Quantum model of strategic decision-making in Prisoner s Dilemma game: quantitative predictions and new empirical results

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1 Quantum model of strategic decision-making in Prisoner s Dilemma game: quantitative predictions and new empirical results Jakub Tesař, Charles University in Prague, Institute of Political Studies, jakub.tesar@fsv.cuni.cz Abstract: The Disjunction effect introduced in the famous study by Shafir and Tversky (1992) and confirmed by several following studies remains one of key anomalies for the standard model of the Prisoner s Dilemma game. In the last 1 years, new approaches have appeared that explain this effect with the use of quantum probability theory. But the existing results do not allow parameter-free test of these models. This paper introduces a general quantum model of Prisoner s Dilemma game as well as a new experimental design that enables to test the main quantitative and qualitative predictions of this model. Main findings of the experiment support the viability of the quantum model and challenge the classical solution concept of the Prisoner s Dilemma game. Keywords: Disjunction effect; Prisoner s Dilemma; Quantum probability theory; Order effect 1 Introduction dilemma of cooperation Game theory is one of the key formal tools in many fields of the social science. It aims to represent a broad range of interactions among human (or non-human) agents, using simple formalism of games. Some of these games are used extensively in social sciences to explain human behavior in more complex circumstances. This is also the case of the Prisoner s Dilemma game, which has become a model for individual-level behavior for theories in economy, sociology, political science and many other fields. The Prisoner s Dilemma (PD) game models strategic decision-making of players in the presence of dilemma: whether to cooperate with or to defect the other player. The dilemma stems from the key property of the game the desirable solution of mutual cooperation (which is the Pareto optimal strategy profile) is for both players dominated by the strategy of defection. The solution concepts of standard decision theory pick defection as the only rational strategy in the game. However, as was shown in many studies (for a review see Camerer 23), the considerable percentage of players choose cooperation as their preferred strategy. Since 195, when the game was formalized by Albert Tucker, we learned many important features of PD game. The high level of cooperation was to a large extent explained by an alternation of the utility function due to the other-regarding preferences or the confidence building in the repeated games (the strategy tit for tat and others).

2 But the study by Shafir and Tversky (1992) discovered another anomaly in PD game that challenges the concept of the rational decision-making itself the Disjunction effect. In their study players met PD game in three different conditions. In different rounds they picked their strategy with: a) No information about the opponent, b) the information that their opponent had defected, or c) the information that their opponent had cooperated. The frequencies of cooperation (strategy C) were 37% in the no-information treatment, 16% after the cooperation of the opponent and 3% after his defection. These results contradict one of the key predictions of the rational choice theory, the Savage s sure thing principle. According to the sure thing principle, if under state of the world X you prefer action A over B, and if under the complementary state of the world X you also prefer action A over B, then you should prefer action A over B when you do not know the state of the world. (Busemeyer and Bruza 214, 261). Many players in the study clearly contradict this principle: they preferred defection after opponent s cooperation and defection after opponent s defection, but chose cooperation in no-information treatment. The results were replicated and extended in several following studies (Croson 1999; Li and Taplin 22; Hristova and Grinberg 28; Shu Li et al. 21; see Busemeyer and Bruza 214, for a review), which showed that the PD game produces a robust Disjunction effect. As was already said, this result contradicts not the specific assumption about the utility function, but the logic of rational choice, as defined by the rational choice theory, itself. Tversky and Shafir explain the result with the combination of the wishful thinking when the opponent s response was not known, many subjects preferred to cooperate, perhaps as a way of inducing cooperation from the other. (Shafir and Tversky 1992, 458) and nonconsequential evaluation. Their theory was able to explain a wide range of Disjunction effects but only by constraining the rationality of the players. The wishful thinking and the nonconsequential reasoning were considered as the examples of bounded rationality and expanded the list of possible phenomena that differentiate the formal game theory of fullyrational agent from the behavioral game theory of the real human beings in the laboratory experiments. This view was challenged by the introduction of the quantum models of cognition and decision-making (see below) which offered the alternative concept of reasoning based on the non-commutative logic of quantum probability theory. When applied to the problem of PD game this logic explains the known effects (e.g. the Disjunction effect) not by depriving the agents of their rationality but by changing the concept of rationality itself. Quantum model can be considered as the generalization of the standard model of strategic decision-making. It releases some of the intrinsic assumptions of the standard model (namely the compatibility of perspectives) and provides more degrees of freedom to fit the data. 2

