A Primer on Strategic Games - CWI Amsterdam
A Primer on Strategic Games Krzysztof R. Apt (so not Krzystof and definitely not Krystof)
CWI, Amsterdam, the Netherlands , University of Amsterdam
A Primer on Strategic Games – p. 1/6
Overview Best response, Nash equilibrium, Weak/strict dominance, Iterated elimination of strategies, Mixed strategies, Variations on the definition, Pre-Bayesian games, Mechanism design: implementation in dominant strategies.
A Primer on Strategic Games – p. 2/6
Strategic Games: Definition Strategic game for n ≥ 2 players: (possibly infinite) set Si of strategies, payoff function pi : S1 × . . . × Sn → R, for each player i. Basic assumptions: players choose their strategies simultaneously, each player is rational: his objective is to maximize his payoff, players have common knowledge of the game and of each others’ rationality.
A Primer on Strategic Games – p. 3/6
Three Examples Prisoner’s Dilemma
C D
C 2, 2 3, 0
D 0, 3 1, 1
F 2, 1 0, 0
B 0, 0 1, 2
The Battle of the Sexes
F B Matching Pennies
H T
H 1, −1 −1, 1
T −1, 1 1, −1
A Primer on Strategic Games – p. 4/6
Three Main Concepts Notation: si , s′i ∈ Si , s, s′ , (si , s−i ) ∈ S1 × . . . × Sn . si is a best response to s−i if ∀s′i ∈ Si pi (si , s−i ) ≥ pi (s′i , s−i ). s is a Nash equilibrium if ∀i si is a best response to s−i : ∀i ∈ {1, . . ., n} ∀s′i ∈ Si pi (si , s−i ) ≥ pi (s′i , s−i ).
Intuition: In a Nash equilibrium no player can gain by unilaterally switching to another strategy. s is Pareto efficient if for no s′ ∀i ∈ {1, . . ., n} pi (s′ ) ≥ pi (s), ∃i ∈ {1, . . ., n} pi (s′ ) > pi (s).
A Primer on Strategic Games – p. 5/6
Nash Equlibrium Prisoner’s Dilemma:
1 Nash equlibrium C D C 2, 2 0, 3 D 3, 0 1, 1
The Battle of the Sexes:
F B
2 Nash equlibria F B 2, 1 0, 0 0, 0 1, 2
no Nash equlibrium H T H 1, −1 −1, 1 T −1, 1 1, −1
Matching Pennies:
A Primer on Strategic Games – p. 6/6
Dominance s′i is strictly dominated by si if ∀s−i ∈ S−i pi (si , s−i ) > pi (s′i , s−i ), s′i is weakly dominated by si if ∀s−i ∈ S−i pi (si , s−i ) ≥ pi (s′i , s−i ), ∃s−i ∈ S−i pi (si , s−i ) > pi (s′i , s−i ).
A Primer on Strategic Games – p. 7/6
Prisoner’s Dilemma C D
C 2, 2 3, 0
D 0, 3 1, 1
Why a dilemma? (D, D) is the unique Nash equilibrium,
For each player C is strictly dominated by D, (C, C) is a Pareto efficient outcome in which each player has a > payoff than in (D, D).
A Primer on Strategic Games – p. 8/6
Prisoner’s Dilemma for n Players Assume ki (n − 1) > li > 0 for all i. ( ki |s−i (C)| + li if si = D pi (s) := ki |s−i (C)| if si = C. For n = 2, ki = 2 and li = 1 we get the original Prisoner’s Dilemma game. pi (C n ) = ki (n − 1) > li = pi (Dn ), so for all players C n yields a > payoff than Dn .
For all players strategy C is strictly dominated by D: pi (D, s−i ) − pi (C, s−i ) = li > 0.
A Primer on Strategic Games – p. 9/6
Quiz
H T E
H 1, −1 −1, 1 −1, −1
T −1, 1 1, −1 −1, −1
E −1, −1 −1, −1 −1, −1
What are the Nash equilibria of this game?
A Primer on Strategic Games – p. 10/6
Answer
H T E
H 1, −1 −1, 1 −1, −1
T −1, 1 1, −1 −1, −1
E −1, −1 −1, −1 −1, −1
(E, E) is the only Nash equilibrium.
It is a Nash equilibrium in weakly dominated strategies.
A Primer on Strategic Games – p. 11/6
IESDS: Example 1 L 3, 0 2, 1 0, 1
T C B
M 2, 1 1, 1 0, 1
R 1, 0 1, 0 0, 0
B is strictly dominated by T , R is strictly dominated by M .
By eliminating them we get: T C
L 3, 0 2, 1
M 2, 1 1, 1
A Primer on Strategic Games – p. 12/6
IESDS, Example 1ctd T C
L 3, 0 2, 1
M 2, 1 1, 1
Now C is strictly dominated by T , so we get: L M T 3, 0 2, 1 Now L is strictly dominated by M , so we get: M T 2, 1 We solved the game by IESDS.
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IESDS Theorem If G′ is an outcome of IESDS starting from a finite G, then s is a Nash equilibrium of G′ iff it is a Nash equilibrium of G. If G is finite and is solved by IESDS, then the resulting joint strategy is a unique Nash equilibrium of G. (Gilboa, Kalai, Zemel, ’90) Outcome of IESDS is unique (order independence).
