2.1 Disjunction effect

The prisoner dilemma game is one of the most well known experimental economic games in the field (Rapoport, Reference Rapoport1988). Briefly, there are two players, call them Ann and Bill, and each player can choose to defect or cooperate. In a one-shot game, they play each other only once. The payoffs for the PD game are arranged so that no matter what choice Ann makes, Bill is better off defecting, and likewise no matter what choice Bill makes, Ann is better off defecting. According to the Nash equilibrium both players should defect, even though both would be better off cooperating. Contrary to this, players frequently cooperate (Rapoport, Reference Rapoport1988) producing better outcomes than mutual defection. This deviation from the Nash equilibrium is commonly explained by additional principles like fairness (Fehr & Schmidt, Reference Fehr and Schmidt1999).

Shafir and Tversky (Reference Shafir and Tversky1992) found something more puzzling than cooperative behavior in the PD game – a result that they called the disjunction effect. Their experiments involved three conditions: (1) In one condition, here called the unknown condition, both players moved simultaneously and remained uncertain about the opponent's decision; (2) in a second condition, here called the known defect condition, before her own move, Ann was informed that Bill defected; and (3) and in a third condition, here called the known cooperate condition, before making her move, Ann was informed that Bill cooperated. Their main finding was that the percentage of cooperation in the unknown condition (37%) was significantly greater than the percentage of cooperation in the case when the opponent was known to defect (3%) and in the case when the opponent was known to cooperate (16%). Shafir and Tversky (Reference Shafir and Tversky1992) claimed that this disjunction effect reflected a violation of the sure-thing principle, because 25% of participants chose to defect when the opponent defected and also chose to defect when the opponent cooperated, but then switched and chose to cooperate when the opponent's decision was unknown. Busemeyer and Matthews (Reference Busemeyer, Matthews, Sun and Myake2006) later replicated these findings: the percentage of cooperation in the unknown condition (34%) was greater than that for the case when the opponent was known to defect (9%) and in the case when the opponent was known to cooperate (16%).

The disjunction effect can also be viewed as a violation of a prediction based on the application of the law of total probability. In the unknown condition, there are only two possible moves that the Bill can take. So when Bill's action is unknown, there is some probability, $p\left(C_B\right)$ , she predicts Bill cooperates and some probability $p\left(D_B\right)$ she predicts Bill defects. If she predicts Bill cooperates, then there is a probability, $p\left(C_A\right|C_B)$ , she cooperates; likewise, if Bill defects, then there is a probability, $p\left(C_A\right|D_B)$ , she cooperates. Therefore, the probability Ann cooperates in the unknown condition should equal

(1) $\begin{array}{cc}{p}_T\left(C_A\right)& =p\left(C_B\right)·p\left(C_A\right|C_B)+p\left(D_B\right)·p\left(C_A\right|D_B)\end{array}$

The total probability is a weighted average that must lie in between the two conditional probabilities. Contrary to this prediction, the disjunction effect occurs when the proportion of cooperation in the unknown condition lies above both of the known conditions. In sum, information about the opponent before taking an action reduces the total probability of cooperation.

Shafir and Tversky (Reference Shafir and Tversky1992) explained the results by arguing that the advantages of defection are clear when the opponent's action is known, but these advantages become unclear when the opponent's action is unknown. In the unknown condition, Ann clearly knows that Bill can either defect or cooperate, so why can’t Ann consider the reasons for each case? This finding can be viewed in terms of Feynman's path integral ideas: if there are two paths a particle can traverse to reach a target and the path is unknown, then they can interfere with each other and cancel each other out. Thus, the intuition from quantum theory is that somehow the two thoughts interfere with each other to produce no clear thought when the Bill's action is unknown.

2.2 Interference of predictions on actions

Croson (Reference Croson1999) conducted 2 studies investigating the effects of predictions on actions using a simultaneous play PD game. These experiments involved two conditions. In one condition (predict-act condition), the player (Ann) was asked to predict the opponent's (Bill's) move before making her own decision; in the other condition (act-alone condition), Ann simply made a decision without expressing any prediction about Bill. The two experiments differed with respect to their payoff matrices. These experiments provided another type of test of the predictions using the law of total probability. In the act-alone condition, there are still only two predictions that Ann can make – predict Bill cooperates or predict Bill defects. If we apply Eq. 1 to this experiment, $p\left(C_B\right)$ is the probability that she predicts Bill will cooperate, $p\left(C_A\right|C_B)$ is the conditional probability that Ann cooperates given Bill cooperates, $p\left(D_B\right)$ is the probability that she predicts Bill will defect, $p\left(C_A\right|D_B)$ is the conditional probability that Ann cooperates given Bill defects. So the total probability that Ann cooperates without expressing her prediction, $p\left(C_A\right),$ is expected to equal the left hand side of Eq. 1. The results turned out contrary to this expectation: In the first experiment, 77.5% of the participants in the act-alone chose to cooperate, but this dropped to 55% for the predict-act condition; in a second experiment, 62.5% cooperated in the act-alone condition, which dropped to 42.5% in the predict-act condition. The percentage of cooperative predictions was $45\%$ in the first study and $42\%$ in the second study, and the correlation between predictions and actions was $\varphi =0.48$ in the first study and $\varphi =0.30$ in the second study. In sum, making a prediction before taking an action reduced the total probability of cooperation. Simply asking participants to report their prediction reduced the overall percentage of cooperation.

2.3 Question order effects

Tesar (Reference Tesar2020) conducted an experiment that included: (a) like Shafir and Tversky (Reference Shafir and Tversky1992), they manipulated the information about the other opponent's (Bill's) move before the player (e.g., Ann) made her decision; (b) like Croson (Reference Croson1999), they manipulated whether or not the player predicted the opponent's (Bill) before the player (e.g., Ann) made her decision; and (c) a new third manipulation that changed the order so that the player (e.g., Ann) first decided an action, and then predicted the opponent's (Bill's) decision. Tesar (Reference Tesar2020) replicated both findings by Shafir and Tversky (Reference Shafir and Tversky1992) and Croson (Reference Croson1999), but also found that the percentage of cooperation was 65% when the player (e.g. Ann) made a decision before prediction, but this percentage was significantly reduced to 42% when the player (Ann) predicted the opponent (Bill) first. The percentage of cooperative predictions was higher, $52\%$ , in the act-predict order as compared to the $39\%$ for the predict-act order, and contingency correlation between predictions and actions was lower, $\varphi =0.26,$ in the act-predict order as compared to $\varphi =0.36$ for the predict-act order. In sum, changing the order of predictions and actions changed the probability of predicting cooperation, the probability of acting cooperatively, as well as the correlation between predictions and actions.

2.4 Sequential prisoner dilemma game

Blanco et al. (Reference Blanco, Engelmann, Koch and Normann2014) examined the effects of prediction on actions in a sequential prisoner's dilemma game in which one player (say Ann) moves first and the second player (say Bill) responds to the first players move. If Ann defects, then both players receive a payoff of 10 units. If Ann cooperates, then the payoffs depend on Bill's move: if Bill cooperates both players gain a payoff of 14 units; but if Bill defects, he wins 17 units while Ann receives only 7. The subgame perfect equilibrium is defection for Bill, and therefore defection for Ann. The experiment included three conditions: In the prediction condition, each player predicted how many players (out of a total on nine in a session) would cooperate as second movers; in the feedback group, each player was told the number of others that cooperated as second movers; in the baseline group, no prediction was made nor any feedback provided. The percentage of cooperation on the first move for the prediction condition was was 55%; the percentage of cooperation for the feedback condition was 57%; but the percentage of cooperation for the baseline condition was only 28%. The percentage of players predicted to cooperate in the second stage was 51%, and there was a high, $r=0.87$ , correlation between predictions about second stage and first stage actions. In sum, asking for a prediction in the sequential game increased the probability of cooperation.