3 The Disjunction effect is than explained as a kind of order effect of reasoning in two different (non-compatible) perspectives. But the existing data offer only one orderings of perspectives the studies mentioned above inquire an effect of getting 1 the opponent s strategy on the level of players cooperation. Our study extends previous works by adding reverse order of decisions: how does the players choice of a strategy influence their expectation about the choice of their opponent. The second ordering of questions allows us to test the quantum model with a non-parametric test. Our aim in this paper is threefold: i) to introduce a general quantum model of two-players game, including quantitative and qualitative predictions for the PD game; ii) to present results of the experiment which took place in 216 at the Ohio State University, OH, USA and iii) to test the quantum model against the results of this experiment and compare standard and quantum model in respect to these results. The paper proceeds as follows. In the next part we introduce the quantum model of PD game, in two basic versions of the twodimensional and 4-dimensional model. Then we derive three prediction of a general quantum model. Next we present the results of the PD game experiment and tests the prediction of the model against them. In conclusion, we show what are the consequences of the quantum model for the classical solution concepts of PD game. 2 Quantum model of the Disjunction effect In this section I present a quantum model of decision-making in the PD game. The quantum explanation for this problem was originally suggested by Busemeyer et al. (Busemeyer, Matthews, and Wang 26). Since then many other models have been proposed (Pothos and Busemeyer 29; Khrennikov and Haven 29; Aerts 29; Yukalov and Sornette 211; Accardi, Khrennikov, and Ohya 29; see Busemeyer and Bruza 214 for a review). These model share the same basic assumption that the state of the cognitive system and the procedure of decision-making can be modeled by the quantum probability theory (so called *-algebra) instead of the standard (Kolgomorov) probability theory (σ-algebra). The models differ in many aspect: in the construction of the initial state, dimensionality of the problem, etc. My approach, that I present below, consider the PD game as the case of the sequential measurement of two incompatible perspectives and will use the quantum interference model proposed by Busemeyer et al. (211). This model is in many aspects similar to Pothos and Busemeyer (29), but instead of the use of quantum dynamics it explains the existence of the interference effect by incompatibility of the two different perspectives the player can employ to approach the game. One of the most important theoretical questions regarding the quantum model is the assumption about the dimensionality of the problem. In the following section I present both 1 or guessing in (Croson 1999) 3

4 the two-dimensional (2D) and the four-dimensional (4D) model of the game. I will show different predictions these two models offer and will test them against the results of the experiment in the next section. Initial state Prior any treatment in the experiment a player can be described by some initial state which defines the probabilities of individual actions in the game. In the classical probability theory this state is the probability function that maps elementary events (A i ) into probabilities p(a i ) [,1]. Probabilities for the general events are determined as the sum of the probabilities of the elementary events that are contained in the general event. All the probabilities in the sample space sum to one: p(a i ) = 1. i In the quantum probability theory, the state of the system can be described as a unitary vector in N-dimensional vector space over the field of complex numbers 2 with well-defined inner product (i.e. the Hilbert space). This space is defined by N basis vectors that are mutually orthogonal (their inner-product 3 is zero) and which together span the whole space. The choice of the basis vectors is not unique, spaces with N 2 can always be spanned by different sets of basis vectors. The general quantum state is then given as a linear combination of the basis vector from one orthogonal set. In our 2D quantum model of PD game, we define the initial state of the player as the linear combination of the vectors that reflects two possible choices of strategies C (cooperation) and D (defection). The initial state is then: S = ψ C C + ψ D D 4 (2.1a) Here ψ C and ψ D are the coefficients of the linear combination and terms with the parenthesis are so called ket-vectors in the Dirac symbolic (more about vectors and Dirac symbolic in Appendix ). Equation 2.1a can be rewritten, knowing that the basis vectors are still C and D, in the simpler form of Equation 2.1b. S = ( ψ C ψ D ) (2.1b) Any general quantum system is the linear combination of its basis states. We also use a term of superposition, i.e. the quantum state is in a superposition of its basis states. Not allways are all the basis vectors present in the superpostion. If for example the coefficients in our 2D model were ψ C = 1 and ψ D = than the state of the system is identical with the basis state 2 For the introduction to the complex numbers and their properties, see Appendix B. 3 See Appendix. 4 The introduction to the vector spaces and Dirac symbolic can be found in Appendix. 4

5 C and the state D is missing. Nevertheless, the superposition is a general state of the quantum system and in the 2D model it is given by the equation 2.1a. If we approach a game from the perspective of the other player, we gain a new pair of basis vectors C and D. In this basis the state vector is given by the superposition: S = ψ C C + ψ D D (2.2a) S = ( ψ C ψ D ) (2.2b) The coefficients ψ C and ψ D are generally different from ψ C and ψ D. If they equal, there is no difference between considering the game from the two different perspectives, but we hypothesize that it does not apply to the PD game. Therefore, in each perspective, the initial state is defined by the different pair of complex coefficients. Is there any relationship between them? The coefficients are not entirely independent because they represent the same vector in the same Hilbert space. Their relationship (in arbitrary large N) is given by the unitary matrix 5 U. ( ψ C ψ D ) = U ( ψ C ψ D ) (2.3a) ( ψ C ψ D ) = U ( ψ C ψ D ) (2.3b) U = ( C C C D D C D D ) (2.4) In the two-dimensional model, this matrix can be parametrized 6 as: U = e iθ ( eiμ cos θ e iφ sin θ e iφ sin θ e iμ cos θ ) (2.5) If we compere Equation 2.4 and 2.5, it is clear that theoretically given form of the unitary matrix determines the relationship among the inner products of basis vectors from different perspectives. Particularly, using parametrization from Equation 2.5, these conditions has to be satisfied: D D = e 2iμ C C D C = e 2iφ C D 5 See Appendix D. 6 There are 4 independent parameters (θ, μ, φ, θ) in this parametrization. Nevertheless, the element e iθ only shifts the phase of the resulting vector and therefore has any physical meaning (the results do not depend on θ). 5