A Primer on Strategic Games – p. 14/6
IESDS: Example Location game (Hotelling ’29) 2 companies decide simultaneously their location, customers choose the closest vendor. Example: Two bakeries, one (discrete) street. For instance: 8 3
Then baker1 (3, 8) = 5, baker2 (3, 8) = 6. Where do I put my bakery?
A Primer on Strategic Games – p. 15/6
Answer 6
Then: baker1 (6, 6) = 5.5, baker2 (6, 6) = 5.5. (6, 6) is the outcome of IESDS.
Hence (6, 6) is a unique Nash equilibrium.
A Primer on Strategic Games – p. 16/6
IEWDS Theorem If G′ is an outcome of IEWDS starting from a finite G and s is a Nash equilibrium of G′ , then s is a Nash equilibrium of G. If G is finite and is solved by IEWDS, then the resulting joint strategy is a Nash equilibrium of G. Outcome of IEWDS does not need to be unique (no order independence).
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IEWDS: Example 1 Beauty-contest game (Moulin, ’86) each set of strategies = {1, . . ., 100}, payoff to each player: 1 is split equally between the players whose submitted number is closest to 23 of the average. Example submissions: 29, 32, 29; average: 30, payoffs: 21 , 0, 12 . This game is solved by IEWDS. Hence it has a Nash equilibrium, namely (1, . . ., 1).
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IEWDS: Example 2 The following game has two Nash equilibria: X Y Z A 2, 1 0, 1 1, 0 B 0, 1 2, 1 1, 0 C 1, 1 1, 0 0, 0 D 1, 0 0, 1 0, 0 D is weakly dominated by A, Z is weakly dominated by X .
By eliminating them we get: A B C
X 2, 1 0, 1 1, 1
Y 0, 1 2, 1 1, 0
A Primer on Strategic Games – p. 19/6
Example 2, ctd
A B C
X 2, 1 0, 1 1, 1
Y 0, 1 2, 1 1, 0
Next, we get A B C
X 2, 1 0, 1 1, 1
A
X 2, 1
and finally
A Primer on Strategic Games – p. 20/6
IEWDS: Example 3 T B
L 1, 1 1, 1
R 1, 1 0, 0
L 1, 1
R 1, 1
can be reduced both to T
and to T B
L 1, 1 1, 1
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Infinite Games Consider the game with Si := N, pi (s) := si .
Here every strategy is strictly dominated, in one step we can eliminate all strategies, all 6= 0 strategies, one strategy per player.
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Infinite Games (2) Conclusions For infinite games IESDS is not order independent, definition of order independence has to be modified.
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IENBR: Example 1 A B C
X 2, 1 0, 1 1, 1
Y 0, 0 2, 0 1, 2
No strategy strictly or weakly dominates another one. C is never a best response.
Eliminating it we get A B
X 2, 1 0, 1
Y 0, 0 2, 0
from which in two steps we get X A 2, 1
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IENBR Theorem If G′ is an outcome of IENBR starting from a finite G, then s is a Nash equilibrium of G′ iff it is a Nash equilibrium of G. If G is finite and is solved by IENBR, then the resulting joint strategy is a unique Nash equilibrium of G. (Apt, ’05) Outcome of IENBR is unique (order independence).
A Primer on Strategic Games – p. 25/6
IENBR: Example 2 Location game on the open real interval (0, 100). s3−i − si si + 2 si − s3−i pi (si , s3−i ) := 100 − si + 2 50
if si < s3−i if si > s3−i if si = s3−i
No strategy strictly or weakly dominates another one. Only 50 is a best response to some strategy (namely 50). So this game is solved by IENBR, in one step.
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Mixed Extension of a Finite Game Probability distribution over a finite non-empty set A: π : A → [0, 1]
such that
P
a∈A π(a)
= 1.
Notation: ∆A. Fix a finite strategic game G := (S1 , . . ., Sn , p1 , . . ., pn ). Mixed strategy of player i in G: mi ∈ ∆Si . Joint mixed strategy: m = (m1 , . . ., mn ).
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Mixed Extension of a Finite Game (2) Mixed extension of G: (∆S1 , . . ., ∆Sn , p1 , . . ., pn ),
where m(s) := m1 (s1 ) · . . . · mn (sn )
and pi (m) :=
X
m(s) · pi (s).
s∈S
Theorem (Nash ’50) Every mixed extension of a finite strategic game has a Nash equilibrium.
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Kakutani’s Fixed Point Theorem Theorem (Kakutani ’41) Suppose A is a compact and convex subset of Rn and Φ : A → P(A)
is such that Φ(x) is non-empty and convex for all x ∈ A,
for all sequences (xi , yi ) converging to (x, y) yi ∈ Φ(xi ) for all i ≥ 0,
implies that y ∈ Φ(x).
Then x∗ ∈ A exists such that x∗ ∈ Φ(x∗ ).
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Proof of Nash Theorem Fix (S1 , . . ., Sn , p1 , . . ., pn ). Define besti : Πj6=i ∆Sj → P(∆Si )
by besti (m−i ) := {m′i ∈ ∆Si | pi (m′i , m−i ) attains the maximum}.
Then define best : ∆S1 × . . .∆Sn → P(∆S1 × . . . × ∆Sn )
by best(m) := best1 (m−1 ) × . . . × best1 (m−n ).
