The Intrinsic Quantum Nature of Nash Equilibrium Mixtures

Abstract

In classical game theory the idea that players randomize between their actions according to a particular optimal probability distribution has always been viewed as puzzling. In this paper, we establish a fundamental connection between n-person normal form games and quantum mechanics (QM), which eliminates the conceptual problems of these random strategies. While the two theories have been regarded as distinct, our main theorem proves that if we do not give any other piece of information to a player in a game, than the payoff matrix—the axiom of “no-supplementary data” holds—then the state of mind of a rational player is algebraically isomorphic to a pure quantum state. The “no supplementary data” axiom is captured in a Lukasiewicz’s three-valued Kripke semantics wherein statements about whether a strategy or a belief of a player is rational are initially indeterminate i.e. neither true, nor false. As a corollary, we show that in a mixed Nash equilibrium, the knowledge structure of a player implies that probabilities must verify the standard “Born rule” postulate of QM. The puzzling “indifference condition” wherein each player must be rationally indifferent between all the pure actions of the support of his equilibrium strategy is resolved by his state of mind being described by a “quantum superposition” prior a player is asked to make a definite choice in a “measurement”. Finally, these results demonstrate that there is an intrinsic limitation to the predictions of game theory, on a par with the “irreducible randomness” of quantum physics.

Appendix

Proof of Lemma 0

We shall prove the following statement: Consider a game G where the axiom of no-supplementary data holds. Then, player i can determine σ ⁱ as being a rational choice in G if and only if i determines profile σ ^{− i} as being rational for players j ≠ i with (σ ⁱ, σ ^{− i}) a Nash equilibrium of G in a frame, $\mathcal {F}^{i}_{G}$ , with at least two perspectives i.e. |M| ≥ 2. □

Proof

Let (A ^J) _{J ∈ M} represent the Nash equilibrium profile σ = (σ ^iL) _{L ∈ M} .

Consider a model with a M-self-projective frame $\mathcal {F}^{i}_{G}=\left \langle {w^{i_{J}}},R^{i_{J}}\right \rangle _{J\in M}$ with |M| ≥ 2. Suppose that E ^σ is common knowledge. Thus, we have the set of binary relations, $R^{i_{J}}\in \mathcal {R}^{i}$ such that w ^iJ R ^iJ w ^iK, for all J ≠ K. We must also have a profile of statements, (A ^L) _{L ∈ M} , which has been determined as being relatively true. That is, we have a system of relational valuation mappings,

$$A^{J}\in\arg\max\limits_{A^{'J}\in \mathbb{A}^{J}} V_{w^{{i}_{J}},w^{{i}_{K}}}(A^{'J},A^{K};A^{-J,-K}), \forall (w^{{i}_{J}},w^{{i}_{K}})\in\mathcal{W}\times \mathcal{W}, J\neq K,$$

whose solution, (A ^L) _{L ∈ M} , corresponds to the Nash equilibrium profile of G, σ = (σ ^iL) _{L ∈ M} . The solution of the above system of equations is guaranteed by the standard existence theorem of a Nash equilibrium in finite games (Nash, [3]). Hence, if E ^σ is common knowledge in the canonical model, then we necessarily have a Nash equilibrium with $\mathcal {M}^{i}_{G}, w^{i_{K}} \vDash A^{\sigma ^{J}}, \forall w^{i_{K}}\in \mathcal {W}.$ When this is true for each i ∈ N, this implies that σ is a mixed Nash equilibrium (NE) of the game being played.

