Download Supplementary Material A: A sample Prisoner`s Dilemma payoff

Supplementary Material A:
A sample Prisoner’s Dilemma payoff matrix, agent, and available choices
Supplemental Material B:
Derivation of the transition matrix for the Type-indeterminate model
In order to ensure that the probabilities of being in each state from the end of the promise
game (which sum to 1) transition to a full type distribution in the Prisoner’s Dilemma game
(probabilities in the PD game also must sum to 1), we use a unitary matrix U to represent the
change of bases that occurs in this transition. A unitary matrix may have complex values, as
may a particular state vector, but the TI model is restricted to the real coordinate space
(Busemeyer & Lambert-Mogiliansky, 2009). Since the state vector is not complex, any complex
rotation in the unitary could result in a complex-valued state vector in the Prisoner’s Dilemma.
Therefore, we include the requirement that the unitary matrix have only real-valued elements. In
addition, there are several other constraints on the unitary matrix. Given
𝛿11
𝑈(3) = �𝛿21
𝛿31
𝛿12
𝛿22
𝛿32
𝛿13
𝛿23 �
𝛿33
(11)
We require that squared values of the rows and columns add up to 1; that is,
∃𝑗, ∑𝑖 𝛿𝑖𝑗 2 = 1
∃𝑖, ∑𝑗 𝛿𝑖𝑗 2 = 1
(12)
This is referred to as the double stochasticity constraint. In addition, U must meet the
definition of a unitary matrix, indicating that UU* = U*U = I, where U* is the inverse of the
unitary matrix U and I is the identity matrix.
The full parameterization of a 3x3 unitary uses 16 parameters (see Dita, 1982); fortunately,
due to the constraint of existing in the real space, the unitary matrix reduces to an orthogonal
matrix and can be parameterized using only 3 Euler rotation angles (Hestenes, 1999). These
Euler angles 𝜃1 , 𝜃2 , and 𝜃3 represent rotations around bases that we’re using. In this case, these
are the x, y, and z axes. Note that the final result is also in the standard Cartesian coordinate
space – this is mainly for convenience. In technicality, we could represent the PD game space in
the same coordinates as the promise game space, but this would result in basis vectors that would
be more difficult to interpret (though mathematically sound).
Supplemental Material C:
Data and model predictions based on best-fit parameters
The conditions are described by [Type of Agent in Phase 1] / [Whether a promise was
made to Agent A in Phase 1] / [Type of Agent in Phase 2]. For example, CoopA/NP/OppB (the
second entry) would mean that the participant saw a cooperative agent in phase 1, made no
promise (NP) to that agent, and then played with an opportunistic agent in phase 2, cooperating
at the rate indicated by the bar.
Supplemental Material D, References:
Busemeyer, J. R., & Lambert-Mogiliansky, A. (2009). An exploration of Type Indeterminacy in
strategic Decision-making. In Quantum Interaction (pp. 113-127). Springer Berlin
Heidelberg.
Dita, P. (1982). Parameterisation of unitary matrices. Journal of Physics A: Mathematics,
General 15: 3465-3473.
Hestenes, D. (1999). New Foundations for Classical Mechanics (2nd ed.). Springer.
Supplemental Material E, other related reading:
Cheon, T., & Tsutsui, I. (2006). Classical and quantum contents of solvable game theory on
Hilbert space. Physics Letters A, 348(3–6), 147–152. doi:10.1016/j.physleta.2005.08.066
Khrennikov, A. I. (2004). Information Dynamics in Cognitive, Psychological, Social, and
Anomalous Phenomena (Vol. 138). Kluwer Academic Publishers.
Khrennikov, A. (2009). Quantum-like model of cognitive decision making and information
processing. Biosystems, 95(3), 179–187. doi:10.1016/j.biosystems.2008.10.004