Correlated equilibria are sometimes more efficient than the Nash equilibria of a game without signals. We investigate whether the availability of quantum signals in the context of a classical strategic game may allow the players to achieve even better efficiency than in any correlated equilibrium with classical signals, and find the answer to be positive.strategic games, quantum mechanics, correlated equilibrium, coordination, entanglement, efficiency

Pierfrancesco La Mura

English

Research Papers in Economics

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Correlated Equilibria of Classical Strategic
Games with Quantum Signals
Pierfrancesco La Mura
Leipzig Graduate School of Management
plamura@hhl.de
comments welcome
September 2, 2003
Abstract
Correlated equilibria are sometimes more eﬃcient than the Nash equi-
libria of a game without signals. We investigate whether the availability
of quantum signals in the context of a classical strategic game may al-
low the players to achieve even better eﬃciency than in any correlated
equilibrium with classical signals, and ﬁnd the answer to be positive.
1 Introduction
Until recently, the most notable connection between game theory and quantum
mechanics was the fact that von Neumann pioneered the mathematical foun-
dations of both ﬁelds [10,11]. In recent years a few proposals have appeared
in the literature, mostly coming from the theoretical physics community, aimed
at ﬁnding a common ground for the two theories. Several recent papers have
investigated the connections between games with classical mixed strategies and
games with quantum strategies [3,8]. Generally, these results have had limited
resonance in the game theory community. Unlike the vast majority of real-
life strategic situations, in which the available strategies are predetermined and
classical, a game with quantum strategies needs to be especially designed and
implemented in order to be played. Moreover, it was soon pointed out that
any such quantum game can be simulated by a suitably designed classical game
(in which the players, say, communicate via telephone their choice of quantum
strategy to a center), and hence interest in the ﬁeld of quantum games has been
so far relatively modest.
The aim of the present paper is also to bring together ideas from game theory
and quantum mechanics, but following a signiﬁcantly diﬀerent approach, which
is closest in spirit to [6] and [7]. We take the view that in many games the
strategies available to the players are predetermined and classical, but following
Aumann [1,2] we postulate the availability of a correlating device able to send
1payoﬀ-irrelevant private signals to the players before they play the game. Un-
like Aumann’s device, which can only send classical information, we also allow
for devices able to send sequences of individual quantum signals (qubits) to the
players. Today similar devices are amply available, as many practical imple-
mentations have been designed and demonstrated. A qubit may be variously
encoded in the properties of a physical carrier, including the spin of a neutron,
or the polarization of a photon.
Correlated equilibria are often more eﬃcient than the Nash equilibria of a
game without signals. This is because the correlation of the signals received
by the players may allow them to coordinate towards Pareto-superior expected
payoﬀ combinations. We investigate whether allowing the use of quantum sig-
nals may allow the players to achieve even better coordination than is possible in
any classical correlated equilibrium, and ﬁnd the answer to be positive. This has
some implications on the signiﬁcance of several well-known impossibility results
[9,5,12], which will need to be re-evaluated if quantum correlation is allowed.
The main contribution of this paper is to introduce a novel notion of cor-
related equilibrium in which the signals sent to the players are represented by
quantum bits, and compare its properties with those of classical correlated equi-
librium. Classical correlated equilibrium deals with information states which
take the form of events in a suitably deﬁned Boolean algebra. By contrast, the
information states induced by quantum signals generally do not correspond to
a Boolean algebra, but conform to a more general logic calculus in which the
distributive laws of propositional logics appear in a weakened form.
Because of its unique features, the ﬁeld of quantum information theory is
currently an object of intense study, together with the rising ﬁelds of quantum
computation, encryption and communication. The most distinguishing feature
of quantum theory, which sets it clearly apart from classical mechanics, is the
phenomenon of entanglement [4], by which the observed states of two parti-
cles may exibit correlations which cannot be rationalized in terms of classical
probabilistic behavior. If two entangled particles are kept isolated from their
