Quantum model of strategic decision-making in
Prisoner’s Dilemma game: quantitative predictions
and new empirical results
Jakub Tesař, Charles University in Prague, Institute of Political Studies,
[email protected]
Abstract: The Disjunction effect introduced in the famous study by Shafir and Tversky (1992)
and confirmed by several following studies remains one of key ‘anomalies’ for the standard
model of the Prisoner’s Dilemma game. In the last 10 years, new approaches have appeared
that explain this effect with the use of quantum probability theory. But the existing results do
not allow parameter-free test of these models. This paper introduces a general quantum
model of Prisoner’s Dilemma game as well as a new experimental design that enables to test
the main quantitative and qualitative predictions of this model. Main findings of the experiment support the viability of the quantum model and challenge the classical solution concept
of the Prisoner’s Dilemma game.
Keywords: Disjunction effect; Prisoner’s Dilemma; Quantum probability theory; Order effect
1 Introduction – dilemma of cooperation
Game theory is one of the key formal tools in many fields of the social science. It aims to
represent a broad range of interactions among human (or non-human) agents, using simple
formalism of games. Some of these games are used extensively in social sciences to explain
human behavior in more complex circumstances. This is also the case of the Prisoner’s Dilemma game, which has become a model for individual-level behavior for theories in economy, sociology, political science and many other fields.
The Prisoner’s Dilemma (PD) game models strategic decision-making of players in the presence of dilemma: whether to cooperate with or to defect the other player. The dilemma
stems from the key property of the game – the desirable solution of mutual cooperation
(which is the Pareto optimal strategy profile) is for both players dominated by the strategy of
defection. The solution concepts of standard decision theory pick defection as the only ‘rational’ strategy in the game. However, as was shown in many studies (for a review see
Camerer 2003), the considerable percentage of players choose cooperation as their preferred strategy. Since 1950, when the game was formalized by Albert Tucker, we learned
many important features of PD game. The high level of cooperation was to a large extent
explained by an alternation of the utility function due to the other-regarding preferences or
the confidence building in the repeated games (the strategy ‘tit for tat’ and others).
But the study by Shafir and Tversky (1992) discovered another ‘anomaly’ in PD game that
challenges the concept of the rational decision-making itself – the Disjunction effect. In their
study players met PD game in three different conditions. In different rounds they picked
their strategy with: a) No information about the opponent, b) the information that their opponent had defected, or c) the information that their opponent had cooperated. The frequencies of cooperation (strategy 𝐶) were 37% in the no-information treatment, 16% after
the cooperation of the opponent and 3% after his defection. These results contradict one of
the key predictions of the rational choice theory, the Savage’s sure thing principle. “According to the sure thing principle, if under state of the world X you prefer action A over B, and if
under the complementary state of the world ̅
X you also prefer action A over B, then you
should prefer action A over B when you do not know the state of the world.” (Busemeyer
and Bruza 2014, 261). Many players in the study clearly contradict this principle: they preferred defection after opponent’s cooperation and defection after opponent’s defection, but
chose cooperation in no-information treatment.
The results were replicated and extended in several following studies (Croson 1999; Li and
Taplin 2002; Hristova and Grinberg 2008; Shu Li et al. 2010; see Busemeyer and Bruza 2014,
263–264 for a review), which showed that the PD game produces a robust Disjunction effect.
As was already said, this result contradicts not the specific assumption about the utility function, but the logic of ‘rational choice’, as defined by the rational choice theory, itself. Tversky
and Shafir explain the result with the combination of the wishful thinking – “… when the opponent’s response was not known, many subjects preferred to cooperate, perhaps as a way
of ‘inducing’ cooperation from the other.” (Shafir and Tversky 1992, 458) – and ‘nonconsequential evaluation’. Their theory was able to explain a wide range of Disjunction effects but
only by constraining the rationality of the players. The wishful thinking and the nonconsequential reasoning were considered as the examples of bounded rationality and expanded the list of possible phenomena that differentiate the formal game theory of fullyrational agent from the behavioral game theory of the real human beings in the laboratory
experiments.
This view was challenged by the introduction of the quantum models of cognition and decision-making (see below) which offered the alternative concept of reasoning based on the
non-commutative logic of quantum probability theory. When applied to the problem of PD
game this logic explains the known effects (e.g. the Disjunction effect) not by depriving the
agents of their rationality but by changing the concept of rationality itself.
Quantum model can be considered as the generalization of the standard model of strategic
decision-making. It releases some of the intrinsic assumptions of the standard model (namely the compatibility of perspectives) and provides more degrees of freedom to fit the data.
2
The Disjunction effect is than explained as a kind of order effect of reasoning in two different
(non-compatible) perspectives. But the existing data offer only one orderings of perspectives
– the studies mentioned above inquire an effect of getting1 the opponent’s strategy on the
level of players’ cooperation. Our study extends previous works by adding reverse order of
decisions: how does the players’ choice of a strategy influence their expectation about the
choice of their opponent. The second ordering of questions allows us to test the quantum
model with a non-parametric test.
Our aim in this paper is threefold: i) to introduce a general quantum model of two-players
game, including quantitative and qualitative predictions for the PD game; ii) to present results of the experiment which took place in 2016 at the Ohio State University, OH, USA and
iii) to test the quantum model against the results of this experiment and compare standard
and quantum model in respect to these results. The paper proceeds as follows. In the next
part we introduce the quantum model of PD game, in two basic versions of the twodimensional and 4-dimensional model. Then we derive three prediction of a general quantum model. Next we present the results of the PD game experiment and tests the prediction
of the model against them. In conclusion, we show what are the consequences of the quantum model for the classical solution concepts of PD game.
2 Quantum model of the Disjunction effect
In this section I present a quantum model of decision-making in the PD game. The quantum
explanation for this problem was originally suggested by Busemeyer et al. (Busemeyer, Matthews, and Wang 2006). Since then many other models have been proposed (Pothos and
Busemeyer 2009; Khrennikov and Haven 2009; Aerts 2009; Yukalov and Sornette 2011; Accardi, Khrennikov, and Ohya 2009; see Busemeyer and Bruza 2014 for a review). These model share the same basic assumption that the state of the cognitive system and the procedure
of decision-making can be modeled by the quantum probability theory (so called *-algebra)
instead of the standard (Kolgomorov) probability theory (σ-algebra). The models differ in
many aspect: in the construction of the initial state, dimensionality of the problem, etc. My
approach, that I present below, consider the PD game as the case of the sequential measurement of two incompatible perspectives and will use the quantum interference model
proposed by Busemeyer et al. (2011). This model is in many aspects similar to Pothos and
Busemeyer (2009), but instead of the use of quantum dynamics it explains the existence of
the interference effect by incompatibility of the two different perspectives the player can
employ to approach the game.
One of the most important theoretical questions regarding the quantum model is the assumption about the dimensionality of the problem. In the following section I present both
1
or guessing in (Croson 1999)
3
the two-dimensional (2D) and the four-dimensional (4D) model of the game. I will show different predictions these two models offer and will test them against the results of the experiment in the next section.
Initial state
Prior any treatment in the experiment a player can be described by some initial state which
defines the probabilities of individual actions in the game. In the classical probability theory
this state is the probability function that maps elementary events (𝐴𝑖 ) into probabilities
𝑝(𝐴𝑖 ) ∈ [0,1]. Probabilities for the general events are determined as the sum of the probabilities of the elementary events that are contained in the general event. All the probabilities
in the sample space sum to one: ∑𝑖 𝑝(𝐴𝑖 ) = 1.
In the quantum probability theory, the state of the system can be described as a unitary vector in 𝑁-dimensional vector space over the field of complex numbers2 with well-defined inner product (i.e. the Hilbert space). This space is defined by 𝑁 basis vectors that are mutually
orthogonal (their inner-product3 is zero) and which together span the whole space. The
choice of the basis vectors is not unique, spaces with 𝑁 ≥ 2 can always be spanned by different sets of basis vectors. The general quantum state is then given as a linear combination
of the basis vector from one orthogonal set.
In our 2D quantum model of PD game, we define the initial state of the player as the linear
combination of the vectors that reflects two possible choices of strategies – C (cooperation)
and D (defection). The initial state is then:
|𝑆⟩ = 𝜓𝐶 ∙ |𝐶⟩ + 𝜓𝐷 ∙ |𝐷⟩ 4 (2.1𝑎)
Here 𝜓𝐶 and 𝜓𝐷 are the coefficients of the linear combination and terms with the parenthesis are so called ket-vectors in the Dirac symbolic (more about vectors and Dirac symbolic in
Appendix 0). Equation 2.1a can be rewritten, knowing that the basis vectors are still |𝐶⟩ and
|𝐷⟩, in the simpler form of Equation 2.1b.
𝜓
|𝑆⟩ = ( 𝐶 ) (2.1𝑏)
𝜓𝐷
Any general quantum system is the linear combination of its basis states. We also use a term
of superposition, i.e. the quantum state is in a superposition of its basis states. Not allways
are all the basis vectors present in the superpostion. If for example the coefficients in our 2D
model were 𝜓𝐶 = 1 and 𝜓𝐷 = 0 than the state of the system is identical with the basis state
2
For the introduction to the complex numbers and their properties, see Appendix B.
See Appendix 0.
4
The introduction to the vector spaces and Dirac symbolic can be found in Appendix 0.
3
4
|𝐶⟩ and the state |𝐷⟩ is missing. Nevertheless, the superposition is a general state of the
quantum system and in the 2D model it is given by the equation 2.1a.
If we approach a game from the perspective of the other player, we gain a new pair of basis
vectors |𝐶′⟩ and |𝐷′⟩. In this basis the state vector is given by the superposition:
|𝑆⟩ = 𝜓𝐶′ ∙ |𝐶 ′ ⟩ + 𝜓𝐷′ ∙ |𝐷′ ⟩ (2.2𝑎)
|𝑆⟩ = (
𝜓𝐶′
) (2.2𝑏)
𝜓𝐷′
The coefficients 𝜓𝐶 ′ and 𝜓𝐷′ are generally different from 𝜓𝐶 and 𝜓𝐷 . If they equal, there is
no difference between considering the game from the two different perspectives, but we
hypothesize that it does not apply to the PD game. Therefore, in each perspective, the initial
state is defined by the different pair of complex coefficients. Is there any relationship between them? The coefficients are not entirely independent because they represent the same
vector in the same Hilbert space. Their relationship (in arbitrary large 𝑁) is given by the unitary matrix5 𝑈.