2.5 Promise effects

Kvam et al. (Reference Kvam, Busemeyer, Lambert-Mogiliansky, Atmanspacher, Haven, Kitto and Rene2014) examined a version of the PD game that allowed for “cheap talk.” Each player played several repetitions with computer agents and each repetition involve a sequence of four stages. During stage 1, they were asked to choose whether or not they were willing to promise agent A that they would cooperate in a future PD game with agent A; during stage 2, they played a standard PD game with a different agent B; during stage 3, they played a standard PD game against agent A that they faced in the first stage; and in stage 4, they played several games with other agents without any promises. Half the computer agents had a reputation for cooperation and half had a reputation for defection. The key comparison was the within participant rate of cooperation to agent B in the second stage (immediately following the promise question) as compared to the unrelated fourth stage (which did not include promises). Once again we can apply Eq. 1, in which case we expect the total probability of cooperating on the second stage (pooled across a promise or no promise on stage 1) should equal the probability of cooperating without any promise being considered (stage 4). Kvam et al. (Reference Kvam, Busemeyer, Lambert-Mogiliansky, Atmanspacher, Haven, Kitto and Rene2014) found that there was little difference between cooperation rates when playing an agent with a defecting reputation: in both cases the rate was very low at 16%. However, when playing against agents with a cooperative reputation, the rate of cooperation was significantly higher for the second stage (47%) as compared to the fourth stage (40%).

2.6 Summary of empirical findings

The empirical results reviewed above all indicate violations of expectations based on the application of the law of total probability to choices in the prisoner dilemma game. There are five main findings. First, the disjunction effect refers to the finding that the probability of cooperation when not knowing the action of the opponent falls below both of the conditional probabilities for the known actions of the opponent. Second, the interference of prediction on action refers to the finding that prediction about an opponent changes the later probability of cooperation (pooled over predictions) compared to action without prediction. Third, the direction of the interference changes across types of games: In the simultaneous play version of the PD game, predictions decrease later cooperation, but in the sequential play version, prediction increase later cooperation. Fourth, the question order effect refers to the finding that cooperation is lower when the predictions are made before actions as compared to the.reverse order. Fifth, asking a player to make a promise to one opponent can affect the player's cooperation to later unrelated opponents.

Regarding the direction of interference effects, Blanco et al. (Reference Blanco, Engelmann, Koch and Normann2014) point out that the difference between their results and Croson (Reference Croson1999) was most likely produced by the difference in best response to the opponent's expected move in the two studies. The dominant strategy in the Croson (Reference Croson1999) experiments was to defect, whereas the best response to the expected number of cooperators in the Blanco et al. (Reference Blanco, Engelmann, Koch and Normann2014) study was to cooperate.

Finally, interference effects, defined as the difference between the total probability of action after making a prediction (pooled across predictions) and the the probability of action when no prediction is made, are not limited to the PD game. These effects have also been reported using gambling tasks (Tversky & Shafir, Reference Tversky and Shafir1992; Broekaert et al., Reference Broekaert, Busemeyer and Pothos2020) and categorization-decision tasks (Busemeyer & Lambert-Mogiliansky, Reference Busemeyer and Lambert-Mogiliansky2009).

3.1 Tesar model

Tesar (Reference Tesar2020) proposed a simple two dimensional quantum model for the disjunction effect in the PD game based on the following ideas. First consider the condition in which the player (e.g., Ann) predicts her opponent's (Bill's) action, and then makes her own decision.

Ann's predictions are assumed to be based on a belief state that has a potential ${\alpha }_D$ for predicting defection and another potential ${\alpha }_C$ for predicting cooperation. In general, these potentials can be complex numbers, but their squared magnitudes must sum to one. These two potentials form a vector ${\psi }_U=\left(\begin{array}{c}{\alpha }_D\\ {\alpha }_C\end{array}\right)$ . The vector ${\psi }_U$ represents Ann's initial state when Bill's actions are unknown. To compute the probability that Ann predicts Bill defects, denoted $p\left(D_B\right),$ we first define a projector $P_D=diag\left(\begin{array}{cc}1& 0\end{array}\right)$ that is used to pick the first coordinate in ${\psi }_U$ , and then compute the probability from the squared projection: $p\left(D_B\right)={\left(P_D,·,{\psi }_U\right)}^2={\left({\alpha }_D\right)}^2.$ The probability that Ann predicts Bill cooperates is then computed from $p\left(C_B\right)={\left(P_C,·,{\psi }_U\right)}^2={\left({\alpha }_C\right)}^2,$ where $P_C=I-P_D$ and I is the identity matrix.

If Ann eventually does predict (or is told) that Bill defects, then her belief state is updated according to ${\psi }_D=\frac{P_D·{\psi }_U}{\sqrt{p\left(D_B\right)}}=\left(\begin{array}{c}1\\ 0\end{array}\right)$ . Likewise if Ann eventually does predict (or is told) that Bill cooperates, then her belief state is updated according to ${\psi }_C=\frac{P_C·{\psi }_U}{\sqrt{p\left(D_C\right)}}=\left(\begin{array}{c}0\\ 1\end{array}\right)$ .

To evaluate her own action decision, Ann rotates her state from a prediction point of view to an action point of view based using a $2\times 2$ unitary “rotation” matrix $U=\left(\begin{array}{cc}{u}_{\mathrm{DD}}& u_{\mathrm{DC}}\\ u_{\mathrm{CD}}& u_{\mathrm{CC}}\end{array}\right),$ where for example $u_{\mathrm{DC}}$ represents the transition to action state D from prediction state C. This unitary matrix is a complex valued, ortho-normal matrix, which is length preserving. Ann's new state after evaluation then equals $U·{\psi }_j$ , $j=C,D,U.$

The projectors for Ann deciding to be cooperative versus defective is represented by the same diagonal matrices as before, $P_C=diag\left(\begin{array}{cc}0& 1\end{array}\right)$ , and $P_D=diag\left(\begin{array}{cc}1& 0\end{array}\right)$ , respectively. The probability that Ann chooses the action to cooperate, given that she predicts Bill cooperates (also in the case that she is told Bill cooperated) equals $p\left(C_A\right|C_B)={\left((U^{†}·P_C·U),·,{\psi }_C\right)}^2={\left(u_{\mathrm{CC}}\right)}^2,$ and her probability to cooperate given she predicts Bill defects equals $p\left(C_A\right|D_B)={\left((U^{†}·P_C·U),·,{\psi }_D\right)}^2={\left(u_{\mathrm{CD}}\right)}^2$ .

Based on the above computations, the probability that Ann predicts Bill defects and she decides to cooperate equals

(2) $\begin{array}{cc}p(D_B,C_A)& =p\left(D_B\right)·p\left(C_A\right|D_B)\\ & ={\left((U^{†}·P_C·U),·,P_D,·,{\psi }_U\right)}^2\end{array}$

The effect of question order is produced by reversing the processing order

(3) $\begin{array}{cc}p(C_A,D_B)& ={\left(P_D,·,(U^{†}·P_C·U),·,{\psi }_U\right)}^2\end{array}$

Finally, consider the case when Ann directly decides her own action when she does not predict (or when she remains uncertain about Bill's action). In this case, she starts from the initial superposition state over predictions, ${\psi }_U$ . Then her probability to cooperate for the unknown (or act - only) condition equals

(4) $\begin{array}{cc}p\left(C_A\right|U)& ={\left(P_C,·,U,·,{\psi }_U\right)}^2\\ & ={\left(u_{\mathrm{CD}},·,{\alpha }_D,+,u_{\mathrm{CC}},·,{\alpha }_C\right)}^2\\ & ={\left(u_{\mathrm{CD}},·,{\alpha }_D\right)}^2+{\left(u_{\mathrm{CC}},·,{\alpha }_C\right)}^2+2·Real\left(u_{\mathrm{CC}}^{\ast },·,\alpha _C^{\ast },·,u_{\mathrm{CD}},·,{\alpha }_D\right)\\ & =p\left(D_B\right)·p\left(C_A\right|D_B)+p\left(C_B\right)·p\left(C_A\right|C_B)+In{t}_C.\end{array}$

Note that $P_C^{†}{P}_C+P_D^{†}{P}_D=I$ (the identity), which guarantees that $p\left(C_A\right|U)+p\left(D_A\right|U)=1.$ The term, $In{t}_C,$ is the interference. If $In{t}_C=0,$ then $p\left(C_A\right)$ obeys the law of total probability (Eq. 1). Positive $In{t}_C$ increases cooperation for $p\left(C_A\right)$ above that predicted by the total probability; negative $In{t}_C$ decreases cooperation for $p\left(C_A\right)$ compared to that predicted by the total probability. Choosing a unitary matrix U to produce positive interference can then be used to account for the observed disjunction effect.