6 As we will see in the next section, this determines one of the key quantitative predictions for the conditioned probabilities in the 2D mode: the double stochasticity of 2x2 transition matrix (see below). What is the initial state in the 4D model? What we can choose as two other bases? Every player has only two choices, but the game has 4 possible outcomes that match the combination of the strategies of both players. Therefore, the basis states in the self-perspective are CC, CD, DC and DD, where e.g. basis state CC means that the player is playing strategy C assuming that the game will end up with the strategy profile (C, C ). Again, we can write the initial state in the form of the linear combination of the basis states: S = ψ CC CC + ψ CD CD + ψ DC DC + ψ DD DD (2.6a) ψ CC ψ CD S = ( ) (2.6b) ψ DC ψ DD In the other-perspective, the basis is the set of vectors C C, C D, D C and D D, where e.g. C C means that the player expects strategy C from his opponent and that he expects that the opponent chooses it by assuming the final strategy profile (C, C). ψ C C ψ C D S = ( ) (2.7) ψ D C ψ D D The mutual relationship between the two sets of coefficients is given by the 4x4 unitary matrix U, which is defined by 12 independent parameters 7 (Atmanspacher and Römer 212). In the form of the inner products of the basis vectors it is given by: ψ CC ψ CD ψ C C ψ C D ( ) = U ( ) (2.8) ψ DC ψ D C ψ DD ψ D D CC C C CC C D U = ( CC D C CC D D CD C C CD C D CD D C CD D D DC C C DC C D DC D C DC D D DD C C DD C D ) (2.9) DD D C DD D D 7 For the hyperspherical parametrization of 4D unitary matrix see (Hedemann 213). The possible form of the unitary matrix in the PD game was explored also in (Busemeyer and Bruza 214, chap. 9). 6

7 First decision Knowing the initial state of the decision-maker what does quantum model say about the probabilities of the first question? Quantum probability theory states that the probability to measure the system S in one of its basis states is given by a square length of the orthogonal projection of this vector into the respective basis state. This can be determined in two equivalent ways using the inner product or the projection operator. The first way is straightforward. The orthogonal projection of one vector into another is given directly by their inner product. Starting with the 2D model e.g. the projection of vector S into the vector C (which corresponds to the player choosing cooperation) is given by the inner product C S. Then the probability of choosing strategy C is given by Equation 2.1 and (using the coefficient of the linear combination) by Equation p(c) = C S 2 (2.1) C S = C (ψ C C + ψ D D ) C S = ψ C C C + ψ D C D C S = ψ C + ψ D 1 = ψ C p(c) = ψ C 2 (2.11) Second option is to use the projection operator 8 P. The orthogonal projection is given by the matrix multiplication of the projection operator and the state vector: P C S and probability of such choice as a squared length of the product. p(c) = P C S 2 (2.12) This method naturally leads to the same result (Equation 2.11). The other probabilities can be computed analogically: p(d) = ψ D 2, p(c ) = ψ C 2, p(d ) = ψ D 2 But the power of the projection operator is particularly obvious in the case of a combined event, like the cooperation in the 4D model. To construct the projection operator for the combined event we have to sum the outer product of all element basis vectors that contain the relevant event. Therefore, for cooperation we sum the outer products of vector CC and CD. P C = CC CC + CD CD (2.13) p(c) = ψ CC 2 + ψ CD 2 (2.14) 8 See Appendix Chyba! Nenalezen zdroj odkazů.. 7

8 The overall probability of cooperation is therefore the sum of the probabilities of both events that contain the cooperation: p(c) = p(cc ) + p(cd ). Our quantum model is in the first question equivalent to classical model (where p(c) = p(c and C ) + p(c and D ) ). It is the sequence of the non-compatible decisions that produce the Disjunction effect which is anomalous in the classical model. Second decision To analyze the probabilities of the second question we have to work with the state of the system after the first decision. From the previous section we know that by the first decision the state vector is projected into one of its basis vectors (i.e. the new vector is a respective basis vector multiplied by a (complex) number). Each basis vector is chosen with probability from the interval [, 1]. But right after the projection, the system is in that respective state with probability 1. To secure this, the quantum probability theory states that the state of the quantum system right after the first question is normalized to be a unitary vector, which is reached by division of the projected vector by its length: S C = P C S P C S (2.15) With the use of the update vector (that is either S C or S D after the player s first decision) we can determine the probabilities of the second decision by the projection operators P C and P D. But to use it, the operator has to be defined in the same basis as the updated vector. The relevant basis vector of the opponent s perspective has to be transformed using the unitary matrix U (or U in the opposite direction). The opponent s cooperation is given by a basis vector C = U C and his defection by D = U D. The resulting projection operators are P C = C C and P D = D D. The probability of the second question is determined exactly in the same way as of the first question as the square length of the projection of vector S C or S D into one of the basis states in the opponent s perspective. The sequential probabilities for observing player s strategy and then his guess of opponent s strategy are: p(cc ) = P C S C 2 = P C P C S 2 (2.16a) p(cd ) = P D S C 2 = P D P C S 2 (2.16b) p(dc ) = P C S D 2 = P C P D S 2 (2.16c) p(dd ) = P D S D 2 = P D P D S 2 (2.16d) Analogically, the sequential probabilities in the reverse order of question are given by: p(c C) = P C S C 2 = P C P C S 2 (2.19a) 8