Note m is a Nash equilibrium iff m ∈ best(m). best(·) satisfies the conditions of Kakutani’s Theorem.
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Comments First special case of Nash theorem: Cournot (1838). Nash theorem generalizes von Neumann’s Minimax Theorem (’28). An alternative proof (also by Nash) uses Brouwer’s Fixed Point Theorem. Search for conditions ensuring existence of Nash equilibrium.
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2 Examples Matching Pennies
H T ( 12 · H +
1 2
H 1, −1 −1, 1
· T, 21 · H +
1 2
T −1, 1 1, −1
· T ) is a Nash equilibrium.
The payoff to each player in the Nash equilibrium: 0. The Battle of the Sexes F B F 2, 1 0, 0 B 0, 0 1, 2 (2/3 · F + 1/3 · B, 1/3 · F + 1/3 · B) is a Nash equilibrium.
The payoff to each player in the Nash equilibrium: 2/3.
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Variations on the Definition Strategic games with qualitative preferences (Osborne, Rubinstein ’94) (S1 , . . ., Sn , 1 , . . ., n ), where each i is a preference relation on S1 × . . . × Sn . Strategic games with parametrized preferences (Apt, Rossi, Venable ’08) Each player i has a set of strategies Si and a preference relation (s−i ) on Si parametrized by s−i ∈ S−i . Conversion/preference games (Le Roux, Lescanne, Vestergaard ’08) The game consists of a set S of situations and for each player i a preference relation i on S and a conversion relation → i on S .
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Graphical Games (Kearns, Littman, Singh ’01) Each player i has a set of neighbours neigh(i). Payoff for player i is a function pi : ×j∈neigh(i)∪{i} Sj → R.
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Dominance by a Mixed Strategy Example A B C D
X 2, − 0, − 1, − 1, −
Y 0, − 2, − 1, − 0, −
Z 1, − 1, − 0, − 0, −
D is weakly dominated by A, C is weakly dominated by D is strictly dominated by
1 1 · A + 2 2 · B, 1 1 · A + 2 2 · C.
A Primer on Strategic Games – p. 35/6
Iterated Elimination of Strategies Consider weak dominance by a mixed strategy. X Y Z A 2, 1 0, 1 1, 0 B 0, 1 2, 1 1, 0 C 1, 1 1, 0 0, 0 D 1, 0 0, 1 0, 0 D is weakly dominated by A, Z is weakly dominated by X , C is weakly dominated by
1 2
·A+
1 2
· B.
By eliminating them we get the final outcome: X Y A 2, 1 0, 1 B 0, 1 2, 1
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Relative Strength of Strategy Elimination Weak dominance by a pure strategy is less powerful than weak dominance by a mixed strategy, but iterated elimination using weak dominance by a pure strategy (W ω ) can be more powerful than iterated elimination using weak dominance by a mixed strategy (M W ω ). In general (Apt ’07): ω S
S
W
W
SM WM
ω
SM
ω
ω WM
A Primer on Strategic Games – p. 37/6
Best responses to Mixed Strategies si is a best response to m−i if ∀s′i ∈ Si pi (si , m−i ) ≥ pi (s′i , m−i ). support(mi ) := {a ∈ Si | mi (a) > 0}.
Theorem (Pearce ’84) In a 2-player finite game si is strictly dominated by a mixed strategy iff it is not a best response to a mixed strategy. si is weakly dominated by a mixed strategy iff it is not a best response to a mixed strategy with full support.
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IESDMS Theorem If G′ is an outcome of IESDMS starting from G, then m is a Nash equilibrium of G′ iff it is a Nash equilibrium of G. If G is solved by IESDMS, then the resulting joint strategy is a unique Nash equilibrium of G. (Osborne, Rubinstein, ’94) Outcome of IESDMS is unique (order independence).
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IESDMS: Example Beauty-contest game each set of strategies = {1, . . ., 100}, payoff to each player: 1 is split equally between the players whose submitted number is closest to 23 of the average. This game is solved by IESDMS, in 99 steps. Hence it has a unique Nash equilibrium, (1, . . ., 1).
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IEWDMS Theorem If G′ is an outcome of IEWDMS starting from G and m is a Nash equilibrium of G′ , then m is a Nash equilibrium of G. If G is solved by IEWDMS, then the resulting joint strategy is a Nash equilibrium of G. Outcome of IEWDS does not need to be unique (no order independence). Every mixed extension of a finite strategic game has a Nash equilibrium in which no pure strategy is weakly dominated by a mixed strategy.
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Rationalizable Strategies Introduced in Bernheim ’84 and Pearce ’84. Strategies in the outcome of IENBRM. Subtleties in the definition . . . Theorem (Bernheim ’84) If G′ is an outcome of IENBRM starting from G, then m is a Nash equilibrium of G′ iff it is a Nash equilibrium of G. If G is solved by IESDMS, then the resulting joint strategy is a unique Nash equilibrium of G. (Apt ’05) Outcome of IENBRM is unique (order independence).
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Pre-Bayesian Games (Hyafil, Boutilier ’04, Ashlagi, Monderer, Tennenholtz ’06,) In a strategic game after each player selected his strategy each player knows all the payoffs (complete information). In a pre-Bayesian game after each player selected his strategy each player knows only his payoff (incomplete information). This is achieved by introducing (private) types.