Necessity

We first show that A ^J is true in a M-self-projective model of i only if i determines a Nash equilibrium. By definition, the self-projective model with the single perspective w ^iN cannot yield the determination of a truth $A^{N}\in \mathbb {A}^{N}$ when there is no strongly dominant actions.³⁶ Thus, we must pick a M-self-projective model with at least two perspectives. Suppose that there exists an arbitrary L ∈ M with $R^{i_{L}}\in \mathcal {R}^{i}$ such that w ^iL ¬ R ^iL w ^iK, for some L ≠ K. Then, by definition, $\emptyset \in \arg \max _{A^{'L}\in \mathbb {A}^{L}} V_{w^{{i}_{L}},w^{{i}_{K}}}(A^{'L},A^{K};A^{-L,-K}),~\mathrm { for } L\neq K,$ since $V_{w^{{i}_{L}},w^{{i}_{K}}}(A^{'L},A^{K};A^{-L,-K}) = \texttt {i}$ whenever $\mathcal {M}^{i}_{G}, w^{i_{L}} \vDash \neg \Box ^{i_{L}} A^{\sigma ^{K}}.$ Hence, we have that a statement is determined only if w ^iJ R ^iJ w ^iK,∀J ≠ K. Notice that E ^σ is a self-evident event to every perspective w ^iJ of player i i.e. $E^{\sigma }= K_{i_{J}}E^{\sigma }.$ This follows since, $w^{i_{J}}\in \left \| \Box ^{i_{J}}A^{\sigma ^{-J}}\rightarrow _{\texttt {L}} A^{\sigma ^{-J}}\right \|^{\mathcal {M}^{i}_{G}}.$ ³⁷ Hence $w^{i_{J}}\in \left \|\Box ^{i_{J}}A^{(\sigma ^{i})_{i\in N}}\right \|^{\mathcal {M}^{i}_{G}}$, for all $w^{i_{J}}\in \mathcal {W}$ whenever the profile of propositions, $(A^{J})^{m}_{J =1}$ is the solution of the above set of equations and corresponds, by construction, to a Nash equilibrium of the game G. □

Proof of Lemma 1

We first characterize the elements of $\mathcal {S}^{O}_{i}$. Fix an empirical self-projective Krikpean model for player i in G, $\mathcal {M}^{iO}_{G},$ where σ ⁱ has been determined as being true. By Theorem 1, the indifference condition holds i.e. any s ⁱ ∈ supp(σ ⁱ) is necessarily a best reply to σ ^{− i}. Hence, when i is making a rational choice, we must have $w^{s^{i}}\in \mathcal {S}^{O}_{i}$, such that $\mathcal {M}^{iO}_{G},w^{s^{i}}\vDash \wedge _{L\in M}\Box ^{i_{L}} (A^{s^{i}}\wedge A^{\sigma ^{-i}}).$ The indivisibility axiom requires that event $E^{s^{i}}\wedge E^{\sigma ^{-i}}$ is self-knowledge, which proves that $K_{i}(E^{s^{i}}\wedge E^{\sigma ^{-i}})$ is an element of $\mathcal {S}^{O}_{i}$. Next, we verify that $\mathcal {L}\left (\mathcal {S}^{O}_{i},\vee ,\wedge ,\sim \right )$ is an orthocomplemented non-distributive lattice. By definition of a pure choice, we must have that $w^{s^{i}}\wedge w^{s^{'i}}=\emptyset$, $\forall w^{s^{i}}\neq w^{s^{'i}}$. By definition of a lattice, ∨ and ∧ must be idempotent, commutative, and associative operations such that a ∨ (b ∧ a) = a and a ∧ (b ∨ a) = a, $\forall a\in \mathcal {L}\left (\mathcal {S}^{O}_{i},\vee ,\wedge ,\sim \right ).$ Idempotence and commutativity are obvious. We also note that $w^{s^{i}}\vee (w^{s^{'i}}\wedge w^{s^{i}}) = w^{s^{i}}$ since $w^{s^{'i}}\wedge w^{s^{i}}=\emptyset$ whenever $w^{s^{\prime i}}\neq w^{s^{i}}$ by definition of a pure choice and $w^{s^{i}}\wedge (w^{s^{i}}\vee w^{s^{\prime i}}) = w^{s^{i}}.$ That ∧ is associative follows since $w^{s^{i}}\wedge (w^{s^{\prime i}}\wedge w^{s^{\prime \prime i}}) = (w^{s^{i}}\wedge w^{s^{\prime i}})\wedge w^{s^{\prime \prime i}}=\emptyset$ whenever $w^{s^{i}}\neq w^{s^{'i}}$ and $w^{s^{'i}}\neq w^{s^{\prime \prime i}}.$ Moreover, in equilibrium, the indifference condition implies that any s ⁱ ∈ supp(σ ⁱ) is a best reply. Thus, $\bigvee _{s^{i}\in \texttt {supp}(\sigma ^{i})}w^{s^{i}}=\mathcal {S}^{O}_{i}$ so that o := ∅ and $1:=\mathcal {S}^{O}_{i}\in \mathcal {L}\left (\mathcal {S}^{O}_{i},\vee ,\wedge ,\sim \right )$ are the two identity elements with a ∧ 1 = a and a ∨ o = a. Moreover, the lattice is orthocomplemented under the set-theoretic complement, $a^{\sim }=\mathcal {S}^{O}_{i}\setminus a, \forall a\in \mathcal {L}\left (\mathcal {S}^{O}_{i},\vee ,\wedge ,\sim \right )$ such that a ⊆ b ↔ a ^∼ ⊆ (b)^∼ and the partial ordering ⊆ corresponds to the material implication “ ⇒”. That this lattice is non distributive³⁸ follows by noting that $w^{s^{i}}\wedge (w^{s^{\prime i}}\vee w^{s^{\prime \prime i}})\subseteq w^{s^{i}}\neq \emptyset$ whenever s ⁱ ∈ supp(σ ⁱ). This implies that, $w^{s^{i}}\wedge (w^{s^{\prime i}}\vee w^{s^{\prime \prime i}}) \neq \emptyset$ while ${(w^{s^{i}}\wedge w^{s^{\prime i}})}\vee (w^{s^{\prime i}}\wedge w^{s^{\prime \prime i}}) = \emptyset$ whenever s ⁱ ≠ s ^′i and s ^′i ≠ s ^″i. □

