Game Theory and Finite Strategies: Nash Equilibrium Takes Quantum Computations to Optimality.

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Finite games of strategy, within the framework of noncooperative quantum game theory, can be approached from finite chain categories, where, by finite chain category, it is understood a category C(n;N) that is generated by n objects and N morphic chains, called primitive chains, linking the objects in a specific order, such that there is a single labelling. C(n;N) is, thus, generated by N primitive chains of the form:

x0 →f1 x1 →f2 x1 → … xn-1 →fn xn —– (1)

A finite chain category is interpreted as a finite game category as follows: to each morphism in a chain xi-1 →fi xi, there corresponds a strategy played by a player that occupies the position i, in this way, a chain corresponds to a sequence of strategic choices available to the players. A quantum formal theory, for a finite game category C(n;N), is defined as a formal structure such that each morphic fundament fi of the morphic relation xi-1 →fi xis a tuple of the form:

fi := (Hi, Pi, Pˆfi) —– (2)

where Hi is the i-th player’s Hilbert space, Pi is a complete set of projectors onto a basis that spans the Hilbert space, and Pˆfi ∈ Pi. This structure is interpreted as follows: from the strategic Hilbert space Hi, given the pure strategies’ projectors Pi, the player chooses to play Pˆfi .

From the morphic fundament (2), an assumption has to be made on composition in the finite category, we assume the following tensor product composition operation:

fj ◦ fi = fji —– (3)

fji = (Hji = Hj ⊗ Hi, Pji = Pj ⊗ Pi, Pˆfji = Pˆfj ⊗ Pˆfi) —– (4)

From here, a morphism for a game choice path could be introduced as:

x0 →fn…21 xn —– (5)

fn…21 = (HG = ⊗i=n1 Hi, PG = ⊗i=n1 Pi, Pˆ fn…21 = ⊗i=n1fi) —– (6)

in this way, the choices along the chain of players are completely encoded in the tensor product projectors Pˆfn…21. There are N = ∏i=1n dim(Hi) such morphisms, a number that coincides with the number of primitive chains in the category C(n;N).

Each projector can be addressed as a strategic marker of a game path, and leads to the matrix form of an Arrow-Debreu security, therefore, we call it game Arrow-Debreu projector. While, in traditional financial economics, the Arrow-Debreu securities pay one unit of numeraire per state of nature, in the present game setting, they pay one unit of payoff per game path at the beginning of the game, however this analogy may be taken it must be addressed with some care, since these are not securities, rather, they represent, projectively, strategic choice chains in the game, so that the price of a projector Pˆfn…21 (the Arrow-Debreu price) is the price of a strategic choice and, therefore, the result of the strategic evaluation of the game by the different players.

Now, let |Ψ⟩ be a ket vector in the game’s Hilbert space HG, such that:

|Ψ⟩ = ∑fn…21 ψ(fn…21)|(fn…21⟩ —– (7)

where ψ(fn…21) is the Arrow-Debreu price amplitude, with the condition:

fn…21 |ψ(fn…21)|2 = D —– (8)

for D > 0, then, the |ψ(fn…21)|corresponds to the Arrow-Debreu prices for the game path fn…21 and D is the discount factor in riskless borrowing, defining an economic scale for temporal connections between one unit of payoff now and one unit of payoff at the end of the game, such that one unit of payoff now can be capitalized to the end of the game (when the decision takes place) through a multiplication by 1/D, while one unit of payoff at the end of the game can be discounted to the beginning of the game through multiplication by D.