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Pre-Bayesian Games: Definition Pre-Bayesian game for n ≥ 2 players: (possibly infinite) set Ai of actions, (possibly infinite) set Θi of (private) types, payoff function pi : A1 × . . . × An × Θi → R, for each player i. Basic assumptions: Nature moves first and provides each player i with a θi , players do not know the types received by other players, players choose their actions simultaneously, each player is rational (wants to maximize his payoff), players have common knowledge of the game and of each others’ rationality.
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Nash Equilibrium A strategy for player i: i si (·) ∈ AΘ i .
Joint strategy s( ·) is a Nash equilibrium if each si (·) is a best response to s−i (·): i ∀θ ∈ Θ ∀i ∈ {1, . . ., n} ∀s′i (·) ∈ AΘ i pi (si (θi ), s−i (θ−i ), θi ) ≥ pi (s′i (θi ), s−i (θ−i ), θi ).
Note: For each θ ∈ Θ we have one strategic game. s( ·) is a Nash equilibrium if for each θ ∈ Θ the joint action (s1 (θ1 ), . . ., sn (θn )) is a Nash equilibrium in the θ-game.
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Quiz Θ1 = {U, D}, Θ2 = {L, R}, A1 = A2 = {F, B}. R
L U
F B
D
F B
F 2, 1 0, 1 F 3, 1 5, 1
B 2, 0 2, 1 B 2, 0 4, 1
F B F B
F 2, 0 0, 0 F 3, 0 5, 0
B 2, 1 2, 1 B 2, 1 4, 1
Which strategies form a Nash equilibrium?
A Primer on Strategic Games – p. 46/6
Answer Θ1 = {U, D}, Θ2 = {L, R}, A1 = A2 = {F, B}. R
L U
F B
D
F B
F 2, 1 0, 1 F 3, 1 5, 1
B 2, 0 2, 1 B 2, 0 4, 1
F B F B
F 2, 0 0, 0 F 3, 0 5, 0
B 2, 1 2, 1 B 2, 1 4, 1
Strategies s1 (U) = F, s1 (D) = B , s2 (L) = F, s2 (R) = B form a Nash equilibrium.
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But . . . Nash equilibrium does not need to exist in mixed extensions of finite pre-Bayesian games. Example: Mixed extension of the following game. Θ1 = {U, B}, Θ2 = {L, R}, A1 = A2 = {C, D}. L U
B
R
C D
C 2, 2 3, 0
D 0, 0 1, 1
C D
C 1, 2 0, 0
D 3, 0 2, 1
C D
C 2, 1 3, 0
D 0, 0 1, 2
C D
C 1, 1 0, 0
D 3, 0 2, 2
A Primer on Strategic Games – p. 48/6
Intelligent Design
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Intelligent Design A theory of an intelligently guided invisible hand wins the Nobel prize WHAT on earth is mechanism design? was the typical reaction to this year’s Nobel prize in economics, announced on October 15th. [...] In fact, despite its dreary name, mechanism design is a hugely important area of economics, and underpins much of what dismal scientists do today. It goes to the heart of one of the biggest challenges in economics: how to arrange our economic interactions so that, when everyone behaves in a self-interested manner, the result is something we all like. (The Economist, Oct. 18th, 2007)
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Mechanism Design Decision problem for n players: set D of decisions, for each player i a set Θi of (private) types Θi and a utility function vi : D × Θi → R.
Intuition: vi (d, θi ) represents the benefit to player i of type θi from decision d ∈ D. When Pn the individual types are θ1 , . . ., θn i=1 vi (d, θi ) represents the social welfare from the decision d ∈ D.
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Decision Rules Decision rule is a function f : Θ1 × . . . × Θn → D.
Decision rule f is efficient if n X
vi (f (θ), θi ) ≥
i=1
n X
vi (d, θi )
i=1
for all θ ∈ Θ and d ∈ D. Intuition: f is efficient if it always yields a best decision for the society.
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Set up each player i receives/has a type θi , each player i submits to the central authority a type θi′ , the central authority computes decision d := f (θ1′ , . . ., θn′ ),
and communicates it to each player i.
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Example 1: Sealed-Bid Auction D = {1, . . . , n},
each Θi is R+ , ( θi if d = i vi (d, θi ) := 0 otherwise argsmax θ := µi(θi = maxj∈{1,...,n} θj ).
f (θ) := argsmax θ.
Note: f is efficient.
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Example 2: Public Project Problem c: cost of the public project (e.g., a bridge), D = {0, 1},
each Θi is R+ , vi (d, θi ) := d(θi − nc ), ( Pn 1 if i=1 θi ≥ c f (θ) := 0 otherwise
Note: f is efficient.
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Manipulations An optimal strategy for player i in public project problem: if θi ≥ if θi
n−1 n c n−1 n c n−1 n c n−1 n c
and and and and
Pn j=1 θj Pn j=1 θj Pn j=1 θj Pn j=1 θj
≥c ≥c
CWI, Amsterdam, the Netherlands , University of Amsterdam
A Primer on Strategic Games – p. 1/6
Overview Best response, Nash equilibrium, Weak/strict dominance, Iterated elimination of strategies, Mixed strategies, Variations on the definition, Pre-Bayesian games, Mechanism design: implementation in dominant strategies.