𝜓 ′
𝜓
( 𝐶 ) = 𝑈 ∙ ( 𝐶 ) (2.3𝑎)
𝜓𝐷
𝜓𝐷′
(
𝜓𝐶′
𝜓
) = 𝑈 † ∙ ( 𝐶 ) (2.3𝑏)
𝜓𝐷′
𝜓𝐷
⟨𝐶|𝐶′⟩
𝑈=(
⟨𝐷|𝐶′⟩
⟨𝐶|𝐷′⟩
) (2.4)
⟨𝐷|𝐷′⟩
In the two-dimensional model, this matrix can be parametrized6 as:
𝑖𝜇
cos 𝜃
𝑈 = 𝑒 𝑖𝜗 ( 𝑒 −𝑖𝜑
−𝑒
sin 𝜃
𝑒 𝑖𝜑 sin 𝜃 ) (2.5)
𝑒 −𝑖𝜇 cos 𝜃
If we compere Equation 2.4 and 2.5, it is clear that theoretically given form of the unitary
matrix determines the relationship among the inner products of basis vectors from different
perspectives. Particularly, using parametrization from Equation 2.5, these conditions has to
be satisfied:
⟨𝐷|𝐷′⟩ = 𝑒 −2𝑖𝜇 ∙ ⟨𝐶|𝐶′⟩
⟨𝐷|𝐶′⟩ = −𝑒 −2𝑖𝜑 ∙ ⟨𝐶|𝐷′⟩
5
See Appendix D.
There are 4 independent parameters (𝜃, 𝜇, 𝜑, 𝜗) in this parametrization. Nevertheless, the element 𝑒 𝑖𝜗 only
shifts the phase of the resulting vector and therefore has any physical meaning (the results do not depend on
𝜗).
6
5
As we will see in the next section, this determines one of the key quantitative predictions for
the conditioned probabilities in the 2D mode: the double stochasticity of 2x2 transition matrix (see below).
What is the initial state in the 4D model? What we can choose as two other bases? Every
player has only two choices, but the game has 4 possible outcomes that match the combination of the strategies of both players. Therefore, the basis states in the self-perspective are
|𝐶𝐶′⟩, |𝐶𝐷′⟩, |𝐷𝐶′⟩ and |𝐷𝐷′⟩, where e.g. basis state |𝐶𝐶′⟩ means that the player is playing
strategy 𝐶 assuming that the game will end up with the strategy profile (𝐶, 𝐶’). Again, we
can write the initial state in the form of the linear combination of the basis states:
|𝑆⟩ = 𝜓𝐶𝐶′ ∙ |𝐶𝐶 ′ ⟩ + 𝜓𝐶𝐷′ ∙ |𝐶𝐷′ ⟩ + 𝜓𝐷𝐶′ ∙ |𝐷𝐶′⟩ + 𝜓𝐷𝐷′ ∙ |𝐷𝐷′⟩ (2.6𝑎)
𝜓𝐶𝐶′
𝜓
|𝑆⟩ = ( 𝐶𝐷′ ) (2.6𝑏)
𝜓𝐷𝐶′
𝜓𝐷𝐷′
In the other-perspective, the basis is the set of vectors |𝐶′𝐶⟩, |𝐶′𝐷⟩, |𝐷′𝐶⟩ and |𝐷′𝐷⟩, where
e.g. |𝐶′𝐶⟩ means that the player expects strategy 𝐶′ from his opponent and that he expects
that the opponent chooses it by assuming the final strategy profile (𝐶′, 𝐶).
𝜓𝐶′𝐶
𝜓
|𝑆⟩ = ( 𝐶′𝐷 ) (2.7)
𝜓𝐷′𝐶
𝜓𝐷′𝐷
The mutual relationship between the two sets of coefficients is given by the 4x4 unitary matrix 𝑈, which is defined by 12 independent parameters7 (Atmanspacher and Römer 2012). In
the form of the inner products of the basis vectors it is given by:
𝜓𝐶𝐶 ′
𝜓𝐶 ′ 𝐶
𝜓𝐶𝐷′
𝜓𝐶 ′ 𝐷
(
)=𝑈∙(
) (2.8)
𝜓𝐷𝐶 ′
𝜓𝐷 ′ 𝐶
𝜓𝐷𝐷′
𝜓𝐷′ 𝐷
⟨𝐶𝐶′|𝐶′𝐶⟩
⟨𝐶𝐶′|𝐶′𝐷⟩
𝑈=(
⟨𝐶𝐶′|𝐷′𝐶⟩
⟨𝐶𝐶′|𝐷′𝐷⟩
⟨𝐶𝐷′|𝐶′𝐶⟩
⟨𝐶𝐷′|𝐶′𝐷⟩
⟨𝐶𝐷′|𝐷′𝐶⟩
⟨𝐶𝐷′|𝐷′𝐷⟩
⟨𝐷𝐶′|𝐶′𝐶⟩
⟨𝐷𝐶′|𝐶′𝐷⟩
⟨𝐷𝐶′|𝐷′𝐶⟩
⟨𝐷𝐶′|𝐷′𝐷⟩
7
⟨𝐷𝐷′|𝐶′𝐶⟩
⟨𝐷𝐷′|𝐶′𝐷⟩
) (2.9)
⟨𝐷𝐷′|𝐷′𝐶⟩
⟨𝐷𝐷′|𝐷′𝐷⟩
For the hyperspherical parametrization of 4D unitary matrix see (Hedemann 2013). The possible form of the
unitary matrix in the PD game was explored also in (Busemeyer and Bruza 2014, chap. 9).
6
First decision
Knowing the initial state of the decision-maker what does quantum model say about the
probabilities of the first question? Quantum probability theory states that the probability to
measure the system |𝑆⟩ in one of its basis states is given by a square length of the orthogonal projection of this vector into the respective basis state. This can be determined in two
equivalent ways – using the inner product or the projection operator.
The first way is straightforward. The orthogonal projection of one vector into another is given directly by their inner product. Starting with the 2D model e.g. the projection of vector
|𝑆⟩ into the vector |𝐶⟩ (which corresponds to the player choosing cooperation) is given by
the inner product ⟨𝐶|𝑆⟩. Then the probability of choosing strategy C is given by Equation
2.10 and (using the coefficient of the linear combination) by Equation 2.11.
𝑝(𝐶) = |⟨𝐶|𝑆⟩|2 (2.10)
⟨𝐶|𝑆⟩ = ⟨𝐶| ∙ (𝜓𝐶 ∙ |𝐶⟩ + 𝜓𝐷 ∙ |𝐷⟩)
⟨𝐶|𝑆⟩ = 𝜓𝐶 ∙ ⟨𝐶|𝐶⟩ + 𝜓𝐷 ∙ ⟨𝐶|𝐷⟩
⟨𝐶|𝑆⟩ = 𝜓𝐶 ∙ 0 + 𝜓𝐷 ∙ 1 = 𝜓𝐶
𝑝(𝐶) = |𝜓𝐶 |2 (2.11)
Second option is to use the projection operator8 𝑃. The orthogonal projection is given by the
matrix multiplication of the projection operator and the state vector: 𝑃𝐶 ∙ |𝑆⟩ and probability
of such choice as a squared length of the product.
𝑝(𝐶) = ‖𝑃𝐶 ∙ |𝑆⟩‖2 (2.12)
This method naturally leads to the same result (Equation 2.11). The other probabilities can
be computed analogically:
𝑝(𝐷) = |𝜓𝐷 |2 , 𝑝(𝐶′) = |𝜓𝐶′ |2 , 𝑝(𝐷′) = |𝜓𝐷′ |2
But the power of the projection operator is particularly obvious in the case of a combined
event, like the cooperation in the 4D model. To construct the projection operator for the
combined event we have to sum the outer product of all element basis vectors that contain
the relevant event. Therefore, for cooperation we sum the outer products of vector |𝐶𝐶′⟩
and |𝐶𝐷′⟩.
𝑃𝐶 = |𝐶𝐶′⟩ ∙ ⟨𝐶𝐶′| + |𝐶𝐷′⟩ ∙ ⟨𝐶𝐷′| (2.13)
𝑝(𝐶) = |𝜓𝐶𝐶′ |2 + |𝜓𝐶𝐷′ |2 (2.14)
8
See Appendix Chyba! Nenalezen zdroj odkazů..
7
The overall probability of cooperation is therefore the sum of the probabilities of both
events that contain the cooperation: 𝑝(𝐶) = 𝑝(𝐶𝐶 ′ ) + 𝑝(𝐶𝐷 ′ ). Our quantum model is in
the first question equivalent to classical model (where 𝑝(𝐶) = 𝑝(𝐶 𝑎𝑛𝑑 𝐶 ′ ) + 𝑝(𝐶 𝑎𝑛𝑑 𝐷′ )
). It is the sequence of the non-compatible decisions that produce the Disjunction effect
which is anomalous in the classical model.
Second decision
To analyze the probabilities of the second question we have to work with the state of the
system after the first decision. From the previous section we know that by the first decision
the state vector is projected into one of its basis vectors (i.e. the new vector is a respective
basis vector multiplied by a (complex) number). Each basis vector is chosen with probability
from the interval [0, 1]. But right after the projection, the system is in that respective state
with probability 1. To secure this, the quantum probability theory states that the state of the
quantum system right after the first question is normalized to be a unitary vector, which is
reached by division of the projected vector by its length:
|𝑆𝐶 ⟩ =
𝑃𝐶 ∙ |𝑆⟩
(2.15)
‖𝑃𝐶 ∙ |𝑆⟩‖
With the use of the update vector (that is either |𝑆𝐶 ⟩ or |𝑆𝐷 ⟩ after the player’s first decision)
we can determine the probabilities of the second decision by the projection operators 𝑃𝐶′
and 𝑃𝐷′ . But to use it, the operator has to be defined in the same basis as the updated vector. The relevant basis vector of the opponent’s perspective has to be transformed using the
unitary matrix 𝑈 † (or 𝑈 in the opposite direction). The opponent’s cooperation is given by a
basis vector |𝐶′⟩ = 𝑈 † ∙ |𝐶⟩ and his defection by |𝐷′⟩ = 𝑈 † ∙ |𝐷⟩. The resulting projection
operators are 𝑃𝐶′ = |𝐶′⟩ ∙ ⟨𝐶′| and 𝑃𝐷′ = |𝐷′⟩ ∙ ⟨𝐷′|.