In this case, the unit operator, 1ˆ = ∑fn…21 Pˆfn…21 has a similar profile as that of a bond in standard financial economics, with ⟨Ψ|1ˆ|Ψ⟩ = D, on the other hand, the general payoff system, for each player, can be addressed from an operator expansion:

πiˆ = ∑fn…21 πi (fn…21) Pˆfn…21 —– (9)

where each weight πi(fn…21) corresponds to quantities associated with each Arrow-Debreu projector that can be interpreted as similar to the quantities of each Arrow-Debreu security for a general asset. Multiplying each weight by the corresponding Arrow-Debreu price, one obtains the payoff value for each alternative such that the total payoff for the player at the end of the game is given by:

⟨Ψ|1ˆ|Ψ⟩ = ∑fn…21 πi(fn…21) |ψ(fn…21)|2/D —– (10)

We can discount the total payoff to the beginning of the game using the discount factor D, leading to the present value payoff for the player:

PVi = D ⟨Ψ|πiˆ|Ψ⟩ = D ∑fn…21 π (fn…21) |ψ(fn…21)|2/D —– (11)

, where π (fn…21) represents quantities, while the ratio |ψ(fn…21)|2/D represents the future value at the decision moment of the quantum Arrow- Debreu prices (capitalized quantum Arrow-Debreu prices). Introducing the ket

|Q⟩ ∈ HG, such that:

|Q⟩ = 1/√D |Ψ⟩ —– (12)

then, |Q⟩ is a normalized ket for which the price amplitudes are expressed in terms of their future value. Replacing in (11), we have:

PVi = D ⟨Q|πˆi|Q⟩ —– (13)

In the quantum game setting, the capitalized Arrow-Debreu price amplitudes ⟨fn…21|Q⟩ become quantum strategic configurations, resulting from the strategic cognition of the players with respect to the game. Given |Q⟩, each player’s strategic valuation of each pure strategy can be obtained by introducing the projector chains:

Cˆfi = ∑fn…i+1fi-1…1 Pˆfn…i+1 ⊗ Pˆfi ⊗ Pˆfi-1…1 —– (14)

with ∑fi Cˆfi = 1ˆ. For each alternative choice of the player i, the chain sums over all of the other choice paths for the rest of the players, such chains are called coarse-grained chains in the decoherent histories approach to quantum mechanics. Following this approach, one may introduce the pricing functional from the expression for the decoherence functional:

D (fi, gi : |Q⟩) = ⟨Q| Cˆfi Cgi|Q⟩  —– (15)

we, then have, for each player

D (fi, gi : |Q⟩) = 0, ∀ fi ≠ gi —– (16)

this is the usual quantum mechanics’ condition for an aditivity of measure (also known as decoherence condition), which means that the capitalized prices for two different alternative choices of player i are additive. Then, we can work with the pricing functional D(fi, fi :|Q⟩) as giving, for each player an Arrow-Debreu capitalized price associated with the pure strategy, expressed by fi. Given that (16) is satisfied, each player’s quantum Arrow-Debreu pricing matrix, defined analogously to the decoherence matrix from the decoherent histories approach, is a diagonal matrix and can be expanded as a linear combination of the projectors for each player’s pure strategies as follows:

Di (|Q⟩) = ∑fi D(fi, f: |Q⟩) Pˆfi —– (17)

which has the mathematical expression of a mixed strategy. Thus, each player chooses from all of the possible quantum computations, the one that maximizes the present value payoff function with all the other strategies held fixed, which is in agreement with Nash.

Econophysics: Financial White Noise Switch. Thought of the Day 115.0

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What is the cause of large market fluctuation? Some economists blame irrationality behind the fat-tail distribution. Some economists observed that social psychology might create market fad and panic, which can be modeled by collective behavior in statistical mechanics. For example, the bi-modular distribution was discovered from empirical data in option prices. One possible mechanism of polarized behavior is collective action studied in physics and social psychology. Sudden regime switch or phase transition may occur between uni-modular and bi-modular distribution when field parameter changes across some threshold. The Ising model in equilibrium statistical mechanics was borrowed to study social psychology. Its phase transition from uni-modular to bi-modular distribution describes statistical features when a stable society turns into a divided society. The problem of the Ising model is that its key parameter, the social temperature, has no operational definition in social system. A better alternative parameter is the intensity of social interaction in collective action.

