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 xi is 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=n1Pˆfi) —– (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)|2 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 πi (fn…21) |ψ(fn…21)|2/D —– (11)
, where πi (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| Cˆ†fi 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, fi : |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.