# Bistable Probabilities: A Unified Framework for Studying Rationality and
Irrationality in Classical and Quantum Games

Shahram Dehdashti
School of Information Systems
Queensland University of Technology
Brisbane, Australia
<EMAIL_ADDRESS>Lauren Fell
School of Information Systems
Queensland University of Technology
Brisbane, Australia
<EMAIL_ADDRESS>Abdul Karim Obeid
School of Information Systems
Queensland University of Technology
Brisbane, Australia
<EMAIL_ADDRESS>Catarina Moreira
School of Information Systems
Queensland University of Technology
Brisbane, Australia
<EMAIL_ADDRESS>Peter Bruza
School of Information Systems
Queensland University of Technology
Brisbane, Australia
<EMAIL_ADDRESS>
###### Abstract

This article presents a unified probabilistic framework that allows both
rational and irrational decision making to be theoretically investigated and
simulated in classical and quantum games. Rational choice theory is a basic
component of game-theoretic models, which assumes that a decision maker
chooses the best action according to their preferences. In this article, we
define irrationality as a deviation from a rational choice. Bistable
probabilities are proposed as a principled and straight forward means for
modeling (ir)rational decision making in games. Bistable variants of classical
and quantum Prisoner’s Dilemma, Stag Hunt and Chicken are analysed in order to
assess the effect of (ir)rationality on agent utility and Nash equilibria. It
was found that up to three Nash equilibria exist for all three classical
bistable games and maximal utility was attained when agents were rational. Up
to three Nash equilibria exist for all three quantum bistable games, however,
utility was shown to increase according to higher levels of agent
irrationality.

_K_ eywords game theory, quantum games, irrational decision making, bistable
probabilities

##  1 Introduction

Classical game theory is a mathematical framework that aims to understand the
interactions of decision makers and their joint outcomes. Decision makers
include people, organizations, animals and intelligent machines. Rational
choice theory (RCT) is a basic component of game-theoretic models, which
assumes that a decision maker chooses the best action according to their

preferences. Under certain assumptions, preferences are well behaved, and can
therefore be represented as utility functions. Rationality then equates to
maximizing the decision maker’s utility, or finding the maximum value of the
decision maker’s utility function [1].

Economists, sociologists, and other adherents of RCT typically view human
behavior to be rational when it involves the maximization of the decision
maker’s expected utility. In a somewhat weaker form, rationality is redefined
in terms of choice consistency: The decision maker with a given set of
preferences is rational to the extent that their choices are internally
consistent with one another [2].

Compared to economic theory, the cognitive and psychological sciences show a
much higher variability in their conceptualizations of rationality. Some
scholars adhere to the postulates of RCT, though with some adaptations. These
approaches include risk-weighted expected utility maximization, which captures
the decision maker’s attitudes toward risk [3], the notion that rationality is
local to a decision-making context [4], the rational analysis of cognition
which assumes a function describing optimal behaviour [5], and models of the
maximization of subjective expected utility under cognitive and task
constraints [6]. All these approaches are somewhat aligned with RCT because
they are based on constrained maximization of a target function, which is
determined by the decision maker’s preferences.

Kahneman, Wakker, and Sarin [7] step out of the bounds of RCT as they
differentiate decision utility, stemming from RCT’s assumptions of rationality
and consistency of preferences, from experienced utility, which focuses on the
predicted hedonic reward from an action. This reward can be immediate or
remembered, and is not restricted to personal gain. In their review, Kahneman
and Thaler [8] find multiple examples of biases in predicting future
experience of reward, affecting the maximisation of utility.

In short, the interplay between rationality and decision making has been
broadly debated in literature covering economics, cognitive and social
psychology. Our intention is not to contribute to this debate. Rather, our
intention is to highlight that however rationality is defined, the decision
maker, for whatever reason, may deviate from a rational choice. We term such a
deviation “irrationality". What then are the consequences for game theory?
This article aims to address this question by introducing a deformed
probability; a so-called “bistable probability", as a formal means of
systematically investigating deviations from a rational choice.

The bistable probability framework is used to theoretically analyse three
well-studied games in the literature, i.e. Prisoner’s Dilemma (PD), Stag Hunt
(SH) and Chicken Game (CG). By using these examples, we analyse how
irrationality leads to additional Nash equilibria (NE) when compared to
rational versions of the games. Using an example sourced from the literature,
the effect of irrationality on the relationship between the benefit to cost
ratio and cooperation is also investigated.

Quantum game theory is a natural generalization of game theory [9, 10, 11]. In
quantum games, strategies are defined by Hermitian operators that act on the
initial states of players. The strategy space can be extended by considering
the superposition of strategies and entanglement of players’ choices. A
player’s rational decision in a quantum game corresponds to a sharp projection
in Quantum Mechanics (QM). Positive operator value (POV) projections are un-
sharp projections [12, 13] that serve to extend a complex Hilbert space. We
will show that bistable operators are equal to un-sharp projections, and can
therefore be used to describe deviations from rationality. Further, we
demonstrate that POV projections from quantum mechanics create a suitable
framework to model irrationality across classical and quantum games.
Therefore, this article proposes POV projections as a general framework for
both classical and quantum games involving both rational and irrational
agents. Through analysis of this framework, we show that agent utility
decreases with the presence of irrationality in classical games but that the
converse occurs in quantum irrational games. Finally, we show that non-
factorisable strategies are naturally obtained when we consider a bistable
parameter in a simulated quantum game [14, 11, 15]. We indicate that this
model can provide a framework by which cooperation in non-cooperative games
can be modelled.
The rest of the paper is organized as follows: In section 2, we model
irrationality using POV projections derived from QM. In section 3, a framework
is suggested for irrational classical games; we study three different games:
PD, SH and CG. In section 4, by using bistable Kraus projections, we study the
three above mentioned games in a quantum setting. Section 5 describes details
of the mathematical framework. Finally, section 6 is devoted to some
conclusions and directions for future research.

##  2 Bistable probabilities

The preceding section describes the manifold ways in which rationality is
viewed. In one conceptualisation of rationality, rational agents might be
expected to defect in one-shot Prisoners’ Dilemma games, as this choice
maximizes monetary gain. In another way, a rational agent may be expected to
cooperate based on factors that include social [16, 17], moral [18],
personality [19], or sociocultural preferences[20], which may promote altruism
over self-gain. The focus of the present paper is to model deviations from
rationality, however it may be viewed. To do this, we consider an agent to be
in two minds: one rational, one irrational. One can consider this in terms of
having two sets of preferences inconsistent with one another. We map this
intuition onto a Dual Process account of human decision making in order to
clarify it.

Dual process theories are prominent and widespread within many fields of
psychological science [21]. Whilst there are multiple variations of these
theories, all have in common the thesis that two processes are employed in
human decision-making: System 1 and System 2. System 1 produces an intuitive
approach to decision making which is often characterized by heuristics and
other mental shortcuts [22]. System 1 is fast, requires few cognitive
resources and little effort [23]. System 2 involves controlled analytic
thought, and is often considered to be logical, however requires conscious
activation and is a significant drain on cognitive resources [23].

The preferences associated with each of these systems may be inconsistent with
each other. For example, studies [24, 25, 26, 27] have found that, under
certain conditions [28], decisions of cooperation tend to coincide with System
1, or intuitive, processing, while further deliberation may lead to decisions
of defection. The Social Heuristics Hypothesis [25] was proposed as a means to
explain cooperation and predicts its prevalence when intuitive (System 1)
thinking is employed. Whilst there is evidence to support the claim that
intuition favours cooperation, this appears to be moderated by other factors,
such as gender differences [20]. In addition to this, there are a range of
methodological factors present in the study of dual processes that may also
affect results (see review by [28]). Nevertheless, it is clear that there are
conflicting preferences across System 1 and System 2 which may place an agent
in a game in two minds when making a choice.

To position this in terms of rationality, consider a situation in which the
preferences of an individual employing their deliberative, System 2 process
promote the decision to defect. Their expected utility would be higher if they
chose to defect than to cooperate. To defect in this specific instance, would
therefore be the rational choice. However, this individual’s intuition (System
1) may promote a decision to cooperate. Employing System 1 may see the
expected utility to be higher if the individual chose to cooperate rather than
defect. In this case, cooperating could be considered to be the rational
choice. When there is disagreement between System 1 and System 2, and when the
preferences between the two systems are inconsistent, we can consider the
potential for irrationality because the inconsistency may lead to a deviation
from a rational choice.

Although the bistable probability framework is agnostic to the order of System
1 and 2 processes, as well which system is deemed the rational one, we can
take the following example for ease of exposition. If System 2, and monetary
gain, is seen to be the rational process and choice in a certain situation, we
introduce irrationality as the deviation from that choice if System 1
preferences conflict with System 2. We introduce the bistable parameter $k$ to
represent the level of disagreement between System 1 and System 2 in a similar
way as in [29]. It should be noted at this point that we can just as easily
flip the explanation under the assumption that System 1 produces a choice that
maximises the agent’s utility, and where System 2 is seen to deviate from this
by a factor defined by $k$. In terms of the interaction between System 1 and
System 2, there are various views about how this might occur, for example, in
parallel or with System 1 acting as a default for which System 2 can intervene
[30]. The model proposed in this paper is agnostic towards the order or
dynamics of System 1 and System 2 processes and describes only the deviation
of one from the other. In this way, we present the bistable parameter as a
means of systematically investigating irrationality in games.

Inspired by [31, 32], bistable probabilities are founded on an event space
featuring two probabilities associated with a given event $E$: $p(E)$ and
$p_{k}(E)$ where $p_{k}(E)=1-p-k+2kp$ and $p_{k}(\neg E)=p+k-2kp$. In terms of
the previously mentioned “two minds" metaphor, bistable probabilities can be
interpreted as follows: The rational “mind" (System 1 or System 2, depending
on the prescribed theory of rationality) would make a decision $E$ with
probability $p(E)$, where $p$ may be a classical or quantum probability
depending on whether the game is classical or quantum. The preferences of the
irrational “mind" may deviate from the rational decision with probability
$p_{k}(E)$. This allows irrational decision making to be modelled as a
deviation from the probability ascribed to a rational decision. The deviation
is modelled by the parameter $k$: When $k=1$, the agent can be considered to
be fully rational because the irrational mind does not deviate from the
rational mind’s decision: $p(E)=p$ and $p_{k}(E)=p$.