The probability of the second question is determined exactly in the same way as of the first
question – as the square length of the projection of vector |𝑆𝐶 ⟩ or |𝑆𝐷 ⟩ into one of the basis
states in the opponent’s perspective. The sequential probabilities for observing player’s
strategy and then his guess of opponent’s strategy are:
𝑝(𝐶𝐶 ′ ) = ‖𝑃𝐶′ |𝑆𝐶 ⟩‖2 = ‖𝑃𝐶′ 𝑃𝐶 |𝑆⟩‖2 (2.16𝑎)
𝑝(𝐶𝐷′ ) = ‖𝑃𝐷′ |𝑆𝐶 ⟩‖2 = ‖𝑃𝐷′ 𝑃𝐶 |𝑆⟩‖2 (2.16𝑏)
𝑝(𝐷𝐶 ′ ) = ‖𝑃𝐶′ |𝑆𝐷 ⟩‖2 = ‖𝑃𝐶′ 𝑃𝐷 |𝑆⟩‖2 (2.16𝑐)
𝑝(𝐷𝐷′) = ‖𝑃𝐷′ |𝑆𝐷 ⟩‖2 = ‖𝑃𝐷′ 𝑃𝐷 |𝑆⟩‖2 (2.16𝑑)
Analogically, the sequential probabilities in the reverse order of question are given by:
𝑝(𝐶′𝐶) = ‖𝑃𝐶 |𝑆𝐶′ ⟩‖2 = ‖𝑃𝐶 𝑃𝐶′ |𝑆⟩‖2 (2.19𝑎)
8
𝑝(𝐶′𝐷) = ‖𝑃𝐷 |𝑆𝐶′ ⟩‖2 = ‖𝑃𝐷 𝑃𝐶′ |𝑆⟩‖2 (2.19𝑏)
𝑝(𝐷′𝐶) = ‖𝑃𝐶 |𝑆𝐷′ ⟩‖2 = ‖𝑃𝐶 𝑃𝐷′ |𝑆⟩‖2 (2.19𝑐)
𝑝(𝐷′𝐷) = ‖𝑃𝐷 |𝑆𝐷′ ⟩‖2 = ‖𝑃𝐷 𝑃𝐷′ |𝑆⟩‖2 (2.19𝑑)
Predictions of the model
Now we can derive three key predictions of the general quantum model and one prediction
which is dimensionally specific. First is the non-commutativity of sequential measurements.
If we compare two sequential probabilities that differ only in the order of decisions, e.g.
𝑝(𝐶𝐷′) and 𝑝(𝐷′𝐶) from above, we see that they contain the same projection operators, but
in reverse order: 𝑃𝐷′ 𝑃𝐶 respective 𝑃𝐶 𝑃𝐷′ . The product of two linear operators is again a linear operator, but if they do not commute (𝑃𝐴 𝑃𝐵 ≠ 𝑃𝐵 𝑃𝐴 ), the resulting operator is dependent
on the order of multiplication9. If this applies, the probability depends on the order of measurement, 𝑝(𝐶𝐷′ ) ≠ 𝑝(𝐷′ 𝐶). If the person used the same bases in both perspectives, the
operators would commute (𝑃𝐴 𝑃𝐵 = 𝑃𝐵 𝑃𝐴 ) and the result would be order independent.
Therefore, assuming the non-compatibility of the different perspectives, non-commutativity
of sequential decision-making is the first testable prediction of the quantum model.
Second prediction regards the order effect of reasoning. It is the non-commutativity of the
decision-making that produce so called interference effect. The probability of the player’s
cooperation is given by 𝑝(𝐶) = ‖𝑃𝐶′ 𝑃𝐶 |𝑆⟩‖2 + ‖𝑃𝐷′ 𝑃𝐶 |𝑆⟩‖2 = ‖𝑃𝐶 |𝑆⟩‖2 when asked as the
first question, but as 𝑝𝑇 (𝐶) = ‖𝑃𝐶 𝑃𝐶′ |𝑆⟩‖2 + ‖𝑃𝐶 𝑃𝐷′ |𝑆⟩‖2 if the strategy is chosen after the
player’s guess of opponent’s strategy. The interference effect is the difference between
these two probabilities: 𝐼𝑛𝑡𝐶 = 𝑝(𝐶) − 𝑝𝑇 (𝐶). For the player’s guess of opponent’s strategy
similarly:
𝑝(𝐶′) = ‖𝑃𝐶 𝑃𝐶′ |𝑆⟩‖2 + ‖𝑃𝐷 𝑃𝐶′ |𝑆⟩‖2 (2.20)
𝑝𝑇 (𝐶′) = ‖𝑃𝐶′ 𝑃𝐶 |𝑆⟩‖2 + ‖𝑃𝐶′ 𝑃𝐷 |𝑆⟩‖2 (2.21)
𝑝(𝐶′) = 𝑝𝑇 (𝐶′) + 𝐼𝑛𝑡𝐶′ (2.22)
The presence of the order effect (or interference effect in quantum terms) is the second
prediction of the quantum model flowing from the non-compatibility of perspectives.
Third prediction provides a non-parametric test of the quantum model which can be used to
falsify it in the experiment. Busemeyer and Bruza (Busemeyer and Bruza 2014, 111–14) derived an equation that applies to quantum decision models of any dimensionality. Their solution stems from the property that “[t]he probability of transition |SA ⟩ → |SB ⟩ is the same as
the probability of a transition in the opposite direction |SB ⟩ → |SA ⟩. This property is called
9
Multiplication of two matrices is a non-commutative operation. ∑𝑘 𝐴𝑖𝑘 𝐵𝑘𝑗 ≠ ∑𝑘 𝐵𝑖𝑘 𝐴𝑘𝑗
9
the law of reciprocity in quantum theory.” (Busemeyer and Bruza 2014, 113). From the law
of reciprocity, they derived an invariant that they called a q-test:
𝑞 = [𝑝(𝐶𝐷′ ) + 𝑝(𝐷𝐶 ′ )] − [𝑝(𝐶 ′ 𝐷) + 𝑝(𝐷′ 𝐶)] = 0 (2.23𝑎)
−𝑞 = [𝑝(𝐶𝐶 ′ ) + 𝑝(𝐷𝐷′ )] − [𝑝(𝐷′ 𝐷) + 𝑝(𝐶 ′ 𝐶)] = 0 (2.23𝑏)
“[A] q-statistic can be computed by inserting the observed relative frequencies, and the
above null hypothesis can be statistically tested by using a standard z-test for a difference
between two independent proportions” (ibid.).
Last prediction of our model is dimension-specific. It applies only to the 2D model and can be
used for the comparison of the fitness of our two models. It stems from the theoretical form
of the unitary matrix that is used for the transition between two different sets of basis
states. Recall that in the 2D model the unitary matrix appears in the form:
𝑈=(
⟨𝐶|𝐶′⟩
⟨𝐷|𝐶′⟩
⟨𝐶|𝐷′⟩
)
⟨𝐷|𝐷′⟩
The elements of the matrix are inner products of the vectors from both bases. Recall that the
inner product defines the length of the orthogonal projection and its square the probability
of such projection. If we square every element in the matrix we get a new matrix that is
called a transition matrix and define the probabilities of transition between basis vectors in
relevant cell of the matrix:
|⟨𝐶|𝐶′⟩|2
𝑇=(
|⟨𝐷|𝐶′⟩|2
|⟨𝐶|𝐷′⟩|2
) (2.24)
|⟨𝐷|𝐷′⟩|2
For example, the first cell in the first column defines the probability of transition between
basis vectors |𝐶⟩ and |𝐶′⟩. This also equals the conditioned probability 𝑝(𝐶′|𝐶) because:
𝑝(𝐶𝐶 ′ ) = 𝑝(𝐶) ∙ 𝑝(𝐶′|𝐶) = ‖𝑃𝐶 |𝑆⟩‖2 ∙ ‖𝑃𝐶′ |𝐶⟩‖2
𝑝(𝐶′|𝐶) = ‖𝑃𝐶′ |𝐶⟩‖2 = |⟨𝐶|𝐶′⟩|2 (2.25)
Due to the properties of the unitary matrix, transition matrix is a double stochastic matrix,
i.e. both the rows and the columns sum to one. In particular, using the same parametrization
as in Equation 2.5, we get a matrix that is given by a single parameter 𝜃.
2
𝑇 = (cos2 𝜃
sin 𝜃
sin2 𝜃 ) (2.26)
cos 2 𝜃
From that follows that in the 2D model the conditioned probabilities has to fulfill the double
stochasticity, namely that 𝑝(𝐶′|𝐶) = 𝑝(𝐷′|𝐷) and 𝑝(𝐶′|𝐷) = 𝑝(𝐷′|𝐶). The same derivation
10
applies also for the reverse order of questions and therefore 𝑝(𝐶|𝐶′) = 𝑝(𝐷|𝐷′) and
𝑝(𝐶|𝐷′) = 𝑝(𝐷|𝐶′).
In the 4D model the transition matrix is also double stochastic (it stems from the unitary matrix that determines 𝑇), but as there are 4 conditioned probabilities in each row and each
column of the matrix, the relationship is more complex. In the next section we only use transition matrix as a test of the appropriateness of the 2D model.
3
Methods
Experimental design
The experiment which tested the prediction of the quantum model consists of several
rounds of two-players game and followed the design of the original study by Shafir and
Tversky (1992) with some important modifications.
In the original study (Shafir and Tversky 1992) the participants played several two-players
games, some of them were Prisoner’s Dilemma games (payoff matrix of the original study in
Table 3.1). Every participant met PD game in three different conditions. In different games
he received: a) No information about the opponent, b) the information that his opponent
had cooperated (𝐶′), or c) the information that his opponent had defected (𝐷′). The frequencies of cooperation (strategy 𝐶) were 37% in the no-information treatment, 16% after the
cooperation of the opponent and 3% after his defection. Moreover, studying the choices of
the individual players, authors showed that in 25% of the triads individual players exhibited
an inconsistency in their preferences: they choose strategy 𝐷 after 𝐶′, and strategy 𝐷 after
𝐷′, but strategy 𝐶 in the unknown condition, which violates the Savage’s sure thing principle.
Prisoner’s Dilemma
𝐶′
𝐷′
75, 75
25, 85
85, 25
𝐷
Table 3.1
30, 30
𝐶
To test the quantum model of PD we modified the original experiment in three important
ways. We replaced the information about the opponent’s strategy by guessing of it, we added the opposite order of decision-making (choosing own strategy first and then guessing the
opponent’s strategy), and we added the modified game, which test the preference among
outcomes with no influence of the strategic thinking. What is the rationale for these changes?
