We show that quantum game theory offers solution to the famous Newcomb's paradox (free will problem). Divine foreknowledge is not necessary for successful completion of the game because quantum theory offers a way to discern human intentions in such way that the human retain her/his free will but cannot profit from changing decision. Possible interpretation in terms of quantum market games is proposed.

Edward W. Piotrowski

Jan Sladkowski

English

Research Papers in Economics

Quantum solution to the Newcomb’s
paradox
(RePEc:sla:eakjkl:10 14-II-2002)
Edward W. Piotrowski
Institute of Theoretical Physics, University of Białystok,
Lipowa 41, Pl 15424 Białystok, Poland
e-mail: ep@alpha.uwb.edu.pl
Jan Sładkowski
Institute of Physics, University of Silesia,
Uniwersytecka 4, Pl 40007 Katowice, Poland
e-mail: sladk@us.edu.pl
Abstract
We show that quantum game theory oﬀers solution to the famous
Newcomb’s paradox (free will problem). Divine foreknowledge is not
necessary for successful completion of the game because quantum
theory oﬀers a way to discern human intentions in such way that
the human retain her/his free will but cannot proﬁt from changing
decision. Possible interpretation in terms of quantum market games is
proposed.
PACS numbers: 02.50.Le, 03.67.-a, 03.65.Bz
1I n t r o d u c t i o n
There is a common belief that the characteristic size of the brain’s integral
parts is too big to allow for quantum eﬀects being important [1]. But re-
cent experiments show that separated objects of the size of a golf ball can
1form quantum entangled states even in a room temperature [2]. Physicists
successfully apply quantum mechanics to describe a lot of complex system
that may have in principle arbitrary size, including black holes or even
the whole Universe. Are there any reasons for quantum modeling of phe-
nomena related to brain activity, consciousness or social behaviour? One
can give an answer to this question only after construction and thorough
veriﬁcation of respective models [3]. Below we consider a problem easily
susceptible of modeling as a quantum game that should shed some light on
the solutions that quantum theory may oﬀer.
In 1960 William Newcomb, a physicist, intrigued the philosopher Robert
Nozick with a claim that in an elementary game characterized by the matrix
M
M
:
=
￿
$
1
0
0
0
$
1
0
0
1
0
0
0
0
$
1
0
0
0
0
0
0
￿
(1)
g i v i n gt h ep a y - o ﬀo ft h ep l a y e r1i na l lp o s s i b l es i t u a t i o n s ,t h ep l a y e r1i s
not able to chose his strategy without having any measure of occurring a
posteriori of any of the four possible events. Rows correspond the player
1’s strategies: feminine
j
0
i
1 and masculine
j
1
i
1
1 and columns to opponent’s
strategies
j
0
i
2
;
j
1
i
2. It so happens even despite the fact that the feminine
strategy dominates the masculine one (that is the pay-oﬀ is greater regard-
less of the opponents strategy). The choice of the masculine strategy
j
1
i
1 is
more proﬁtable when the event corresponding to the oﬀ-diagonal elements
of
M do not occur and the rest have almost equal probabilities. This might
happen if the opponent is able to foresee the player 1 moves. Due to this
paradoxical property the above game with indeﬁnite (hidden) set of oc-
currences became for philosophers, economists and theologians a graceful
theme of speculations about free will and its consequences [4, 5]. The dis-
putes, often referred to as newcombmania [6], deserve a thorough analyzis
from the quantum game theory point of view [7]-[11]. The development of
the probability theory provide us with many intriguing examples where
ambiguous speciﬁcation of the appropriate probability measures resulted
in contradiction (Bertrand [13] and Banach-Tarski [14] paradoxes are the
most famous ones). One can still ﬁnd people who regardless of this facts
continue philosophical disputes while ignoring the necessity of precise deﬁ-
1The use of the adjectives feminine and masculine to underline the character of the
strategies will be explained later, see also the Gardner book
2nition of the probabilistic measures in their models. We would like to show
that quantum theory may be of help in settling the ambiguities.
2 Quantum description of the game
Quantum game theory exploits the formalism of quantum mechanics in or-
der to oﬀer the players new classes of strategies. Interesting generalization
of well known classical games have been put forward [7, 8]. There are argu-
ments that quantum strategies may oﬀer extraordinary tools for biologists
[15]-[17]. Economics being the theater of various games and conﬂicts should
