Abstract
We develop a basic framework encoding preference relations on the set of possible strategies in a quantum-like fashion. The Type Indeterminacy model introduces quantum-like uncertainty affecting preferences. The players are viewed as systems subject to measurements. The decision nodes are, possibly non-commuting, operators that measure preferences modulo strategic reasoning. We define a Hilbert space of types and focus on pure strategy TI games of maximal information. Preferences evolve in a non-deterministic manner with actions along the play: they are endogenous to the interaction. We propose the Type Indeterminate Nash Equilibrium as a solution concept relying on best-replies at the level of eigentypes.
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Notes
- 1.
- 2.
This is an idealized model for agents whose type may change in the decision-making process, formalizing the idea that in these situations – of violation of procedural invariance – observed preferences are not simply read off from some master list; they are actually constructed in the elicitation process, as argued by Kahneman and Tversky (2000, p. 504).
- 3.
They are inspired by the Principle of Superposition and the Postulate of Measurement and Observables in Quantum Mechanics, respectively.
- 4.
Under some simplifying conditions, a TI model can yield the same predictions as a ‘classical’ Bayesian framework.
- 5.
For a more general discussion in terms of ortholattices, see Danilov and Lambert-Mogiliansky (2008). We restrict ourselves to separable and finite-dimensional Hilbert spaces defined over the field of real numbers.
- 6.
Given two elements \(o_j,\,o_k\in O\), \(o_j\sim o_k\) denotes the indifference between both of them defined as \(o_j\succsim _io_k\) and \(o_k\succsim _io_j\), and \(o_j\succ _io_k\) denotes the strict preference for \(o_j\) defined as \(o_j\succsim _io_k\) and not \(o_k\succsim _i o_j\). A utility function \(u_i\) representing \(\succsim _i\) is a real-valued function defined over \(O\) and satisfying \( u_i(o_1)\ge u_i(o_2)\,\Leftrightarrow \, o_1\succsim _i o_2\;\;\forall \,o_1,o_2\in O\).
- 7.
As a simplifying remark, a preference relation defined over a set of \(M\) alternatives is, generally, composed of a list of the order of \(C_{M,2}\) elements, the combinatorial number counting how many different pairs we can form with the collection of \(M\) elements. Imagine a decision-maker \(i\) who can conceive only two monotonic ways of ranking the elements: either in a ‘positive’ way \(M\succsim ^{(+)}_i\ldots \succsim ^{(+)}_i 1\) or in a ‘negative’ way \(1\succsim ^{(-)}_i\ldots \succsim ^{(-)}_i M\). Then, the number of eigentypes required to describe our thought-agent is just \(N=2\), because there are only two possible preference relations even though each of these relations contains many pairwise comparisons between the elements in the outcome set. Please note that due to the interactive nature of the game-theoretical settings, we will usually find more complex representations.
- 8.
All the elements of the form \(\lambda t_i\) (with \(\lambda \ne 0\)) represent the same state of the system as \(t_i\), with \(\Vert t_i\Vert =1\).
- 9.
At this initial stage of our research, we restrict the model to pure types. We do not consider mixed types (represented by density matrices) that are expression of incomplete information. We deal exclusively with situations of maximal information, where all the uncertainty is intrinsic.
- 10.
We below return in more detail to the distinction between the observed choice and the underlying preferences.
- 11.
This follows in the spirit of Lüders’ postulate, a generalization of the so-called von Neumann’s Postulate for Pure States, which applies only for nondegenerate spectrums of observables. We consider our manuscript presents a self-contained exposition of the use of the Hilbert spaces for describing the space of types in Game Theory. For the reader deeply interested in the interpretation of the measurement postulate in Quantum Mechanics, we suggest Chap. 8 of The Logic of Quantum Mechanics by E.G. Beltrametti and G. Cassinelli (1981), as well as A. Khrennikov (2009), Int. J. of Quant. Inf., 7, 1303-1311.
- 12.
The row-vector \(t_m^T\) is the transposition of \(t_m\in \mathcal {T}_i\), and \(P_{a_i(d)}\) is a \(N\times N\) matrix.
- 13.
We shall deal with the problem of coarse measurement in the future.
- 14.
Note that commuting decision situations do not preclude statistical correlations.
- 15.
Just for notation, \(A_{-i}(d)\) represents the actions available for all the players other than \(i\) that are involved in the decision-situation \(d\). Then, \(u_i\big [\theta _i(d),\,a_i(d),a_{-i}(d)\big ]\) gives the utility associated to eigenpreferences \(\theta _i(d)\) when facing an interaction \((a_i,a_{-i})\) in the decision-situation \(d\).
- 16.
The particular exception \(t_i^{(\text {after}\,d)}=t_i^{(\text {before}\;d')}\) is found if \(d\) strictly precedes \(d'\) in the path, or if all of the intermediate decision-situations between \(d\) and \(d'\) are trivially associated to the identity projector.
- 17.
Note that for this first work we consider only pure strategies.
- 18.
The situations that can be analyzed with the notion of Nash Equilibrium are contained in the TI framework as an oversimplified case where only one eigentype exists for each player and therefore, the type and the eigentype trivially coincide.
- 19.
Cashing-on-the-go has been introduced to deal with the fact that the initial preferences of TI players are, in general, altered along the path of the game as already discussed.
- 20.
Note that the utility defined above is linear in the probabilistic content of the outcoming preferences (compare \(c_m'^2\) to Definition 6).
- 21.
When some paths become ‘branches’ including moves of chance, the utility shall be considered in expected terms.
- 22.
The notation \(\prod _{d_l\in \sigma _l}P_{a_i^*(d_l)}t_i^0\) for the subsequent application of projectors has to be understood in the sense of the composition of operators, \(P_{a_i^*(d_L)}\cdots P_{a_i^*(d_2)}P_{a_i^*(d_1)}t_ii^0\).
- 23.
We assume the open-mindness and the taste for personal challenge are not the same psychological features but they are somehow related.
- 24.
We shall understand this two different aspects of Bob’s personality as two orthogonal eigentypes of \(\theta \)-preferences. See Definition 2.
- 25.
The extension from pure states to the formulation in terms of density matrices has been excluded from this paper to avoid computational complexity, in order to keep the discussion as fundamental and formal as possible according to our best.
- 26.
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Lambert-Mogiliansky, A., Martínez-Martínez, I. (2015). Games with Type Indeterminate Players. In: Atmanspacher, H., Bergomi, C., Filk, T., Kitto, K. (eds) Quantum Interaction. QI 2014. Lecture Notes in Computer Science(), vol 8951. Springer, Cham. https://doi.org/10.1007/978-3-319-15931-7_18
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