9 p(c D) = P D S C 2 = P D P C S 2 (2.19b) p(d C) = P C S D 2 = P C P D S 2 (2.19c) p(d D) = P D S D 2 = P D P D S 2 (2.19d) Predictions of the model Now we can derive three key predictions of the general quantum model and one prediction which is dimensionally specific. First is the non-commutativity of sequential measurements. If we compare two sequential probabilities that differ only in the order of decisions, e.g. p(cd ) and p(d C) from above, we see that they contain the same projection operators, but in reverse order: P D P C respective P C P D. The product of two linear operators is again a linear operator, but if they do not commute (P A P B P B P A ), the resulting operator is dependent on the order of multiplication 9. If this applies, the probability depends on the order of measurement, p(cd ) p(d C). If the person used the same bases in both perspectives, the operators would commute (P A P B = P B P A ) and the result would be order independent. Therefore, assuming the non-compatibility of the different perspectives, non-commutativity of sequential decision-making is the first testable prediction of the quantum model. Second prediction regards the order effect of reasoning. It is the non-commutativity of the decision-making that produce so called interference effect. The probability of the player s cooperation is given by p(c) = P C P C S 2 + P D P C S 2 = P C S 2 when asked as the first question, but as p T (C) = P C P C S 2 + P C P D S 2 if the strategy is chosen after the player s guess of opponent s strategy. The interference effect is the difference between these two probabilities: Int C = p(c) p T (C). For the player s guess of opponent s strategy similarly: p(c ) = P C P C S 2 + P D P C S 2 (2.2) p T (C ) = P C P C S 2 + P C P D S 2 (2.21) p(c ) = p T (C ) + Int C (2.22) The presence of the order effect (or interference effect in quantum terms) is the second prediction of the quantum model flowing from the non-compatibility of perspectives. Third prediction provides a non-parametric test of the quantum model which can be used to falsify it in the experiment. Busemeyer and Bruza (Busemeyer and Bruza 214, ) derived an equation that applies to quantum decision models of any dimensionality. Their solution stems from the property that [t]he probability of transition S A S B is the same as the probability of a transition in the opposite direction S B S A. This property is called 9 Multiplication of two matrices is a non-commutative operation. k A ik B kj 9 k B ik A kj

10 the law of reciprocity in quantum theory. (Busemeyer and Bruza 214, 113). From the law of reciprocity, they derived an invariant that they called a q-test: q = [p(cd ) + p(dc )] [p(c D) + p(d C)] = (2.23a) q = [p(cc ) + p(dd )] [p(d D) + p(c C)] = (2.23b) [A] q-statistic can be computed by inserting the observed relative frequencies, and the above null hypothesis can be statistically tested by using a standard z-test for a difference between two independent proportions (ibid.). Last prediction of our model is dimension-specific. It applies only to the 2D model and can be used for the comparison of the fitness of our two models. It stems from the theoretical form of the unitary matrix that is used for the transition between two different sets of basis states. Recall that in the 2D model the unitary matrix appears in the form: U = ( C C C D D C D D ) The elements of the matrix are inner products of the vectors from both bases. Recall that the inner product defines the length of the orthogonal projection and its square the probability of such projection. If we square every element in the matrix we get a new matrix that is called a transition matrix and define the probabilities of transition between basis vectors in relevant cell of the matrix: T = ( C C 2 C D 2 D C 2 D D 2) (2.24) For example, the first cell in the first column defines the probability of transition between basis vectors C and C. This also equals the conditioned probability p(c C) because: p(cc ) = p(c) p(c C) = P C S 2 P C C 2 p(c C) = P C C 2 = C C 2 (2.25) Due to the properties of the unitary matrix, transition matrix is a double stochastic matrix, i.e. both the rows and the columns sum to one. In particular, using the same parametrization as in Equation 2.5, we get a matrix that is given by a single parameter θ. T = ( cos2 θ sin 2 θ sin 2 θ cos 2 θ ) (2.26) From that follows that in the 2D model the conditioned probabilities has to fulfill the double stochasticity, namely that p(c C) = p(d D) and p(c D) = p(d C). The same derivation 1

11 applies also for the reverse order of questions and therefore p(c C ) = p(d D ) and p(c D ) = p(d C ). In the 4D model the transition matrix is also double stochastic (it stems from the unitary matrix that determines T), but as there are 4 conditioned probabilities in each row and each column of the matrix, the relationship is more complex. In the next section we only use transition matrix as a test of the appropriateness of the 2D model. 3 Methods Experimental design The experiment which tested the prediction of the quantum model consists of several rounds of two-players game and followed the design of the original study by Shafir and Tversky (1992) with some important modifications. In the original study (Shafir and Tversky 1992) the participants played several two-players games, some of them were Prisoner s Dilemma games (payoff matrix of the original study in Table 3.1). Every participant met PD game in three different conditions. In different games he received: a) No information about the opponent, b) the information that his opponent had cooperated (C ), or c) the information that his opponent had defected (D ). The frequencies of cooperation (strategy C) were 37% in the no-information treatment, 16% after the cooperation of the opponent and 3% after his defection. Moreover, studying the choices of the individual players, authors showed that in 25% of the triads individual players exhibited an inconsistency in their preferences: they choose strategy D after C, and strategy D after D, but strategy C in the unknown condition, which violates the Savage s sure thing principle. Prisoner s Dilemma C D C 75, 75 25, 85 D 85, 25 3, 3 Table 3.1 To test the quantum model of PD we modified the original experiment in three important ways. We replaced the information about the opponent s strategy by guessing of it, we added the opposite order of decision-making (choosing own strategy first and then guessing the opponent s strategy), and we added the modified game, which test the preference among outcomes with no influence of the strategic thinking. What is the rationale for these changes? 11