not despise these new ideas [18, 12]. We will describe player’s strategies
as vectors (often referred to as states) in Hilbert spaces
H
i where the sub-
scripts
i
=
1
;
2 distinguish between the player 1 and 2. It is convenient to
deﬁne the strategy density operator
W
W
=
2
X
r
;
s
=
1
W
r
s
j
r
￿
1
i
1
j
s
￿
1
i
2
1
h
r
￿
1
j
2
h
s
￿
1
j
where
(
W
r
s
) is a matrix with nonnegative entries such that
P
r
;
s
W
r
s
=
1
and
j
r
i
1
j
s
i
2
1
h
r
j
2
h
s
j,
r
;
s
2
f
0
;
1
g are projective operators on the states of
the game,
j
r
i
1
j
s
i
2
2
H
1
￿
H
2. For our aims it will be suﬃcient to use two
dimensional Hilbert spaces for the players’ strategies. The states of the
classical setting (mixed strategies) are represented by a diagonal matrix
(
W
r
s
). Non-diagonal elements of
(
W
r
s
) describe situations (strategies) that
are out of the reach for classical players. Following the classical terminology
we will call the pay-oﬀ observable
M a Hermitian operator corresponding
to the matrix
(1
):
M
:
=
2
X
r
;
s
=
1
M
r
s
j
r
￿
1
i
1
j
s
￿
1
i
2
1
h
r
￿
1
j
2
h
s
￿
1
j
:
Therefore, according to the classical interpretation of the game, the player
1’s expected pay-oﬀ is equal to the sum of diagonal elements (trace) of the
product of
M and the transpose of
W:
E
(
M
)
:
=Tr
M
W
=
X
r
s
M
r
s
W
r
s
= Tr
M
W
T
:
33N e w c o m b ’ s p a r a d o x
M. Gardner proposed the following fabulous description of a game with pay-
oﬀ given by the matrix
(1
) [4]. An alien Omega (or Alf?) being a omniscent
representative of alien civilization (player 2) oﬀers a human (player 1) a
choice between two boxes. The player 1 can take the content of both boxes
or only the content of the second one. The ﬁrst one is transparent and
contains $1000. Omega declares to have put into the second box that is
not transparent $1000000 (strategy
j
1
i
2) but only if he foresaw that the
player 1 decided to take only the content of that box (
j
1
i
1). A male player 1
thinks: If Omega knows what I am going to do then I have the choice
between $1000 and $1000000. Therefore I take the $1000000 (strategy
j
1
i
1). A female player 1 thinks: Its obvious that I want to take the only
the content of the second box therefore Omega foresaw it and put the
$1000000 into the box. So the one million dollar is in the second box.
Why should I not take more – I take the content of both boxes (strategy
j
0
i
1). The question is whose strategy, male’s or female’s, is better? One
cannot give unambiguous answer to this question without precise deﬁnition
of the measures of the events relevant for the pay-oﬀ.
4 Human’s and Omega’s strategies
Omega as representative of an advanced alien civilization is certainly aware
of quantum properties of the Universe that are still obscure or mysterious
to humans. The boxes containing pay-oﬀs are probably coupled. One can
suspect this because the human cannot take content of the transparent box
only ($1000). The female player is sceptical about the possibility of realiza-
tion of the Omega’s scenario for the game. She thinks that the choice of the
male strategy results in Omega putting the one million dollar in the sec-
ond box, and after this being done no one can prevent from her taking the
content of the both boxes in question (ie $1001000). But Meyer proposed a
quantum tactics [7] that, if adopted by Omega, allows Omega to accomplish
his scenario. Let us note that Omega may not be able to foresee the future
[4]. For it aims it is suﬃcient that it is able to discern human intentions re-
gardless of their will or feelings on the matter. The obstacles to this implied
by the no-cloning theorem can be overcome by means of teleportation [19]:
Omega has must be able to intercept and then return human’s strategies.
4The presented below manipulations leading to thwarting humans are fea-
sible with contemporary technologies. The course of the game may look
as follows. At the starting-point, the density operator
W acting on
H
1
￿
H
2
describes the human’s intended strategy and the Omega’s strategy based
on its prediction of human’s intentions. The actual game must be carried on
according to quantum rules that is players are allowed to change the state
of the game by unitary action on
W [7, 8]. The human player can only act
on her/his
q-bit Hilbert space
H
1. Omega’s tactics must not depend on the
actual move performed by the human player (it may not be aware of the
human strategy): its moves are performed by automatic device that couples
the boxes. The Meyer’s recipe leads to:
1. Just before the human’s move, Omega set the automatic devise ac-
cording to its knowledge of human’s intention. The device executes
the tactics
F
￿
I,w h e r e
I is the identity transform (Omega cannot
change its decision) and
F is the well known Hadamard transform
frequently used in quantum algorithms:
F
:
=
1
p
2
￿
1
1
1
￿
1
￿
.
2. The human player with the probability
w uses the female tactics
N
￿
I ,
where
N is the negation operator2 and with the probability
1
￿
w the
male tactics
I
￿
I .
3. At the ﬁnal step the boxes are being opened and the built-in coupling
mechanism performs once more the transform
F
￿
Iand the game
is settled.
5 The course of the game and its result
Let us analyze the evolution of the density operator
W. The players’ tactics,
by deﬁnition, could have resulted in changes in the (sub-)space
H
1 only
therefore it suﬃces to analyze the human’s strategies. In a general case the
human can use a mixed strategy: the female one with the probability
v and
the male one with the probability
1
￿
v. Let us begin with the extreme values
of
v (pure strategies). If the human decided to use the female strategy (
v
=
1)
or the male one (
v
=
0) then the matrices
W
i,
i
=
0
;
1 corresponding to the
2
N
j
0
i
=
j
1
i,
N
j
1
i
=
j
0
i
5density operators
W
0
=
2
X
r
;
s
=
1
W
0
r
s
j
r
￿
1
i
1
j
0
i
2
1
h
s
￿
1
j
2
h
0
j
and
W
1
=
2
X
r
;
s
=
1
W
1
r
s
j
r
￿
1
i
1
j
1
i
2
1
h
s
￿
1
j
2
h
1
j
are calculated as follows:
￿
v
0
0
1
￿
v
￿
￿
!
1
2
￿
1
1
1
￿
1
￿
￿
v
0
0
1
￿
v
￿
￿
1
1
1
￿
1
￿
=
1
2
￿
1
2
v
￿
1
2
v
￿
1
1
￿
￿
!
w
2
￿
0
1
1
0
￿
￿
1
2
v
￿
1
2
v
￿
1
1
￿
￿
0
1
1
0
￿
+
1
￿
w
2
￿
1
2
v
￿
1
2
v
￿
1
1
￿
=
1
2
￿
1
2
v
￿
1
2
v
￿
1
1
￿
￿
!
1
4
￿
1
1
1
￿
1
￿
￿
1
2
v
￿
1
2
v
￿
1
1
￿
￿
1
1
1
￿
1
￿
=
￿
v
0
0
1
￿
v
￿
:
It is obvious that independently of the used tactics, human’s strategy takes
the starting form. For the mixed strategy the course of the game is described
by the density operator
W
=
v
W
0
+
(
1
￿
v
)
W
1
which also has the same diagonal form at the beginning and at the end of
the game:
W
=
v
j
0
i
1
j
0
i
2
1
h
0
j
2
h
0
j
+
(
1
￿
v
)
j
1
i
1
j
1
i
2
1
h
1
j
2
h
1
j
:
Therefore the change of mind resulting from the female strategy cannot
lead to any additional proﬁts. If the human using the female tactics (that
is changes his/her mind) begins the game with the female strategy then at
the end the untransparent box will be empty and he/she will not get the
content of the transparent box: the pay-oﬀ will be minimal (0). If the human
acts just the opposite the transparent box must not be opened but never-
theless the pay-oﬀ will be maximal ($100000). Only if the human begins
with the female strategy and then applies the male tactics the content of
the transparent box is accessible. If restricted to the classical game theory
Omega would have to prevent humans from changing their minds. In the
6quantum domain the pay-oﬀ
M
2
1 (female strategy and tactics) is possible
(the phrase la donna mobile gets a quantum context): humans regain their
free will but they have to remember that Omega has (quantum) means to
prevent humans from proﬁting from altering their decisions. In that way
quantum approach allows to remove the paradox from the rationally de-
ﬁned dilemma. One can also consider games with more alternatives for
the human player. The respective larger pay-oﬀ matrices would oﬀer even
more sophisticated versions of the Newcomb’s observation. But even then
there is a quantum protocol that guarantees that Omega keeps its promises
(threats) [21].
6 Market interpretation of the game
It is obvious that the above scenario cannot be realized if the actual con-
ditions would diﬀer from Omega’s promises. For example, Omega may not
be able to predict humans intentions or its understanding of the rules of
the game diﬀers from that implied by their expression in human language
(cultural diﬀerences). There may be much dispute over the question what
Omega really has in mind? We would like to consider one of the variant
that may be interesting in the context of quantum market games [12, 20]. This
may result from pondering over the meaning of the term Omega adopts
t h es a m es t r a t e g y .
Players in a quantum market game sometimes buy and sometimes sell. A
demand representation of the player’s strategy is a Fourier transform of his
strategy used while supplying the goods [12, 11]. In a simpliﬁed model where
player’s strategies span a ﬁnite dimensional Hilbert space we should apply
discrete Fourier transform which transforms the demand representation
of the strategy, being
m-tuple of complex numbers
h
d
j
 