Proof of Theorem 1

We first construct the isomorphism between the complex Hilbert space structure of QM together with its postulates and the self-projective graph induced by a canonical model i.e. a model whose M-frame has exactly two perspectives. Then, in a second step, we will use this result as a lemma to show that any other model with more than two perspectives (i.e. the M-frame is such that |M| > 2) does not induce a well-defined empirical distribution e ⁱ in an experiment. From this we will conclude that the only possible self-projective model is the canonical model. The isomorphism between the canonical model and the QM structure then terminates to demonstrate that the mental state space of a rational player in the classical game model is always isomorphic to the state-space structure of QM. □

The derivation of the QM formalism of Theorem 1 builds on the following result.

Proposition 1

Fix the canonical self-projective Krikpean model for player i in G, $\mathcal {M}^{iO}_{G},$ where σ ⁱ has been determined as being true at w ^† . There exists a faithful representation $\mathfrak {g}$ of the knowledge structure of player i in ${{\Gamma }^{i}_{G}}$ if the set $(\mathfrak {g},\wedge )$ is generated by the space

$$\mathfrak{g}=\left\{\left|\omega_{{e}_{-i,i}}\right\rangle,\omega_{{e}_{i,-i}}:=\left\langle\omega_{{e}_{-i,i}} \right|\right\} $$

over the complex field 𝕂 = ℂ such that:

• (1) $\omega _{e_{J,-J}}(s^{i})\omega _{e_{-J,J}}(s^{i}) = \omega _{e_{J,J}}(s^{i}), \forall s^{i}\in$ supp (σ ⁱ);

• (2) $\left \langle \omega _{{e}_{-i,i}}\right |\left .\omega _{{e}_{-i,i}}\right \rangle =1$ is the identity element i.e. $a\wedge 1=a, \forall a\in (\mathfrak {g},\wedge );$

• (3) The multiplicative operation ∧ is such that $\left \langle \omega _{{e}_{-i,i}} \right |\wedge \left |\omega _{{e}_{-i,i}}\right \rangle =\left \langle \omega _{{e}_{-i,i}}\right |\left .\omega _{{e}_{-i,i}}\right \rangle$ and $\left |\omega _{{e}_{-i,i}}\right \rangle \wedge \left \langle \omega _{{e}_{-i,i}} \right |=\left |{\omega }_{{e}_{-i,i}} \right \rangle \left \langle {\omega }_{{e}_{-i,i}} \right |$ is an orthogonal operator;

• (4) $\sigma ^{i}(s^{i}) = \left |\omega _{(w,w^{{\dag }})}(s^{i})\right |^{2}, \forall s^{i}$ (Born rule) with $\omega _{(w^{\prime },w^{\prime })}=\sigma ^{i}\in {\Omega }\setminus \mathfrak {g}$ , for w′=w,w ^†.

Proof

We first state the following lemma.

Lemma 3

Fix the empirical canonical self-projective Krikpean model for player i in G, $\mathcal {M}^{iO}_{G},$ where σ ⁱ has been determined as being true. Then the set (R ⁱ, ∧) generated by the transitive closure of $R^{i}:=R^{i_{i}}\bigcup R^{i_{-i}}$ must have the multiplicative operation ∧ := ”and” such that:

• 1. (wR ⁱ w) ∧ (wR ⁱ w) = (wR ⁱ w) (idempotence);

• 2. (w′ R ⁱ w) ∧ (wR ⁱ w′ ) ⇒ (w′ R ⁱ w′) (transitivity).