12 We added the guess of the opponent s strategy player that replaces the information the player gets from the experimenter. This solution was introduced by Croson (1999) and, in our experiment, it helps to solve two difficulties. One of them is the uncertainty about the information the player receives. He can doubt that the information is correct or hesitate to exploit it for his own profit as an unfair advantage in an otherwise symmetric game. By guessing the decision of his opponent, the player is forced to consider the perspective of the opponent and act accordingly without introducing any external element that could disturb the game. The second reason is that guessing the opponent s strategy enables us to switch the order of questions to measure the order effect (see below). To test the non-commutativity of the model we added the reverse order of questions. Half of the respondents guesses the opponent s strategy after they have picked their own strategy (the self-perspective) whereas the second half guesses the opponent s strategy first (the other-perspective). To motivate players to think out their tips properly, they will receive a bonus if they guess correctly. Last modification intends to test the level of cooperation without the effect of strategic thinking. As we know from the previous research (e.g. Camerer 23), players are not entirely self-interested in selecting the strategies. Even in the Dictator game when the outcome of the game depends entirely on the decision of a single player, the noticeable share of the players (2-4%) decides to cooperate. To estimate the level of this inclination in our population we added a modified game that preserves the structure of the payoffs in the PD game but leaves the decision entirely on the single player. This was done by transformation of the 2x2 payoff matrix of PD game 1 (Table 3.2) into 4x1 payoff matrix of the Dictator game (Table 3.3 and Table 3.4). In the self-perspective, the outcome of the Dictator game stems from the decision of a single player (the row player in Table 3.3) and is a measurement of his preferences with no effect of strategic thinking. Similarly, in the other-perspective (Table 3.4), players guess the strategy of their opponent without considering their own move (they have only one choice in the game which is to accept whatever their opponent picks). 1 We used the payoffs in the range -5 that directly match the reward in dollars (in the pilot study) or reward in dollars with coefficient.5 (in the MTurk study). In order to make the game more real for the players (they gamble for money directly), we deviated from the original study, but the structure of the game remains the same and the particular numbers in the payoff matrix have no impact on the model. 12

13 Prisoner s Dilemma game 11 C D C 3, 3, 5 D 5,, Table 3.2 Dictator game: self-perspective a a 3, 3 b, 5 c 5, d 1, 1 Table 3.3 Dictator game: other-perspective a b c d a 3, 3 5,, 5 1, 1 Table 3.4 Participants The study was piloted with the undergraduate students at the Ohio State University (N=2), and then run online through the Amazon Mechanical Turk (MTurk) with total number of N=284 participants. The results presented below are the results of the online study on the Amazon MTurk. In the pilot study students played the game in the laboratory settings on the computer with an unknown opponent. Participants were payed $5 for participating and $-6 according to their performance in the study. The mean reward payed in the pilot study was $9.2. In the online study, participants were MTurk users from the eastern Midwest of USA (states of Illinois, Indiana, Michigan, Ohio and Wisconsin). Participants were payed $2.5 for finishing the study and $-3 according to their performance in the game ($-2.5 as a payoff from one randomly chosen game and $.5 as a possible bonus for guessing correctly the strategy of their opponent in the same game). The mean reward payed through the Amazon MTurk was $3.91. We use sex, age, native language, education, population of the city and the 11 Here I use C (or C ) for cooperation and D (D ) for defection (as in the model above). In the experiment, the two strategies were labelled a and b (respective a and b ) to make sure they are value neutral. 13

14 knowledge of the game theory concepts (Game Theory, Pareto Efficiency, Dominated Strategy, Nash Equilibrium) as the control variables. Procedure The experiment was implemented in the Qualtrics survey form (Qualtrics, Provo, UT). The participants were instructed in the rules of the game and, after they correctly answered the control questions that checked their understanding of the instructions, they played ten different games, four of them were the experimental games (Prisoners Dilemma game or the Dictator game). Other games, with different payoffs structures, were interspersed among the experimental games in order to force the participants to consider the experimental games anew. Players were told that they play against a randomly chosen opponent currently present in the game (strategy of their opponent was in fact randomly chosen as the cooperation or defection with probability.5 each). They learned the strategy of their opponent at the end of the game, they had no feedback in between the games. Results presented below are the results of the second round of the games when participants encounter the experimental game for the first time 12, so they are the results of the one-shot Prisoners Dilemma game or the one-shot Dictator game. There were six different treatments (games) the different groups of players encountered in the second round: Dictator game Prisoner s Dilemma game Table 3.5 Selfperspective Otherperspective Selfperspective Otherperspective Players choose their strategy in the DG. Players guess the strategy of their opponent in DG. Players choose their strategy and then guess the strategy of their opponent. Guessing Players guess the strategy of their opponent and then treatment choose their own strategy. Players get the information about the strategy of Bonus their opponent (C or D ) and then choose their own treatment strategy. Variables of the game In sum, the experiment enables to measure several different types of probabilities. Starting with PD game and the self-perspective (choosing own strategy and then guessing the opponent s strategy) we get the sequential probabilities p(cc ), p(cd ), p(dc ) and p(dd ). Here the plain letters denote the player s strategy (C for cooperation and D for defection), primed letters denote the guess of the opponent s strategy (C for opponent s cooperation and D for his defection), and the order matches the order of decision-making. From the se- 12 The data from the other rounds exhibits a strong consistency bias. Players often follow the strategy they have chosen in their first experimental game. 14