Although this model contains several unknown parameters, it does make two very strong a priori predictions. First of all, the unitary nature of the rotation matrix U requires double stochasticity: the squared lengths of the rows equal one and so do the squared lengths of the columns. In the case of a $2\times 2$ doubly stochastic matrix, this implies $p\left(C_A\right|D_B)=p\left(D_A\right|C_B)$ . Second the reversed operations used to produce the question order effect require satisfaction of the QQ equality (see Busemeyer & Bruza, Reference Busemeyer and Bruza2012, Ch. 3): $QQ=p(C_A,D_B)+p(D_A,C_B)=p(D_B,C_A)+p(C_B,D_A)$ . Both of these predictions were supported in the data reported by Tesar (Reference Tesar2020). However, the double stochasticity condition is strongly violated by the results of Shafir and Tversky (Reference Shafir and Tversky1992) where they found $p\left(C_A\right|D_B)=0.03$ and $p\left(D_A\right|C_B)=0.83.$

3.2 Yukalov and Sornette model

Yukalov and Sornette (Reference Yukalov and Sornette2011, Reference Yukalov and Sornette2014) proposed a four dimensional model to explain the disjunction effect summarized as follows. They begin by assuming a (e.g., Ann's) cognitive state represented by a $4\times 1$ unit length vectorFootnote ²

(5) $\begin{array}{c}\psi =\left(\begin{array}{cccc}\psi _{\mathrm{DD}}^{\ast }& \psi _{\mathrm{DC}}^{\ast }& \psi _{\mathrm{CD}}^{\ast }& \psi _{\mathrm{CC}}^{\ast }\end{array}\right)_{}^{†}.\end{array}$

The first index stands for the Ann's prediction, and the second index stands for Ann's action. For example, ${\psi }_{\mathrm{CD}}$ is the potential to predict Bill cooperates but Ann defects.

The key idea of this model is that Ann's decision to cooperate in the unknown (or the act-alone) condition is not based only on the cognitive state $\psi$ . Instead it is generated from another vector ${\pi }_C=\frac{1}{\sqrt{K}}{\left(\begin{array}{cccc}0& {\varphi }_1^{\ast }& 0& {\varphi }_2^{\ast }\end{array}\right)}^{†},$ called the prospect for cooperation. The probability to cooperate for the unknown (or act-only) condition is computed from the squared magnitude of the inner product

(6) $\begin{array}{cc}& p\left(C_A\right|U)={\left({\pi }_C^{†},·,\psi \right)}^2\\ & =\left(\varphi _1^{\ast },·,{\psi }_{\mathrm{DC}},+,\varphi _2^{\ast },·,{\psi }_{\mathrm{CC}}\right)_{}^2\\ & =\left(\varphi _1^{\ast },·,{\psi }_{\mathrm{DC}}\right)_{}^2+\left(\varphi _2^{\ast },·,{\psi }_{\mathrm{CC}}\right)_{}^2\\ & +2·Real\left(\varphi _1^{\ast },·,{\psi }_{\mathrm{DC}},·,{\varphi }_2,·,\psi _{\mathrm{CC}}^{\ast }\right)\\ & =p_{\mathrm{DC}}+p_{\mathrm{CC}}+In{t}_C\end{array}$

with $p_{\mathrm{DC}}=\left(\varphi _1^{\ast },·,{\psi }_{\mathrm{DC}}\right)_{}^2$ and $p_{\mathrm{CC}}=\left(\varphi _2^{\ast },·,{\psi }_{\mathrm{CC}}\right)_{}^2$ .

Ann's decision to defect in the unknown (or act-only) condition is generated from the cognitive state by using another vector called the prospect for defection defined by ${\pi }_D=\frac{1}{\sqrt{K}}{\left(\begin{array}{cccc}{\phi }_1^{\ast }& 0& {\phi }_2^{\ast }& 0\end{array}\right)}^{†}$ . The probability to defect during the unknown (or act-alone) condition is then computed from the squared magnitude of the inner product

(7) $\begin{array}{cc}& p\left(D_A\right|U)={\left({\pi }_D^{†},·,\psi \right)}^2\\ & =\left(\phi _1^{\ast },·,{\psi }_{\mathrm{DD}},+,\phi _2^{\ast },·,{\psi }_{\mathrm{CD}}\right)_{}^2\\ & =\left(\phi _1^{\ast },·,{\psi }_{\mathrm{DD}}\right)_{}^2+\left(\phi _2^{\ast },·,{\psi }_{\mathrm{CD}}\right)_{}^2\\ & +2·Real\left(\phi _1^{\ast },·,{\psi }_{\mathrm{DD}},·,{\phi }_2,·,\psi _{\mathrm{CD}}^{\ast }\right)\\ & =p_{\mathrm{DD}}+p_{\mathrm{CD}}+In{t}_D\end{array}$

where we have defined $p_{\mathrm{DD}}=\left(\phi _1^{\ast },·,{\psi }_{\mathrm{DD}}\right)_{}^2$ , $p_{\mathrm{CD}}=\left(\phi _2^{\ast },·,{\psi }_{\mathrm{CD}}\right)_{}^2$ .

The normalization factor K is needed to satisfy $p\left(C_A\right|U)+p\left(D_A\right|U)=1$ ; i.e., K is selected to satisfy ${\left({\pi }_C^{†},·,\psi \right)}^2+{\left({\pi }_D^{†},·,\psi \right)}^2=1.$ Note that the normalization factor K depends on the state $\psi .$ Again the empirical fact that $p\left(C_A\right|U)+p\left(D_A\right|U)=1$ implies that $In{t}_C+In{t}_D=0$ , which then implies that $p_{\mathrm{CC}}+p_{\mathrm{DC}}+p_{\mathrm{CD}}+p_{\mathrm{DD}}=1.$

Thus, the distribution $(p_{\mathrm{CC}},p_{\mathrm{DC}},p_{\mathrm{CD}},p_{\mathrm{DD}})$ can be interpreted as a classical joint probability distribution. These joint probabilities are factored into marginal and conditional probabilities so that $p_{\mathrm{ij}}=p\left(i\right)·p\left(j\right|i),$ where p(i) is interpreted as the probability that Bill takes action i, and p(j|i) is interpreted as the probability that Ann takes action j given Bill takes action i. This model is then applied to the results by setting p(i) equal to the observed proportion that the opponent (e.g., Bill) is predicted (or is told) to take action i, and setting p(j|i) equal to the observed proportions of the player's (e.g. Ann's) action given the opponent's action.