Proof

We first check that any $(w,w^{\prime })\in \mathcal {W}\times \mathcal {W}$ lies in R ⁱ. By Theorem 1, in equilibrium, there is common knowledge, $K^{i}_{*}E^{\sigma }=\left \{w^{i_{i}},w^{i_{-i}}\right \},$ which amounts to saying that the transitive closure $R^{i}:=R^{i_{i}}\cup R^{i_{-i}}=\mathcal {W}\times \mathcal {W}$. Hence, by definition of the transitive closure, this implies that the set R ⁱ is equipped with the non-commutative operation $\wedge :\mathcal {W}\times \mathcal {W}\rightarrow \mathcal {W}$ such that (1)-(2) hold. □

The above lemma shows that any $(w,w^{\prime })\in \mathcal {W}\times \mathcal {W}$ lies in R ⁱ. Next, we prove the existence of a faithful representation $\mathfrak {g}$ together with its characterization.

We construct an isomorphism h for the canonical model with $h:(R^{i},\wedge )\rightarrow (\mathfrak {g},\wedge )$ as follows. Let $\omega _{(w,w^{\dag })}:S^{i}\rightarrow \mathbb {C}^{m_{i}}$ with h((w R ⁱ w ^†))(s ⁱ) = ω _{(w, w†)}(s ⁱ), ∀s ⁱ so that h((w R ⁱ w ^†)) = |w〉, h((w ^† R ⁱ w)) = 〈w|, and the multiplicative map ∧ over $\mathfrak {g}$ is defined such that h((w ^† R ⁱ w) ∧ (w R ⁱ w ^†)) = h(w ^† R ⁱ w)) ∧ h((w R ⁱ w ^†)) = 〈w||w〉 and h((w R ⁱ w ^†) ∧ (w ^† R ⁱ w)) = h(w R ⁱ w ^†)) ∧ h((w ^† R ⁱ w)) = |w〉〈w| and we set h((w ^† R ⁱ w ^†)) = 1, which implies that $h((w R^{i} w)) = {\Pi }_{\omega _{(w,w^{\dag })}}$ is an orthogonal projector.

We check idempotence (1) as follows:

$$h((w R^{i} w)\wedge (wR^{i} w)) = \left|w\right\rangle\underbrace{\left\langle w\right|\wedge\left|w\right\rangle}_{1}\left\langle w\right|=\left|w\right\rangle\left\langle w\right|$$

and h((w ^† R ⁱ w ^†) ∧ (w ^† R ⁱ w ^†)) = 〈w||w〉 ∧ 〈w||w〉 = 1 since 1 is the identity element. Thus, we have constructed an isomorphism that induces a set $\mathfrak {g}$ with: $\omega _{((w^{\prime },w)\wedge (w,w^{\prime }))}(s^{i}) = \omega _{((w^{\prime },w))}(s^{i})\omega _{((w,w^{\prime }))}(s^{i}), \forall s^{i},$ whenever w′ ≠ w, where $\omega _{((w^{\prime },w))}(s^{i})$ is the complex conjugate of $\omega _{((w,w^{\prime }))}(s^{i}).$ The condition that h((w ^† R ⁱ w ^†)) = 1 implies that ω _{((w, w†))} must lie in the complex unit sphere and that the induced vectors, $(\omega _{((w^{\prime },w)\wedge (w,w^{\prime }))}(s^{i}))_{s^{i}\in S^{i}}=(\omega _{((w^{\prime },w^{\prime }))}(s^{i}))_{s^{i}\in S^{i}}$, are in the unit simplex Δⁱ,∀w′=w ^†, w. This means that the property $\omega _{((w^{\prime },w^{\prime }))}:=(\omega _{((w^{\prime },w^{\prime }))}(s^{i}))_{s^{i}\in S^{i}}$ with $\omega _{((w^{\prime },w^{\prime }))}=\sigma ^{i}, \forall w^{\prime }$ is met, which immediately implies the Born rule (4). □

Necessity of a faithful representation in the canonical model

Next, we show that the existence of an empirical distribution e ⁱ in the canonical model implies necessarily the faithful representation given in Proposition 2. We first start with a Lemma which proves that the Indivisibility Axiom is equivalent to the “Projection postulate” of QM.