A degree of decreasing rationality is determined by the range $0\leq k<1$.
When $k=0.5$, $p_{k}(E)=0.5$ irrespective of the rational mind’s judgement of
the probability of $E$. This reflects the situation in which the agent is
caught between two minds with no means of resolving the conflict between the
two, so the ultimate decision is random. Finally, $k=0$, $p(E)=p$ and
$p_{k}(E)=1-p$ expresses full irrationality as the irrational mind contradicts
the judgement of the rational mind. Note that both $k=0$ and $k=0.5$ can be
considered irrational, depending on the theory of rationality to which one
might subscribe. If an agent’s choice is essentially random ($k=0.5$), this
may describe an internal irrationality associated with their decision. If
normative measures of irrationality are considered, such that there is an
objectively rational choice, then a final decision which departs completely
from the rational choice ($k=0$) can define a maximum level of irrationality.
Regardless, irrationality increases as values of $k$ decrease.

We do not claim that defining irrationality in the preceding way presents a
complete picture. In fact, irrationality can be considered in different ways,
e.g., in relation to ideas and theories about anti-realism [33] to some
theories about biases and perceptions in cognitive science [31, 32, 34, 35,
36]. However, by parameterizing irrationality by a bistable parameter, this
method allows degrees of irrationality to be investigated in a systematic way.

Figure 1: (a) Agents, e.g. Alice and Bob, choose rationally ideal strategies,
Y and X, while bistable parameters deform these decisions to the alternative
strategy with probabilities $k$ and $k^{\prime}$. Probabilities
$\epsilon_{i}=\epsilon_{i}(p,q;k,k^{\prime})$, $i=1,\cdots,4$, denote
probabilities of outcomes of different strategies subject to bistable
parameters. (b) Alice’s payoffs for choosing $(Y,Y)$, $(Y,X)$, $(X,Y)$ and
$(X,X)$ are respectively $\alpha$, $\beta$, $\gamma$ and $\delta$, while for
Bob are given by $\alpha$, $\gamma$, $\beta$ and $\delta$.

Consider a game, such as the one defined in Fig. 1 with two players, Alice and
Bob. Alice and Bob must decide on a strategic option $X$ or $Y$ leading to
four possible strategies after a single move $(X,X),(X,Y),(Y,X)$ and $(Y,Y)$,
where the first component in the ordered pair represents Alice’s decision and
the second represents Bob’s decision. Ideally assuming Alice and Bob to be
rational, the probability of each choosing option $Y$ is denoted by $p$ and
$q$ respectively. However, considering irrationality, a given agent may
deviate from the rational choice. Bistability captures this, and the
corresponding bistable probabilities are denoted $p_{k}$ and $q_{k^{\prime}}$
where the bistable parameter $k$ models the (ir)rationality of Alice’s
decision making and $k^{\prime}$ models the (ir)rationality of Bob’s decision
making. The probabilities of the four possible strategies are denoted
$\epsilon_{i}=\epsilon_{i}(p,q;k,k^{\prime})$, $i=1,\cdots,4$.

##  3 Classical bistable game

We consider a scenario in which two agents, e.g. Alice and Bob, cannot
communicate with each other. This is equal to no-signaling in physics [14]. In
any round of the game, either agent can choose one of two possible options, X
and Y, as part of their strategy (i.e. X and Y in Fig. 1(a)). Experimental
results are recorded for a large number of individual trials so that each
experimental trial comprises two choices, denoted $+1$ and $-1$. Payoffs are
awarded depending on agent decisions over many trials, the matrix of the game
they play and the statistics of the measurement outcomes. It turns out that
when agents are influenced by the bistable parameter, the probabilities
$\epsilon_{i}$’s are non-factorisable, and therefore a bistable game simulates
a quantum game [14, 11]. To study the impact of bistablity, three different
scenarios are considered. In the first, we consider $k^{\prime}=k$, which
means the level of (ir)rationality in both agents is equal. In the second, we
consider situations in which one agent is irrational ($k<1$), while the other
is not ($k^{\prime}=1$). The third scenario considers situations in which both
payers are influenced by complementary levels of (ir)rationality,
$k^{\prime}=1-k$. Moreover, when $k=k^{\prime}=0.5$, a NE is obtained,
regardless of the payoffs. The curious fact in this case is that the outcome
of the game does not depend on the strategies of the respective agents.
In a two-agent, two-choice game, there can be up to five possible NEs based on
the payoffs. Fig. 2, Table 1. specifies the conditions which have to be
fulfilled in order for NEs to be obtained. The number of NE depend on both the
payoffs and probability of options chosen by the agents. Note that the first
four NE, i.e. $(0,0)$, $(0,1)$, $(1,0)$ and $(1,1)$, are pure strategies, as
that they can be determined in a single round. The last NE $(P^{*},Q^{*})$
denotes a mixed strategy, meaning that it can only be determined through a
sample of plays taken from the game, and it is presented in Table 2.

Figure 2: Table 1 illustrates all possible NEs and their associated conditions
for four different scenarios; the rational case in which $k^{\prime}=k=1$, the
symmetrical irrational case, i.e., $k^{\prime}=k$ (such that $k\neq 1$), the
case where one agent is irrational, $k^{\prime}=1$ and finally the case of
complementary irrationality, in which $k^{\prime}=1-k$. Tables 2-4 show the
NEs for respectively the PD, SH and CG.

###  3.1 Prisoner’s Dilemma

PD is an example of a non-zero sum game. Contrary to zero sum games in which
agents’ gains or losses are exactly balanced between them, in non-zero sum
games, both agents may not necessarily be engaged and benefiting from strictly
opposing strategies, but rather benefit from mutual cooperation.

In PD, each agent must independently choose between two actions: to cooperate
(C) or to defect (D). In the classic version of the game, ’independent’ means
that there is no way for the two agents to communicate with each other, so one
agent is never aware of the strategy of the other. Agents are assumed to rely
on the reward (or payoff) associated with each strategy when making a
decision, with the objective of maximizing their individual payoffs.

In a classical setting, the PD game has one single dominant strategy: to
defect. The dilemma in the game comes from the fact that if both agents choose
the dominant strategy, which is to defect, then they perform substantially
worse than if they had both chosen to cooperate. In game theory, this strategy
also coincides with the NE. When both agents choose to defect, then it seems
that there is no other possible strategy that could bring a higher individual
payoff to the agents. With the study of bistable games, one can avoid this
situation where agents are attached to a single NE corresponding to the defect
strategy by providing agents alternative NEs with which they can engage in
their strategies.

In a traditional PD game, a payoff matrix is associated to each of the agents’
actions, as demonstrated in Fig. 3(b). For this specific game, the payoffs of
the payers need to satisfy the following rules: $\delta>\alpha>\gamma>\beta$
and $2\alpha>\beta+\gamma$ [37]. To expand on the scenarios that result in
these payoffs, when both agents cooperate, they both have a payoff of
$\delta$, when they both defect, they have a payoff of $\alpha$ and when they
engage in different strategies, the agent choosing to cooperate receives
$\gamma$, while the agent who chooses to defect gets $\beta$.

Figure 3: Plot (a) indicates the utility of Alice for different values of the
bistable parameter, while the bistable parameter is the same for both agents,
$k=k^{\prime}$. The plot (b) presents the utility in the case in which
$k^{\prime}=1$. Finally, the plot (c) depicts the situation in which
$k=1-k^{\prime}$. For all plots: $\alpha=3,\beta=0,\gamma=1$ and $\delta=5$.

In order to analyze the impact of varying degrees of irrationality, the three
previously mentioned scenarios are investigated: a) $k=k^{\prime}$, b)
$k^{\prime}=1$ and c) $k^{\prime}=1-k$, where $k$ is the bistable parameter
that models the (ir)rationality of Alice and $k^{\prime}$ models the
(ir)rationality of Bob.

Table (2), in Figure 2, illustrates NEs in four different scenarios. Despite
the fact that in case $k^{\prime}=k=1$, which corresponds to the usual PD
where both agents are rational and where there is only one NE, in other cases,
in which at least one agent is influenced by the bistable parameter, the
number of NEs increase up to the maximum of three. For example, in the
scenario in which $k=k^{\prime}$, based on the payoffs, either $(0,0)$,
$(1,1)$ and $(P^{\ast},Q^{\ast})$ or $(0,1)$, $(1,0)$ and
$(P^{\ast},Q^{\ast})$ can be NEs. In the case of $k^{\prime}=1$, based on the
order of payoffs in the PD, only $(0,0)$, $(1,1)$ and $(P^{\ast},Q^{\ast})$
are potentially NEs. In the last scenario, in contrast with the PD without any
bistablity, $(0,0)$ and $(1,1)$, definitely cannot be NEs.

We summarize the impacts of the bistable parameter on the PD game as the
following: (i) Mixed strategy $(P^{\ast},Q^{\ast})$ can yield a NE; (ii) The
number of NEs depends on the values of the payoffs and the level of
irrationality. The maximum number of NEs is three.

Plots in Fig. 3 (a) show Alice’s utility in a symmetrical bistable PD game,
i.e., $k=k^{\prime}$ with $\alpha=3$, $\beta=0$, $\gamma=1$ and $\delta=5$ for
different levels of irrationality for Alice $k=0.5,\cdots,1$. These plots
indicate that by increasing the level of the agents’ irrationality, their
utility becomes increasingly independent of the rational decision with regards
to a move. Finally, their utility becomes completely independent from their
rational decisions when the bistable parameter equals $0.5$. Plots in Fig. 3
(b) indicate Alice’s utility using the same parameters but where Bob is
completely rational ($k^{\prime}=1$). Drawing a comparison between plots in
(a) and (b) demonstrates that Alice’s utility can be at a maximum whenever Bob
acts rationally, $k^{\prime}=1$. Finally, plots in (c) illustrate the scenario
where Alice and Bob have complementary levels of irrationality, i.e.,
$k=1-k^{\prime}$. Despite the fact that defection/cooperation is the NE, the
maximum payoff for Alice is obtained whenever she cooperates with Bob. Also,
the maximum utility is less than two previous scenarios (a) and (b).