local environment they will remain in a correlated state until some operation is
performed on one of them (say, the spin is observed), in which case the states
of both particles will change in a predictably correlated way, even if they have
been meanwhile separated in space and time. This phenomenon is non-local:
the states of the two particles are jointly aﬀected, even though only one of them
is observed.
2 An example
Consider the following game with imperfect information. Two players are ran-
domly drawn from an i.i.d., uniform distribution over a set of three types,
{A, B,C}. They privately observe their own type, then simultaneously decide
whether to say “yes” or “no”. If two players of the same type agree, i.e. they
both say yes or they both say no, each of them receives a payoﬀ of −900. If two
players of diﬀerent types agree they both receive a payoﬀ of 9. If the two players
2disagree they both get zero. Due to the catastrophic consequences of agreeing
when of the same type, there are no eﬃcient Nash equilibria which give positive
probability to those events. All the possible pure strategy combinations which
do not lead to catastrophes are (denoting “no” by 0 and “yes” by 1)
player 1 A B C player2 A B C
a 0 0 0 1 1 1
b 0 0 1 1 1 0
c 0 1 0 1 0 1
d 0 1 1 1 0 0
e 1 0 0 0 1 1
f 1 0 1 0 1 0
g 1 1 0 0 0 1
h 1 1 1 0 0 0
Strategy combinations a and h lead to an expected payoﬀ of zero. Any of
the other strategy combinations leads to an expected payoﬀ of 4. This game has
175 Nash equilibria, 8 of which are in pure strategies and correspond to strategy
combinations a,b,...,h above. The only eﬃcient equilibria are in pure strategies,
and correspond to strategy combinations b,c,...,g. Furthermore, (4,4) are not
only the eﬃcient Nash equilibrium payoﬀs, but also the eﬃcient correlated equi-
librium payoﬀs. In fact, it is well known that the set of correlated equilibrium
payoﬀs which can be obtained with public signals is the convex hull of the set
of Nash equilibrium payoﬀs. But this is a game of pure coordination, and hence
any correlated equilibrium with private signals is still an equilibrium if the sig-
nals are made public (as the players can always ignore the extra information).
It follows that (4,4) are also the eﬃcient correlated equilibrium payoﬀs.
Now suppose that each player receives one half of a fully entangled qubit pair,
and that three diﬀerent measurements, x,y and z, are possible (say, spin mea-
surements along three commonly known orthogonal axes). Then the following
equilibrium improves on (4,4). If of type A, choose measurement x. Say “yes”
if and only if the result is “up”, and say “no” otherwise. Behave analogously, if
of type B, after applying measurement y, and if of type C after applying mea-
surement z. It follows from the rules of quantum mechanics that the conditional
probabilities of agreement, depending on the combination of types, are those
represented in the following table.
A B C
A 0 3/4 3/4
B 3/4 0 3/4
C 3/4 3/4 0
It is easily checked that the strategy proﬁle described above is an equilibrium,
yielding an expected payoﬀ of 6/9 ∗ 3/4 ∗ 9 = 9/2 > 4 to both players.
Some remarks are in order at this point.
3• The game is constructed in order to take advantage of the violation of
one of Bell’s inequalities by the entangled qubit pair. No distribution of
classical signals can induce the conditional probabilities of agreement in
the table above, and this the reason why no classical correlated equilibrium
can achieve the same expected payoﬀs.
• In general, better coordination than in classical correlated equilibrium
may ensue due to the rich patterns of positive and negative correlation
which are allowed by the rules of quantum mechanics.
• Note that no actual information is exchanged between the two players.
From the point of view of each player the probability that the opponent is
of any given type is still 1/3, and the probability that he will say “yes” is
still 1/2, even after performing the observation. Yet, the strategies of the
two players have become eﬀectively entangled, by making them contingent
on the choice of measurement and on the observed state of the qubit pair.
• The correlated equilibrium described above makes use of a single entangled
qubit pair, but possibly even better coordination may be achieved by
transmitting ﬁnite sequences of entangled qubit pairs (q-bytes).
• Finally, note that in contrast with the classical case here observing is
explicitly modelled as an act. This is necessary in the quantum-theoretic
framework, where any observation intrinsically aﬀects the state of the
observed system.
3 Game-theoretic setup and a formal deﬁnition
The example described in the previous section can be conveniently modelled as
a Bayesian game. In this section we introduce some standard game-theoretic
notation and deﬁnitions, and then give a formal deﬁnition of quantum correlated
equilibrium. We start with the deﬁnition of a strategic game.