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Strategic Games: Definition Strategic game for n ≥ 2 players: (possibly infinite) set Si of strategies, payoff function pi : S1 × . . . × Sn → R, for each player i. Basic assumptions: players choose their strategies simultaneously, each player is rational: his objective is to maximize his payoff, players have common knowledge of the game and of each others’ rationality.
A Primer on Strategic Games – p. 3/6
Three Examples Prisoner’s Dilemma
C D
C 2, 2 3, 0
D 0, 3 1, 1
F 2, 1 0, 0
B 0, 0 1, 2
The Battle of the Sexes
F B Matching Pennies
H T
H 1, −1 −1, 1
T −1, 1 1, −1
A Primer on Strategic Games – p. 4/6
Three Main Concepts Notation: si , s′i ∈ Si , s, s′ , (si , s−i ) ∈ S1 × . . . × Sn . si is a best response to s−i if ∀s′i ∈ Si pi (si , s−i ) ≥ pi (s′i , s−i ). s is a Nash equilibrium if ∀i si is a best response to s−i : ∀i ∈ {1, . . ., n} ∀s′i ∈ Si pi (si , s−i ) ≥ pi (s′i , s−i ).
Intuition: In a Nash equilibrium no player can gain by unilaterally switching to another strategy. s is Pareto efficient if for no s′ ∀i ∈ {1, . . ., n} pi (s′ ) ≥ pi (s), ∃i ∈ {1, . . ., n} pi (s′ ) > pi (s).
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Nash Equlibrium Prisoner’s Dilemma:
1 Nash equlibrium C D C 2, 2 0, 3 D 3, 0 1, 1
The Battle of the Sexes:
F B
2 Nash equlibria F B 2, 1 0, 0 0, 0 1, 2
no Nash equlibrium H T H 1, −1 −1, 1 T −1, 1 1, −1
Matching Pennies:
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Dominance s′i is strictly dominated by si if ∀s−i ∈ S−i pi (si , s−i ) > pi (s′i , s−i ), s′i is weakly dominated by si if ∀s−i ∈ S−i pi (si , s−i ) ≥ pi (s′i , s−i ), ∃s−i ∈ S−i pi (si , s−i ) > pi (s′i , s−i ).
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Prisoner’s Dilemma C D
C 2, 2 3, 0
D 0, 3 1, 1
Why a dilemma? (D, D) is the unique Nash equilibrium,
For each player C is strictly dominated by D, (C, C) is a Pareto efficient outcome in which each player has a > payoff than in (D, D).
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Prisoner’s Dilemma for n Players Assume ki (n − 1) > li > 0 for all i. ( ki |s−i (C)| + li if si = D pi (s) := ki |s−i (C)| if si = C. For n = 2, ki = 2 and li = 1 we get the original Prisoner’s Dilemma game. pi (C n ) = ki (n − 1) > li = pi (Dn ), so for all players C n yields a > payoff than Dn .
For all players strategy C is strictly dominated by D: pi (D, s−i ) − pi (C, s−i ) = li > 0.
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Quiz
H T E
H 1, −1 −1, 1 −1, −1
T −1, 1 1, −1 −1, −1
E −1, −1 −1, −1 −1, −1
What are the Nash equilibria of this game?
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Answer
H T E
H 1, −1 −1, 1 −1, −1
T −1, 1 1, −1 −1, −1
E −1, −1 −1, −1 −1, −1
(E, E) is the only Nash equilibrium.
It is a Nash equilibrium in weakly dominated strategies.
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IESDS: Example 1 L 3, 0 2, 1 0, 1
T C B
M 2, 1 1, 1 0, 1
R 1, 0 1, 0 0, 0
B is strictly dominated by T , R is strictly dominated by M .
By eliminating them we get: T C
L 3, 0 2, 1
M 2, 1 1, 1
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IESDS, Example 1ctd T C
L 3, 0 2, 1
M 2, 1 1, 1
Now C is strictly dominated by T , so we get: L M T 3, 0 2, 1 Now L is strictly dominated by M , so we get: M T 2, 1 We solved the game by IESDS.
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IESDS Theorem If G′ is an outcome of IESDS starting from a finite G, then s is a Nash equilibrium of G′ iff it is a Nash equilibrium of G. If G is finite and is solved by IESDS, then the resulting joint strategy is a unique Nash equilibrium of G. (Gilboa, Kalai, Zemel, ’90) Outcome of IESDS is unique (order independence).
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IESDS: Example Location game (Hotelling ’29) 2 companies decide simultaneously their location, customers choose the closest vendor. Example: Two bakeries, one (discrete) street. For instance: 8 3
Then baker1 (3, 8) = 5, baker2 (3, 8) = 6. Where do I put my bakery?
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Answer 6
Then: baker1 (6, 6) = 5.5, baker2 (6, 6) = 5.5. (6, 6) is the outcome of IESDS.
Hence (6, 6) is a unique Nash equilibrium.
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IEWDS Theorem If G′ is an outcome of IEWDS starting from a finite G and s is a Nash equilibrium of G′ , then s is a Nash equilibrium of G. If G is finite and is solved by IEWDS, then the resulting joint strategy is a Nash equilibrium of G. Outcome of IEWDS does not need to be unique (no order independence).