In order to illustrate how the bistable probability framework relates to
published experimental findings, we apply it to the findings in [16]. This
study utilised a continuous strategy Prisoner’s Dilemma paradigm where
participants were given a choice of varying levels of cooperation, rather than
a binary choice between cooperate and defect. The effect of a benefit/cost
($b/c$) ratio was investigated by varying a constant that multiplied the
amount that the player’s partner gained by the amount a participant chose to
donate to them. The study found that the majority of participants fell under
three categories: maximum cooperation, maximum defection (i.e. no
cooperation), and $50$% cooperation. Increasing the $b/c$ ratio resulted in an
increase to maximal cooperation and a decrease to maximal defection.

Figure 4: Motivation to cooperate $\Delta_{m}$ as a function of bistable
parameter $k$ and cost-benefit ratio $b/c$.

The $b/c$ ratios on motivation to cooperate were varied within the bistable
framework, and compared to the findings presented in [16]. In order to do
this, the payoffs of each strategy are defined as follows:

$\displaystyle\alpha=b-c,\ \beta=-c,\ \gamma=0,\ \delta=b,$ (1)

which means an agent pays a cost $c$ with respects to benefit $b$ ($b>c$).
This is therefore the difference between the gain that the agent is able to
achieve:

$\displaystyle\Delta_{m}$ $\displaystyle=$ $\displaystyle
GainExpectationofCooperation-GainExpectationofDefection$ (2) $\displaystyle=$
$\displaystyle\left[p_{k}q_{k^{\prime}}(b-c)-cp_{k}(1-q_{k^{\prime}})\right]-\left[b(1-p_{k})(1-q_{k^{\prime}})\right],$

which can be considered as a motivating factor for cooperation among agents.
In Fig. 4, the motivation to cooperate $\Delta_{m}$ is illustrated as function
of two parameters, i.e., the bistable parameter $k$ and the $b/c$ ratio. When
$k=1$ (i.e., a player is rational) an increase in $b/c$ ratio coincides with
an increased motivation to cooperate, which is consistent with results
presented in [16]. However, as $k$ decreases (i.e., the the deviation from
rationality increases), the overall motivation to cooperate decreases, which
corresponds to an increased likelihood to defect. An irrational agent can
therefore be seen to become less sensitive to the $b/c$ ratio, and become
increasingly motivated to defect regardless of $b/c$. This finding can be
placed in the context of the cognitive bias of risk aversion, where agents
tend to weigh risk and reward asymmetrically. Although the above does not
provide a direct explanation for the subset of participants in [16] who chose
50% cooperation regardless of the $b/c$ ratio, it does illustrate an
insensitivity to changes in a $b/c$ ratio for irrational players, which could
be explained in terms of heuristics.

###  3.2 Stag Hunt

SH is a prototype of a social contract between two parties, originally
proposed by the philosopher Jean-Jacques Rousseau to question the degree to
which people are faithful to their contracts. The traditional version of the
game puts two agents in a hunting scenario. Each of the agents has the choice
of pursuing either a rabbit or a stag. The stag draws the largest reward,
however the rabbit is enough to satisfy an individual agent. What Jean-Jacques
Rousseau questioned is that if both agents want to hunt the stag, then they
need to be faithful to their posts and strategy. But if a rabbit happens to
pass within an agent’s reach, they face a choice to either remain at their
post, or break the contract and pursue the rabbit for individual satisfaction.
In this sense, the SH game suggests a scenario where the two agents can
jointly cooperate to get the largest reward, or be selfish and get a
satisfactory individual reward.
The payoffs for this game conform to the following rules:
$\alpha>\gamma\geq\delta>\beta$ and $\gamma+\delta>\alpha+\beta$, where
$\alpha$ is the maximum payoff that Alice and Bob can obtain in a mutual
cooperation strategy, $\gamma$ corresponds to the payoff when Alice decides to
break the contract and pursue the rabbit, $\delta$ is the payoff when Bob
decides to break the contract and pursue the rabbit, and finally, $\beta$ is
the payoff that Alice and Bob both obtain if they both break the contract and
decide to hunt the rabbit.
The order of payoffs in this game leads to two NE states: (1) both agents
choose to hunt the stag and (2) both agents choose to hunt the rabbit. By
exploring bistable parameters, one can find additional strategies that can
release agents from these two NE for means of engaging in alternative
strategies.
Table (3), in Fig. 2 represents NEs in the three different scenarios. In the
first scenario a) in which $k=k^{\prime}=1$, there are up to three different
NEs. In fact, two pure strategies $(0,0)$ and $(1,1)$ are always fulfilled by
NE inequalities, while the mixed strategy $(P^{\ast},Q^{\ast})$ as the third
NE depends on values of payoffs. In the second scenario, in which
$k=k^{\prime}\neq 1$, there is a maximum of three NEs. In the scenario b) in
which $k^{\prime}=1$, except the point $(0,1)$, all other points can
potentially yield NE, based on specific values of payoffs. In the last
scenario c), there is a maximum of three NEs, namely points $(0,1)$, $(1,0)$
and $(P^{\ast},Q^{\ast})$.

###  3.3 Chicken Game

CG is another type of game that models coordination strategies between two
agents. The game describes two agents who are heading towards each other. If
one of the agents deviates their path in order to avoid a collision, then this
agent loses and is labeled the ’chicken’ (implied as the defeated), while the
other is considered the victor.

Under this scenario, the strategies for NEs are straightforward. If Alice
keeps heading straight and Bob swerves, then Alice is engaged in a dominant
strategy where, regardless of the action she chooses, it will not bring a
higher payoff. Consequently, this obtains a NE. The same NE can be found if
Bob continues straight and Alice deviates. The payoffs for this game have the
following order: $\gamma>\alpha>\beta\gg\delta$ subject to the following
condition: $0<\beta<(\gamma+\beta)$. Under this order, the parameters $\gamma$
and $\alpha$ respectively correspond to the payoffs of Alice and Bob when they
engage in the dominant strategy.

In the first scenario a), in which $k=k^{\prime}=1$, there are maximum up to
three NEs. In this scenario, while differences in payoffs matter for
calculating the mixed strategy NE, i.e. $(P^{\ast},Q^{\ast})$, only the order
of payoff matters for determining regular NEs, i.e., $(0,1)$ and $(1,0)$. In
the second scenario b), $k=k^{\prime}\neq 1$, there is a maximum of three NEs.
Based on the payoffs, either points $(0,0)$, $(1,1)$ and $(P^{\ast},Q^{\ast})$
or $(0,1)$, $(1,0)$ and $(P^{\ast},Q^{\ast})$ are NEs. In the scenario which
$k^{\prime}=1$, there is a maximum of three NEs, depending on the payoffs. In
the last scenario c) in which $k^{\prime}=1-k$, subject to payoffs, up to
three NEs are determined, i.e., points $(0,1)$, $(1,0)$, and
$(P^{\ast},Q^{\ast})$, or $(0,0)$, $(1,1)$ and $(P^{\ast},Q^{\ast})$.

##  4 Quantum Bistable games

Quantum game theory has been suggested as a way to generalize game theory by
exploiting concepts inherent to quantum theory such as entanglement. [38, 9,
39]. Quantum games are based on each agent having a qubit. A quantum move in
the game is modelled by applying a unitary matrix to the given agent’s qubit.
If agents’ qubits become entangled, there are hidden channels of
communication. Due to the fact that accessible information in a quantum system
is greater than in a classical system [40, 41, 42], a quantum game can feature
more strategies. One potential result of this fact is that the number of NEs
can differ from the classical variant of the game. Notwithstanding the extra
features that quantum games provide, the quantum game-theoretic framework
still assumes rationality, e.g. agents focus on maximizing their expected
utility. We begin by observing that the utilities of agents in a classical
game can be formalized as expectation values of sharp projections when the
initial states of agents are modelled by a two dimensional real Hilbert space.
This observation opens the door to generalise strategies in classical games by
means of unitary operators. A second observation is that unsharp projections
as positive-operator values (POV) projections is a means to generalize
measurement in quantum mechanics. Both observations allow irrational quantum
games to be modelled.
In these games, Bob’s and Alice’s strategies are defined by unitary operators
in a complex Hilbert space and their associated expected utilities are given
by expectation values computed by means of bistable Kraus (POV) projections.
We assume that Alice’s and Bob’s initial states are described by a a Bell
state of the form $|\psi_{i}\rangle=(|0,0\rangle+i|1,1\rangle)\sqrt{2}$, which
is a maximally entangled state. In this section, quantum bistable variants of
PD, SH and CG are studied.

Figure 5: Utility functions of Alice for different operators
$\hat{U}(\theta_{i},\phi_{i}=0)$, $i=a,b$, are plotted in plots (a), (b) and
(c) for different games, respectively PD, ST and CG.

###  4.1 Prisoner’s Dilemma

In this subsection, we study the effect of bistable parameters $k$ and
$k^{\prime}$ on the quantum PD game. When $0.5<k=k^{\prime}\leq 1$, the point
$(\theta_{a}^{\ast},\theta_{b}^{\ast})=(\pi/2,\pi/2)$ corresponds to a NE.
When $0.5<k=1-k^{\prime}\leq 1$, the point
$(\theta_{a}^{\ast},\theta_{b}^{\ast})=(0,0)$ is a NE.
Also, we study the effects of irrationality on the utility of agents. Plots
(a) in Fig. 5 demonstrate the utility function of Alice for different values
of bistability when Alice and Bob are equally irrational, i.e.,
$k=k^{\prime}$. These plots illustrate that increasing the bistable parameter
causes the utility to increase. In other words, the more irrational Alice
becomes, the greater her utility. Recall in the classical variant of the
bistable game when $k=k^{\prime}=0.5$ the utilities of Alice and Bob are
independent of their strategies. However, in the quantum variant the
strategies of the agents do have an effect on their utilities. In Alice’s
case, this effect is due to the initial entangled state and the phase factor
$\phi_{a}$ associated with her strategy.