Finally, in order to make a prediction about the probability to cooperate when Bill's action is unknown, Yukalov and Sornette (Reference Yukalov and Sornette2011) use what they call the interference quarter law: assuming a flat prior across interference terms, the expected interference is $\pm \frac{1}{4}$ , and only the sign of the interference needs to be determined. So finally, the probability to cooperate in the unknown condition equals

$\begin{array}{c}p\left(C_A\right|U)=p\left(C\right)·p\left(C\right|C)+p\left(D\right)·p\left(C\right|D)\pm \frac{1}{4},\end{array}$

If $In{t}_C=+.25,$ , then the model produces cooperation for $p\left(C_A\right|U)$ above the total probability, and if $In{t}_C=-.25,$ , then the model produces cooperation for $p\left(C_A\right|U)$ below the total probability. For example, to account for the Shafir and Tversky (Reference Shafir and Tversky1992) results, the interference needs to be positive.

The use of the quarter law makes this theory highly testable with respect to predicting the size of the interference. However, there are several issues with this model. One problem is that the model does not attempt to predict the marginal or conditional response probabilities, and instead it simply estimates them from the data. Another problem is that the model does not explain the change in the sign of the interference across studies. A third is that the interference term is theoretically bounded by the observed choice probabilities (see Appendix A), but the quarter law ignores this bound. In fact, a violation of the bound occurs when applying the quarter law to the disjunction effects Shafir and Tversky (Reference Shafir and Tversky1992); Busemeyer and Matthews (Reference Busemeyer, Matthews, Sun and Myake2006) (see Appendix A). A fourth problem is that the interference is not consistently well approximated by the quarter law. For example, considering the study by Kvam et al. (Reference Kvam, Busemeyer, Lambert-Mogiliansky, Atmanspacher, Haven, Kitto and Rene2014), the empirically observed interference equals .07 which is far below the quarter law. Finally, the model has not been developed to account for question order effects.

3.3 Pothos and Busemeyer model

Pothos and Busemeyer (Reference Pothos and Busemeyer2009) also proposed a 4 dimensional quantum model. The player's (e.g., Ann's) cognitive state is again represented as a 4 dimensional unit length vector generally described by Eq. 5. However, Pothos and Busemeyer (Reference Pothos and Busemeyer2009) assumed the player (e.g. Ann) starts out the game in an initial belief state, ${\psi }_U=\frac{1}{\sqrt{2}}\left(\begin{array}{cccc}\varphi _1^{\ast }& \varphi _1^{\ast }& \varphi _2^{\ast }& \varphi _2^{\ast }\end{array}\right)_{}^{†}.$ The parameter ${\varphi }_1$ represents the potential for Ann to predict Bill defects, and ${\varphi }_2$ is the potential to predict Bill cooperates, and $|{\varphi }_1{|}^2+{\left|{\varphi }_2\right|}^2=1.$ This initial state begins by assigning equal potentials, $\frac{1}{\sqrt{2}}$ for Ann to take the cooperation or defection actions.

The projector for predicting Bill cooperates is represented by a diagonal matrix $P_{\mathrm{PC}}=diag\left(\begin{array}{cccc}0& 0& 1& 1\end{array}\right)$ , which picks out the potentials for predicting Bill cooperates; the projector for predicting Bill defects is the complement $P_{\mathrm{PD}}=I-P_{\mathrm{PC}},$ where I is the identity. The probability that Ann predicts Bill cooperates then equals $p\left(C_B\right)={\left(P_{\mathrm{PC}},·,{\psi }_U\right)}^2={\left|{\varphi }_2\right|}^2$ and the probability she predicts Bill defects is $p\left(D_B\right)={\left(P_{\mathrm{PD}},·,{\psi }_U\right)}^2={\left|{\varphi }_1\right|}^2.$

If Bill is known or predicted to cooperate, then Ann's belief state is projected to ${\psi }_C=\frac{P_{\mathrm{PC}}·\psi }{\sqrt{p\left(C_D\right)}}=\frac{1}{\sqrt{2}}{\left(\begin{array}{cccc}0& 0& 1& 1\end{array}\right)}^{†}$ . Likewise, if Bill is known or predicted to defect, then Ann's initial belief state is projected to ${\psi }_D=\frac{1}{\sqrt{2}}{\left(\begin{array}{cccc}1& 1& 0& 0\end{array}\right)}^{†}$ . If Bill's action is unknown or not predicted, then Ann remains in the initial unknown belief state, ${\psi }_U$ but also note that ${\psi }_U=\left({\varphi }_1,·,{\psi }_C,+,{\varphi }_2,·,{\psi }_D\right)$ .

Before an action can be chosen, Ann needs to evaluate the actions based on the payoffs of the game. This evaluation is represented by a unitary transformation, U, of the current state ${\psi }_j,j=C,D,U$ to produce a new action state $U·{\psi }_j.$ We describe how U is built later in this subsection.

A decision is generated probabilistically from the new action state by using a projector that picks out the potentials associated with an action. The projector for Ann to cooperate is given by $P_{\mathrm{AC}}=diag\left(\begin{array}{cccc}0& 1& 0& 1\end{array}\right)$ , and the projector for Ann to defect equals $P_{\mathrm{AD}}=diag\left(\begin{array}{cccc}1& 0& 1& 0\end{array}\right).$ The probability for Ann to cooperate then equals the squared magnitude of the projection of the action state on cooperative actions $p\left(C_A\right|j)={\left(P_{\mathrm{AC}},·,U,·,{\psi }_j\right)}^2,$ and the probability for Ann to defect equals the squared magnitude of the projection of the action state on defection actions $p\left(D_A\right|j)={\left(P_{\mathrm{AD}},·,U,·,{\psi }_j\right)}^2.$ More specifically, if Ann predicted (or was told) Bill cooperated, then her probability to cooperate is $p\left(C_A\right|C_B)={\left(P_{\mathrm{AC}},·,U,·,{\psi }_C\right)}^2$ ; and if Ann predicted (or was told) Bill defected then her probability to cooperate is $p\left(C_A\right|D_B)={\left(P_{\mathrm{AC}},·,U,·,{\psi }_D\right)}^2;$

If Bill's action remains uncertain, then

(8) $\begin{array}{cc}p\left(C_A\right|U)& ={\left(P_{\mathrm{AC}},·,U,·,{\psi }_U\right)}^2\\ & ={\left(P_{\mathrm{AC}},·,U,·,\left({\varphi }_1,·,{\psi }_C,+,{\varphi }_2,·,{\psi }_D\right)\right)}^2\\ & ={\left({\varphi }_1,·,P_{\mathrm{AC}},·,U,·,{\psi }_C,+,{\varphi }_2,·,P_{\mathrm{AC}},·,U,·,{\psi }_D\right)}^2\\ & ={\varphi }_1^2·{\left(P_{\mathrm{AC}},·,U,·,{\psi }_C\right)}^2+{\varphi }_2^2·{\left(P_{\mathrm{AC}},·,U,·,{\psi }_D\right)}^2\\ & +Real\left({\psi }_C^{†},·,U^{†},·,P_{\mathrm{AC}}^{†},·,\varphi _1^{\ast },·,{\varphi }_2,·,P_{\mathrm{AC}},·,U,·,{\psi }_D\right)\\ & =\frac{1}{2}p\left(C_A\right|C_P)+\frac{1}{2}p\left(C_A\right|D_P)+In{t}_C.\end{array}$

Note that $P_{\mathrm{AC}}^{†}{P}_{\mathrm{AC}}+P_{\mathrm{AD}}^{†}{P}_{\mathrm{AD}}=I$ (the identity), which guarantees that $p\left(C_A\right|U)+p\left(D_A\right|U)=1.$ Once again the interference can be positive or negative with this model, depending on the unitary transformation U.