11
We added the guess of the opponent’s strategy player that replaces the information the
player gets from the experimenter. This solution was introduced by Croson (1999) and, in
our experiment, it helps to solve two difficulties. One of them is the uncertainty about the
information the player receives. He can doubt that the information is correct or hesitate to
exploit it for his own profit as an ‘unfair’ advantage in an otherwise symmetric game. By
guessing the decision of his opponent, the player is forced to consider the perspective of the
opponent and act accordingly without introducing any external element that could disturb
the game. The second reason is that guessing the opponent’s strategy enables us to switch
the order of questions to measure the order effect (see below).
To test the non-commutativity of the model we added the reverse order of questions. Half
of the respondents guesses the opponent’s strategy after they have picked their own strategy (the self-perspective) whereas the second half guesses the opponent’s strategy first (the
other-perspective). To motivate players to think out their tips properly, they will receive a
bonus if they guess correctly.
Last modification intends to test the level of cooperation without the effect of strategic
thinking. As we know from the previous research (e.g. Camerer 2003), players are not entirely self-interested in selecting the strategies. Even in the Dictator game when the outcome of
the game depends entirely on the decision of a single player, the noticeable share of the
players (20-40%) decides to cooperate. To estimate the level of this inclination in our population we added a modified game that preserves the structure of the payoffs in the PD game
but leaves the decision entirely on the single player. This was done by transformation of the
2x2 payoff matrix of PD game10 (Table 3.2) into 4x1 payoff matrix of the Dictator game (Table
3.3 and Table 3.4). In the self-perspective, the outcome of the Dictator game stems from the
decision of a single player (the row player in Table 3.3) and is a measurement of his preferences with no effect of strategic thinking. Similarly, in the other-perspective (Table 3.4),
players guess the strategy of their opponent without considering their own move (they have
only one “choice” in the game which is to accept whatever their opponent picks).
10
We used the payoffs in the range 0-5 that directly match the reward in dollars (in the pilot study) or reward
in dollars with coefficient 0.5 (in the MTurk study). In order to make the game more real for the players (they
gamble for money directly), we deviated from the original study, but the structure of the game remains the
same and the particular numbers in the payoff matrix have no impact on the model.
12
Prisoner’s Dilemma game11
𝐶′
𝐷′
𝐶
3, 3
0, 5
𝐷
Table 3.2
5, 0
0, 0
Dictator game:
self-perspective
𝑎′
𝑎
3, 3
𝑏
0, 5
𝑐
5, 0
𝑑
Table 3.3
1, 1
Dictator game: other-perspective
𝑎
Table 3.4
𝑎′
𝑏′
𝑐′
𝑑′
3, 3
5, 0
0, 5
1, 1
Participants
The study was piloted with the undergraduate students at the Ohio State University (N=20),
and then run online through the Amazon Mechanical Turk (MTurk) with total number of
N=284 participants. The results presented below are the results of the online study on the
Amazon MTurk.
In the pilot study students played the game in the laboratory settings on the computer with
an unknown opponent. Participants were payed $5 for participating and $0-6 according to
their performance in the study. The mean reward payed in the pilot study was $9.20.
In the online study, participants were MTurk users from the eastern Midwest of USA (states
of Illinois, Indiana, Michigan, Ohio and Wisconsin). Participants were payed $2.5 for finishing
the study and $0-3 according to their performance in the game ($0-2.5 as a payoff from one
randomly chosen game and $0.5 as a possible bonus for guessing correctly the strategy of
their opponent in the same game). The mean reward payed through the Amazon MTurk was
$3.91. We use sex, age, native language, education, population of the city and the
11
Here I use 𝐶 (or 𝐶′) for cooperation and 𝐷 (𝐷′) for defection (as in the model above). In the experiment, the
two strategies were labelled 𝑎 and 𝑏 (respective 𝑎′ and 𝑏′) to make sure they are value neutral.
13
knowledge of the game theory concepts (Game Theory, Pareto Efficiency, Dominated Strategy, Nash Equilibrium) as the control variables.
Procedure
The experiment was implemented in the Qualtrics survey form (Qualtrics, Provo, UT). The
participants were instructed in the rules of the game and, after they correctly answered the
control questions that checked their understanding of the instructions, they played ten different games, four of them were the experimental games (Prisoners’ Dilemma game or the
Dictator game). Other games, with different payoffs structures, were interspersed among
the experimental games in order to force the participants to consider the experimental
games anew. Players were told that they play against a randomly chosen opponent currently
present in the game (strategy of their opponent was in fact randomly chosen as the cooperation or defection with probability 0.5 each). They learned the strategy of their opponent at
the end of the game, they had no feedback in between the games. Results presented below
are the results of the second round of the games when participants encounter the experimental game for the first time12, so they are the results of the one-shot Prisoners’ Dilemma
game or the one-shot Dictator game. There were six different treatments (games) the different groups of players encountered in the second round:
Dictator
game
Selfperspective
Otherperspective
Selfperspective
Players choose their strategy in the DG.
Players guess the strategy of their opponent in DG.
Players choose their strategy and then guess the strategy of their
opponent.
Prisoner’s
Guessing Players guess the strategy of their opponent and then
Dilemma
treatment choose their own strategy.
Othergame
Players get the information about the strategy of
perspective Bonus
their opponent (C’ or D’) and then choose their own
treatment
strategy.
Table 3.5
Variables of the game
In sum, the experiment enables to measure several different types of probabilities. Starting
with PD game and the self-perspective (choosing own strategy and then guessing the opponent’s strategy) we get the sequential probabilities 𝑝(𝐶𝐶′), 𝑝(𝐶𝐷′), 𝑝(𝐷𝐶′) and 𝑝(𝐷𝐷′).
Here the plain letters denote the player’s strategy (𝐶 for cooperation and 𝐷 for defection),
primed letters denote the guess of the opponent’s strategy (𝐶′ for opponent’s cooperation
and 𝐷′ for his defection), and the order matches the order of decision-making. From the se12
The data from the other rounds exhibits a strong consistency bias. Players often follow the strategy they
have chosen in their first experimental game.
14
quential probabilities we know also the overall cooperation of the player: 𝑝(𝐶) = 𝑝(𝐶𝐶′) +
𝑝(𝐶𝐷′ ), level of cooperation the player expects conditioned by his cooperation: 𝑝(𝐶′|𝐶) =
𝑝(𝐶𝐶 ′ )
𝑝(𝐷𝐶 ′ )
𝑝(𝐶)
𝑃(𝐷)
, or by his defection: 𝑝(𝐶′|𝐷) =
, and also the overall expectation of the oppo-
nent’s cooperation: 𝑝𝑇 (𝐶 ′ ) = 𝑝(𝐶𝐶 ′ ) + 𝑝(𝐷𝐶 ′ ). Here the subscript 𝑇 denotes the total
probability that stems from the classical law of total probability. The probabilities of defection are the corresponding complements into 1 (e.g. 𝑝(𝐷) = 1 − 𝑝(𝐶) = 𝑝(𝐷𝐶′) + 𝑝(𝐷𝐷′)).
By the same token the reverse order of questions enables to measure sequential probabilities 𝑝(𝐶′𝐶), 𝑝(𝐶′𝐷), 𝑝(𝐷′𝐶) and 𝑝(𝐷′𝐷) and the derived probabilities 𝑝(𝐶 ′ ), 𝑝(𝐶|𝐶′),
𝑝(𝐶|𝐷′), and 𝑝𝑇 (𝐶).
From the Dictator game we get the probabilities of individual strategies 𝑝(𝑎), 𝑝(𝑏), 𝑝(𝑐),
𝑝(𝑑) in the self-perspective and 𝑝(𝑎′), 𝑝(𝑏′), 𝑝(𝑐′), 𝑝(𝑑′) in the other-perspective.
4
Results of the experiment
Replication of previous studies
One of the treatments in the experiment was a replication of the original treatment by
Tversky and Shafir (1992). When presented the PD game, the participants were said to be in
a ‘bonus’ group which receive the information about the choice of their opponent. Participants were instructed to use the information freely “to help you choose your own strategy”.
They were assured that their strategy will not be revealed to anyone playing with them. This
treatment was included to the experiment to verify that it follows the main patterns of the
original study, namely that the conditioned probabilities in both treatments exhibit the Disjunction effect. Table 4.1 compares the results from the bonus group treatment and the
guessing treatment with the comparable results by Shafir and Tversky (1992) and Croson
(1999).
bonus
treatment
Shafir &
Tversky
(bonus
group)
guessing
treatment
Croson
(guessing)
𝑃(𝐶)
0.649
0.37
0.649
0.78
𝑝(𝐶|𝐶′)
0.212
0.16
0.630
0.83
𝑝(𝐶|𝐷′)
Table 4.1
0.086
0.03
0.286
0.68
It is clear that both treatment exhibits a Disjunction effect and the conditioned probabilities
replicate the results of previous studies (violation of the Sure thing principle). Our group (to15
gether with the different framing of the experiment) is more cooperative than in the Shafir
and Tversky (1992) study but the Disjunction effect is very strong: 8.6% (N=35) of participants chose cooperation after opponent’s defection, 21.2% (N=33) after opponent’s cooperation, and 64.9% (N=77) cooperated in the no-information treatment. The conditioned cooperation is about three times higher in the guessing treatment (28.6%, N=27 respective
63.0%, N=42) but still lower than the level of cooperation without guessing (64.9%).
Compatibility of perspectives
First hypothesis of the quantum model regards the compatibility of two different perspectives the player can use to approach the game: The self-perspective and the otherperspective (which is the perspective of unknown opponent). In order to be responsible for
the Disjunction effect these two perspectives has to produce different probabilities. The
measured frequencies of played and guessed strategies for the Dictator game and the PD
game are in Table 4.2 and Table 4.3.
In the Dictator game (Table 4.2) only two strategies seem to be reasonable for the vast majority of players (94.3% of all players choose one of them). In the self-perspective 48.6% of
players chooses an even distribution of the points (strategy 𝑎), whereas 51.4% keep the
maximum of available points for themselves (strategy 𝑐). In the other-perspective, 20.0% of
the players guess opponent’s decision to evenly distribute the points (strategy 𝑎′) and 68.6%
guess that their opponent chooses to keep all the points for himself (strategy 𝑐′). The chisquared test of independence of proportions using only these two strategies returns
𝜒 2 =4.799 (p=0.028). Based on this result we can say that the two perspectives differ in the
Dictator game.