###  4.2 Stag Hunt

In the case of $0.5<k=k^{\prime}\leq 1$,
$(\theta_{a}^{\ast},\theta_{b}^{\ast})=(\pi/2,\pi/2)$ and
$(\theta_{a}^{\ast},\theta_{b}^{\ast})=(0,0)$ are two pure strategies that
yield NEs, while
$(\theta_{a}^{\ast},\theta_{b}^{\ast})=(\cos^{-1}\sqrt{(\delta-\beta)/\Gamma},\cos^{-1}\sqrt{(\delta-\beta)/\Gamma})$
is a mixed NE. In addition, The scenario in which $0.5<k=1-k^{\prime}\leq 1$,
points $(\pi/2,0)$, $(0,\pi/2)$ and
$(\cos^{-1}\sqrt{(\delta-\beta)/\Gamma},\cos^{-1}\sqrt{(\delta-\beta)/\Gamma})$
are NEs with $\Gamma=\alpha-\beta+\delta-\gamma$. Plots in Fig. 5 (b)
demonstrate the effect of the bistable parameter $k=k^{\prime}$ on Alice’s
utility. Drawing a comparison between plots in (b), indicates maximum utility
is obtained when both agents’ strategies approach $(\pi/2,\pi/2)$. Like the
previous case when Alice and Bob were equally irrational, the level of utility
increases in accordance with increased levels of irrationality i.e., $k$
decreases.

Finally, when $k=k^{\prime}=0.5$ the utility depends on the respective
strategies, which is in direct contrast to the same scenario in the classical
bistable game, where utility was shown to be independent of the given
strategy.

###  4.3 Chicken Game

When $0.5<k=1-k^{\prime}\leq 1$ the quantum bistable CG yields up to three
NEs, i.e., $(\theta_{a}^{\ast},\theta_{b}^{\ast})=(0,\pi/2)$,
$(\theta_{a}^{\ast},\theta_{b}^{\ast})=(\pi/2,0)$ and
$(\theta_{a}^{\ast},\theta_{b}^{\ast})=(\cos^{-1}\sqrt{(\delta-\beta)/\Gamma},\cos^{-1}\sqrt{(\delta-\beta)/\Gamma})$.

When another scenario is considered in which $0.5<k=1-k^{\prime}\leq 1$, up to
three NEs can be found, but the pure NEs turn out to differ from the previous
case, namely $(0,0)$, and $(\pi/2,\pi/2)$, while the mixed NE is similar with
the previous scenario. Fig. 5-Plots in (c) illustrates the impact of the
bistable parameter on the utility. Once again, the level of utility increases
in accordance with increased levels of irrationality i.e., $k$ decreases.
Also, the maximum utility for Alice is obtained whenever one of the agents’
strategies is determined by an operator in the $z$-direction, i.e.,
$\theta=0$.

##  5 Methods

The following provides the theoretical details of both classical and quantum
bistable games.

###  5.1 Nash equilibria for a general classical bistable game

Utilities for Alice and Bob are specified as follows:

$\displaystyle\Pi_{A}(p,q;k)$ $\displaystyle=$
$\displaystyle\alpha\epsilon_{1}+\beta\epsilon_{2}+\gamma\epsilon_{3}+\delta\epsilon_{4},$
(3) $\displaystyle\Pi_{B}(p,q;k)$ $\displaystyle=$
$\displaystyle\alpha\epsilon_{1}+\gamma\epsilon_{2}+\beta\epsilon_{3}+\delta\epsilon_{4}.$
(4)

in which $\epsilon_{1}=p_{k}q_{k^{\prime}}$,
$\epsilon_{2}=p_{k}(1-q_{k^{\prime}})$, $\epsilon_{3}=(1-p_{k})q_{k^{\prime}}$
and $\epsilon_{4}=(1-p_{k})(1-q_{k^{\prime}})$. It is interesting that the
probability distributions $\epsilon_{i}$’s are naturally non-factorizable and
can therefore generate a quantum-like game [14, 11]. According to the
definition of NE, i.e.,

$\displaystyle\Pi_{A}(p^{\ast},q^{\ast})-\Pi_{A}(p^{\ast},q)\geq 0,\hskip
5.69046pt\Pi_{B}(p^{\ast},q^{\ast})-\Pi_{B}(p,q^{\ast})\geq 0,$ (5)

and using relations (3) and (4), the Nash strategies are given by

$\displaystyle(2k^{\prime}-1)(q^{\ast}-q)\Big{[}\left(\alpha-\beta+\delta-\gamma\right)p_{k}^{\ast}+\beta-\delta\Big{]}\geq
0,$ (6)
$\displaystyle(2k-1)(p^{\ast}-p)\Big{[}\left(\alpha-\beta+\delta-\gamma\right)q_{k^{\prime}}^{\ast}+\beta-\delta\Big{]}\geq
0.$ (7)

###  5.2 Nash Equilibria for a general quantum bistable game

In bistable quantum games Bob’s and Alice’s strategies are modelled by a
unitary operator. Their irrational decisions ($k<1$) are defined by a bistable
projection operator:

$\displaystyle\mathbb{P}_{\pm}(\hat{n};k)=(1-k)\mathbb{I}_{2\times
2}+(2k-1)\pi_{\pm}^{\hat{n}}$ (8)

in which $k$ is the bistable parameter in the interval $0\leq k\leq 1$,
$\mathbb{I}_{2\times 2}$ and $\pi_{\pm}^{\hat{n}}$ are respectively the
identity matrix and projection operator in the $\hat{n}$-direction,

$\displaystyle\mathbb{I}_{2\times 2}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}1&0\\\ 0&1\end{array}\right),$ (11)
$\displaystyle\pi_{+}^{\hat{n}}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}\cos^{2}\frac{\vartheta}{2}&e^{-i\varphi}\sin\frac{\vartheta}{2}\cos\frac{\vartheta}{2}\\\
e^{i\varphi}\sin\frac{\vartheta}{2}\cos\frac{\vartheta}{2}&\sin^{2}\frac{\vartheta}{2}\end{array}\right),$
(14) $\displaystyle\pi_{-}^{\hat{n}}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}\sin^{2}\frac{\vartheta}{2}&-e^{-i\varphi}\sin\frac{\vartheta}{2}\cos\frac{\vartheta}{2}\\\
-e^{i\varphi}\sin\frac{\vartheta}{2}\cos\frac{\vartheta}{2}&\cos^{2}\frac{\vartheta}{2}\end{array}\right),$
(17)

where
$\hat{n}=(\sin\vartheta\cos\varphi,\sin\vartheta\sin\varphi,\cos\vartheta)$.
Note that in quantum mechanics, the bistable parameter $k$ (which in physics
might be analogous with a parameter to model noise111Note that this analogy is
imprecise as we do not expect the effect of noise to be comparable with the
main system., see for example [43]) is defined by a number between $0.5$ and
$1$.
Now, if we consider the bistable projections in direction of $z$-axis,

$\displaystyle\mathbb{P}_{+}(\hat{z};k)=\left(\begin{array}[]{cc}k&0\\\
0&1-k\end{array}\right),\
\mathbb{P}_{-}(\hat{z};k)=\left(\begin{array}[]{cc}1-k&0\\\
0&k\end{array}\right).$ (22)

by considering a binary distribution, i.e.,
$|\psi\rangle=(\sqrt{p},\sqrt{1-p})^{T}$, the probability of finding any
option is obtained by expectation values of the relations (22),
$\langle\psi|\mathbb{P}_{\pm}(\hat{z};k)|\psi\rangle=P_{k}(\pm)$, i.e.

$\displaystyle P_{k}(+)$ $\displaystyle=$ $\displaystyle 1-p-k+2kp$ (27)
$\displaystyle=$ $\displaystyle\left(\sqrt{p}\
\sqrt{1-p}\right)\left(\begin{array}[]{cc}k&0\\\
0&1-k\end{array}\right)\left(\begin{array}[]{c}\sqrt{p}\\\
\sqrt{1-p}\end{array}\right),$ $\displaystyle P_{k}(-)$ $\displaystyle=$
$\displaystyle p+k-2kp$ (32) $\displaystyle=$ $\displaystyle\left(\sqrt{p}\
\sqrt{1-p}\right)\left(\begin{array}[]{cc}1-k&0\\\
0&k\end{array}\right)\left(\begin{array}[]{c}\sqrt{p}\\\
\sqrt{1-p}\end{array}\right).$

We consider a quantum game in which the initial state is maximally entangled,
i.e., a Bell state, $|\psi_{i}\rangle=(|0,0\rangle+i|1,1\rangle)\sqrt{2}$.
Alice and Bob choose their own quantum strategy which is defined by the
following unitary operator,

$\displaystyle\hat{U}=\hat{U}_{A}\otimes\hat{U}_{B}$ (33)

in which

$\displaystyle U_{A}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}e^{i\phi_{a}}\cos\frac{\theta_{a}}{2}&\sin\frac{\theta_{a}}{2}\\\
-\sin\frac{\theta_{a}}{2}&e^{-i\phi_{a}}\cos\frac{\theta_{a}}{2}\end{array}\right),$
(36) $\displaystyle U_{B}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}e^{i\phi_{b}}\cos\frac{\theta_{b}}{2}&\sin\frac{\theta_{b}}{2}\\\
-\sin\frac{\theta_{b}}{2}&e^{-i\phi_{b}}\cos\frac{\theta_{b}}{2}\end{array}\right),$
(39)

where $\theta_{a},\theta_{b}\in[0,\pi]$ and $\phi_{a},\phi_{b}\in[0,\pi/2]$.
Hence, the final state is given by $|\psi_{f}\rangle=\hat{U}|\psi_{i}\rangle$,
i.e.,
$|\psi_{f}\rangle=\left(\begin{array}[]{cccc}\psi_{1}&\psi_{2}&\psi_{3}&\psi_{4}\end{array}\right)^{T}/\sqrt{2}$,
in which