Similar to the Tesar model, this model predicts order effects by reversing the order of the projections

(9) $\begin{array}{cc}p(D_B,C_A)& ={\left(\left(U^{†},·,P_{\mathrm{AC}},·,U\right),·,P_{\mathrm{PD}},·,{\psi }_U\right)}^2\\ p(C_A,D_B)& ={\left(P_{\mathrm{PD}},·,\left(U^{†},·,P_{\mathrm{AC}},·,U\right),·,{\psi }_U\right)}^2.\end{array}$

Pothos and Busemeyer (Reference Pothos and Busemeyer2009) construct the unitary rotation, U, from the payoffs for each action as follows. The unitary transformation was based on the matrix exponential, $U=exp\left(-,i,·,\frac{\pi }{2},·,H\right)$ of a Hamiltonian built from two parts

$\begin{array}{cc}H& =H_1+H_2\\ H_1& =\left(\begin{array}{cccc}\frac{{\mu }_1}{\sqrt{1+{\mu }_1^2}}& 1& 0& 0\\ 1& \frac{-{\mu }_1}{\sqrt{1+{\mu }_1^2}}& 0& 0\\ 0& 0& \frac{{\mu }_2}{\sqrt{1+{\mu }_2^2}}& 1\\ 0& 0& 1& \frac{-{\mu }_2}{\sqrt{1+{\mu }_2^2}}\end{array}\right),H_2=\frac{-\gamma }{\sqrt{2}}\left(\begin{array}{cccc}1& 0& 1& 0\\ 0& -1& 0& 1\\ 1& 0& -1& 0\\ 0& 1& 0& 1\end{array}\right).\end{array}$

The first part $H_1$ is determined by the game payoffs. The upper left $2\times 2$ sub-matrix of $H_1$ rotates the potentials for the actions in the directions determined by the utility parameter ${\mu }_1$ , with ${\mu }_1$ an increasing function of the advantage for defecting if the opponents defects; the lower right $2\times 2$ sub-matrix of $H_1$ rotates the potential for the actions in directions determined by the utility parameter ${\mu }_2$ , with ${\mu }_2$ an increasing function of the advantage for defecting if the opponents cooperates.

The second part $H_2$ implements what Shafir and Tversky (Reference Shafir and Tversky1992) called “wishful thinking’ – Ann wishes that the own strategy is same with Bill's strategy. It coordinates actions with beliefs by increasing the potentials for ${\psi }_{\mathrm{CC}},{\psi }_{\mathrm{DD}}$ and decreasing the potentials for ${\psi }_{\mathrm{CD}},{\psi }_{\mathrm{DC}}.$ The second part of the Hamiltonian is critical for producing interference effects: the sign of the interference effect depends on the sign of $\gamma$ , and if $\gamma =0$ then no interference effects can occur Pothos and Busemeyer (Reference Pothos and Busemeyer2009). This model also accounts for order effects using the same reversal of order of projections as used in the Tesar model. In sum, this model accounts for the interference effects, and it also accounts for question order effects.

Pothos and Busemeyer (Reference Pothos and Busemeyer2009) applied the model to the disjunction effect using only two parameters. They fixed the initial belief state parameters to ${\varphi }_1={\varphi }_2=\sqrt{.5}$ , and they constrained ${\mu }_1={\mu }_2=\mu$ , because the advantage for defecting was the same amount regardless of whether the opponent cooperated or defected in these studies. Using the estimates $\mu =0.5263$ and $\gamma =2.2469$ fit to the experiment by Busemeyer and Matthews (Reference Busemeyer, Matthews, Sun and Myake2006) produces the following model predictions: $p\left(C_A\right|C_B)=0.28,p\left(C_A\right|D_B)=0.18,p\left(C_A\right|U)=0.35$ . The model is able to produce a disjunction effect, but not as extreme as the results of the Busemeyer and Matthews (Reference Busemeyer, Matthews, Sun and Myake2006) experiment.

3.4 Denolf and Martínez model

Denolf et al. (Reference Denolf, Martínez-Martínez, Josephy and Barque-Duran2016) developed a quantum model for the Blanco et al. (Reference Blanco, Engelmann, Koch and Normann2014) experiment. They proposed a simple 2-dimensional model for this experiment. Accordingly, Ann's initial state is a $2\times 1$ column vector $\psi ={\left(\begin{array}{cc}{\psi }_C& {\psi }_D\end{array}\right)}^{†}.$ For simplicity, they assumed that these potentials are real valued. The initial state has unit length so that ${\psi }_D^2=1-{\psi }_C^2$ . If no prediction is made (i.e., the baseline condition), then the probability that Ann cooperates on the first stage simply equals $p\left(C_A\right)={\left({\psi }_C\right)}^2$ .

Recall that in the prediction condition of this experiment, the player (e.g., Ann) had to predict how many of the other 9 other players would cooperate. To represent the 10 possible predictions about the number of players, they propose that each prediction corresponds to a projector $P_j=B_j·B_j^{†}$ for prediction $j=0,1,\dots 9$ , where $B_j$ is a unit length $2\times 1$ vector.The two extreme belief states (0, 9) are assumed to be orthogonal $\left(B_0^{†},·,B_9,=,0\right)$ . The intermediate belief states are spaced at equal increments in angles between these two extremes. The probability of predicting $j=0,\dots ,9$ is determined from the squared length of the projection $\psi$ on vector $B_j$ , which equals ${\left(P_j,·,\psi \right)}^2={\left(B_j^{†},·,\psi \right)}^2$ . However, ${\sum }_j{P}_j^{†}·P_j\ne I$ so that $K={\sum }_k{\left(B_j^{†},·,\psi \right)}^2\ne 1$ and the squared projections need to be divided by a normalization factor K, which depends on the state $\psi$ , to obtain the probability: $p\left(j\right|\psi )=\frac{1}{K}·{\left(B_j^{†},·,\psi \right)}^2.$

Ann's state after predicting j is reduced to the unit vector $B_j$ . The event representing Ann's decision to cooperate on the first stage is represented by a projector $P_C=A·A^{†}$ , where A is a unit length $2\times 1$ vector. The event representing Ann's decision to defect on the first stage is the complement $P_D=I-P_C.$ The probability that Ann decides to cooperate on the first stage after making the prediction j is then equal to the squared length of the projection of her current state $B_j$ on the event representing cooperation, A, which equals $p\left(C_A\right|j)={\left(P_C,·,B_j\right)}^2={\left(A^{†},·,B_j\right)}^2$ . Likewise, the probability that Ann defects after making the prediction j equals $p\left(D_A\right|j)={\left(P_D,·,B_j\right)}^2$ , and note $P_C^{†}{P}_C+P_D^{†}{P}_D=I$ so that $p\left(C_A\right|j)+p\left(D_A\right|j)=1$ .

An interference effect occurs because

(10) $\begin{array}{cc}{\left({\psi }_C\right)}^2& \ne \sum _{j}p\left(C_A\right|j)·p\left(j\right|\psi )\\ & =\frac{1}{K}\sum _j{\left(A^{†},·,B_j\right)}^2·{\left(B_j^{†},·,\psi \right)}^2.\end{array}$

The difference between the left and right hands sides of Eq. 10 can determine the interference effect.

The Denolf et al. (Reference Denolf, Martínez-Martínez, Josephy and Barque-Duran2016) model includes as free parameters two angles to determine cosines between the belief vector $B_0$ and $\psi$ and $B_0$ and A; plus the potential ${\psi }_C$ in $\psi$ representing the initial state. After fitting these parameters to the data, the model closely reproduce all the results of Blanco et al. (Reference Blanco, Engelmann, Koch and Normann2014). However, this model has never been used to account for the disjunction effect or question order effects.

3.5 Kvam and Mogiliansky model

Kvam et al. (Reference Kvam, Busemeyer, Lambert-Mogiliansky, Atmanspacher, Haven, Kitto and Rene2014) developed a “type indeterminacy” model (Lambert-Mogiliansky et al., Reference Lambert-Mogiliansky, Zamir and Zwirn2009) for the effects of promises on playing the PD game. They postulated three types of tendencies that a person could have in a PD game: a tendency $T_1$ to cooperate, a tendency $T_3$ to defect, and a moderate tendency $T_2$ to cooperate if the agent cooperates and defect if the agent defects. A person (e.g., Ann) is not exactly any one of these types, and instead she is superposed over the types. During the first stage, these types have different probabilities to choose to promise or not promise cooperation.