A difficult issue in business cycle theory is how to explain the recurrent feature of business cycles that is widely observed from macro and financial indexes. The problem is: business cycles are not strictly periodic and not truly random. Their correlations are not short like random walk and have multiple frequencies that changing over time. Therefore, all kinds of math models are tried in business cycle theory, including deterministic, stochastic, linear and nonlinear models. We outline economic models in terms of their base function, including white noise with short correlations, persistent cycles with long correlations, and color chaos model with erratic amplitude and narrow frequency band like biological clock.

 

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The steady state of probability distribution function in the Ising Model of Collective Behavior with h = 0 (without central propaganda field). a. Uni-modular distribution with low social stress (k = 0). Moderate stable behavior with weak interaction and high social temperature. b. Marginal distribution at the phase transition with medium social stress (k = 2). Behavioral phase transition occurs between stable and unstable society induced by collective behavior. c. Bi-modular distribution with high social stress (k = 2.5). The society splits into two opposing groups under low social temperature and strong social interactions in unstable society. 

Deterministic models are used by Keynesian economists for endogenous mechanism of business cycles, such as the case of the accelerator-multiplier model. The stochastic models are used by the Frisch model of noise-driven cycles that attributes external shocks as the driving force of business fluctuations. Since 1980s, the discovery of economic chaos and the application of statistical mechanics provide more advanced models for describing business cycles. Graphically,

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The steady state of probability distribution function in socio-psychological model of collective choice. Here, “a” is the independent parameter; “b” is the interaction parameter. a Centered distribution with b < a (denoted by short dashed curve). It happens when independent decision rooted in individualistic orientation overcomes social pressure through mutual communication. b Horizontal flat distribution with b = a (denoted by long dashed line). Marginal case when individualistic orientation balances the social pressure. c Polarized distribution with b > a (denoted by solid line). It occurs when social pressure through mutual communication is stronger than independent judgment. 

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Numerical 1 autocorrelations from time series generated by random noise and harmonic wave. The solid line is white noise. The broken line is a sine wave with period P = 1. 

Linear harmonic cycles with unique frequency are introduced in business cycle theory. The auto-correlations from harmonic cycle and white noise are shown in the above figure. Auto-correlation function from harmonic cycles is a cosine wave. The amplitude of cosine wave is slightly decayed because of limited data points in numerical experiment. Auto-correlations from a random series are an erratic series with rapid decade from one to residual fluctuations in numerical calculation. The auto-regressive (AR) model in discrete time is a combination of white noise term for simulating short-term auto-correlations from empirical data.

The deterministic model of chaos can be classified into white chaos and color chaos. White chaos is generated by nonlinear difference equation in discrete-time, such as one-dimensional logistic map and two-dimensional Henon map. Its autocorrelations and power spectra look like white noise. Its correlation dimension can be less than one. White noise model is simple in mathematical analysis but rarely used in empirical analysis, since it needs intrinsic time unit.

Color chaos is generated by nonlinear differential equations in continuous-time, such as three-dimensional Lorenz model and one-dimensional model with delay-differential model in biology and economics. Its autocorrelations looks like a decayed cosine wave, and its power spectra seem a combination of harmonic cycles and white noise. The correlation dimension is between one and two for 3D differential equations, and varying for delay-differential equation.

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History shows the remarkable resilience of a market that experienced a series of wars and crises. The related issue is why the economy can recover from severe damage and out of equilibrium? Mathematically speaking, we may exam the regime stability under parameter change. One major weakness of the linear oscillator model is that the regime of periodic cycle is fragile or marginally stable under changing parameter. Only nonlinear oscillator model is capable of generating resilient cycles within a finite area under changing parameters. The typical example of linear models is the Samuelson model of multiplier-accelerator. Linear stochastic models have similar problem like linear deterministic models. For example, the so-called unit root solution occurs only at the borderline of the unit root. If a small parameter change leads to cross the unit circle, the stochastic solution will fall into damped (inside the unit circle) or explosive (outside the unit circle) solution.