$\displaystyle\psi_{1}$ $\displaystyle=$ $\displaystyle
e^{i(\phi_{a}+\phi_{b})}\cos\frac{\theta_{a}}{2}\cos\frac{\theta_{b}}{2}+i\sin\frac{\theta_{a}}{2}\sin\frac{\theta_{b}}{2},$
$\displaystyle\psi_{2}$ $\displaystyle=$
$\displaystyle-e^{i\phi_{a}}\cos\frac{\theta_{a}}{2}\sin\frac{\theta_{b}}{2}+ie^{-i\phi_{b}}\sin\frac{\theta_{a}}{2}\cos\frac{\theta_{b}}{2},$
$\displaystyle\psi_{3}$ $\displaystyle=$
$\displaystyle-e^{i\phi_{b}}\sin\frac{\theta_{a}}{2}\cos\frac{\theta_{b}}{2}+ie^{-i\phi_{a}}\cos\frac{\theta_{a}}{2}\sin\frac{\theta_{b}}{2},$
$\displaystyle\psi_{4}$ $\displaystyle=$
$\displaystyle\sin\frac{\theta_{a}}{2}\sin\frac{\theta_{b}}{2}+ie^{-i(\phi_{a}+\phi_{b})}\cos\frac{\theta_{a}}{2}\cos\frac{\theta_{b}}{2}.$

By using bistable projection (8), bistable Kraus projections are given by:

$\displaystyle\mathbb{P}_{cc}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(\begin{array}[]{cccc}a_{1}&0&0&-ia_{3}\\\
0&a_{2}&0&0\\\ 0&0&a_{2}&0\\\
ia_{3}&0&0&a_{1}\end{array}\right)\mathbb{P}_{cd}=\frac{1}{2}\left(\begin{array}[]{cccc}a_{2}&0&0&0\\\
0&a_{1}&ia_{3}&0\\\ 0&-ia_{3}&a_{1}&0\\\ 0&0&0&a_{2}\end{array}\right)$ (48)
$\displaystyle\mathbb{P}_{dc}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(\begin{array}[]{cccc}a_{2}&0&0&0\\\
0&a_{1}&-ia_{3}&0\\\ 0&ia_{3}&a_{1}&0\\\
0&0&0&a_{2}\end{array}\right)\mathbb{P}_{dd}=\frac{1}{2}\left(\begin{array}[]{cccc}a_{1}&0&0&ia_{3}\\\
0&a_{2}&0&0\\\ 0&0&a_{2}&0\\\ -ia_{3}&0&0&a_{1}\end{array}\right)$ (57)

in which

$\displaystyle a_{1}$ $\displaystyle=$ $\displaystyle
kk^{\prime}+(1-k)(1-k^{\prime})$ (58) $\displaystyle a_{2}$ $\displaystyle=$
$\displaystyle k^{\prime}(1-k)+k(1-k^{\prime})$ (59) $\displaystyle a_{3}$
$\displaystyle=$ $\displaystyle(2k-1)(2k^{\prime}-1)$ (60)

Note that by considering $k=k^{\prime}=1$ in the preceding relations i.e.,
Alice and Bob are rational, Kraus projections are respectively given by:

$\displaystyle\pi_{cc}=|\psi_{cc}\rangle\langle\psi_{cc}|,\hskip
14.22636pt|\psi_{cc}\rangle=\frac{|0\rangle_{a}\otimes|0\rangle_{b}+i|1\rangle_{a}\otimes|1\rangle_{b}}{\sqrt{2}},$
(61) $\displaystyle\pi_{cd}=|\psi_{cd}\rangle\langle\psi_{cd}|,\hskip
14.22636pt|\psi_{cd}\rangle=\frac{|0\rangle_{a}\otimes|1\rangle_{b}-i|1\rangle_{a}\otimes|0\rangle_{b}}{\sqrt{2}},$
(62) $\displaystyle\pi_{dc}=|\psi_{dc}\rangle\langle\psi_{dc}|,\hskip
14.22636pt|\psi_{dc}\rangle=\frac{|1\rangle_{a}\otimes|0\rangle_{b}-i|0\rangle_{a}\otimes|1\rangle_{b}}{\sqrt{2}},$
(63) $\displaystyle\pi_{dd}=|\psi_{dd}\rangle\langle\psi_{dd}|,\hskip
14.22636pt|\psi_{dd}\rangle=\frac{|1\rangle_{a}\otimes|1\rangle_{b}+i|0\rangle_{a}\otimes|0\rangle_{b}}{\sqrt{2}}.$
(64)

The final state $|\psi\rangle_{f}$ together with bistable Kraus projections
define the agent’s utilities as follows:

$\displaystyle\Pi_{A}(\hat{U}_{a},\hat{U}_{b};k)$ $\displaystyle=$
$\displaystyle\alpha\langle\psi_{f}|\mathbb{P}_{cc}|\psi_{f}\rangle+\beta\langle\psi_{f}|\mathbb{P}_{cd}|\psi_{f}\rangle+\gamma\langle\psi_{f}|\mathbb{P}_{dc}|\psi_{f}\rangle+\delta\langle\psi_{f}|\mathbb{P}_{dd}|\psi_{f}\rangle$
(65) $\displaystyle\Pi_{B}(\hat{U}_{a},\hat{U}_{b};k)$ $\displaystyle=$
$\displaystyle\alpha\langle\psi_{f}|\mathbb{P}_{cc}|\psi_{f}\rangle+\gamma\langle\psi_{f}|\mathbb{P}_{cd}|\psi_{f}\rangle+\beta\langle\psi_{f}|\mathbb{P}_{dc}|\psi_{f}\rangle+\delta\langle\psi_{f}|\mathbb{P}_{dd}|\psi_{f}\rangle$
(66)

in which

$\displaystyle\langle\psi_{f}|\mathbb{P}_{cc}|\psi_{f}\rangle$
$\displaystyle=$
$\displaystyle\cos^{2}\frac{\theta_{a}}{2}\cos^{2}\frac{\theta_{b}}{2}\cos^{2}(\phi_{a}+\phi_{b})$
(67) $\displaystyle+$
$\displaystyle\frac{k+k^{\prime}-2kk^{\prime}}{2}\Big{[}\sin^{2}\frac{\theta_{a}}{2}$
$\displaystyle+$
$\displaystyle\cos^{2}\frac{\theta_{a}}{2}\left(1-4\cos^{2}\frac{\theta_{b}}{2}\cos^{2}(\phi_{a}+\phi_{b})\right)\Big{]},$
$\displaystyle\langle\psi_{f}|\mathbb{P}_{cd}|\psi_{f}\rangle$
$\displaystyle=$
$\displaystyle\cos^{2}\frac{\theta_{a}}{2}\sin^{2}\frac{\theta_{b}}{2}\cos^{2}\phi_{a}+\sin^{2}\frac{\theta_{a}}{2}\cos^{2}\frac{\theta_{b}}{2}\sin^{2}\phi_{b}$
(68) $\displaystyle-$ $\displaystyle
2\sin\theta_{a}\sin\theta_{b}\sin\phi_{a}\cos\phi_{b}$ $\displaystyle+$
$\displaystyle\frac{k+k^{\prime}-2kk^{\prime}}{2}\Big{[}\sin^{2}\frac{\theta_{a}}{2}\left(1-4\cos^{2}\frac{\theta_{b}}{2}\sin^{2}\phi_{b}\right)$
$\displaystyle+$
$\displaystyle\cos^{2}\frac{\theta_{a}}{2}\left(1-4\sin^{2}\frac{\theta_{b}}{2}\cos^{2}\phi_{a}\right)$
$\displaystyle+$ $\displaystyle
2\sin\theta_{a}\sin\theta_{b}\sin\phi_{b}\cos\phi_{a}\Big{]},$
$\displaystyle\langle\psi_{f}|\mathbb{P}_{dc}|\psi_{f}\rangle$
$\displaystyle=$
$\displaystyle\cos^{2}\frac{\theta_{a}}{2}\sin^{2}\frac{\theta_{b}}{2}\sin^{2}\phi_{a}+\sin^{2}\frac{\theta_{a}}{2}\cos^{2}\frac{\theta_{b}}{2}\cos^{2}\phi_{b}$
(69) $\displaystyle-$ $\displaystyle
2\sin\theta_{a}\sin\theta_{b}\sin\phi_{a}\cos\phi_{b}$ $\displaystyle+$
$\displaystyle\frac{k+k^{\prime}-2kk^{\prime}}{2}\Big{[}\cos^{2}\frac{\theta_{a}}{2}\left(1-4\sin^{2}\frac{\theta_{b}}{2}\sin^{2}\phi_{a}\right)$
$\displaystyle+$
$\displaystyle\sin^{2}\frac{\theta_{a}}{2}\left(1-4\cos^{2}\frac{\theta_{b}}{2}\cos^{2}\phi_{b}\right)$
$\displaystyle+$ $\displaystyle
2\sin\theta_{a}\sin\theta_{b}\sin\phi_{b}\cos\phi_{a}\Big{]},$
$\displaystyle\langle\psi_{f}|\mathbb{P}_{dd}|\psi_{f}\rangle$
$\displaystyle=$
$\displaystyle\cos^{2}\frac{\theta_{a}}{2}\cos^{2}\frac{\theta_{b}}{2}\sin^{2}(\phi_{a}+\phi_{b})+\sin^{2}\frac{\theta_{a}}{2}\sin^{2}\frac{\theta_{b}}{2}$
(70) $\displaystyle+$
$\displaystyle\frac{1}{2}\sin\theta_{a}\sin\theta_{b}\sin(\phi_{a}+\phi_{b})$
$\displaystyle+$
$\displaystyle\frac{k+k^{\prime}-2kk^{\prime}}{2}\Big{[}\sin^{2}\frac{\theta_{a}}{2}\left(1-4\sin^{2}\frac{\theta_{b}}{2}\right)$
$\displaystyle+$
$\displaystyle\cos^{2}\frac{\theta_{a}}{2}\left(1-4\cos^{2}\frac{\theta_{b}}{2}\sin^{2}(\phi_{a}+\phi_{b})\right)$
$\displaystyle-$
$\displaystyle\sin\theta_{a}\sin\theta_{b}\sin(\phi_{a}+\phi_{b})\Big{]}$