They represent the Ann's state at stage 1 as a $6\times 1$ unit length column vector

$\begin{array}{c}\psi ={\left(\begin{array}{cccccc}{\psi }_1·{\alpha }_{P|1}& {\psi }_1·{\alpha }_{N|1}& {\psi }_2·{\alpha }_{P|2}& {\psi }_2·{\alpha }_{N|2}& {\psi }_3·{\alpha }_{P|3}& {\psi }_3·{\alpha }_{N|3}\end{array}\right)}^{†}.\end{array}$

(This model is restricted to real values). For example, ${\psi }_2^2$ is the probability that Ann becomes a type 2 agent, ${\alpha }_{P|2}^2$ is the probability that Ann promises if she is a type 2 agent, and ${\alpha }_{N|2}^2$ is the probability that Ann does not promise if she is a type 2 agent. The potentials of this state vector depend on the reputation (cooperative or not cooperative) of the computer agent.

Let's consider the situation of an agent with a cooperative reputation. In this case, they assume that ${\alpha }_{P|1}=1$ , ${\alpha }_{P|2}=1$ , ${\alpha }_{N|3}=1$ (for an agent that has a non-cooperative reputation, ${\alpha }_{P|2}=0)$ . So when Ann plays a cooperative agent, denoted here as CA, her initial state is ${\psi }_{\mathrm{CA}}={\left(\begin{array}{cccccc}{\psi }_1& 0& {\psi }_2& 0& 0& {\psi }_3\end{array}\right)}^{†}.$

Choosing to promise is represented by the projector $P_P=diag\left(\begin{array}{cccccc}1& 0& 1& 0& 1& 0\end{array}\right)$ , and choosing not to promise is represented by $P_N=I-P_P$ , where I is the identity. The probability that Ann promises to cooperate to a cooperative agent equals $p\left(P\right)={\left(P_P,·,{\psi }_{\mathrm{CA}}\right)}^2={\psi }_1^2+{\psi }_2^2$ ; and if Ann chooses to promise, then her new conditional state equals ${\psi }_P=\frac{P_P·{\psi }_{\mathrm{CA}}}{\sqrt{p\left(P\right)}}.$ The probability that Ann does not promise to a cooperative agent equals $p\left(N\right)={\left(P_N,·,{\psi }_{\mathrm{CA}}\right)}^2={\psi }_3^2$ ; and if Ann chooses not to promise, then her new conditional state equals ${\psi }_N=\frac{P_N·{\psi }_{\mathrm{CA}}}{\sqrt{p\left(N\right)}}.$ If no promise is made, then Ann remains in the initial state ${\psi }_{\mathrm{CA}}$ for a cooperative agent.

The decision to cooperate or not with agent B in the second stage requires an evaluation about the actions. This evaluation is represented by a $6\times 6$ rotation matrix U. The rotation matrix has only three rotation parameters, because it only operates on the subspace containing potentials $({\psi }_1,{\psi }_2,{\psi }_3)$ . The new state after evaluation is $U·{\psi }_i$ , where $i=P,N,CA$ . Using the evaluation basis, the decision to cooperate is represented by a projector $P_C=diag\left(\begin{array}{cccccc}1& 0& 1& 0& 1& 0\end{array}\right)$ , and the projector for the decision to defect equals $P_D=I-P_C.$ The probability that Ann cooperates after promising to cooperate equals $p\left(C\right|P)={\left(P_C,·,U,·,{\psi }_P\right)}^2;$ the probability that Ann cooperates after not promising to cooperate equals $p\left(C\right|N)={\left(P_C,·,U,·,{\psi }_N\right)}^2;$

The total probability that Ann's cooperates at stage 2, after choosing to promise or not, then equals $p_T\left(C\right)=p\left(P\right)·p\left(C\right|P)+p\left(N\right)·p\left(C\right|N)$ . The probability that Ann cooperates at stage 4 when there are no promises to make is equal to $p\left(C\right)={\left(P_C,·,U,·,{\psi }_{\mathrm{CA}}\right)}^2,$ which can be decomposed into the total probability plus interference in the same way as Eq. 8. The interference term then accounts for the effect of promises made of stage 1 to decisions made on stage 4. Although this model was not applied to question order effects, it could be applied in the same way as the Tesar and Pothos-Busemeyer models by reversing the order of projection operations (as in Eq. 9).

Kvam et al. (Reference Kvam, Busemeyer, Lambert-Mogiliansky, Atmanspacher, Haven, Kitto and Rene2014) estimated $\psi$ for each type of agent from the data for each computer agent reputation. The unitary matrix U was derived from three rotation parameters also estimated from the data. These parameters were then used to predict the cooperation data from stages 2 and 4 for each computer agent reputation, which provided an accurate fit the data. However, this model has never been used to account for the Shafir and Tversky (Reference Shafir and Tversky1992) disjunction effect.

3.6 Asano Ohya Tanaka Basieva Khrennikov model

Asano et al. (Reference Asano, Ohya, Tanaka, Basieva and Khrennikov2011b) and later Asano et al. (Reference Asano, Basieva, Khrennikov, Ohya and Tanaka2012) proposed a dynamic quantum operations model for the PD game. The state (e.g., Ann's state) regarding each action is initially represented by a $2\times 1$ unit length vector $\psi ={\left(\begin{array}{cc}{\psi }_C& {\psi }_D\end{array}\right)}^{†}$ . This vector is used to form what is called a density matrix in quantum theory, denoted $\rho$ , which is defined by the outer product: $\rho \left(0\right)=\psi ·{\psi }^{†}.$ The diagonal values of $\rho$ , denoted ${\rho }_C$ for the upper diagonal and ${\rho }_D$ for the lower diagonal, contain the probabilities for Ann to cooperate and defect, respectively. It is hypothesized that Ann vacillates between thinking about cooperating and defecting, and during this deliberation, the density matrix evolves over time according to the following quantum updating operation ( $\tau$ represents a small time increment for each update)

(11) $\begin{array}{cc}\rho (t+\tau )& ={\Gamma }_1\left(\tau \right)·\rho \left(t\right)·{\Gamma }_1^{†}\left(\tau \right)+{\Gamma }_2\left(\tau \right)·\rho \left(t\right)·{\Gamma }_2^{†}\left(\tau \right),\\ {\Gamma }_1\left(\tau \right)& =\left(\begin{array}{cc}\sqrt{1-{\Delta }_D\left(\tau \right)}& 0\\ 0& \sqrt{1-{\Delta }_C\left(\tau \right)}\end{array}\right),\\ {\Gamma }_2\left(\tau \right)& =\left(\begin{array}{cc}0& \sqrt{{\Delta }_C\left(\tau \right)}·e^{i{\theta }_2}\\ \sqrt{{\Delta }_D\left(\tau \right)}·e^{i{\theta }_1}& 0\end{array}\right).\end{array}$

Note that ${\Gamma }_1^{†}{\Gamma }_1+{\Gamma }_2^{†}{\Gamma }_2=I$ , which guarantees that ${\rho }_C\left(t\right)+{\rho }_D\left(t\right)=1$ across time. Assuming $0<{\Delta }_i<1$ , $i=1,2$ , then the off diagonal elements of $\rho \left(t\right)$ eventually decay toward zero, and the diagonal elements converge toward the asymptote

$\begin{array}{cc}{\rho }_C& =\frac{{\Delta }_C}{{\Delta }_D+{\Delta }_C}\\ {\rho }_D& =\frac{{\Delta }_D}{{\Delta }_D+{\Delta }_C}.\end{array}$

The updating weights, ${\Delta }_i,i=C,D$ are determined by two amplitudes, ${\mu }_C$ and ${\mu }_D$ as follows:

$\begin{array}{cc}{\Delta }_C\left(\tau \right)& =\tau ·\frac{{\left({\mu }_C\right)}^2}{{\left({\mu }_C\right)}^2+{\left({\mu }_D\right)}^2},{\Delta }_D\left(\tau \right)=\tau ·\frac{{\left({\mu }_D\right)}^2}{{\left({\mu }_C\right)}^2+{\left({\mu }_D\right)}^2}.\end{array}$

The two amplitudes, ${\mu }_C$ and ${\mu }_D$ , are in turn derived from the beliefs about Bill's moves and the utilities of the payoffs to Ann.