i to the supply
representation given by
h
s
j
 
i
=
1
p
m
m
￿
1
X
d
=
0
e
2
￿i
m
s
d
h
d
j
 
i
:
If
m
=
2 then the discrete Fourier transform reduces to the Hadamard
transform
F which we have already met. In our case the Hadamrd trans-
form switches maximally localized strategies with the the maximally indef-
inite strategies and vice versa, eg.
h
d
j
 
i
=
[
d
=
0
]
F
￿
!
h
s
j
 
i
=
1
p
2 (the Iverson
7notation [22]i su s e d :
[
e
x
p
r
e
s
s
i
o
n
] denotes the logical value (1 or 0) of the
sentence
e
x
p
r
e
s
s
i
o
n).
We introduce the nonhomogeneous complex coordinate
z
2
C to pa-
rameterize player’s strategies
H
3
j
 
z
i
:
=
j
0
i
+
z
j
1
i
:
If the ”buying human” decides to use the strategy
j
 
z
i
1 and the other side
of the bargain (Omega) want to play in the same way and therefore uses the
supply representation of human’s strategy setting its
q-bit to
F
(
j
0
i
2
+
z
j
1
i
2
)
=
j
0
i
2
+
1
￿
z
1
+
z
j
1
i
2 then the quantum state of the game takes the form3
W
z
=
(
1
+
z
)
(
1
+
z
)
2
(
1
+
z
z
)
2
￿
j
0
i
1
+
z
j
1
i
1
￿
￿
1
h
0
j
+
z
1
h
1
j
￿
￿
j
0
i
2
+
1
￿
z
1
+
z
j
1
i
2
￿
￿
2
h
0
j
+
1
￿
z
1
+
z
2
h
1
j
￿
:
Therefore, as in the previous discussion, the female strategy gives not
-1
0
1 Re z
-1
0
1
Im z
0
1000500
-1
0
1 Re z
Figure 1: The average human pay-oﬀ Tr
M
W
z in the market version of the
Newcomb game.
higher a pay-oﬀ. The expectation value of the human’s pay-oﬀ, Tr
M
W
z,
3Cases when the player may in fact play also ”against himself” often happen in mar-
ket description: demand or supply result from self-consistent strategy of all players. For
example in a stock exchange one big bid or transaction can inﬂuence the whole market
8is maximal for a superposition of male and female strategies with phase
shifted by
￿ (i.e. for
z
=
￿
1). In this case the human is better oﬀ than in
the previous case but she or he must be cautious because the phase shift
by
￿ (
z
=
1) does not change the respective probabilities but result in the
lowest expectation value of the pay-oﬀ ($500). Classical human’s strategies
correspond to
z
=
0(female) and
z
=
￿
1 (male). The expectation values
of the human pay-oﬀ with respect to the adopted strategy are presented in
Figure 1.
Enthusiasts for newcombmania will certainly ﬁnd a lot of new quantum
solutions to the Newcomb game.
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10

A note on

Darwinism in quantum systems?,

Entaglment and Dynamic Stability of Nash Equilibria in a Symmetric Game,

Evolutionary stable strategies in quantum games,

G r a h a m ,D .E .K n u t h ,a n dO .P a t a s h n i k ,Concrete Mathematics,

Geometric probability,G r i ﬃ n ,L o n -don

Newcomb’s problem and two principles of choice in Essays in Honour of Carl Hempel,E d s . :N .R e s c h e re ta l . ,D .R e i d e l ,

Paradoxes to puzzle and delight, Freeman & Co,

Playing by Quantum Rules,

Quantum Mechanics gives Stability to a Nash Equilibrium,

Quantum roulette: an extended quantum strategy,

Quantum-like approach to ﬁ-nancial risk: quantum anthropic principle,

Starway to the Mind,

The Feynman Processor: Quantum Entaglement and the Computing Revolution, Perseus Book Group,

Trading by Quantum Rules – Quantum Anthropic Principle, submitted to Economics Letters;

Quantum solution to the Newcomb's paradox

http://alpha.uwb.edu.pl/ep/RePEc/sla/eakjkl/10.pdf

Quantum solution to the Newcomb's paradox

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