Dictator game
self
other
′
𝑝(𝑎)
0.486
𝑝(𝑎 )
0.200
𝑝(𝑏)
0.000
𝑝(𝑏 ′ )
0.114
𝑝(𝑐)
0.514
𝑝(𝑐 ′ )
0.686
𝑝(𝑑)
0.000
𝑝(𝑑 ′ )
0.000
𝑁
35
𝑁
35
4.799*13
𝜒2
Table 4.2
If we include also the strategy 𝑏’, which was guessed by 11.4% of players in the otherperspective, we have two choices. We can add these responses to the cooperative strategies
𝑎′ because in the analogy of PD game this strategy means opponent’s cooperation even in
13
This number compares only the relative frequency of two strategies (𝑎 and 𝑐) in the two perspectives.
16
the presence of my (hypothetical) defection. In other words, guessing this strategy, players
expect that their opponent will be altruistic and give all the points to them and keep nothing
for themselves. In this case, 31.4% of players expect cooperative choice from their opponent
which is still lower than the willingness to cooperate in the self-perspective (48.6%), but the
difference is no longer significant (𝜒 2 =2.143, p=0.143).
The second option is to count the 𝑏’ strategies among the ‘selfish’ 𝑐’ strategies. The rationale
for this is that we can expect that when the players encounter the second-perspective for
the first time some of them can choose it by mistake because they have not switched their
view and consider strategy 𝑏’ as opponent’s selfish strategy. If we accept this explanation
than the level of expected cooperation in the other-perspective drops to 20.0% and the difference between the two perspectives is consequently bigger (𝜒 2 =6.341, p=0.012).
From the data it is not obvious, which of the two presented explanations is correct. It is also
possible that the rationale for the choice 𝑏’ differs among the cases (4 players guessed this
strategy). Therefore, we decided to not include these cases in our analysis and the test presented in the Table 4.2 is the test of the relative frequency of only 2 strategies 𝑎 and 𝑐 (respective 𝑎′ and 𝑐′).
Table 4.3 includes the sequential probabilities in the PD game. E.g. in the first row (𝐶𝐶′), the
self-perspective means that the player chooses a strategy 𝐶 and then guess opponent’s
strategy 𝐶′. In other-perspective (𝐶′𝐶), the player guesses the opponent’s strategy 𝐶′ and
then he chooses his own strategy 𝐶.
PD game: sequential probabilities
self
′
other
𝑝(𝐶𝐶 )
0.403
𝑝(𝐶′𝐶)
0.246
𝑝(𝐶𝐷′)
0.247
𝑝(𝐶′𝐷)
0.145
𝑝(𝐷𝐶′)
0.117
𝑝(𝐷′𝐶)
0.174
𝑝(𝐷𝐷′)
0.234
𝑝(𝐷′𝐷)
0.435
𝑁
77
𝑁
69
9.896*
𝜒2
Table 4.3
The data reveals a strong tendency toward the wishful thinking described by Tversky and
Shafir (1992). In both orderings players match their strategies with their guesses (and vice
versa) in about two thirds of all cases (exact numbers are in Table 5.1). E.g. after playing the
strategy 𝐶, 62.0% of players guess opponent’s strategy 𝐶′, similarly, after choosing 𝐷 66.7%
17
of players expect opponent’s strategy 𝐷′. Nevertheless, this is not enough to bring a consistence among the sequential probabilities. Players choose cooperation more often then they
expect it from their opponents and the frequencies of sequential decision differ substantially
(𝜒 2 =9.896, p=0.019). We can conclude that the two perspectives differ also in the PD game.
Order effect of sequential decision-making
From the previous we know that the sequential probabilities of choosing and guessing the
strategy in the PD game differ. Does it apply also for the frequencies of choices presented as
the first versus the second question? The overall frequencies of strategies 𝐶 (choosing cooperation) and 𝐶′ (guessing cooperation) measured in the experiment are summarized in Table
4.4.
PD game: total probabilities
𝑝(𝐶)
0.649
𝑝(𝐶′)
0.391
𝑝𝑇 (𝐶)
0.420
𝑝𝑇 (𝐶′)
0.520
𝐼𝑛𝑡 𝐶
-0.229
𝐼𝑛𝑡 𝐶′
0.128
𝜒2
2.408
7.689**
𝜒2
Table 4.4
The frequencies are derived from the data above as 𝑝(𝐶) = 𝑝(𝐶𝐶 ′ ) + 𝑝(𝐶𝐷′ ), 𝑝𝑇 (𝐶) =
𝑝(𝐶 ′ 𝐶) + 𝑝(𝐷′ 𝐶) etc. Player’s willingness to cooperate is significantly higher when presented as the first question when compared to be presented as the second question (𝜒 2 =7.689,
p=0.006). In particular, 64.9% of players choose to cooperate when deciding their strategy
first, and this rate drops to 42.0% when they consider strategy of their opponent first. The
guess of opponent’s cooperation shows a similar, but reverse effect. Guessing opponent’s
strategy first, 39.1% of players expect cooperation, and this number increases to 52.0%
when players have chosen their own strategy before their guess. Nevertheless, the difference between 𝑝(𝐶′) and 𝑝𝑇 (𝐶′) is not significant (𝜒 2 =2.408, p=0.121).
If we analyze this result using the framework of the quantum model, we see that the sequential decision making in PD game exhibits a considerable interference effect. The effect is
negative for the player’s own strategy – the level of cooperation decreases by 22.9% if the
choice of strategy is presented as the second question, compared to the first-question
choice. When guessing opponent’s strategy first, the effect is positive. The level of expected
cooperation increases by 12.8% if preceded by the choice of player’s own strategy.
The interference term is a source of the Disjunction effect described by Tversky and Shafir
(1992) and also an important obstacle for the classical solution concept. Recall the classical
solution by the dominance equilibrium: The defection is picked as the dominant strategy
because it is a preferred strategy after both opponent’s cooperation and his defection. But if
18
the two questions exhibit strong order effect, how do the decisions after some question determine the decision before that question? Classical solution concept avoids this problem by
the intrinsic assumption of the commutativity of the sequential decision-making which leads
to the ‘anomaly’ of Disjunction effect. In the quantum model the order effect is expected
given the non-compatibility of the perspectives. Does it mean that the frequencies of sequential decision-makings are entirely contingent? We can test the consistency of the decision-making in the quantum model with a q-test.
Q-test: quantum law of reciprocity
The last test available is the q-test which explores if the data meets the criteria of the quantum model given by Equation 3.24a (respective 3.24b).
We can check the q-test in two equivalent ways. In Table 4.5 there are the sums of frequencies of sequential probabilities in which players’ chosen strategy and their guess of the opponent’s strategy match together. The sum of all matching choices equals 63.6% in the
choosing-guessing treatment, whereas the reverse order gives 68.1% of such choices. The
difference (and the value of the q-test) is 4.5%. Using the z-test for a difference between two
independent proportions we cannot say that the frequencies differ significantly (z=-0.5695,
p=0.569).
q-test
′)
𝑝(𝐶𝐶 + 𝑝(𝐷𝐷′ )
′
′
0.636
𝑝(𝐶 𝐶) + 𝑝(𝐷 𝐷)
0.681
𝑞
-0.045
𝑧
Table 4.5
-0.5695
The results of the three tests above reveal that even if the frequencies of the individual or
the sequential decisions differ and the same apply also to the frequencies of strategies chosen as the first or the second question, the sum of all matching choices remains invariant
among the different orderings. This non-parametric test indicates that there is the consistency among the players’ strategies that was predicted by the quantum model.
Comparing the PD game and the Dictator game
The Dictator game had two different roles in the experiment. We used it to test the compatibility of two different perspectives (presented above). Besides this, we propose it as a test
of the players’ preferences among the possible outcomes of the game with no effect of the
strategic thinking. How does the level of cooperation differ from the PD game?
19
We assume that the framework of the Dictator game sets a different initial state vector,
therefor results of the PD game does not match the Dictator game directly. Nevertheless, we
assume that the vectors in both games are close enough to produce similar effects. In particular, we assume that level of cooperation in the PD game (self-perspective) corresponds
to the level of cooperative choice (the strategy 𝑎) in the Dictator game and the same apply
also to the players’ guesses of the opponent’s cooperation. The results for both types of
games are summarized in Table 4.6.
PD game
𝑝(𝐶)
Dictator game14
0.649
𝑝(𝑎)
0.486
0.391
𝑝(𝐶 ′ )
Table 4.6
𝑝(𝑎′ )
0.226
In both games, the level of cooperation is significantly higher than expected cooperation
from the opponents (see the exact tests in Table 4.2 and Table 4.3). In the Dictator game the
willingness to cooperate and the expected cooperation is lower than in the PD game – both
by about 16.5%. That means that also the two perspectives in both games differ by a similar
percentage of choices (25.8% versus 26.0%). The results of the Dictator game differ in important aspects but corroborate one of the key findings of the PD game: the players expect
much lower cooperation of their opponents then they exhibit in the same game.
5
Discussion
Comparison of the models
In the previous section we presented results that tested several independent features of the
quantum model of the PD game. What do these results imply for the model in general and
how does it relate to the standard model of PD game?
Non-compatibility of perspectives is problematic for the standard model because it undermines one of the basic assumption of its solution concepts. Recall that e.g. iterated dominance equilibrium or Nash equilibrium are based on the assumption that players are aware
not only their own preferences among the outcomes of the game but also the preferences of
their opponents. As is shown in Table 4.4, this is not the case. Majority of players (64.9%)
choose cooperation but only 39.1% of them expect it from the others. That means, statistically speaking, that only 46.8%15 of players will guess the strategy of their opponent rightly
The value 𝑝(𝑎′ ) is computed as the ratio of 𝑎’ strategies among all 𝑎’ and 𝑐’ strategies, 𝑏’ strategies are omitted (see rationale in the second section of results).
15
𝑝(𝑟𝑖𝑔ℎ𝑡 𝑔𝑢𝑒𝑠𝑠) = 𝑝(𝐶) ∙ 𝑝(𝐶 ′ ) + 𝑝(𝐷) ∙ 𝑝(𝐷′ )
14
20
and in only 21.9%16 of hypothetical pairs both players will be right about their opponent.
This is not a good basis for an equilibrium state17. In the quantum model the noncompatibility of perspectives is possible and, given by the fact that the order effect is present, even expected. In this aspect, the experimental data support qualitative prediction of
the quantum model.
The presence of the order effect is even more critical for the standard model. Standard model, based on the classical probability theory which is commutative, cannot explain different
probabilities caused simply by reverse order of question. For quantum model it is always an
option flowing from the fact that matrix multiplication (used as a projector operator, see
Appendix Chyba! Nenalezen zdroj odkazů.) is a non-commutative operation.