The basic intuition is that Ann imagines the advantages produced by different possible changes. The notation ${\mu }_{CC\to CD}$ symbolizes the potential for the strategies to change from the CC (Bill cooperates, Ann cooperates) to the CD (Bill cooperates, Ann defects) pair, and the magnitude of this potential is an increasing function of the payoff advantage to Ann for making this change. Likewise, ${\mu }_{DC\to DD}$ is the potential for changing from the CD to DD, ${\mu }_{DC\to CD}$ is the potential for changing from DC to CD, and ${\mu }_{DD\to CC}$ is the potential for changing from the DD to the CC. Then the potentials ${\mu }_D$ and ${\mu }_C$ are defined as

(12) $\begin{array}{cc}{\mu }_D& ={\alpha }_1·{\mu }_{CC\to CD}+{\alpha }_2·{\mu }_{DC\to DD}+{\alpha }_3·{\mu }_{DC\to CD}\\ {\mu }_C& ={\alpha }_4·u_{DD\to CC.}\end{array}$

The coefficients, ${\alpha }_i,i=1,\dots ,4$ , represents beliefs that these transitions occur.

The disjunction effect can be obtained from this model because the potentials ${\mu }_D$ and ${\mu }_C$ change for known and unknown conditions. In particular, the “wishful thinking” transition $DD\to CC$ is only possible if Ann is uncertain about Bill's action.

The main advantage of this model is that it provides a description of the dynamic evolution of preference during Ann's decision. However, this model has not been developed to account for predictions that a player might make concerning an opponent's play, and consequently, it is unable to account for question order effects.

3.7 Martínez Sánchez-Burillo model

Martínez-Martínez and Sánchez-Burillo (Reference Martínez-Martínez and Sánchez-Burillo2016) also proposed a dynamic quantum model for the PD game. Once again, they begin by assuming a cognitive state (e.g., Ann's) represented by a $4\times 1$ unit length vector $\psi$ shown in Eq. 5.

As mentioned above, states in quantum theory can be represented by a vector, such as $\psi$ , or by what is called a density matrix. The density operator produced by the state vector $\psi$ is formed from the outer product of the cognitive state: $\rho =\psi ·{\psi }^{†}$ . The diagonals of the density matrix contain the action probabilities.

The key new idea is that the density matrix evolves across time according to the following quantum master equation

(13) $\begin{array}{cc}\frac{d}{\mathrm{dt}}\rho \left(t\right)& =-i·\left(1,-,\alpha \right)·\left(H,,,\rho \right)+\alpha ·L\left(\rho \right),\\ L\left(\rho \right)& =\sum {\gamma }_{\mathrm{ij}}·\left(L_{\mathrm{ij}},·,\rho ,·,L_{\mathrm{ij}}^{†},-,\frac{1}{2},\left(L_{\mathrm{ij}}^{†},·,L_{\mathrm{ij}},,,\rho \right)\right),\\ \left(H,,,\rho \right)& =H·\rho -\rho ·H,\\ \left(\left(L_{\mathrm{ij}}^{†},·,L_{\mathrm{ij}}\right),,,\rho \right)& =\left(L_{\mathrm{ij}}^{†},·,L_{\mathrm{ij}}\right)·\rho +\rho ·\left(L_{\mathrm{ij}}^{†},·,L_{\mathrm{ij}}\right)\end{array}$

where $L_{\mathrm{ij}}$ is a $4\times 4$ matrix with a one located at cell (i, j) and zeros elsewhere. The master equation describes the dynamics of what is called an open quantum system.

The first part, $\left(H,,,\rho \right)$ , of the sum in Eq. 13 represents von Neumann dynamics produced by a pure quantum system. The parameter, $h_{\mathrm{ij}}$ , in the $4\times 4$ Hermitian matrix $H=H^{†}$ determines the rate of change in transition to state i from another state j. For simplicity, they set

$\begin{array}{c}H=\left(\begin{array}{cccc}1& 1& 0& 0\\ 1& 1& 0& 0\\ 0& 0& 1& 1\\ 0& 0& 1& 1\end{array}\right).\end{array}$

The second part, $L\left(\rho \right),$ of the sum in Eq. 13 represents the Lindblad dynamics of a Markov system. The parameters ${\gamma }_{\mathrm{ij}},i,j=1,\dots ,4$ form a $4\times 4$ Markov transition matrix, $\Gamma ,$ with ${\gamma }_{\mathrm{ij}}$ equal to the probability of transiting to state i from another state j. These parameters were determined from the payoffs to the player (e.g., Ann) depending on her anticipation regarding the opponent's (e.g., Bill's) move. At each moment in time, there is some probability $\varphi$ that Ann switches from contemplating that Bill cooperates to defecting and visa versa. When contemplating that Bill cooperates, Ann considers defecting with a probability based on the payoff advantage for defecting when Bill cooperates; when contemplating that Bill defects, Ann considers defecting with a probability based on the payoff advantage for defecting when Bill defects.

Martínez-Martínez and Sánchez-Burillo (Reference Martínez-Martínez and Sánchez-Burillo2016) applied the model to the disjunction effect as follows. When told Bill cooperated, Ann's initial state is set equal to ${\rho }_{11}={\rho }_{22}=1/2$ , and ${\rho }_{33}={\rho }_{44}=0$ ; and it evolves for some period of time t using Eq. 13. During this evolution, the switching parameter $\varphi$ fixed to zero so the Ann remains contemplating Bill to cooperate. Then at time t, using the projection matrix $P_C=diag\left(\begin{array}{cccc}0& 1& 0& 1\end{array}\right)$ , the probability that Ann cooperates equals the trace: $p\left(C_A\right|C_B)=Tr\left(P_C,·,{\rho }_C,\left(t\right)\right).$ Likewise, when told Bill defected, Ann's initial state is set to equal ${\rho }_{33}={\rho }_{44}=1/2$ , and ${\rho }_{11}={\rho }_{22}=0$ . Again this state evolves with the switching probability fixed to zero so that Ann remains contemplating Bill to defect. The probability that Ann cooperates under this condition equals the trace: $p\left(C_A\right|D_B)=Tr\left(P_C,·,{\rho }_D,\left(t\right)\right).$ However, when Bill's move is unknown, the initial state remains equal to ${\rho }_{33}={\rho }_{44}=1/4$ , and ${\rho }_{11}={\rho }_{22}=1/4$ ., and it evolves with the switching probability free so that Ann's attention can switch back and forth between Bill cooperating and Bill defecting. Then probability that Ann cooperates equals the trace: $p\left(C_A\right|U)=Tr\left(P_C,·,{\rho }_U,\left(t\right)\right).$

This open system model provides a good way to combine quantum and Markov systems. The system starts out operating in the quantum regime to produce a coherent (quantum) density matrix with non-zero off-diagonal cells, but later the system ends in the Markov regime to produce a de-coherent (classical) density matrix with zero off-diagonal cells.