15 quential probabilities we know also the overall cooperation of the player: p(c) = p(cc ) + p(cd ), level of cooperation the player expects conditioned by his cooperation: p(c C) = p(cc ), or by his defection: p(c D) = p(dc ), and also the overall expectation of the opponent s cooperation: p T (C ) = p(cc ) + p(dc ). Here the subscript T denotes the total p(c) P(D) probability that stems from the classical law of total probability. The probabilities of defection are the corresponding complements into 1 (e.g. p(d) = 1 p(c) = p(dc ) + p(dd )). By the same token the reverse order of questions enables to measure sequential probabilities p(c C), p(c D), p(d C) and p(d D) and the derived probabilities p(c ), p(c C ), p(c D ), and p T (C). From the Dictator game we get the probabilities of individual strategies p(a), p(b), p(c), p(d) in the self-perspective and p(a ), p(b ), p(c ), p(d ) in the other-perspective. 4 Results of the experiment Replication of previous studies One of the treatments in the experiment was a replication of the original treatment by Tversky and Shafir (1992). When presented the PD game, the participants were said to be in a bonus group which receive the information about the choice of their opponent. Participants were instructed to use the information freely to help you choose your own strategy. They were assured that their strategy will not be revealed to anyone playing with them. This treatment was included to the experiment to verify that it follows the main patterns of the original study, namely that the conditioned probabilities in both treatments exhibit the Disjunction effect. Table 4.1 compares the results from the bonus group treatment and the guessing treatment with the comparable results by Shafir and Tversky (1992) and Croson (1999). bonus treatment Shafir & Tversky (bonus group) guessing treatment Croson (guessing) P(C) p(c C ) p(c D ) Table 4.1 It is clear that both treatment exhibits a Disjunction effect and the conditioned probabilities replicate the results of previous studies (violation of the Sure thing principle). Our group (to- 15

16 gether with the different framing of the experiment) is more cooperative than in the Shafir and Tversky (1992) study but the Disjunction effect is very strong: 8.6% (N=35) of participants chose cooperation after opponent s defection, 21.2% (N=33) after opponent s cooperation, and 64.9% (N=77) cooperated in the no-information treatment. The conditioned cooperation is about three times higher in the guessing treatment (28.6%, N=27 respective 63.%, N=42) but still lower than the level of cooperation without guessing (64.9%). Compatibility of perspectives First hypothesis of the quantum model regards the compatibility of two different perspectives the player can use to approach the game: The self-perspective and the otherperspective (which is the perspective of unknown opponent). In order to be responsible for the Disjunction effect these two perspectives has to produce different probabilities. The measured frequencies of played and guessed strategies for the Dictator game and the PD game are in Table 4.2 and Table 4.3. In the Dictator game (Table 4.2) only two strategies seem to be reasonable for the vast majority of players (94.3% of all players choose one of them). In the self-perspective 48.6% of players chooses an even distribution of the points (strategy a), whereas 51.4% keep the maximum of available points for themselves (strategy c). In the other-perspective, 2.% of the players guess opponent s decision to evenly distribute the points (strategy a ) and 68.6% guess that their opponent chooses to keep all the points for himself (strategy c ). The chisquared test of independence of proportions using only these two strategies returns χ 2 =4.799 (p=.28). Based on this result we can say that the two perspectives differ in the Dictator game. Dictator game self other p(a).486 p(a ).2 p(b). p(b ).114 p(c).514 p(c ).686 p(d). p(d ). N 35 N 35 χ * 13 Table 4.2 If we include also the strategy b, which was guessed by 11.4% of players in the otherperspective, we have two choices. We can add these responses to the cooperative strategies a because in the analogy of PD game this strategy means opponent s cooperation even in 13 This number compares only the relative frequency of two strategies (a and c) in the two perspectives. 16

17 the presence of my (hypothetical) defection. In other words, guessing this strategy, players expect that their opponent will be altruistic and give all the points to them and keep nothing for themselves. In this case, 31.4% of players expect cooperative choice from their opponent which is still lower than the willingness to cooperate in the self-perspective (48.6%), but the difference is no longer significant (χ 2 =2.143, p=.143). The second option is to count the b strategies among the selfish c strategies. The rationale for this is that we can expect that when the players encounter the second-perspective for the first time some of them can choose it by mistake because they have not switched their view and consider strategy b as opponent s selfish strategy. If we accept this explanation than the level of expected cooperation in the other-perspective drops to 2.% and the difference between the two perspectives is consequently bigger (χ 2 =6.341, p=.12). From the data it is not obvious, which of the two presented explanations is correct. It is also possible that the rationale for the choice b differs among the cases (4 players guessed this strategy). Therefore, we decided to not include these cases in our analysis and the test presented in the Table 4.2 is the test of the relative frequency of only 2 strategies a and c (respective a and c ). Table 4.3 includes the sequential probabilities in the PD game. E.g. in the first row (CC ), the self-perspective means that the player chooses a strategy C and then guess opponent s strategy C. In other-perspective (C C), the player guesses the opponent s strategy C and then he chooses his own strategy C. PD game: sequential probabilities self other p(cc ).43 p(c C).246 p(cd ).247 p(c D).145 p(dc ).117 p(d C).174 p(dd ).234 p(d D).435 N 77 N 69 χ * Table 4.3 The data reveals a strong tendency toward the wishful thinking described by Tversky and Shafir (1992). In both orderings players match their strategies with their guesses (and vice versa) in about two thirds of all cases (exact numbers are in Table 5.1). E.g. after playing the strategy C, 62.% of players guess opponent s strategy C, similarly, after choosing D 66.7% 17