According to definition, the NEs for the quantum bistable game are obtained
from the following inequalities:

$\displaystyle\Pi_{A}(\hat{U}_{A}^{\ast},\hat{U}_{B}^{\ast};k)-\Pi_{A}(\hat{U}_{A},\hat{U}_{B}^{\ast};k)\geq
0,$ (71)
$\displaystyle\Pi_{B}(\hat{U}_{A}^{\ast},\hat{U}_{B}^{\ast};k)-\Pi_{B}(\hat{U}_{A}^{\ast},\hat{U}_{B};k)\geq
0.$ (72)

Explicitly, in the case in which $\phi_{a}=\phi_{b}=0$, the potential NEs are
given by

$\displaystyle
f(k,k^{\prime})\left(\cos^{2}\frac{\theta_{a}^{\ast}}{2}-\cos^{2}\frac{\theta_{a}}{2}\right)\left[\left(\alpha-\beta+\delta-\gamma\right)\cos^{2}\frac{\theta_{b}^{\ast}}{2}+\beta-\delta\right]\geq
0,$ (73) $\displaystyle
f(k,k^{\prime})\left(\cos^{2}\frac{\theta_{b}^{\ast}}{2}-\cos^{2}\frac{\theta_{b}}{2}\right)\left[\left(\alpha-\beta+\delta-\gamma\right)\cos^{2}\frac{\theta_{a}^{\ast}}{2}+\beta-\delta\right]\geq
0,$ (74)

in which $f(k,k^{\prime})=1-2\left(k+k^{\prime}-2kk^{\prime}\right)$. As is
evident, the NE relations are independent of the bistable parameters $k$ and
$k^{\prime}$ . Moreover, in the case in which $k=k^{\prime}=1/2$, the NE
relations are in fact independent of any choices made by Bob or Alice.

##  6 Conclusion

In this article, the irrationality of agents in both classical and quantum
game theory has been modelled using bistable probabilities which allow
irrationality to be modelled as deviations from a rational choice. A bistable
probability framework for both classical and quantum games was presented using
positive operator value (POV) projections borrowed from quantum mechanics,
which are used to model noise. However, in the framework presented in this
article, the bistable parameter should not be construed as noise, but as a
means of systematically investigating deviations from rationality. This
general framework was used to theoretically analyse classical and quantum
variants of three well known games: Prisoner’s Dilemma, Stag Hunt and Chicken.
In each of these, Nash equilibria were shown to be a function of bistability,
i.e., the level of irrationality of an agent. As would be expected, the
utilities of the payers in classical games decreased as the level of
irrationality increased.
To specify the outcomes of quantum strategies in a range of quantum games, we
introduced bistable Kraus projections as a general formalism. A surprising
result was seen in quantum games where the initial states of agents are
maximally entangled and strategies include a phase factor. In contrast to the
results seen in classical bistable games, the utility increases as the level
of irrationality increases. Whenever the phase factor is excluded,
calculations revealed that NEs in irrational quantum games mimic those in
rational classical games and are independent of the bistability parameter,
i.e., the degree of deviation from a rational choice. In addition, it is
worthwhile to mention that application of quantum games in cognitive science
have been generalized to contextuality scenarios, for example see [44, 45, 46,
47], in which intransitive and transitive preferences have been studied. In
fact, the bistable framework exhibits some potential to model contextuality
[29] and future work will be directed to the modelling of contextuality in
games. Another future direction involves extending our quantum treatment of
incentivised games beyond two agents which opens the possibility of modelling
agents with varying levels of entanglement.

In summary, by defining irrationality as a deviation from a rational choice,
we provide a more realistic model of decision-making in a range of game
situations because both rationality and irrationality can be systematically
examined in a general, unified setting.

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# Novel aspects of particle production
in ultra-peripheral collisions

F.C. Sobrinho1 L.M. Abreu1,2 C.A. Bertulani3,4 I. Danhoni4 F.S. Navarra1

1Instituto de Física  Universidade de São Paulo  Rua do Matão 1371  Cidade
Universitária  \- 05508-090
São Paulo  SP  Brazil
2Instituto de Física  Universidade Federal da Bahia  Campus Ondina - 40170-115
Salvador  Bahia  Brazil
3Department of Physics and Astronomy  Texas A&M University-Commerce
Commerce  Texas 75429  USA
4Institut für Kernphysik  Technische Universität Darmstadt  64289 Darmstadt
Germany

###### Abstract

One of the hot topics in hadron physics is the study of the new exotic
charmonium states and the determination of their internal struture. Another
important topic is the search for effects of the magnetic field created in
high energy nuclear collisions. In this note we show that we can use ultra-
peripheral collisions to address both topics. We compute the cross section for
the production of the $D^{+}D^{-}$ molecular bound state in $\gamma-\gamma$
collisions. We also show how the magnetic field of the projectile can induce
pion production in the target. Both processes have sizeable cross sections and
their measurement would be very useful in the study of the topics mentioned
above.

###### keywords:

ultra-peripheral collisions,exotic charmonium,strong magnetic field

##  1 Introduction

In ultra-peripheral collisions (UPCs) target and projectile do not overlap and
stay intact. As a consequence only few particles are produced, the background
is reduced and we can study more carefully specific particle production
processes, such as those addressed here. In UPCs the elementary processes
which contribute to particle production are photon-photon, photon-Pomeron and
Pomeron-Pomeron fusion. They are a good environment to search for particles
which are more difficult to identify in central collisions [1].

In this work we discuss two processes of particle production, which may be
studied in UPCs: production of $D^{+}D^{-}$ meson molecules and production of
forward pions. In the first we can gain some insight on the nature of these
exotic charmonium states and in the second we can measure the magnetic field
produced by relativistic heavy ions.

##  2 Production of charm meson molecules

One important research topic in modern hadron physics is the study of the
exotic charmonium states [2]. These new mesonic states are not conventional
$c\bar{c}$ configurations and their minimum quark content is
$c\bar{c}q\bar{q}$. The main question in the field is: are these multiquark
states compact tetraquarks or are they large and loosely bound meson
molecules? Perhaps the largest fraction of the community tends to believe that
they are molecules. One of the frequently invoked arguments is that the masses
of almost all these states are very close to thresholds, i.e. to the sum of
the masses of two well known meson states [2, 3, 4]. A genuine tetraquark
state could in principle have any mass, including masses far from thresholds.
Besides, some problems have been detected in the calculation of tetraquark
masses with QCD sum rules [5, 6]. Nevertheless, so far there is no conclusive
answer.

The production of hadron molecules has been discussed in the context of B
decays [3], in $e^{+}e^{-}$ collisions, in proton-proton [4, 7, 8], in proton-
nucleus, in central nucleus-nucleus collisions [9] and also in UPCs [10]. In
this section we focus on the $D^{+}D^{-}$ molecule production in UPCs, but the
method employed here is applicable to all molecular states.

The $D^{+}D^{-}$ pair is produced from two photons. This process can be
described by well known hadronic effective Lagrangians, from which we obtain
the pair production amplitude. This amplitude is subsequently projected onto
the amplitude for bound state formation. If the properties of the bound state
are known, the only unknown in this formalism is the form factor, which must
be attached to the vertices to account for the finite size of the hadrons.

We will study the process $\gamma\gamma\to D^{+}D^{-}$ with the Lagrangian
densities [11]

$\mathcal{L}=(D_{\mu}\phi)^{*}(D^{\mu}\phi)-m_{D}^{2}\phi^{*}\phi-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}\,,$
(1)

and

$\mathcal{L}=-ig_{\gamma
D^{+}D^{*-}}F_{\mu\nu}\epsilon^{\mu\nu\alpha\beta}(D^{*-}_{\alpha}\partial_{\beta}D^{+}-\partial_{\beta}D^{*-}_{\alpha}D^{+}+D^{-}\partial_{\beta}D^{*+}_{\alpha}-\partial_{\beta}D^{-}D^{*+}_{\alpha})\,,$
(2)

where

$D_{\mu}\phi=\partial_{\mu}\phi+ieA_{\mu}\phi\,\,,\hskip
28.45274ptF_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}\,,$

and $\phi$, $D^{*}$ and $A_{\mu}$ represent the pseudoscalar charm meson (with
mass $m_{D}$), the vector charm meson (with mass $m_{D}^{*}$) and the photon
field, respectively. The Feynman rules can be derived from the interaction
terms and they yield the Feynman diagrams for the process $\gamma\gamma\to
D^{+}D^{-}$ shown in Fig. 1a. In the figure we also show the quadrimomenta of
the incoming photons $k^{\mu}=(E_{p},0,0,{\bf k})$,
${k^{\prime}}^{\mu}=(E_{k^{\prime}},0,0,{\bf k^{\prime}})$ and of the outgoing
mesons $p^{\mu}=(E_{p},0,0,{\bf p})$,
${p^{\prime}}^{\mu}=(E_{p^{\prime}},0,0,{\bf p^{\prime}})$. The scattering
amplitude can be derived from the Feynman rules.

|
---|---
(a) | (b)
Figure 1: a) Feynman diagrams for the process $\gamma\gamma\rightarrow
D^{+}D^{-}$. b) Energy dependence of the $e^{+}e^{-}\to e^{+}e^{-}c\bar{c}$
cross section. Data come from several collaborations from LEP and were taken
from arXiv:hep-ex/0010060 [hep-ex]. Lines represent the results obtained with
Eq.(14) adapted to $e^{+}e^{-}\to e^{+}e^{-}D^{+}D^{-}$.