To account for the disjunction effect, Martínez-Martínez and Sánchez-Burillo (Reference Martínez-Martínez and Sánchez-Burillo2016), set the initial conditions as follows. They used three free parameters: one for the probability to cooperate based on the payoffs, on for the switching probability, and the third was the parameter $0\le \alpha \le 1$ that determines the weight of the contribution of each dynamic process. They were able to fit parameters that could very accurately reproduce the disjunction effect results: $p\left(C_A\right|C_B)=0.16,p\left(C_A\right|D_B)=0.09,p\left(C_A\right|U)=0.34$ Also their analyses show that it is necessary for $\left(1,-,\alpha \right)>0$ to reproduce the disjunction effect (the quantum dynamics are essential). To fit the results of Shafir and Tversky (Reference Shafir and Tversky1992), their estimate of the weight equaled $\alpha =0.81.$ However, this model has not been developed to account for predictions that a player might make concerning an opponent's play, and consequently, it is unable to account for question order effects.

3.8 Tanaka Umegaki Nishiyama Kitoh-Nishioka model

Tanaka et al. (Reference Tanaka, Umegaki, Nishiyama and Kitoh-Nishioka2022) proposed another type of dynamic open system model of the PD game. When applied to the disjunction effect, they also use a four dimensional state represented by Eq. 5, which they convert into a density matrix $\rho =\psi ·{\psi }^{†}.$ For the known and unknown conditions, they used the same initial conditions as used by Martínez-Martínez and Sánchez-Burillo (Reference Martínez-Martínez and Sánchez-Burillo2016).

They assume that the state $\rho$ evolves across time according to Eq. 13, but with $\alpha =0$ . Instead of using a Markov process, they assume that the cognitive state $\psi$ is influenced by many unknown (noisy) environmental influences, which they directly model using Hamiltonians. The total Hamiltonian for Eq. 13 (with $\alpha =0$ ) is composed of two parts: $H=H_S+H_E$ , where $H_S$ is the Hamiltonian for the system represented by $\psi$ , and $H_E$ is the environmental noise. The environment is modeled by a set of quantum-mechanical harmonic oscillators that produce noise with an amount controlled by a temperature parameter T. The environmental influences from $H_E$ eventually stabilize the state toward an equilibrium state, and without the latter, the system would oscillate indefinitely, The Hamiltonian $H_S$ is the key for producing the disjunction effect. They define the system Hamiltonian as

(14) $\begin{array}{c}{H}_S=\left(\begin{array}{cccc}{h}_{\mathrm{DD}}-\delta & \Delta & -\delta & 0\\ \Delta & h_{\mathrm{DC}}+\delta & 0& -\delta \\ -\delta & 0& h_{\mathrm{CD}}+\delta & \Delta \\ 0& -\delta & \Delta & h_{\mathrm{CC}}-\delta \end{array}\right)\end{array}$

where the diagonal elements are determined by the payoffs to Ann for each pair of actions. For example, $h_{\mathrm{CD}}$ is determined by the payoff Ann receives when Bill cooperates and she defects. The parameter $\delta$ is used to produce an effect related to what Shafir and Tversky (Reference Shafir and Tversky1992) called “wishful thinking.” When the opponent's move is known, they set $\delta =0$ , and when the opponent's move is unknown they allow $\delta$ to be a free parameter to be estimate from the data. The value of parameter $\delta$ changes the payoffs of the game, that is, it increases the payoffs for both cooperating and both defecting and it decreases the payoffs for one player cooperating while the other player defects.

Using the two parameters, $\alpha ,\delta$ to determine the system Hamiltonian, $H_S$ , and an additional parameter, T, for the environmental noise in $H_E$ , they were able to approximately reproduce the disjunction effect: $p\left(C_A\right|C_B)=0.22,p\left(C_A\right|D_B)=0.22,p\left(C_A\right|U)=0.44$ . However, this model does provide any account for predictions that a player might make concerning an opponent's play, and consequently, it is unable to account for question order effects.

3.9 Summary of models

Table 1 provides a summary comparison of the eight models with the five findings. Of course, the responses in the table are based on the authors’ judgments and the creators of the models might have a different opinion. At least the table reflects the finding that was the target of the each model. There are, of course, advantages and disadvantages of each model that are not reflected in the table. In particular, some of the models are designed to account for both predictions as well as actions (Tesar, Reference Tesar2020; Pothos & Busemeyer, Reference Pothos and Busemeyer2009; Denolf et al., Reference Denolf, Martínez-Martínez, Josephy and Barque-Duran2016; Kvam et al., Reference Kvam, Busemeyer, Lambert-Mogiliansky, Atmanspacher, Haven, Kitto and Rene2014) while others have not developed this capability (Yukalov & Sornette, Reference Yukalov and Sornette2014; Asano et al., Reference Asano, Ohya, Tanaka, Basieva and Khrennikov2011b; Martínez-Martínez & Sánchez-Burillo, Reference Martínez-Martínez and Sánchez-Burillo2016; Tanaka et al., Reference Tanaka, Umegaki, Nishiyama and Kitoh-Nishioka2022). Some of the models (Pothos & Busemeyer, Reference Pothos and Busemeyer2009; Asano et al., Reference Asano, Ohya, Tanaka, Basieva and Khrennikov2011b; Martínez-Martínez & Sánchez-Burillo, Reference Martínez-Martínez and Sánchez-Burillo2016; Tanaka et al., Reference Tanaka, Umegaki, Nishiyama and Kitoh-Nishioka2022) specify how the payoffs of the game map into the Hamiltonian used to predict the choice probabilities in the game, and some of the models don’t explicitly describe any mapping (Tesar, Reference Tesar2020; Yukalov & Sornette, Reference Yukalov and Sornette2014; Denolf et al., Reference Denolf, Martínez-Martínez, Josephy and Barque-Duran2016; Kvam et al., Reference Kvam, Busemeyer, Lambert-Mogiliansky, Atmanspacher, Haven, Kitto and Rene2014). Some of the models (Asano et al., Reference Asano, Ohya, Tanaka, Basieva and Khrennikov2011b; Martínez-Martínez & Sánchez-Burillo, Reference Martínez-Martínez and Sánchez-Burillo2016; Tanaka et al., Reference Tanaka, Umegaki, Nishiyama and Kitoh-Nishioka2022) are specifically designed to describe the dynamics of the decision in the game. Another important issue concerns differences among the models with respect to complexity and number of parameters. Finally, some quantum models not reviewed here (Bagarello et al., Reference Bagarello, Haven and Khrennikov2017) have been applied to the PD game but not applied to the findings reviewed in this article. All of these models can of course be modified to address some of their limitations, and so this theory development is still under construction.

Table 1 Comparison of models with findings

	Finding
Model	Disjunction	Interference	Question order	Sequential	Promise
Tesar	N	Y	Y	N	?
Yukalov	Y	N	N	N	N
Pothos	Y	Y	Y	N	?
Denolf	N	?	N	Y	N
Kvam	?	?	?	N	Y
Asano	Y	N	N	N	N
Martinez	Y	N	N	N	N
Tanaka	Y	N	N	N	N

Y indicates the existing model can reproduce the finding, ? indicates that the existing model is capable but has not been applied to this finding, N indicates the existing model is not yet capable (but could be modified in the future) to reproduce the finding. Tesar refers to (Tesar, Reference Tesar2020), Yukalov refers to (Yukalov & Sornette, Reference Yukalov and Sornette2014), Pothos refers to (Pothos & Busemeyer, Reference Pothos and Busemeyer2009), Denolf refers to (Denolf et al., Reference Denolf, Martínez-Martínez, Josephy and Barque-Duran2016), Kvam refers to (Kvam et al., Reference Kvam, Busemeyer, Lambert-Mogiliansky, Atmanspacher, Haven, Kitto and Rene2014), Asano refers to (Asano et al., Reference Asano, Ohya, Tanaka, Basieva and Khrennikov2011b), Martinez refers to (Martínez-Martínez & Sánchez-Burillo, Reference Martínez-Martínez and Sánchez-Burillo2016), Tanaka refers to (Tanaka et al., Reference Tanaka, Umegaki, Nishiyama and Kitoh-Nishioka2022)