18 of players expect opponent s strategy D. Nevertheless, this is not enough to bring a consistence among the sequential probabilities. Players choose cooperation more often then they expect it from their opponents and the frequencies of sequential decision differ substantially (χ 2 =9.896, p=.19). We can conclude that the two perspectives differ also in the PD game. Order effect of sequential decision-making From the previous we know that the sequential probabilities of choosing and guessing the strategy in the PD game differ. Does it apply also for the frequencies of choices presented as the first versus the second question? The overall frequencies of strategies C (choosing cooperation) and C (guessing cooperation) measured in the experiment are summarized in Table 4.4. PD game: total probabilities p(c).649 p(c ).391 p T (C).42 p T (C ).52 Int C Int C.128 χ ** χ Table 4.4 The frequencies are derived from the data above as p(c) = p(cc ) + p(cd ), p T (C) = p(c C) + p(d C) etc. Player s willingness to cooperate is significantly higher when presented as the first question when compared to be presented as the second question (χ 2 =7.689, p=.6). In particular, 64.9% of players choose to cooperate when deciding their strategy first, and this rate drops to 42.% when they consider strategy of their opponent first. The guess of opponent s cooperation shows a similar, but reverse effect. Guessing opponent s strategy first, 39.1% of players expect cooperation, and this number increases to 52.% when players have chosen their own strategy before their guess. Nevertheless, the difference between p(c ) and p T (C ) is not significant (χ 2 =2.48, p=.121). If we analyze this result using the framework of the quantum model, we see that the sequential decision making in PD game exhibits a considerable interference effect. The effect is negative for the player s own strategy the level of cooperation decreases by 22.9% if the choice of strategy is presented as the second question, compared to the first-question choice. When guessing opponent s strategy first, the effect is positive. The level of expected cooperation increases by 12.8% if preceded by the choice of player s own strategy. The interference term is a source of the Disjunction effect described by Tversky and Shafir (1992) and also an important obstacle for the classical solution concept. Recall the classical solution by the dominance equilibrium: The defection is picked as the dominant strategy because it is a preferred strategy after both opponent s cooperation and his defection. But if 18

19 the two questions exhibit strong order effect, how do the decisions after some question determine the decision before that question? Classical solution concept avoids this problem by the intrinsic assumption of the commutativity of the sequential decision-making which leads to the anomaly of Disjunction effect. In the quantum model the order effect is expected given the non-compatibility of the perspectives. Does it mean that the frequencies of sequential decision-makings are entirely contingent? We can test the consistency of the decision-making in the quantum model with a q-test. Q-test: quantum law of reciprocity The last test available is the q-test which explores if the data meets the criteria of the quantum model given by Equation 3.24a (respective 3.24b). We can check the q-test in two equivalent ways. In Table 4.5 there are the sums of frequencies of sequential probabilities in which players chosen strategy and their guess of the opponent s strategy match together. The sum of all matching choices equals 63.6% in the choosing-guessing treatment, whereas the reverse order gives 68.1% of such choices. The difference (and the value of the q-test) is 4.5%. Using the z-test for a difference between two independent proportions we cannot say that the frequencies differ significantly (z=-.5695, p=.569). q-test p(cc ) + p(dd ).636 p(c C) + p(d D).681 q -.45 z Table 4.5 The results of the three tests above reveal that even if the frequencies of the individual or the sequential decisions differ and the same apply also to the frequencies of strategies chosen as the first or the second question, the sum of all matching choices remains invariant among the different orderings. This non-parametric test indicates that there is the consistency among the players strategies that was predicted by the quantum model. Comparing the PD game and the Dictator game The Dictator game had two different roles in the experiment. We used it to test the compatibility of two different perspectives (presented above). Besides this, we propose it as a test of the players preferences among the possible outcomes of the game with no effect of the strategic thinking. How does the level of cooperation differ from the PD game? 19