Conjuncted: Internal Logic. Thought of the Day 46.1

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So, what exactly is an internal logic? The concept of topos is a generalization of the concept of set. In the categorial language of topoi, the universe of sets is just a topos. The consequence of this generalization is that the universe, or better the conglomerate, of topoi is of overwhelming amplitude. In set theory, the logic employed in the derivation of its theorems is classical. For this reason, the propositions about the different properties of sets are two-valued. There can only be true or false propositions. The traditional fundamental principles: identity, contradiction and excluded third, are absolutely valid.

But if the concept of a topos is a generalization of the concept of set, it is obvious that the logic needed to study, by means of deduction, the properties of all non-set-theoretical topoi, cannot be classic. If it were so, all topoi would coincide with the universe of sets. This fact suggests that to deductively study the properties of a topos, a non-classical logic must be used. And this logic cannot be other than the internal logic of the topos. We know, presently, that the internal logic of all topoi is intuitionistic logic as formalized by Heyting (a disciple of Brouwer). It is very interesting to compare the formal system of classical logic with the intuitionistic one. If both systems are axiomatized, the axioms of classical logic encompass the axioms of intuitionistic logic. The latter has all the axioms of the former, except one: the axiom that formally corresponds to the principle of the excluded middle. This difference can be shown in all kinds of equivalent versions of both logics. But, as Mac Lane says, “in the long run, mathematics is essentially axiomatic.” (Mac Lane). And it is remarkable that, just by suppressing an axiom of classical logic, the soundness of the theory (i.e., intuitionistic logic) can be demonstrated only through the existence of a potentially infinite set of truth-values.

We see, then, that the appellation “internal” is due to the fact that the logic by means of which we study the properties of a topos is a logic that functions within the topos, just as classical logic functions within set theory. As a matter of fact, classical logic is the internal logic of the universe of sets.

Another consequence of the fact that the general internal logic of every topos is the intuitionistic one, is that many different axioms can be added to the axioms of intuitionistic logic. This possibility enriches the internal logic of topoi. Through its application it reveals many new and quite unexpected properties of topoi. This enrichment of logic cannot be made in classical logic because, if we add one or more axioms to it, the new system becomes redundant or inconsistent. This does not happen with intuitionistic logic. So, topos theory shows that classical logic, although very powerful concerning the amount of the resulting theorems, is limited in its mathematical applications. It cannot be applied to study the properties of a mathematical system that cannot be reduced to the system of sets. Of course, if we want, we can utilize classical logic to study the properties of a topos. But, then, there are important properties of the topos that cannot be known, they are occult in the interior of the topos. Classical logic remains external to the topos.

Dialectics: Mathematico-Philosophical Sequential Quantification. Drunken Risibility.

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Figure: Graphical representation of the quantification of dialectics.

A sequence S of P philosophers along a given period of time would incorporate the P most prominent and visible philosophers in that interval. The use of such a criterion to build the time-sequence for the philosophers implies in not necessarily uniform time-intervals between each pair of subsequent entries.

The set of C measurements used to characterize the philosophers define a C−dimensional feature space which will be henceforth referred to as the philosophical space. The characteristic vector v⃗i of each philosopher i defines a respective philosophical state in the philosophical space. Given a set of P philosophers, the average state at time i, i ≤ P, is defined as

a⃗i = 1/i ∑k=1i v⃗k

The opposite state of a given philosophical state v⃗i is defined as:

r⃗i = v⃗i +2(a⃗i −v⃗i) = 2a⃗i − v⃗i

The opposition vector of philosophical state v⃗i is given by D⃗i = r⃗i − v⃗i. The opposition amplitude of that same state is defined as ||D⃗i||.

An emphasis move taking place from the philosophical state v⃗i is any displacement from v⃗i along the direction −r⃗i. A contrary move from the philosophical state v⃗i is any displacement from v⃗i along the direction r⃗i.

Given a time-sequence S of P philosophers, the philosophical move implied by two successive philosophers i and j corresponds to the M⃗i,j vector extending from v⃗to v⃗j , i.e.