The significance of the q-test is somehow different from the previous features of the model.
The q-test should be met both in the classical model and the quantum model. But while the
standard model predicts that all the sequential probabilities (Table 4.3) should be same in
both perspectives (they are not), quantum model predicts that only certain combination of
the probabilities should remain invariant. Therefore, the q-test controls if there is any consistency among the probabilities, it is a necessary condition for both models, but cannot be
used to adjudicate between the standard and quantum model.
In sum, the data reveals, that the strategies of the players fulfill the q-test, which is the minimum requirement of both models. The models differ in the prediction of the presence of
the order effect and non-compatibility of perspectives, which are both present in the data.
This supports the quantum model in place of the standard model of PD game.
Double stochasticity
So far the results meet the qualitative and quantitative predictions of the quantum model.
But which model matches the results better? Is the simpler 2D model satisfying, or is it inappropriate and more complex 4D model is needed? To answer this question, we should explore the transition matrix of the experimental game.
Recall that the transition matrix contains the probabilities of choosing a particular strategy
conditioned by previous guess of opponent’s strategy (and vice versa):
|⟨𝐶|𝐶′⟩|2
𝑇=(
|⟨𝐷|𝐶′⟩|2
|⟨𝐶|𝐷′⟩|2
𝑝(𝐶|𝐶′) 𝑝(𝐶|𝐷′)
)=(
)
2
𝑝(𝐷|𝐶′) 𝑝(𝐷|𝐷′)
|⟨𝐷|𝐷′⟩|
Above we showed that the transition matrix has to be a double stochastic matrix, i.e. each of
its rows and columns has to sum to one. We can test this condition by constructing the tran𝑝(𝑏𝑜𝑡ℎ 𝑟𝑖𝑔ℎ𝑡) = [𝑝(𝑟𝑖𝑔ℎ𝑡 𝑔𝑢𝑒𝑠𝑠)]2
Choosing and guessing the strategy with a toss of coin would bring higher probability of success:
𝑝𝑐𝑜𝑖𝑛 (𝑏𝑜𝑡ℎ 𝑟𝑖𝑔ℎ𝑡) = 0.25.
16
17
21
sition matrix of the 2D model in both orderings. Individual data can be computed from Table
4.3 as the conditioned probability of individual choices, e.g.:
𝑝(𝐶|𝐶′) =
𝑝(𝐶 ′ 𝐶)
𝑝(𝐶 ′ 𝐶) + 𝑝(𝐶 ′ 𝐷)
The probabilities are summarized in the Table 5.1 and the resulting transition matrix can be
seen below (Equation 5.1a and 5.1b).
PD game - conditioned choice
𝑝(𝐶|𝐶′)
0.630
𝑝(𝐶′|𝐶)
0.620
𝑝(𝐷|𝐶′)
0.370
𝑝(𝐷′|𝐶)
0.380
𝑝(𝐶|𝐷′)
0.286
𝑝(𝐶′|𝐷)
0.333
𝑝(𝐷|𝐷′)
Table 5.1
0.714
𝑝(𝐷′|𝐷)
0.667
𝑝(𝐶′|𝐶) 𝑝(𝐶′|𝐷)
0.620
(
)=(
0.380
𝑝(𝐷′|𝐶) 𝑝(𝐷′|𝐷)
0.333
) (5.1𝑎)
0.667
𝑝(𝐶|𝐶′)
(
𝑝(𝐷|𝐶′)
0.286
) (5.1𝑏)
0.714
𝑝(𝐶|𝐷′)
0.630
)=(
0.370
𝑝(𝐷|𝐷′)
From the transition matrices we see that the condition of double stochasticity is not met
precisely. If the matrices had been double stochastic then the rows would sum to one, and
equivalently, the elements in the corners of the matrices would equals: 𝑝(𝐶′|𝐶) −
𝑝(𝐷′ |𝐷) = 0 and 𝑝(𝐶|𝐶′) − 𝑝(𝐷|𝐷′ ) = 0. But these terms return a value of -0.047, respective -0.085, i.e. the player’s own defection ‘attracts’ his guess of opponent’s defection more
than his cooperation ‘attracts’ his guess off opponent’s cooperation. In the opposite order,
the player’s guess of opponent’s defection ‘attracts’ his own defection more than his guess
of cooperation ‘attracts’ his cooperation. We can test these differences by a 𝜒 2 -test:
𝜒 2 =0.165 (p=0.685) respective 𝜒 2 =0.542 (p=0.461). The differences are not big enough to
reject the 2D model directly, but the data also does not support the model entirely. Especially the second matrix (Equation 5.1b) exhibits non-marginal discrepancy from the double stochasticity.
To decide whether the 2D model is appropriate for the PD game, we can explore the possibility that both matrices are double stochastic but they differ from each other. Assume that
matrix 𝐴 is given by a parameter 𝑎, and the matrix 𝐵 by parameter 𝑏.
(
𝑝(𝐶′|𝐶)
𝑝(𝐷′|𝐶)
𝑝(𝐶′|𝐷)
𝑎
)=(
1−𝑎
𝑝(𝐷′|𝐷)
22
1−𝑎
) (5.2𝑎)
𝑎
(
𝑝(𝐶|𝐶′)
𝑝(𝐷|𝐶′)
𝑝(𝐶|𝐷′)
𝑏
)=(
𝑝(𝐷|𝐷′)
1−𝑏
1−𝑏
) (5.2𝑏)
𝑏
Both matrices are double stochastic but the individual frequencies differ. Is it compatible
with the quantum model? We can test this possibility by the q-test for it must be satisfied by
a quantum model of any dimension. If we rewrite the Equation 3.24b as below
𝑝(𝐶𝐶 ′ ) + 𝑝(𝐷𝐷′ ) = 𝑝(𝐶 ′ 𝐶) + 𝑝(𝐷 ′ 𝐷)
𝑝(𝐶) ∙ 𝑝(𝐶 ′ |𝐶) + 𝑝(𝐷) ∙ 𝑝(𝐷′ |𝐷) = 𝑝(𝐶′) ∙ 𝑝(𝐶|𝐶′) + 𝑝(𝐷′) ∙ 𝑝(𝐷|𝐷′)
𝑝(𝐶) ∙ 𝑝(𝐶 ′ |𝐶) + [1 − 𝑝(𝐶)] ∙ 𝑝(𝐷′ |𝐷) = 𝑝(𝐶 ′ ) ∙ 𝑝(𝐶|𝐶 ′ ) + [1 − 𝑝(𝐶 ′ )] ∙ 𝑝(𝐷|𝐷′)
And if we replace the conditional probabilities with the parameters 𝑎 and 𝑏, we get:
𝑝(𝐶) ∙ 𝑎 + [1 − 𝑝(𝐶)] ∙ 𝑎 = 𝑝(𝐶 ′ ) ∙ 𝑏 + [1 − 𝑝(𝐶 ′ )] ∙ 𝑏
𝑎 + [𝑝(𝐶) − 𝑝(𝐶)] ∙ 𝑎 = 𝑏 + [𝑝(𝐶 ′ ) − 𝑝(𝐶 ′ )] ∙ 𝑏
𝑎=𝑏
For the q-test to be satisfied, the coefficient 𝑎 and 𝑏 must be equal, i.e. in the 2D model not
only 𝑝(𝐶′|𝐶) = 𝑝(𝐷′|𝐷) but also 𝑝(𝐶′|𝐶) = 𝑝(𝐶|𝐶′) = 𝑝(𝐷|𝐷′) = 𝑝(𝐷′|𝐷)18. With this
result, we can improve our test of 2D model by including all 4 frequencies 𝑝(𝐶′|𝐶), 𝑝(𝐷′|𝐷),
𝑝(𝐶|𝐶′), and 𝑝(𝐷|𝐷′).The 𝜒 2 -test of the proportions shows this difference to not be significant (𝜒 2 =1.017, p=0.797).
We can also test the condition of double stochasticity with other known results. In the study
by Croson (1999) players choose their strategy with no instruction or after they have
guessed a strategy of their opponent. Level of cooperation in the ‘no-guessing’ treatment
was 77.5%, whereas 𝑝(𝐶|𝐶′)=83% and 𝑝(𝐶|𝐷′)=68%. Here the discrepancy from the double
stochasticity is even bigger but due to the small number of players (N=40) the 𝜒 2 -test of difference is not significant (𝜒 2 =1.212, p=0.271).
Based on the results presented above, we suppose that the 2D model is not sufficient for the
PD game. Nevertheless, as it can produce an interference effect which in size is close to the
experimental results, we can accept it as a first approximation on the way from the classical
model to the fully equipped 4D model.
The four-dimensional model has five degrees of freedom. From the eight sequential probabilities (Table 4.3) only six are independent because each column (each ordering of questions) have sum to 1. The remaining equation which specifies a model and makes it testable
18
Moreover, if the coefficients equal, then the q-test (Equation 3.24a and 3.24b) is satisfied for any values of
𝑝(𝐶) and 𝑝(𝐶′).
23
is the q-test. We have already test it by z-test for difference and showed that 4D model
match the empirical results.
Control variables: men and women play the game differently
We used six control variables in the game: sex, age, native language, education, population
of the city and the knowledge of the game theory concepts (Game Theory, Pareto Efficiency,
Dominated Strategy, Nash Equilibrium). The multilinear regression of PD game reveals that
only first category has a predictive power: women choose cooperation more often (k=-0.22*,
se=0.12) and expect cooperation from their opponent less often (k=0.26**, se=0.12) than
men. What does it mean for the quantum model?
Table 5.2 summarizes the main results with respect to the sex of the participants and reveals
that both groups play the game differently. Women choose cooperation more often
(𝜒 2 =5.013, p=0.026), and expect cooperation less often (𝜒 2 =4.168, p=0.042) than men.
Whereas men are consistent among perspectives (𝜒 2 =0.006, p=0.940), and the order effect
is mild (16.9% in their strategy versus 8.8% in their expectation), women differ largely
(𝜒 2 =18.202, p=0.002) and exhibit a strong order effect (in a size of about 30% in both choosing and guessing). Women meet the q-test very well (q=-0.033, p=0.761), men are far from
the equilibrium (q=0.145, p=0.218). Moreover, women meet the requirement of double stochasticity (𝜒 2 =0.172, p=0.98219).