As usual, we include form factors, $F(q)$, in the vertices of the amplitudes.
We shall follow [12] and use the monopole form factor given by

$F(q^{2})=\frac{\Lambda^{2}-m^{2}_{D^{(*)}}}{\Lambda^{2}-q^{2}}\,,$ (3)

where $q$ is the 4-momentum of the exchanged meson and $\Lambda$ is a cut-off
parameter. This choice has the advantage of yielding automatically
$F(m^{2}_{D})=1$ and $F(m^{2}_{D^{*}})=1$ when the exchanged meson is on-
shell. The above form is arbitrary but there is hope to improve this
ingredient of the calculation using QCD sum rules to calculate the form
factor, as done in [13], thereby reducing the uncertainties. Taking the square
of the amplitude and the average over the photon polarizations it is
straigthforward to calculate the cross section:

$\sigma=\frac{1}{64\pi^{2}}\frac{1}{\hat{s}}\sqrt{1-\frac{4m_{D}^{2}}{\hat{s}}}\int|M(\gamma\gamma\rightarrow
D^{+}D^{-})|^{2}d\Omega\,.$ (4)

where $\hat{s}=(k+k^{\prime})^{2}$. We emphasize that the only unknown in our
calculation is the cut-off parameter $\Lambda$. In what follows, we will
determine it fitting our cross section to the LEP data on the process
$e^{+}e^{-}\to e^{+}e^{-}c\bar{c}$.

From the $D^{+}D^{-}$ pair we can construct a bound state (denoted $B$). As in
[4], we impose phase space constraints on the mesons, forcing them to be
“close together”. Here we do this through the prescription discussed in [14].
The bound state $|B\rangle$ is defined as

$\frac{|B\rangle}{\sqrt{2E_{B}}}\equiv\int\frac{d^{3}q}{(2\pi)^{3}}\tilde{\psi^{*}}({\bf
q})\frac{1}{\sqrt{2E_{q}}}\frac{1}{\sqrt{2E_{-q}}}|{{\bf q},-{\bf q}\rangle},$
(5)

where $E_{B}$ is the bound state energy, ${\bf q}$ is the relative three
momentum between $D^{+}$ and $D^{-}$ in the state $B$, $E_{\pm q}$ are the
energies of $D^{+}$ and $D^{-}$ and $\tilde{\psi}({\bf q})$ is the bound state
wave function in momentum space. From Eq. (5), we can write the following
relation between the amplitudes:

$\frac{M(\gamma\gamma\rightarrow
B)}{\sqrt{2E_{B}}}=\int\frac{d^{3}q}{(2\pi)^{3}}\tilde{\psi}^{*}({\bf
q})\frac{1}{\sqrt{2E_{D^{+}}}}\frac{1}{\sqrt{2E_{D^{-}}}}M(\gamma\gamma\rightarrow
D^{+}D^{-}),$ (6)

We assume that the ${\bf p}\simeq{\bf p^{\prime}}$ and hence $E_{D^{+}}\simeq
E_{D^{-}}=E_{D}$. Consequently, the relative momentum ${\bf q}={\bf p}-{\bf
p^{\prime}}$ is close to zero. Therefore the energy $E_{D}$ and the amplitude
$M(\gamma\gamma\rightarrow D^{+}D^{-})$ depend only weakly on ${\bf q}$ and
can be taken out of the integral. Moreover, since the binding energy is small
we have $E_{B}\simeq 2E_{D}$ and hence

$M(\gamma\gamma\rightarrow
B)=\psi^{*}(0)\sqrt{\frac{2}{E_{B}}}M(\gamma\gamma\rightarrow D^{+}D^{-})\,.$
(7)

With the amplitude above we calculate the cross section for bound state
production:

$d\sigma=\frac{1}{H}\frac{d^{3}p_{B}}{(2\pi)^{3}}\frac{1}{2E_{B}}(2\pi)^{4}\delta^{(4)}(k+k^{\prime}-p_{B})|M(\gamma\gamma\rightarrow
B)|^{2},$ (8)

where $p_{B}$ is the momentum of the produced bound state and $H$ is the flux
factor. In the center of mass frame of the $AA\to AAB$ collision, we have

$k=(\omega_{1},0,0,\omega_{1})\,,\hskip
28.45274ptk^{\prime}=(\omega_{2},0,0,-\omega_{2})\,,\hskip
28.45274ptp_{B}\equiv p+p^{\prime}=(E_{B},0,0,\omega_{1}-\omega_{2})\,,$ (9)

where $E_{B}=\sqrt{(\omega_{1}-\omega_{2})^{2}+m_{B}^{2}}$ and $\omega_{1}$
and $\omega_{2}$ are the energies of the colliding photons. The flux factor is
then given by $H=8\omega_{1}\omega_{2}$. The integrated cross section reads:

$\sigma(\omega_{1},\omega_{2})=\frac{2\pi}{2(4\omega_{1}\omega_{2})}\int\frac{d^{3}p_{B}}{2E_{B}}\delta(E_{CM}-E_{B})\delta^{(3)}({\bf
k}+{\bf k^{\prime}}-{\bf
p}_{B})\left[\frac{2}{E_{B}}|\psi(0)|^{2}|M(\gamma\gamma\rightarrow
D^{+}D^{-})|^{2}\right]$ (10)

where $E^{2}_{CM}=4\omega_{1}\omega_{2}$. To complete the calculation we need
the wave function of the bound state. In [15] a similar particle made of open
charm mesons was studied with the Bethe-Salpeter equation and an expression
for the wave function was derived. Here we will just quote the final
expression needed to calculate $\psi(0)$, which is given by:

$\psi(0)=\frac{-8\mu\pi g}{(2\pi)^{3/2}}\left(\Lambda_{0}-\sqrt{2\mu
E_{b}}\arctan\left(\frac{\Lambda_{0}}{\sqrt{2\mu
E_{b}}}\right)\right)\,,\hskip 28.45274ptg^{2}=\frac{\sqrt{2\mu
E_{b}}}{8\pi\mu^{2}(\arctan(\frac{\Lambda_{0}}{\sqrt{2\mu
E_{b}}})-\frac{\sqrt{2\mu E_{b}}\Lambda_{0}}{2\mu E_{b}+\Lambda_{0}^{2}})}.$
(11)

In the above expressions $\mu$ is the reduced mass ($\mu=m_{D}/2$),
$\Lambda_{0}$ is a cut-off parameter and $E_{b}$ is the binding energy. We
shall follow [16] and assume that $\Lambda_{0}=1$ GeV. From [16] we see that
one can compute the (dynamically generated) mass of a bound state and then
determine its binding energy. Knowing $\mu$, $E_{b}$ and fixing $\Lambda_{0}$,
we can use (11) to calculate $\psi(0)$. In what follows our reference value
will be obtained using $m_{D}=1870$ MeV and the mass of the bound state equal
to $M_{B}=3723$ MeV, as found in [16]. With these numbers we get $E_{b}=17$
MeV and $|\psi(0)|^{2}=0.008$ GeV3.

The equivalent photon approximation is well known and it is described in
several papers [17]. In general, when the photon source is a nucleus one has
to use form factors and the calculation becomes somewhat complicated. Here we
will follow [18] and define an UPC in momentum space. The momentum
distribution of the equivalent photons created by a source with charge $Ze$ is
[18]:

$n(\vec{q})=\frac{Z^{2}\alpha}{\pi^{2}\omega}\frac{(\vec{q})^{2}}{\left((\vec{q})^{2}+(\omega/\gamma)^{2}\right)^{2}}$
(12)

where $\vec{q}$ is the photon transverse momentum, $\omega$ its energy and
$\gamma$ is given by $\gamma=\sqrt{s}/2m_{p}$, where $m_{p}$ is the proton
mass. In order to obtain the energy spectrum, one has to integrate this
expression over the transverse momentum up to some value $\hat{q}$. The value
of $\hat{q}$ is given by $\hat{q}=\hbar c/2R$, where $R$ is the radius of the
projetile. For Pb, $R\approx 7$ fm and hence $\hat{q}\approx 0.014$ GeV. After
the integration over the photon transverse momentum the photon energy
distribution is given by:

$n(\omega)=\frac{2Z^{2}\alpha}{\pi}\ln\left(\frac{\hat{q}\gamma}{\omega}\right)\frac{1}{\omega},$
(13)

Because of the approximations [18] the above distribution is valid when the
condition $\omega\ll\hat{q}\gamma$ is fullfiled. Using Eq. (13) we can compute
the cross sections of free pair production, $\sigma_{P}$, and of bound state
production, $\sigma_{B}$. They are given by:

$\sigma_{P}(A\,A\rightarrow
A\,A\,D^{+}D^{-})=\int\limits_{m_{D}^{2}/\hat{q}\gamma}^{\hat{q}\gamma}d\omega_{1}\int\limits_{m_{D}^{2}/\omega_{1}}^{\hat{q}\gamma}d\omega_{2}\,\sigma_{P}(\omega_{1},\omega_{2})\,n(\omega_{1})\,n(\omega_{2}),$
(14) $\sigma_{B}(A\,A\rightarrow
A\,A\,B)=\int\limits_{m_{D}^{2}/\hat{q}\gamma}^{\hat{q}\gamma}d\omega_{1}\int\limits_{m_{D}^{2}/\omega_{1}}^{\hat{q}\gamma}d\omega_{2}\,\sigma_{B}(\omega_{1},\omega_{2})\,n(\omega_{1})\,n(\omega_{2}),$
(15)

where $\sigma_{P}(\omega_{1},\omega_{2})$ and
$\sigma_{B}(\omega_{1},\omega_{2})$ are given by Eqs. (4) (with
$\hat{s}=4\omega_{1}\omega_{2}$) and (10) respectively.

|
---|---
(a) | (b)
Figure 2: Cross sections for $D^{+}D^{-}$ bound state production as a function
of the energy $\sqrt{s}$. a) Dependence on $\hat{q}$ for fixed $\Lambda$. b)
Dependence on $\Lambda$ for fixed $\hat{q}$.