20 We assume that the framework of the Dictator game sets a different initial state vector, therefor results of the PD game does not match the Dictator game directly. Nevertheless, we assume that the vectors in both games are close enough to produce similar effects. In particular, we assume that level of cooperation in the PD game (self-perspective) corresponds to the level of cooperative choice (the strategy a) in the Dictator game and the same apply also to the players guesses of the opponent s cooperation. The results for both types of games are summarized in Table 4.6. PD game Dictator game 14 p(c).649 p(a).486 p(c ).391 p(a ).226 Table 4.6 In both games, the level of cooperation is significantly higher than expected cooperation from the opponents (see the exact tests in Table 4.2 and Table 4.3). In the Dictator game the willingness to cooperate and the expected cooperation is lower than in the PD game both by about 16.5%. That means that also the two perspectives in both games differ by a similar percentage of choices (25.8% versus 26.%). The results of the Dictator game differ in important aspects but corroborate one of the key findings of the PD game: the players expect much lower cooperation of their opponents then they exhibit in the same game. 5 Discussion Comparison of the models In the previous section we presented results that tested several independent features of the quantum model of the PD game. What do these results imply for the model in general and how does it relate to the standard model of PD game? Non-compatibility of perspectives is problematic for the standard model because it undermines one of the basic assumption of its solution concepts. Recall that e.g. iterated dominance equilibrium or Nash equilibrium are based on the assumption that players are aware not only their own preferences among the outcomes of the game but also the preferences of their opponents. As is shown in Table 4.4, this is not the case. Majority of players (64.9%) choose cooperation but only 39.1% of them expect it from the others. That means, statistically speaking, that only 46.8% 15 of players will guess the strategy of their opponent rightly 14 The value p(a ) is computed as the ratio of a strategies among all a and c strategies, b strategies are omitted (see rationale in the second section of results). 15 p(right guess) = p(c) p(c ) + p(d) p(d ) 2

21 and in only 21.9% 16 of hypothetical pairs both players will be right about their opponent. This is not a good basis for an equilibrium state 17. In the quantum model the noncompatibility of perspectives is possible and, given by the fact that the order effect is present, even expected. In this aspect, the experimental data support qualitative prediction of the quantum model. The presence of the order effect is even more critical for the standard model. Standard model, based on the classical probability theory which is commutative, cannot explain different probabilities caused simply by reverse order of question. For quantum model it is always an option flowing from the fact that matrix multiplication (used as a projector operator, see Appendix Chyba! Nenalezen zdroj odkazů.) is a non-commutative operation. The significance of the q-test is somehow different from the previous features of the model. The q-test should be met both in the classical model and the quantum model. But while the standard model predicts that all the sequential probabilities (Table 4.3) should be same in both perspectives (they are not), quantum model predicts that only certain combination of the probabilities should remain invariant. Therefore, the q-test controls if there is any consistency among the probabilities, it is a necessary condition for both models, but cannot be used to adjudicate between the standard and quantum model. In sum, the data reveals, that the strategies of the players fulfill the q-test, which is the minimum requirement of both models. The models differ in the prediction of the presence of the order effect and non-compatibility of perspectives, which are both present in the data. This supports the quantum model in place of the standard model of PD game. Double stochasticity So far the results meet the qualitative and quantitative predictions of the quantum model. But which model matches the results better? Is the simpler 2D model satisfying, or is it inappropriate and more complex 4D model is needed? To answer this question, we should explore the transition matrix of the experimental game. Recall that the transition matrix contains the probabilities of choosing a particular strategy conditioned by previous guess of opponent s strategy (and vice versa): T = ( C C 2 C D 2 p(c D ) D C 2 D D 2) = (p(c C ) p(d C ) p(d D ) ) Above we showed that the transition matrix has to be a double stochastic matrix, i.e. each of its rows and columns has to sum to one. We can test this condition by constructing the tran- 16 p(both right) = [p(right guess)] 2 17 Choosing and guessing the strategy with a toss of coin would bring higher probability of success: p coin (both right) =

22 sition matrix of the 2D model in both orderings. Individual data can be computed from Table 4.3 as the conditioned probability of individual choices, e.g.: p(c C ) = p(c C) p(c C) + p(c D) The probabilities are summarized in the Table 5.1 and the resulting transition matrix can be seen below (Equation 5.1a and 5.1b). PD game - conditioned choice p(c C ).63 p(c C).62 p(d C ).37 p(d C).38 p(c D ).286 p(c D).333 p(d D ).714 p(d D).667 Table 5.1 p(c C) p(c D).333 ( ) = (.62 p(d C) p(d D) ) (5.1a) p(c C ) p(c D ).286 ( ) = (.63 p(d C ) p(d D ) ) (5.1b) From the transition matrices we see that the condition of double stochasticity is not met precisely. If the matrices had been double stochastic then the rows would sum to one, and equivalently, the elements in the corners of the matrices would equals: p(c C) p(d D) = and p(c C ) p(d D ) =. But these terms return a value of -.47, respective -.85, i.e. the player s own defection attracts his guess of opponent s defection more than his cooperation attracts his guess off opponent s cooperation. In the opposite order, the player s guess of opponent s defection attracts his own defection more than his guess of cooperation attracts his cooperation. We can test these differences by a χ 2 -test: χ 2 =.165 (p=.685) respective χ 2 =.542 (p=.461). The differences are not big enough to reject the 2D model directly, but the data also does not support the model entirely. Especially the second matrix (Equation 5.1b) exhibits non-marginal discrepancy from the double stochasticity. To decide whether the 2D model is appropriate for the PD game, we can explore the possibility that both matrices are double stochastic but they differ from each other. Assume that matrix A is given by a parameter a, and the matrix B by parameter b. p(c C) p(c D) a 1 a ( ) = ( p(d C) p(d D) 1 a a ) (5.2a) 22