) did not hold, i.e.,
Eθ-i[ui(si(θ’i), s-i(θ-i), θ’i) | θ’i)] <
Eθ-i[ui(si’, s-i(θ-i), θ’i) | θ’i)]
→ Some player I for whom θ’i Є Θihappens with a
positive probability, is better off by changing its
strategy and using si’ instead of si(θ’i)
→ (s1(.
), s2(.
), s3(.
), ….sI(.
)) is not a Bayesian Nash equilibrium
→ Contradiction
21. Outline of Proof (by contradiction)
Reverse direction:
Eθ-i[ui(si(θ’i), s-i(θ-i), θ’i) | θ’i)] >=
Eθ-i[ui(si’, s-i(θ-i), θ’i) | θ’i)] holds for
all θ’i Є Θioccurring with positive probability
→ player i cannot improve on the payoff received by
playing strategy si(.
)
→ si(.
) constitutes a Nash equilibrium
23. Solution to Modified DA’s Brother
Prisoner 2, type I plays C with probability 1 (dominant
strategy)
Prisoner 2, type II plays DC with probability 1(dominant
strategy)
Expected payoff to P1 with strategy DC: -10µ +0(1-µ)
Expected payoff to P1 with strategy C: -5µ -1(1-µ)
Player 1:
Play DC when -10µ +0(1-µ) > -5µ -1(1-µ)
or, µ<1/6
Play C when µ>1/6
µ=1/6 makes player 1 indifferent
24. Weakly Dominated Strategy
Problem of weakly dominated strategy:
for some strategy of the opponent the weakly dominant
and the weakly dominated strategy give exactly same
payoff
player CAN play weakly dominated strategy for some
strategy of the opponent
25. U
D
(D,R) is a Nash equilibrium involving a play
of weakly dominated strategies
26. Perturbed Game
Start with normal form game
ΓN=[I, {∆(Si)}, {ui(.
)}]
Define a perturbed game
Γε= [I, {∆ε(Si)}, {ui(.
)}]
where ∆ε (Sj)={(σi: σi> εi(si) for all si Є Siand Σ siЄ Siσi (si)=1}
εi(si) denotes the minimum probability of player I playing
strategy si
27. Perturbed Game
Start with normal form game
ΓN=[I, {∆(Si)}, {ui(.
)}]
Define a perturbed game
Γε= [I, {∆ε(Si)}, {ui(.
)}]
where ∆ε (Sj)={(σi: σi> εi(si) for all si Є Siand Σ siЄ Siσi (si)=1}
εi(si) denotes the minimum probability of player I playing
strategy si
εi(si) denotes the unavoidable probability of playing
strategy siby mistake
28. Definition: Nash equilibrium in
Trembling Hand Perfect Game
A Nash equilibrium of a game ΓN=[I, {∆(Si)}, {ui(.
)}] is
(normal form) trembling hand perfect if there is
some sequence of perturbed games {Γεk }k=1
∞
that
converges to ΓN [in the sense that lim k ∞→ (εi
k
(si) )=0 for
all I and si Є Si] for which there is some
associated sequence of Nash equilibria {σk
} k=1
∞
that
converges to σ (i.e. such that lim k ∞→ σk
= σ)
29. Proposition 1
A Nash equilibrium of a game ΓN=[I, {∆(Si)}, {ui(.
)}] is
(normal form) trembling hand perfect if and only if
there is some sequence of totally mixed strategies
{σk
} k=1
∞
such that lim k ∞→ σk
= σ and σi is a best response
to every element of sequence {σk
-i} k=1
∞
for all i=1…I.
30. Proposition 2
If σ = (σ1, σ2, σ3,.. σI) is a normal form trembling hand
perfect Nash equilibrium then is not a weakly
dominated strategy for any i=1…I. Hence, in any
normal form trembling hand perfect Nash
equilibrium, no weakly dominated pure strategy can
be played with positive probability.