A Primer on Strategic Games – p. 17/6
IEWDS: Example 1 Beauty-contest game (Moulin, ’86) each set of strategies = {1, . . ., 100}, payoff to each player: 1 is split equally between the players whose submitted number is closest to 23 of the average. Example submissions: 29, 32, 29; average: 30, payoffs: 21 , 0, 12 . This game is solved by IEWDS. Hence it has a Nash equilibrium, namely (1, . . ., 1).
A Primer on Strategic Games – p. 18/6
IEWDS: Example 2 The following game has two Nash equilibria: X Y Z A 2, 1 0, 1 1, 0 B 0, 1 2, 1 1, 0 C 1, 1 1, 0 0, 0 D 1, 0 0, 1 0, 0 D is weakly dominated by A, Z is weakly dominated by X .
By eliminating them we get: A B C
X 2, 1 0, 1 1, 1
Y 0, 1 2, 1 1, 0
A Primer on Strategic Games – p. 19/6
Example 2, ctd
A B C
X 2, 1 0, 1 1, 1
Y 0, 1 2, 1 1, 0
Next, we get A B C
X 2, 1 0, 1 1, 1
A
X 2, 1
and finally
A Primer on Strategic Games – p. 20/6
IEWDS: Example 3 T B
L 1, 1 1, 1
R 1, 1 0, 0
L 1, 1
R 1, 1
can be reduced both to T
and to T B
L 1, 1 1, 1
A Primer on Strategic Games – p. 21/6
Infinite Games Consider the game with Si := N, pi (s) := si .
Here every strategy is strictly dominated, in one step we can eliminate all strategies, all 6= 0 strategies, one strategy per player.
A Primer on Strategic Games – p. 22/6
Infinite Games (2) Conclusions For infinite games IESDS is not order independent, definition of order independence has to be modified.
A Primer on Strategic Games – p. 23/6
IENBR: Example 1 A B C
X 2, 1 0, 1 1, 1
Y 0, 0 2, 0 1, 2
No strategy strictly or weakly dominates another one. C is never a best response.
Eliminating it we get A B
X 2, 1 0, 1
Y 0, 0 2, 0
from which in two steps we get X A 2, 1
A Primer on Strategic Games – p. 24/6
IENBR Theorem If G′ is an outcome of IENBR starting from a finite G, then s is a Nash equilibrium of G′ iff it is a Nash equilibrium of G. If G is finite and is solved by IENBR, then the resulting joint strategy is a unique Nash equilibrium of G. (Apt, ’05) Outcome of IENBR is unique (order independence).
A Primer on Strategic Games – p. 25/6
IENBR: Example 2 Location game on the open real interval (0, 100). s3−i − si si + 2 si − s3−i pi (si , s3−i ) := 100 − si + 2 50
if si < s3−i if si > s3−i if si = s3−i
No strategy strictly or weakly dominates another one. Only 50 is a best response to some strategy (namely 50). So this game is solved by IENBR, in one step.
A Primer on Strategic Games – p. 26/6
Mixed Extension of a Finite Game Probability distribution over a finite non-empty set A: π : A → [0, 1]
such that
P
a∈A π(a)
= 1.
Notation: ∆A. Fix a finite strategic game G := (S1 , . . ., Sn , p1 , . . ., pn ). Mixed strategy of player i in G: mi ∈ ∆Si . Joint mixed strategy: m = (m1 , . . ., mn ).
A Primer on Strategic Games – p. 27/6
Mixed Extension of a Finite Game (2) Mixed extension of G: (∆S1 , . . ., ∆Sn , p1 , . . ., pn ),
where m(s) := m1 (s1 ) · . . . · mn (sn )
and pi (m) :=
X
m(s) · pi (s).
s∈S
Theorem (Nash ’50) Every mixed extension of a finite strategic game has a Nash equilibrium.
A Primer on Strategic Games – p. 28/6
Kakutani’s Fixed Point Theorem Theorem (Kakutani ’41) Suppose A is a compact and convex subset of Rn and Φ : A → P(A)
is such that Φ(x) is non-empty and convex for all x ∈ A,
for all sequences (xi , yi ) converging to (x, y) yi ∈ Φ(xi ) for all i ≥ 0,
implies that y ∈ Φ(x).
Then x∗ ∈ A exists such that x∗ ∈ Φ(x∗ ).
A Primer on Strategic Games – p. 29/6
Proof of Nash Theorem Fix (S1 , . . ., Sn , p1 , . . ., pn ). Define besti : Πj6=i ∆Sj → P(∆Si )
by besti (m−i ) := {m′i ∈ ∆Si | pi (m′i , m−i ) attains the maximum}.
Then define best : ∆S1 × . . .∆Sn → P(∆S1 × . . . × ∆Sn )
by best(m) := best1 (m−1 ) × . . . × best1 (m−n ).
Note m is a Nash equilibrium iff m ∈ best(m). best(·) satisfies the conditions of Kakutani’s Theorem.
A Primer on Strategic Games – p. 30/6
Comments First special case of Nash theorem: Cournot (1838). Nash theorem generalizes von Neumann’s Minimax Theorem (’28). An alternative proof (also by Nash) uses Brouwer’s Fixed Point Theorem. Search for conditions ensuring existence of Nash equilibrium.
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2 Examples Matching Pennies
H T ( 12 · H +
1 2
H 1, −1 −1, 1
· T, 21 · H +
1 2
T −1, 1 1, −1
· T ) is a Nash equilibrium.