PD game: women
self
PD game: men
other
self
other
𝑝(𝐶𝐶 ′ )
0.513
𝑝(𝐶′𝐶)
0.195
𝑝(𝐶𝐶 ′ )
0.289
𝑝(𝐶′𝐶)
0.321
𝑝(𝐶𝐷′)
0.256
𝑝(𝐶′𝐷)
0.098
𝑝(𝐶𝐷′)
0.237
𝑝(𝐶′𝐷)
0.214
𝑝(𝐷𝐶′)
0.077
𝑝(𝐷′𝐶)
0.268
𝑝(𝐷𝐶′)
0.158
𝑝(𝐷′𝐶)
0.036
𝑝(𝐷𝐷′)
0.154
𝑝(𝐷′𝐷)
0.439
𝑝(𝐷𝐷′)
0.316
𝑝(𝐷′𝐷)
0.429
𝑝(𝐶)
0.769
𝑝(𝐶′)
0.293
𝑝(𝐶)
0.526
𝑝(𝐶′)
0.536
𝑝𝑇 (𝐶)
0.463
𝑝𝑇 (𝐶′)
0.590
𝑝𝑇 (𝐶)
0.357
𝑝𝑇 (𝐶′)
0.447
𝐼𝑛𝑡 𝐶
-0.306
𝐼𝑛𝑡 𝐶′
0297
𝐼𝑛𝑡 𝐶
-0.169
𝐼𝑛𝑡 𝐶′
-0.088
𝑁
Table 5.2
39
𝑁
41
𝑁
38
𝑁
28
In sum, the data reveals that women follow the predictions of the quantum (2D) model very
closely and they are their choices which are for the most part responsible that also aggregate data follow the requirement of the model very well. The data from the male sample are
ambiguous, they are closer to the 4D model but the q-test reveals some deviation from the
Or 𝜒 2 =0.112, p=0.989 with the Yates correction for the small number of cases in some cells of the matrix that
nevertheless, in this case strengthen the finding.
19
24
general quantum model. Nevertheless, especially due to the smaller sample in the otherperspective, we cannot reject the possibility that the q-test is met even in the sample of
men.
6 Conclusions
In this paper we introduced the results of the PD game and the statistical tests of the quantum model of strategic decision-making in this game. The results replicate the findings of
Tversky and Shafir (1992) and exhibit the Disjunction effect in both tested treatments
(guessing condition and the bonus group). The predictions of the quantum model were
largely confirmed. The assumption of the non-compatibility of perspectives was met in both
the Dictator game and PD game. The levels of players’ cooperation and their expectation of
the opponent’s cooperation exhibit a strong order effect which is qualitatively close to the
findings from the Dictator game. And most importantly, the sequential probabilities met the
general criterion of the q-test, which has to be satisfied in the quantum model of any dimensionality.
From the two models that has been presented above, we explored the basic condition of 2D
model – the double stochasticity of the 2x2 transition matrix. Empirical evidence for the
simpler model is ambiguous. Deeper analysis showed that women meet the requirement of
the double stochasticity closely whereas men deviate from the 2D model. More data are
needed to explore the exact form of the difference between men and women in the PD
game.
With regard to the presented results, what can be said about the dilemma the players face in
the PD game? There are two main conclusions that can be made. The first relates to the classical solution concept based on the dominant strategy. As has been said, this solution stems
from the commutative logic of classical probability which claims that if the player prefers
defection after opponent’s cooperation and also defection after opponent’s defection, then
he has to prefer defection also in the standing-alone decision. Quantum logic rejects this
kind of reasoning. If the player prefers defection after opponent’s decision, this tendency
can manifest itself in the opposite ordering even by an increased cooperation! The choice of
a strategy conditioned on strategy of the opponent (known or guessed) gives no hints for the
choice in the reverse order of questions. In other words, we cannot infer the level of cooperation in the self-perspective from its level in the other-perspective and the classical solution
concept loses its grounds.
The second main conclusion regards the wishful thinking introduced by Tversky and Shafir
(1992). In our study the relative stable share of participants matched their strategies and
their expectations in the PD game. By this, we replicated the patterns of the previous studies. But while this was considered to be an evidence of the bounded rationality of players in
25
the classical game theory, the quantum model offers this as a possible (in the 4D model) or
even predicted (in 2D) feature of any two-player experiment which is required by a theoretically given form of the unitary matrix.
Both conclusions challenge the classical concept of rationality based on the commutative
logic of classical probability theory. Quantum model provides an alternative approach that
can challenge our understanding of many aspects of social world. It does not offer a unique
prediction that could replace classical model as a non-parametric solution whose predictions
are context-independent, nevertheless it fits the known experimental data and is falsifiable
in the form of the q-test. We should test this model in other social situation that correspond
to the settings of the two-players game and improve our understanding of strategic decisionmaking accordingly.
7 Acknowledgment
I would like to thank professors Alexander Wendt and Joyce Wang for their comments and
Thomas Nelson for his universal support during the preparation and conduct of the experiment. The study was supported by a research grant from the Mershon Center for International Security Studies, the Ohio State University.
The data for this paper was collected with Qualtrics software, Version 05/2015. Copyright ©
2015 Qualtrics. Qualtrics and all other Qualtrics product or service names are registered
trademarks or trademarks of Qualtrics, Provo, UT, USA. http://www.qualtrics.com
8
References
Accardi, Luigi, Andrei Khrennikov, and Masanori Ohya. 2009. “Quantum Markov Model for
Data from Shafir-Tversky Experiments in Cognitive Psychology.” Open Systems & Information Dynamics 16 (4): 371–85.
Aerts, Diederik. 2009. “Quantum Structure in Cognition.” Journal of Mathematical Psychology, Special Issue: Quantum Cognition, 53 (5): 314–48. doi:10.1016/j.jmp.2009.04.005.
Atmanspacher, H., and H. Römer. 2012. “Order Effects in Sequential Measurements of NonCommuting Psychological Observables.” Journal of Mathematical Psychology 56 (4):
274–80. doi:10.1016/j.jmp.2012.06.003.
Busemeyer, Jerome R., and Peter D. Bruza. 2014. Quantum Models of Cognition and Decision. Cambridge: Cambridge University Press.
Busemeyer, Jerome R., M. Matthews, and Zheng Wang. 2006. “A Quantum Information Processing Explanation of Disjunction Effects.” In The 29th Annual Conference of the
26
Cognitive Science Society and the 5th International Conference of Cognitive Science,
131–35. Mahwah, NJ. Erlbaum.
Busemeyer, Jerome R., Emmanuel M. Pothos, Riccardo Franco, and Jennifer S. Trueblood.
2011. “A Quantum Theoretical Explanation for Probability Judgment Errors.” Psychological Review 118 (2): 193–218. doi:10.1037/a0022542.
Camerer, Colin. 2003. Behavioral Game Theory: Experiments in Strategic Interaction. New
York, N.Y.; Princeton, N.J.: Russell Sage Foundation; Princeton University Press.
Croson, Rachel T. A. 1999. “The Disjunction Effect and Reason-Based Choice in Games.” Organizational Behavior and Human Decision Processes 80 (2): 118–33.
doi:10.1006/obhd.1999.2846.
Hedemann, S. R. 2013. “Hyperspherical Parameterization of Unitary Matrices.”
arXiv:1303.5904 [Quant-Ph], March. http://arxiv.org/abs/1303.5904.
Hristova, Evgenia, and Maurice Grinberg. 2008. “Disjunction Effect in Prisoner’s Dilemma:
Evidences from an Eye-Tracking Study.” In Proceedings of the 30th Annual Conference
of the Cognitive Science Society, 1225–30.
Khrennikov, Andrei Yu., and Emmanuel Haven. 2009. “Quantum Mechanics and Violations of
the Sure-Thing Principle: The Use of Probability Interference and Other Concepts.”
Journal of Mathematical Psychology, Special Issue: Quantum Cognition, 53 (5): 378–
88. doi:10.1016/j.jmp.2009.01.007.
Li, S., and J. Taplin. 2002. “Examining Whether There Is a Disjunction Effect in Prisoner’s Dilemma Games.” https://digital.library.adelaide.edu.au/dspace/handle/2440/3240.
Pothos, Emmanuel M., and Jerome R. Busemeyer. 2009. “A Quantum Probability Explanation
for Violations of ‘rational’ Decision Theory.” Proceedings of the Royal Society B: Biological Sciences, March, rspb.2009.0121. doi:10.1098/rspb.2009.0121.
Shafir, Eldar, and Amos Tversky. 1992. “Thinking through Uncertainty: Nonconsequential
Reasoning and Choice.” Cognitive Psychology 24 (4): 449–74. doi:10.1016/00100285(92)90015-T.
Shu Li, Zuo-Jun Wang, Li-Lin Rao, and Yan-Mei Li. 2010. “Is There a Violation of Savage’s
Sure-Thing Principle in the Prisoner’s Dilemma Game?” Adaptive Behaviour 18 (3–4):
3–4.
Tversky, Amos, and Eldar Shafir. 1992. “The Disjunction Effect in Choice under Uncertainty.”
Psychological Science 3 (5): 305–9.
Yukalov, V., I., and D. Sornette I. 2011. “Decision Theory with Prospect Interference and Entanglement.” Theory and Decision 70 (3): 283–328. doi:10.1007/s11238-010-9202-y.
27
9
Appendices
A. Notation
𝐶, 𝐷 … possible strategies in the PD game (cooperation, defection)
a, b, c, d … possible strategies in the Dictator game
𝑝(𝐶) … probability of playing cooperation
𝑝(𝐶′) … probability of guessing cooperation
𝑝(𝐶𝐷′ ) … sequential probability of playing cooperation and guessing defection of the opponent
𝑝(𝐶│𝐷′) … conditioned probability of playing cooperation after guessing defection of the
opponent
|𝑆⟩ … ket-vector in the Dirac symbolic (see Appendix 0)
⟨𝑆| … bra-vector in the Dirac symbolic (see Appendix 0)
𝜓𝑖 … coefficient in the linear combination in the quantum superposition
𝐼, 𝑈, 𝑇 … identity/unitary/transition matrix
𝑃𝐶 … projection matrix into the basis vector |𝐶⟩
B. Complex numbers
A complex number is a number in a form 𝑧 = 𝑎 + 𝑏𝑖, where 𝑎 is the real part and 𝑏 is the
imaginary part of the complex number and 𝑖 is the imaginary unit which satisfied the equation:
𝑖 2 = −1, or equivalently 𝑖 = √−1
There are two basic features of the complex numbers that are of the great importance for
the quantum model. Every complex number has its complex conjugate, denoted by 𝑧 ∗ , defined as 𝑧 ∗ = 𝑎 − 𝑏𝑖. Complex number which equals its own complex conjugate is a real
number: 𝑏 = 0.