M⃗i,j = v⃗j – v⃗i

In principle, an innovative or differentiated philosophical move would be such that it departs substantially from the current philosophical state v⃗i. Decomposing innovation moves into two main subtypes: opposition and skewness.

The opposition index Wi,j of a given philosophical move M⃗i,j is defined as

Wi,j = 〈M⃗i,j, D⃗i〉/  ||D⃗i||2

This index quantifies the intensity of opposition of that respective philosophical move, in the sense of having a large projection along the vector D⃗i. It should also be noticed that the repetition of opposition moves lead to little innovation, as it would imply in an oscillation around the average state. The skewness index si,j of that same philosophical move is the distance between v⃗j and the line L defined by the vector D⃗i, and therefore quantifies how much the new philosophical state departs from the respective opposition move. Actually, a sequence of moves with zero skewness would represent more trivial oscillations within the opposition line Li.

We also suggest an index to quantify the dialectics between a triple of successive philosophers i, j and k. More specifically, the philosophical state v⃗i is understood as the thesis, the state j is taken as the antithesis, with the synthesis being associated to the state v⃗k. The hypothesis that k is the consequence, among other forces, of a dialectics between the views v⃗i and v⃗j can be expressed by the fact that the philosophical state v⃗k be located near the middle line MLi,j defined by the thesis and antithesis (i.e. the points which are at an equal distance to both v⃗i and v⃗j) relatively to the opposition amplitude ||D⃗i||.

Therefore, the counter-dialectic index is defined as

ρi→k = di→k /||M⃗i,j||

where di→k is the distance between the philosophical state v⃗k and the middle-line MLi,j between v⃗i and v⃗j. Note that 0 ≤ di→k ≤ 1. The choice of counter-dialectics instead of dialectics is justified to maintain compatibility with the use of a distance from point to line as adopted for the definition of skewness….

Are Categories Similar to Sets? A Folly, if General Relativity and Quantum Mechanics Think So.

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The fundamental importance of the path integral suggests that it might be enlightening to simplify things somewhat by stripping away the knot observable K and studying only the bare partition functions of the theory, considered over arbitrary spacetimes. That is, consider the path integral

Z(M) = ∫ DA e (i ∫M S(A) —– (1)

where M is an arbitrary closed 3d manifold, that is, compact and without boundary, and S[A] is the Chern-Simons action. Immediately one is struck by the fact that, since the action is topological, the number Z(M) associated to M should be a topological invariant of M. This is a remarkably efficient way to produce topological invariants.

Poincaré Conjecture: If M is a closed 3-manifold, whose fundamental group π1(M), and all of whose homology groups Hi(M) are equal to those of S3, then M is homeomorphic to S3.

One therefore appreciates the simplicity of the quantum field theory approach to topological invariants, which runs as follows.

  1. Endow the space with extra geometric structure in the form of a connection (alternatively a field, a section of a line bundle, an embedding map into spacetime)
  2. Compute a number from this manifold-with-connection (the action)
  3. Sum over all connections.

This may be viewed as an extension of the general principle in mathematics that one should classify structures by the various kinds of extra structure that can live on them. Indeed, the Chern-Simons Lagrangian was originally introduced in mathematics in precisely this way. Chern-Weil theory provides access to the cohomology groups (that is, topological invariants) of a manifold M by introducing an arbitrary connection A on M, and then associating to A a closed form f(A) (for instance, via the Chern-Simons Lagrangian), whose cohomology class is, remarkably, independent of the original arbitrary choice of connection A. Quantum field theory takes this approach to the extreme by being far more ambitious; it associates to a connection A the actual numerical value of the action (usually obtained by integration over M) – this number certainly depends on the connection, but field theory atones for this by summing over all connections.