The payoff to each player in the Nash equilibrium: 0. The Battle of the Sexes F B F 2, 1 0, 0 B 0, 0 1, 2 (2/3 · F + 1/3 · B, 1/3 · F + 1/3 · B) is a Nash equilibrium.
The payoff to each player in the Nash equilibrium: 2/3.
A Primer on Strategic Games – p. 32/6
Variations on the Definition Strategic games with qualitative preferences (Osborne, Rubinstein ’94) (S1 , . . ., Sn , 1 , . . ., n ), where each i is a preference relation on S1 × . . . × Sn . Strategic games with parametrized preferences (Apt, Rossi, Venable ’08) Each player i has a set of strategies Si and a preference relation (s−i ) on Si parametrized by s−i ∈ S−i . Conversion/preference games (Le Roux, Lescanne, Vestergaard ’08) The game consists of a set S of situations and for each player i a preference relation i on S and a conversion relation → i on S .
A Primer on Strategic Games – p. 33/6
Graphical Games (Kearns, Littman, Singh ’01) Each player i has a set of neighbours neigh(i). Payoff for player i is a function pi : ×j∈neigh(i)∪{i} Sj → R.
A Primer on Strategic Games – p. 34/6
Dominance by a Mixed Strategy Example A B C D
X 2, − 0, − 1, − 1, −
Y 0, − 2, − 1, − 0, −
Z 1, − 1, − 0, − 0, −
D is weakly dominated by A, C is weakly dominated by D is strictly dominated by
1 1 · A + 2 2 · B, 1 1 · A + 2 2 · C.
A Primer on Strategic Games – p. 35/6
Iterated Elimination of Strategies Consider weak dominance by a mixed strategy. X Y Z A 2, 1 0, 1 1, 0 B 0, 1 2, 1 1, 0 C 1, 1 1, 0 0, 0 D 1, 0 0, 1 0, 0 D is weakly dominated by A, Z is weakly dominated by X , C is weakly dominated by
1 2
·A+
1 2
· B.
By eliminating them we get the final outcome: X Y A 2, 1 0, 1 B 0, 1 2, 1
A Primer on Strategic Games – p. 36/6
Relative Strength of Strategy Elimination Weak dominance by a pure strategy is less powerful than weak dominance by a mixed strategy, but iterated elimination using weak dominance by a pure strategy (W ω ) can be more powerful than iterated elimination using weak dominance by a mixed strategy (M W ω ). In general (Apt ’07): ω S
S
W
W
SM WM
ω
SM
ω
ω WM
A Primer on Strategic Games – p. 37/6
Best responses to Mixed Strategies si is a best response to m−i if ∀s′i ∈ Si pi (si , m−i ) ≥ pi (s′i , m−i ). support(mi ) := {a ∈ Si | mi (a) > 0}.
Theorem (Pearce ’84) In a 2-player finite game si is strictly dominated by a mixed strategy iff it is not a best response to a mixed strategy. si is weakly dominated by a mixed strategy iff it is not a best response to a mixed strategy with full support.
A Primer on Strategic Games – p. 38/6
IESDMS Theorem If G′ is an outcome of IESDMS starting from G, then m is a Nash equilibrium of G′ iff it is a Nash equilibrium of G. If G is solved by IESDMS, then the resulting joint strategy is a unique Nash equilibrium of G. (Osborne, Rubinstein, ’94) Outcome of IESDMS is unique (order independence).
A Primer on Strategic Games – p. 39/6
IESDMS: Example Beauty-contest game each set of strategies = {1, . . ., 100}, payoff to each player: 1 is split equally between the players whose submitted number is closest to 23 of the average. This game is solved by IESDMS, in 99 steps. Hence it has a unique Nash equilibrium, (1, . . ., 1).
A Primer on Strategic Games – p. 40/6
IEWDMS Theorem If G′ is an outcome of IEWDMS starting from G and m is a Nash equilibrium of G′ , then m is a Nash equilibrium of G. If G is solved by IEWDMS, then the resulting joint strategy is a Nash equilibrium of G. Outcome of IEWDS does not need to be unique (no order independence). Every mixed extension of a finite strategic game has a Nash equilibrium in which no pure strategy is weakly dominated by a mixed strategy.
A Primer on Strategic Games – p. 41/6
Rationalizable Strategies Introduced in Bernheim ’84 and Pearce ’84. Strategies in the outcome of IENBRM. Subtleties in the definition . . . Theorem (Bernheim ’84) If G′ is an outcome of IENBRM starting from G, then m is a Nash equilibrium of G′ iff it is a Nash equilibrium of G. If G is solved by IESDMS, then the resulting joint strategy is a unique Nash equilibrium of G. (Apt ’05) Outcome of IENBRM is unique (order independence).
A Primer on Strategic Games – p. 42/6
Pre-Bayesian Games (Hyafil, Boutilier ’04, Ashlagi, Monderer, Tennenholtz ’06,) In a strategic game after each player selected his strategy each player knows all the payoffs (complete information). In a pre-Bayesian game after each player selected his strategy each player knows only his payoff (incomplete information). This is achieved by introducing (private) types.