Deﬁnition 1 A strategic game consists of a ﬁnite set N (the set of players),
and for each player i ∈ N
• a set Ai (the set of actions available to i)
• a function ui : A → R, where A = ×j∈NAj (the payoﬀ of player i).
Denote by Σi the set of probability measures over Ai. These are player
i’s mixed strategies. We use the suﬃx −i to denote all players except i. For
instance, a proﬁle a−i ∈ A−i speciﬁes an action for all players except i. A Nash
equilibrium is a combination of probabilistic decisions (mixed strategies), one
for each player, such that no single player can get a better expected payoﬀ by
choosing a diﬀerent strategy.
4Deﬁnition 2 A (mixed-strategy) Nash equilibrium of a strategic game (N,(Ai),(ui))
is a vector (π1,...,πn), with πi ∈ Σi for all i ∈ N, such that
X
a∈A
πi(ai)π−i(a−i)ui(a−i,ai) ≥
X
a∈A
ρi(ai)π−i(a−i)ui(a−i,ai)
for all ρi ∈ Σi and for all i ∈ N.
Pure-strategy Nash equilibria are those which only involve degenerate mixed
strategies. In a Bayesian game, each player can make its choice of available
action contingent on the observation of a private signal.
Deﬁnition 3 A Bayesian game (in strategic form) consists of:
• a ﬁnite set N (the set of players)
• a ﬁnite set Ω (the set of states)
and for each player i ∈ N
• a set Ai (the set of actions available to i)
• a ﬁnite set Ti (the set of signals that may be observed by player i) and a
function τi : Ω → Ti (the signal function of player i)
• a probability measure pi on Ω (the prior belief of player i) for which
pi(τ
−1
i (ti)) > 0 for all ti ∈ Ti
• a function ui : A × Ω → R, where A = ×j∈NAj (the payoﬀ of player i).
Any Bayesian game can be reduced to a suitably deﬁned strategic game.
The Nash equilibria of a Bayesian game are deﬁned as the Nash equilibria of
the corresponding strategic game.
Deﬁnition 4 A Nash equilibrium of a Bayesian game (N,Ω,(Ai),(Ti),(τi),(pi),(ui))
is a Nash equilibrium of the strategic game deﬁned as follows.
• The set of players is the set of pairs (i,ti) for i ∈ N and ti ∈ Ti.
• The set of actions of each player (i,ti) is Ai.
• The payoﬀ function u(i,ti) of each player (i,ti) is given by u(i,ti)(a) =
E[ui(a,ω)|ti] (where E[ ] denotes the expectation operator)
While a Nash equilibrium captures the notion of simultaneous, independent
decisions, in a correlated equilibrium the players can make their choice of strat-
egy conditional on some privately observed classical signals. Even though the
signals are payoﬀ-irrelevant, they inﬂuence the expected payoﬀ by changing the
players’ information states.
5Deﬁnition 5 A correlated equilibrium of a strategic game (N,(Ai),(ui)) con-
sists of
• a ﬁnite probability space (Ω,π) (Ω is the set of states and π is a probability
measure on Ω)
• for each player i ∈ N a partition Pi of Ω (player i’s information partition)
• for each player i ∈ N a function σi : Ω → Ai with σi(ω) = σi(ω′) whenever
ω ∈ Pi and ω′ ∈ Pi for some Pi ∈ Pi such that
X
ω∈Ω
π(ω)ui(σ−i(ω),σi(ω)) ≥
X
ω∈Ω
π(ω)ui(σ−i(ω),ρi(ω))
for any other function ρi : Ω → Mi×Ai such that ρi(ω) = ρi(ω′) whenever
ω ∈ Pi and ω′ ∈ Pi for some Pi ∈ Pi.
As for the case of Nash equilibria, the correlated equilibria of a Bayesian
game are simply the correlated equilibria of the associated strategic game.
We are now in the position to give a formal deﬁnition of quantum correlated
equilibrium in a strategic game. The deﬁnition is similar to that of correlated
equilibrium, but now the choice of measurement is explicitly modelled as an act.
Deﬁnition 6 A quantum correlated equilibrium of a strategic game (N,(Ai),(ui))
consists of
• a ﬁnite probability space (Ω,π)
• for each player i ∈ N, a set Mi (a subset of the set of Hermitian op-
erators on a complex, separable Hilbert space) representing the available
measurements
• for each player i ∈ N, and for each mi ∈ Mi, a partition P
mi
i of Ω
representing player i’s information
• for each player i ∈ N, a function σi : Ω → Mi × Ai, with σa
i (ω) = σa
i (ω′)
whenever σm
i (ω) = σm
i (ω′) and ω ∈ Pi and ω′ ∈ Pi for some Pi ∈ P
σ
m
i (ω)
i
(a strategy of player i), such that for any other strategy ρi
X
ω∈Ω
π(ω)ui(σa
−i(ω),σa
i (ω)) ≥
X
ω∈Ω
π(ω)ui(σa
−i(ω),ρa
i(ω)),
where σa
i (ω) and σm
i (ω) denote, respectively, the measurement and the
action selected by strategy σi at state ω.
6Observe that the choice of measurement does not appear in the payoﬀ func-
tions ui, but inﬂuences the expected payoﬀ via the players’ information parti-
tions. The deﬁnition of quantum correlated equilibrium insures that there is no
proﬁtable unilateral strategy deviation after a choice of measurement has been
performed and its result observed by a player, and also insures that there is no
other choice of measurement which may lead to higher expected payoﬀs. Any
correlated equilibrium is a quantum correlated equilibrium (with just a single
measurement available to each player). The converse is not true in general,
unless all the available measurements commute.
Note that a quantum correlated equilibrium always exists. In fact, if all the