Quantum field theory is however, in its path integral manifestation, far more than a mere machine for computing numbers associated with manifolds. There is dynamics involved, for the natural purpose of path integrals is not to calculate bare partition functions such as equation (1), but rather to express the probability amplitude for a given field configuration to evolve into another. Thus one considers a 3d manifold M (spacetime) with boundary components Σ1 and Σ2 (space), and considers M as the evolution of space from its initial configuration Σ1 to its final configuration Σ2:

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This is known mathematically as a cobordism from Σ1 to Σ2. To a 2d closed manifold Σ we associate the space of fields A(Σ) living on Σ. A physical state Ψ corresponds to a functional on this space of fields. This is the Schrödinger picture of quantum field theory: if A ∈ A(Σ), then Ψ(A) represents the probability that the state known as Ψ will be found in the field A. Such a state evolves with time due to the dynamics of the theory; Ψ(A) → Ψ(A, t). The space of states has a natural basis, which consists of the delta functionals  – these are the states satisfying ⟨Â|Â′⟩ = δ(A − A′). Any arbitrary state Ψ may be expressed as a superposition of these basis states. The path integral instructs us how to compute the time evolution of states, by first expanding them in the  basis, and then specifying that the amplitude for a system in the state Â1 on the space Σ1 to be found in the state Â2 on the space Σ2 is given by:

〈Â2|U|Â1〉= ∫A | ∑2 = A2 A | ∑1 = A1 DA e i S[A] —– (2)

This equation is the fundamental formula of quantum field theory: ‘Perform a weighted sum over all possible fields (connections) living on spacetime that restrict to A1 and A2 on Σ1 and Σ2 respectively’. This formula constructs the time evolution operator U associated to the cobordism M.

In this way we see that, at the very heart of quantum mechanics and quantum field theory, is a formula which associates to every space-like manifold Σ a Hilbert space of fields A(Σ), and to every cobordism M from Σ1 to Σ2 a time evolution operator U(M) : Σ1 – Σ2. To specify a quantum field theory is nothing more than to give rules for constructing the Hilbert spaces A(Σ) and the rules (correlation functions) for calculating the time evolution operators U(M). This is precisely the statement that a quantum field theory is a functor from the cobordism category nCob to the category of Hilbert spaces Hilb.

A category C consists of a collection of objects, a collection of arrows f:a → b from any object a to any object b, a rule for composing arrows f:a → b and g : b → c to obtain an arrow g f : a → c, and for each object A an identity arrow 1a : a → a. These must satisfy the associative law f(gh) = (fg)h and the left and right unit laws 1af = f and f1a = f whenever these composites are defined. In many cases, the objects of a category are best thought of as sets equipped with extra structure, while the morphisms are functions preserving the structure. However, this is neither true for the category of Hilbert spaces nor for the category of cobordisms.

The fundamental idea of category theory is to consider the ‘external’ structure of the arrows between objects instead of the ‘internal’ structure of the objects themselves – that is, the actual elements inside an object – if indeed, an object is a set at all : it need not be, since category theory waives its right to ask questions about what is inside an object, but reserves its right to ask how one object is related to another.

A functor F : C → D from a category C to another category D is a rule which associates to each object a of C an object b of D, and to each arrow f :a → b in C a corresponding arrow F(f): F(a) → F(b) in D. This association must preserve composition and the units, that is, F(fg) = F(f)F(g) and F(1a) = 1F(a).

1. Set is the category whose objects are sets, and whose arrows are the functions from one set to another.

2. nCob is the category whose objects are closed (n − 1)-dimensional manifolds Σ, and whose arrows M : Σ1 → Σ2 are cobordisms, that is, n-dimensional manifolds having an input boundary Σ1 and an output boundary Σ2.

3. Hilb is the category whose objects are Hilbert spaces and whose arrows are the bounded linear operators from one Hilbert space to another.

The ‘new philosophy’ amounts to the following observation: The last two categories, nCob and Hilb, resemble each other far more than they do the first category, Set! If we loosely regard general relativity or geometry to be represented by nCob, and quantum mechanics to be represented by Hilb, then perhaps many of the difficulties in a theory of quantum gravity, and indeed in quantum mechanics itself, arise due to our silly insistence of thinking of these categories as similar to Set, when in fact the one should be viewed in terms of the other. That is, the notion of points and sets, while mathematically acceptable, might be highly unnatural to the subject at hand!