The absolute value of the complex number (it’s “length” if we imagine the number as a vector in the two-dimensional plane where real part of the complex number gives its horizontal
coordinate and the imaginary part gives the vertical coordinate) is given by |𝑧| = √𝑧 ∙ 𝑧 ∗ =
√𝑎2 + 𝑏 2 .
28
C. Vectors in the Dirac symbolic, the inner product
The use of Dirac symbolic is common in the quantum theory: The ket-vector (denoted |𝑆⟩)
represents a column vectors whereas the bra-vector (denoted ⟨𝑆|) represents a row vector
with elements that are complex conjugate20 to the elements of the ket-vector. Together they
form a bra-ket complex ⟨𝑆|𝑆⟩ that denotes the inner product of these two vectors.
To see how this works, consider the following example. Let the vector |𝑆⟩ be given in the
basis |𝐶⟩ and |𝐷⟩ by the coefficients 𝜓𝐶 =
1
√6
∙ (1 − 𝑖) and 𝜓𝐷 =
1
√6
∙ 2𝑖. Then, in this basis,
we can write it as a column vector:
𝜓𝐶
)
𝜓𝐷
1
1−𝑖
|𝑆⟩ =
∙(
)
2𝑖
√6
|𝑆⟩ = (
The complementary bra-vector is a row vector with complex conjugate elements.
⟨𝑆| = (𝜓𝐶∗
⟨𝑆| =
1
√6
𝜓𝐷∗ )
∙ (1 + 𝑖
−2𝑖 )
And the inner product (bra-ket) of these two vectors is:
⟨𝑆|𝑆⟩ = (𝜓𝐶∗
⟨𝑆|𝑆⟩ =
1
√6
∙ (1 + 𝑖
⟨𝑆|𝑆⟩ =
𝜓
𝜓𝐷∗ ) ∙ ( 𝐶 )
𝜓𝐷
−2𝑖 ) ∙
1
√6
∙(
1−𝑖
)
2𝑖
1
(2 + 4) = 1
6
From the example above we can see three important features of the quantum model. First,
even if the state vector has complex coefficients the inner product of the vector with itself is
always a real number and represents the length of the vector. Second we can also observe
the role of the number 1⁄√6 which is the factor that normalizes the vector to be of a unitary
length. And third, even if the dimension of our model is 𝑁 = 2, the vector is given by 2 complex parameters, that is the sum of 4 real numbers that define the vector. Using the condi-
20
More about complex numbers and the operation of the complex conjugation in Appendix B.
29
tion of unitary length only 3 of them are independent variables21 but we can see that the
degree of freedom of the quantum state exceeds the dimensionality of the model.
D. Unitary matrix
The unitary matrix is a 𝑁-dimensional matrix that satisfies the condition 𝑈𝑈 † = 𝑈 † 𝑈 = 𝐼,
where 𝑈 † is a conjugate transpose of the matrix 𝑈 and 𝐼 the identity matrix. This condition is
equivalent to the condition that row vectors |𝑅𝑖 ⟩ and the column vectors |𝐶𝑖 ⟩ of the matrix
form on orthogonal basis in the vector space. That means that the inner product of every
row or column vector with itself is equal to 1 and the inner product of every row (respective
column) vector with all others row (column) vectors yield 0. Mathematically speaking:
⟨𝑅𝑖 |𝑅𝑗 ⟩ = 𝛿𝑖𝑗 ; ⟨𝐶𝑖 |𝐶𝑗 ⟩ = 𝛿𝑖𝑗 ; 𝛿𝑖𝑗 = {
1
0
𝑖𝑓 𝑖 = 𝑗
𝑖𝑓 𝑖 ≠ 𝑗
This condition guarantees that the unitary vector transformed by this matrix remains unitary
and therefore the probabilities defined by the state vector still sums to 1.
E. Projection operator
The projection operator is the 𝑁 × 𝑁 matrix that is constructed as an outer product of the
respective basis state 𝑃𝐵𝑖 = |𝐵𝑖 ⟩ ∙ ⟨𝐵𝑖 |. E.g. the projection operator 𝑃𝐶 that projects the state
|𝑆⟩ into basis vector |𝐶⟩ can be computed as:
𝑃𝐶 = |𝐶⟩ ∙ ⟨𝐶|
1
𝑃𝐶 = ( ) ∙ (1
0
𝑃𝐶 ∙ |𝑆⟩ = (
1
0
1 0
)
0) = (
0 0
𝜓
𝜓
0
) ∙ ( 𝐶 ) = ( 𝐶)
𝜓𝐷
0
0
𝑝(𝐶) = (𝜓𝐶
0)∗ ∙ (𝜓𝐶 )
0
𝑝(𝐶) = 𝜓𝐶∗ ∙ 𝜓𝐶 = |𝜓𝐶 |2
In the 4D model is the construction of the operator analogic, we start again by the outer
product of this basis vector we would like to project into. E.g. the projection into the basis
vector |𝐶𝐷′⟩ (that corresponds to the situation where the player chooses cooperation expecting defection of the opponent) is given by:
21
The 2-dimensional vector over the field of complex numbers has 3 degrees of freedom, but only 2 of them
are physically relevant. See Bloch sphere representation of the two-level quantum systems.
30
𝑃𝐶𝐷′ = |𝐶𝐷′⟩ ∙ ⟨𝐶𝐷′|
0
0) = (0
0
0
0
𝑃𝐶 = (1) ∙ (0 1 0
0
0
0
𝑃𝐶𝐷′ ∙ |𝑆⟩ = (0
0
0
0
1
0
0
0
1
0
0
0
0
0
0
0
0)
0
0
𝜓𝐶𝐶′
0
0
𝜓
0) ∙ ( 𝐶𝐷′ ) = (𝜓𝐶𝐷′ )
0
𝜓𝐷𝐶′
0
0
𝜓𝐷𝐷′
0
0
0
0
0
𝑝(𝐶𝐷′) = ‖𝑃𝐶𝐷′ ∙ |𝑆⟩‖2
0
𝜓
0 0)∗ ∙ ( 𝐶𝐷′ )
0
0
𝑝(𝐶𝐷′) = (0 𝜓𝐶𝐷′
∗
𝑝(𝐶𝐷′) = 𝜓𝐶𝐷′
∙ 𝜓𝐶𝐷′ = |𝜓𝐶𝐷′ |2
Projection operator of the combined event is than given as the sum of the operators of the
individual events. E.g. the operator that corresponds to the situation that player chooses
cooperation regardless of his expectation is given as a sum of 𝑃𝐶𝐷′ and 𝑃𝐶𝐶′ .
𝑃𝐶 = 𝑃𝐶𝐷′ + 𝑃𝐶𝐶′
1
𝑃𝐶 = (0
0
0
0
0
0
0
0
0
0
0
0
0
0) + (0
0
0
0
0
1
𝑃𝐶 ∙ |𝑆⟩ = (0
0
0
0
1
0
0
𝑝(𝐶) = (𝜓𝐶𝐶′
0
0
0
0
0
1
0
0
0
0
0
0
0
1
0) = (0
0
0
0
0
0
1
0
0
0
0
0
0
𝜓𝐶𝐶′
𝜓𝐶𝐶′
0
𝜓
0) ∙ ( 𝐶𝐷′ ) = (𝜓𝐶𝐷′ )
0
𝜓𝐷𝐶′
0
0
𝜓𝐷𝐷′
0
𝜓𝐶𝐷′
𝜓𝐶𝐶′
𝜓
0 0)∗ ∙ ( 𝐶𝐷′ )
0
0
𝑝(𝐶) = |𝜓𝐶𝐶′ |2 + |𝜓𝐶𝐷′ |2
31
0
0)
0
0
F. Normalization procedure
After the projection of the vector into one of the basis state, the system is in the particular
basis state with probability p=1. What does the new vector look like? When we turn back to
the example from the Appendix C, how will the initial state |𝑆⟩ change after the projection
into the cooperation basis state |𝐶⟩?
|𝑆⟩ =
1
√6
∙(
1−𝑖
)
2𝑖
The projection into basis vector |𝐶⟩ is given by a projection operator 𝑃𝐶 .
1
1−𝑖
1
1−𝑖
0
)∙
∙(
)=
∙(
)
0 √6
2𝑖
0
√6
1
0
𝑃𝐶 ∙ |𝑆⟩ = (
‖𝑃𝐶 ∙ |𝑆⟩‖2 =
1
√6
∙ (1 + 𝑖
0) ∙
1
1−𝑖
1
1
∙(
)= ∙2=
0
6
3
√6
And the length of the projected vector is:
‖𝑃𝐶 ∙ |𝑆⟩‖ =
1
√3
Now we get a normalized by dividing the projected vector 𝑃𝐶 ∙ |𝑆⟩ by its length:
|𝑆𝐶 ⟩ =
|𝑆𝐶 ⟩ =
𝑃𝐶 ∙ |𝑆⟩
‖𝑃𝐶 ∙ |𝑆⟩‖
1
1−𝑖
1
1
1−𝑖
√3 1 − 𝑖
∙(
)⁄ =
∙(
)=
∙(
)
0
0
0
√6
√3 √6
√2
To verify that this operation returns a vector with desired features, i.e. a vector which a) is of
the unitary length and b) which projects into basis vector |𝐶⟩ with probability 1, we can
compute the relevant inner products:
⟨𝑆𝐶 |𝑆𝐶 ⟩ =
1
√2
∙ (1 + 𝑖
⟨𝑆𝐶 |𝑆𝐶 ⟩ =
0) ∙
1
√2
∙(
1−𝑖
)
0
1
∙ (2 + 0) = 1
2
|⟨𝑆𝐶 |𝑆𝐶 ⟩| = 1
1
⟨𝐶|𝑆𝐶 ⟩ = ⟨𝐶| ∙ ( ∙ (1 + 𝑖) ∙ |𝐶⟩ + 0 ∙ |𝐷⟩)
√2
32
⟨𝐶|𝑆𝐶 ⟩ =
1
√2
∙ (1 + 𝑖) ∙ ⟨𝐶|𝐶⟩ + 0 ∙ ⟨𝐶|𝐷⟩ =
|⟨𝐶|𝑆𝐶 ⟩|2 =
1
√2
∙ (1 + 𝑖) ∙
1
√2
33
∙ (1 − 𝑖) =
1
√2
∙ (1 + 𝑖)
1
∙2=1
2