A Primer on Strategic Games – p. 43/6
Pre-Bayesian Games: Definition Pre-Bayesian game for n ≥ 2 players: (possibly infinite) set Ai of actions, (possibly infinite) set Θi of (private) types, payoff function pi : A1 × . . . × An × Θi → R, for each player i. Basic assumptions: Nature moves first and provides each player i with a θi , players do not know the types received by other players, players choose their actions simultaneously, each player is rational (wants to maximize his payoff), players have common knowledge of the game and of each others’ rationality.
A Primer on Strategic Games – p. 44/6
Nash Equilibrium A strategy for player i: i si (·) ∈ AΘ i .
Joint strategy s( ·) is a Nash equilibrium if each si (·) is a best response to s−i (·): i ∀θ ∈ Θ ∀i ∈ {1, . . ., n} ∀s′i (·) ∈ AΘ i pi (si (θi ), s−i (θ−i ), θi ) ≥ pi (s′i (θi ), s−i (θ−i ), θi ).
Note: For each θ ∈ Θ we have one strategic game. s( ·) is a Nash equilibrium if for each θ ∈ Θ the joint action (s1 (θ1 ), . . ., sn (θn )) is a Nash equilibrium in the θ-game.
A Primer on Strategic Games – p. 45/6
Quiz Θ1 = {U, D}, Θ2 = {L, R}, A1 = A2 = {F, B}. R
L U
F B
D
F B
F 2, 1 0, 1 F 3, 1 5, 1
B 2, 0 2, 1 B 2, 0 4, 1
F B F B
F 2, 0 0, 0 F 3, 0 5, 0
B 2, 1 2, 1 B 2, 1 4, 1
Which strategies form a Nash equilibrium?
A Primer on Strategic Games – p. 46/6
Answer Θ1 = {U, D}, Θ2 = {L, R}, A1 = A2 = {F, B}. R
L U
F B
D
F B
F 2, 1 0, 1 F 3, 1 5, 1
B 2, 0 2, 1 B 2, 0 4, 1
F B F B
F 2, 0 0, 0 F 3, 0 5, 0
B 2, 1 2, 1 B 2, 1 4, 1
Strategies s1 (U) = F, s1 (D) = B , s2 (L) = F, s2 (R) = B form a Nash equilibrium.
A Primer on Strategic Games – p. 47/6
But . . . Nash equilibrium does not need to exist in mixed extensions of finite pre-Bayesian games. Example: Mixed extension of the following game. Θ1 = {U, B}, Θ2 = {L, R}, A1 = A2 = {C, D}. L U
B
R
C D
C 2, 2 3, 0
D 0, 0 1, 1
C D
C 1, 2 0, 0
D 3, 0 2, 1
C D
C 2, 1 3, 0
D 0, 0 1, 2
C D
C 1, 1 0, 0
D 3, 0 2, 2
A Primer on Strategic Games – p. 48/6
Intelligent Design
A Primer on Strategic Games – p. 49/6
Intelligent Design A theory of an intelligently guided invisible hand wins the Nobel prize WHAT on earth is mechanism design? was the typical reaction to this year’s Nobel prize in economics, announced on October 15th. [...] In fact, despite its dreary name, mechanism design is a hugely important area of economics, and underpins much of what dismal scientists do today. It goes to the heart of one of the biggest challenges in economics: how to arrange our economic interactions so that, when everyone behaves in a self-interested manner, the result is something we all like. (The Economist, Oct. 18th, 2007)
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Mechanism Design Decision problem for n players: set D of decisions, for each player i a set Θi of (private) types Θi and a utility function vi : D × Θi → R.
Intuition: vi (d, θi ) represents the benefit to player i of type θi from decision d ∈ D. When Pn the individual types are θ1 , . . ., θn i=1 vi (d, θi ) represents the social welfare from the decision d ∈ D.
A Primer on Strategic Games – p. 51/6
Decision Rules Decision rule is a function f : Θ1 × . . . × Θn → D.
Decision rule f is efficient if n X
vi (f (θ), θi ) ≥
i=1
n X
vi (d, θi )
i=1
for all θ ∈ Θ and d ∈ D. Intuition: f is efficient if it always yields a best decision for the society.
A Primer on Strategic Games – p. 52/6
Set up each player i receives/has a type θi , each player i submits to the central authority a type θi′ , the central authority computes decision d := f (θ1′ , . . ., θn′ ),
and communicates it to each player i.
A Primer on Strategic Games – p. 53/6
Example 1: Sealed-Bid Auction D = {1, . . . , n},
each Θi is R+ , ( θi if d = i vi (d, θi ) := 0 otherwise argsmax θ := µi(θi = maxj∈{1,...,n} θj ).
f (θ) := argsmax θ.
Note: f is efficient.
A Primer on Strategic Games – p. 54/6
Example 2: Public Project Problem c: cost of the public project (e.g., a bridge), D = {0, 1},
each Θi is R+ , vi (d, θi ) := d(θi − nc ), ( Pn 1 if i=1 θi ≥ c f (θ) := 0 otherwise
Note: f is efficient.
A Primer on Strategic Games – p. 55/6
Manipulations An optimal strategy for player i in public project problem: if θi ≥ if θi
n−1 n c n−1 n c n−1 n c n−1 n c
and and and and
Pn j=1 θj Pn j=1 θj Pn j=1 θj Pn j=1 θj
≥c ≥c