players choose to ignore the results of measurement and play a Nash equilibrium
of the classical game this is always a quantum correlated equilibrium according
to the above deﬁnition. More remarkably, the set of payoﬀ combinations sus-
tained by quantum correlated equilibria can be strictly larger than the set of
classical correlated equilibrium payoﬀs, as the example in the previous section
demonstrates.
It is not diﬃcult to check that the equilibrium with two entangled qubits
described in the example satisﬁes the deﬁnition of quantum correlated equilib-
rium. In the example, the set of states can be identiﬁed with the set of all
combinations of type, measurement and action by each player. The informa-
tion partition of player i captures which measurement (x,y or z) was chosen by
i, and its outcome (“up” or “down”). A strategy speciﬁes which of the three
measurements is chosen, and restricts the subsequent choice of classical action
(announcing “yes” or “not”) to depend only on the type of the player, on the
choice of measurement and on its outcome. In equilibrium, with probability
(6/9)∗(3/4) the joint strategy yields a payoﬀ of 9, and with the complementary
probability yields a payoﬀ of zero. Note that the equilibrium is strict: any other
choice of action (for any given type, measurement and observation), as well as
any other choice of measurement, would lead to a stricly lower expected payoﬀ.
4 Discussion
We introduced a novel notion of correlated equilibrium which takes into explicit
account the fact that any information received by the players prior to playing the
game must be embodied in a physical carrier, and showed that the availability
of a correlating device able to send quantum signals to the players may allow
for equilibria which are more eﬃcient than any classical equilibrium. One of the
two approaches in the literature which comes closest to ours is the one in [7],
which deﬁnes a class of games (‘Quantum Correlation Games’, or QCG) in which
quantum correlation is exploited to yield more eﬃcient equilibria if entangled
quantum signals are available to the players. Yet, in QCGs the only available
actions are choices of measurement, which makes QCGs a rather limited class
of games. Unlike [7], we proposed an equilibrium notion which applies to any
7classical strategic-form game, provided that quantum signals can be sent to the
players and observed prior to their choice of classical action. Also, unlike [7],
in our approach departures from classical correlated equilibrium are directly
related to violations of Bell’s inequalities.
The other approach which comes closest to ours is the one in [6], which
also postulates the availability of quantum signals and demonstrates how en-
tanglement can induce correlation in the players’ strategies but fails to relate
these ideas with the notion of correlated equilibrium, and to realize that quan-
tum correlated equilibria can be strictly more eﬃcient than classical correlated
equilibria. In particular, in the game used in [6] to demonstrate quantum corre-
lation, correlated equilibria based on classical private signals would be equally
eﬃcient. Unlike [6], we esplicitly related the notion of entanglement with that
of correlated equilibrium in strategic games, and showed that in some games
quantum correlated equilibria can be strictly more eﬃcient than classical ones.
5 References
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[3] J. Eisert, M. Wilkens, and M. Lewenstein. “Quantum games and quan-
tum strategies”, Physical Review Letters 83, 3077-3080, 1999.
[4] A. Einstein, B. Podolsky, and N. Rosen. “Can Quantum-Mechanical
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[5] A. Gibbard. “Manipulation of voting schemes: a general result”, Econo-
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[6] B. A. Huberman and T. Hogg, “Quantum Solution of Coordination Prob-
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[8] D. A. Meyer, “Quantum strategies”, Physical Review Letters 82, 1052–
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8

Can Quantum-Mechanical Description of Physical Reality be Considered Complete?”,

Correlated equilibrium as an expression of Bayesian rationality.

Eﬃcient mechanisms for bilateral trading”,

Manipulation of voting schemes: a general result”,

Mathematical Foundations of Quantum Mechanics,

Quantum Correlation Games”,

Quantum games and quantum strategies”,

Quantum strategies”,

Strategy-Proofness and Arrow’s Conditions: Existence and Correspondence Theorem for Voting Procedures and Social Welfare Functions”,

Subjectivity and correlation in randomized strategies.

Theory of Games and Economic Behavior.

Correlated Equilibria of Classical Strategic Games with Quantum Signals

http://128.118.178.162/eps/game/papers/0309/0309001.pdf

Correlated Equilibria of Classical Strategic Games with Quantum Signals

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