In Fig. 1b we show the cross sections for free pair production and compare it
to the existing experimental data from LEP [19]. In fact, the LEP data are for
$e^{+}\,e^{-}\to e^{+}\,e^{-}\,c\,\bar{c}$, i.e., the measured final states
are $D^{+}D^{-}$ and $D^{0}\bar{D}^{0}$. We assume that these two final states
have the same cross section and, in order to compare with the data, we
multiply our cross section $\sigma(e^{+}\,e^{-}\to
e^{+}\,e^{-}\,D^{+}\,D^{-})$ by a factor two. In order to fit these data we
will adapt expression (14) to electron-positron collisions. The
$\gamma\gamma\to D^{+}D^{-}$ cross section is the same but the photon flux
from the electron (and also from the positron) and the integration limits are
different [1, 17, 18]. Comparing our formula with these data, we determine the
only parameter in the calculation, which is the cut-off $\Lambda$. In the
figure, the curves are obtained substituting Eqs. (4) and (13) into (14). In
the latter $\hat{q}=m_{e}$. The band is defined by the choice of two limiting
values of $\Lambda$. In what follows we will use these values to estimate the
uncertainty of our results.

In Fig. 2 shows the cross section for bound state production cross section and
its dependence on $\hat{q}$ (Fig. 2a) and on $\Lambda$ (Fig. 2b). It is
encouraging to see that at $\sqrt{s_{NN}}\approx 5.02$ TeV we have:

$\sigma(Pb\,Pb\rightarrow Pb\,Pb\,B)=3.0^{+0.8}_{-1.2}\,\,\mu b$ (16)

This number should be compared with results found in [10] and in [20]. In
those papers, the production cross section of scalar states $X(3940)$ and
$X(3915)$ in $Pb-Pb$ UPCs at $\sqrt{s_{NN}}=5.02$ TeV were calculated and the
results were in the range

$5\leq\sigma(Pb\,Pb\rightarrow Pb\,Pb\,R)\leq 11\,\,\mu b$ (17)

where $R$ stands for $X(3940)$ or $X(3915)$. In both papers the X states were
treated as meson molecules, as in the present work. It is reassuring to see
that, in spite of the differences, the obtained cross sections are not so
different. The cross section (16) should also be compared with the results
obtained in [21] for the production of the same state treated as a tetraquark.
Interestingly in that work the authors find $\sigma(Pb\,Pb\rightarrow
Pb\,Pb\,B)=0.18\,\,\mu b$, more than one order of magnitude smaller than (16).
The existence of this significant difference is auspicious for our scientific
goal, namely, to use UPCs to discriminate between hadron molecules and
tetraquarks. For more information about the material presented in this section
we refer the reader to the article [22].

##  3 Production of very forward pions

Some time ago, several calculations [23, 24] have shown that in high energy
nuclear collisions a very strong magnetic field is produced. Since then, the
effects of these fields have been looked for in different physical processes.
Perhaps the most famous one is the chiral magnetic effect [25]. Another effect
was investigated in [26]. In that work it was argued that in an ultra-
peripheral collision magnetic excitation (ME) can lead to pion production. In
a ME the projectile creates a $\vec{B}$ field which causes a “splin-flip” in a
nucleon in the target, resulting in the process $N\to\Delta$. After the
excitation, the $\Delta$ decays into pions: $\Delta\to N+\pi$. The pions
produced in this way have extremely large rapidities. This is in contrast to
all other particle production processes in UPC, in which the produced
particles have small rapidities. Hence the appearance of very forward pions
would be a confirmation of ME and would be a clear manifestation of the
magnetic field. In [26] and [27] this process was treated in two different
ways.

Figure 3: a) Classical magnetic transition: a moving projectile creates a
magnetic field $\vec{B}$ which acts on the target at rest (at the origin of
coordinates) flipping its spin. b) Quantum version of the same transition.

In what folows we will briefly review the two calculations and compare them.

Let us start considering the $Pb-p$ collision depicted in Fig. 3a, where the
proton is at rest. The incoming nucleus creates a magnetic field which
converts the proton into a $\Delta$. As an example, let us focus on the
transition $|p\uparrow\rangle\to|\Delta^{+}\uparrow\rangle$. The corresponding
amplitude reads [26]:

$a_{fi}=-i\int_{-\infty}^{\infty}e^{iE_{fi}t^{\prime}}\langle\Delta^{+}\uparrow|H(t^{\prime})|p\uparrow\rangle\,dt^{\prime}$
(18)

where $E_{fi}=(m_{\Delta}^{2}-m_{p}^{2})/2m_{p}$, $m_{\Delta}$ is the $\Delta$
mass and $m_{p}$ is the proton mass. The Hamiltonian reads:

$H(t)=-\vec{\mu}.\vec{B}(t)\,\,\,\,\,\,\,\,\,\,\,\,\,\,\mbox{with}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\vec{\mu}=\sum_{i=u,d}\vec{\mu}_{i}=\sum_{i=u,d}\frac{q_{i}}{m_{i}}\vec{S}_{i}$
(19)

where $q_{i}$ and $m_{i}$ are the charge and the mass of the constituent quark
of type $i$ and $\vec{S}_{i}$ is the spin operator acting on the spin state of
this quark.

The system of coordinates is shown in Fig. 3a. The $\vec{B}$ field is given by
[26]:

$B_{z}(t)=\frac{1}{4\pi}\frac{qv\gamma(b-y)}{((\gamma(x-vt))^{2}+(y-b)^{2}+z^{2})^{3/2}}$
(20)

where $v\simeq 1$ and $q=Ze$. The required spin wave functions are:

$|p\uparrow\rangle=\frac{1}{3\sqrt{2}}[udu(\downarrow\uparrow\uparrow+\uparrow\uparrow\downarrow-2\uparrow\downarrow\uparrow)+duu(\uparrow\downarrow\uparrow+\uparrow\uparrow\downarrow-2\downarrow\uparrow\uparrow)+uud(\uparrow\downarrow\uparrow+\downarrow\uparrow\uparrow-2\uparrow\uparrow\downarrow)]$
(21)
$|\Delta^{+}\uparrow\rangle=\frac{1}{3}(uud+udu+duu)(\uparrow\uparrow\downarrow+\uparrow\downarrow\uparrow+\downarrow\uparrow\uparrow)$
(22)

Now we substitute Eq. (20) into Eq. (19) and the latter into Eq. (18). Then,
using the states given above, we calculate the sandwiches of $H$, obtaining
the final amplitude. The cross section for the process $p\to\Delta$ reads:

$\sigma=\int|a_{fi}|^{2}\,d^{2}b=\frac{Z^{2}e^{4}}{9\pi
m^{2}}\left(\frac{E_{fi}}{v\gamma}\right)^{2}\int_{R}^{\infty}\Big{[}K_{1}\Big{(}\frac{E_{fi}b}{v\gamma}\Big{)}\Big{]}^{2}b\,db$
(23)

For our purpose it is sufficient to have only one proton as target.

In the formalism developed in [27] the cross section of reaction depicted in
Fig. 3b reads [1]:

$\sigma=\int\frac{d\omega}{\omega}\,n(\omega)\,\sigma_{\gamma N\rightarrow
N\pi}(\omega)$ (24)

where $n(\omega)$ is given by [1]:

$n(\omega)=\frac{Z^{2}\alpha}{\pi}\Bigg{[}2\xi
K_{0}(\xi)K_{1}(\xi)-\xi^{2}[K_{1}^{2}(\xi)-K_{0}^{2}(\xi)]\Bigg{]},\hskip
56.9055pt\xi=\frac{\omega(R_{1}+R_{2})}{\gamma}$ (25)

In the above expression $\omega$ is the photon energy, $R_{i}$ is the radius
of nucleus $i$. The Lorentz $\gamma$ factor is in the target frame. In the LHC
$\gamma\simeq 1000$.

The cross section of the process $\gamma p\rightarrow p\pi$ can be computed
from the Feynman graph depicted in Fig. 3b. A simple parametrization of the
$\pi^{0}$ photoproduction cross section was introduced in [27]. Knowing
$\sigma_{\gamma N\rightarrow N\pi}$, we use it to calculate the cross section
(24). In Fig. 4 we plot together the obtained quantum (24) (solid lines) and
semi-classical (23) (dashed line) cross sections. The band shows the
uncertainty related to the decay width $\Gamma$ [27]. The difference between
the results obtained with (23) and with (24) is small and reaches 9 % at the
highest energies.

Figure 4: Cross sections for pion production obtained with the semi-classical
formalism, Eq.(23), dashed line, and with the quantum formalism, Eq. (24),
solid line. $E_{Lab}=\gamma\,m_{n}$ is the energy per nucleon in the
laboratory frame.

The measurement of these ultra-forward pions may be challenging, but there is
some hope. In fact, ultra-forward neutral pions have already been detected in
proton-proton and proton-lead collisions at the LHC [28]. Unfortunately, in
those measurements it was not possible to focus only on ultra-peripheral
collisions. We hope that this could be done in the future.

##  4 Conclusion

In this work the cross section for the production of a $D^{+}D^{-}$ molecule
in ultra-peripheral collisions was calculated. It is $\sigma_{B}(AA\to
AAB)=3.0^{+0.8}_{-1.2}\,\,\mu b$ for $\sqrt{s_{NN}}=5.02$ TeV. This number is
consistent with the results obtained for other scalar exotic charmonium
molecules in Ref. [10] and in Ref. [20]. The parameters of the calculation are
the hadronic form factor cut-off, the maximum transverse momentum of an
emitted photon and the binding energy. All these parameters can be constrained
by experimental information and/or by calculations and hence the precision of
our calculation can be increased. The method used here can be easily applied
to other exotic states.

We have also calculated the cross section for the production of very forward
pions. We have used two methods, one with a classical magnetic field and the
other with equivalent photons. Both methods yield a similar result: a quite
large cross section for forward pion production. The neutral pions can in
principle be measured. This would improve our knowledge about the validity of
the classical approximation and about the strength of the magnetic field
created in these collisions.

## Acknowledgments

We are deeply indebted to K. Khemchandani, A. Martinez Torres, A. Szczurek and
A. Esposito for instructive discussions. This work was partially financed by
the Brazilian funding agencies CNPq, CAPES, FAPESP, FAPERGS and INCT-FNA
(process number 464898/2014-5). F.S.N. gratefully acknowledges the support
from the Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP).

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