Proof of Lemma 2

Given $\widehat {S}^{i}\subseteq S^{i},$ let ${{\Pi }_{\widehat {S}^{i}}}:={\sum }_{{s^{i}}\in \widehat {S}^{i}}{\Pi }_{a^{s^{i}}}$ so that

$$H_{{\Pi}_{\widehat{S}^{i}}}=\left\{\Psi\in\mathbb{H}:{{\Pi}_{\widehat{S}^{i}}}{\Psi}={\Psi}\right\}$$

is a closed subspace of $\mathbb {H}$ which corresponds to statement ${\widehat {S}^{i}}:=$“${\widehat {S}^{i}}$ is a rational subset of strategies for player i”. It is well-known that a set of subspaces like $H_{{\Pi }_{\widehat {S}^{i}}}$ ordered by inclusion ⊆, with the complement ∼ defined by orthocomplementation ⊥ and meet and join operations defined by intersection ∩ and direct sum + of subspaces forms an orthocomplemented and non-distributive lattice. Hence, we can also identify the logical connectives of the lattice $\mathcal {L}\left (\mathcal {S}^{O}_{i},\vee ,\wedge ,\sim \right )$ in terms of the lattice of projectors, $\mathcal {P}(\mathbb {H})$, with $H_{{\Pi }_{\widehat {S}^{i}}}\bigcap H_{{\Pi }_{\widehat {S}^{'i}}} \longleftrightarrow {{\Pi }_{\widehat {S}^{i}}}\wedge {{\Pi }_{\widehat {S}^{'i}}}={{\Pi }_{\widehat {S}^{i}}}{{\Pi }_{\widehat {S}^{'i}}}$, $H_{{\Pi }_{\widehat {S}^{i}}}\subseteq H_{{\Pi }_{\widehat {S}^{'i}}} \longleftrightarrow {{\Pi }_{\widehat {S}^{i}}}{{\Pi }_{\widehat {S}^{'i}}}={{\Pi }_{\widehat {S}^{'i}}}{{\Pi }_{\widehat {S}^{i}}}={{\Pi }_{\widehat {S}^{i}}}$, $H_{{\Pi }_{\widehat {S}^{i}}}+ H_{{\Pi }_{\widehat {S}^{'i}}}\longleftrightarrow {{\Pi }_{\widehat {S}^{i}}}+{{\Pi }_{\widehat {S}^{'i}}}-{{\Pi }_{\widehat {S}^{'i}}}{{\Pi }_{\widehat {S}^{'i}}}, (H_{{\Pi }_{\widehat {S}^{i}}})^{\perp }\longleftrightarrow {\Pi }_{(\widehat {S}^{i})^{\sim }}=I-{\Pi }_{\widehat {S}^{i}},$Hence, there exists a one to one mapping,

$$g:\mathcal{P}(\mathbb{H})\rightarrow \mathcal{L}\left(\mathcal{S}^{O}_{i},\vee,\wedge,\sim\right), {\Pi}_{\widehat{S}^{i}}\stackrel{g}{\mapsto}\bigvee_{s^{i}\in\widehat{S}^{i}} K_{i}(E^{s^{i}}\cap E^{\sigma^{-i}}),$$

with ${\Pi }_{a^{s^{i}}}{\Pi }_{a^{s^{'i}}}=o_{2}$, ∀s ⁱ ≠ s ^′i, g(o ₂) = ∅, $g(I) = \mathcal {S}^{O}_{i}$ and ${\sum }_{s^{i}\in \texttt {supp}(\sigma ^{i})}{\Pi }_{a^{s^{i}}}=I.$ This implies the existence of a measure μ such that $e^{i}\circ g({\Pi }_{a^{s^{i}}}) = \mu ({\Pi }_{a^{s^{i}}}).$ □

Proof of Theorem 1 (1-2)

We prove that the set of all equilibrium local intrinsic states of player i, ${\omega }_{(w^{i_{-i}},w^{i_{i}})},$ in a faithful representation $\mathfrak {g}$ of ${{\Gamma }^{i}_{G}}$ are in the complex unit sphere $\mathbb {S}$ and forms a ray. To see that every ${\omega }_{(w^{i_{-i}},w^{i_{i}})}\in \mathfrak {g}$ lies in the unit sphere $\mathbb {S}$ of $\mathbb {H}=\mathbb {C}^{m_{i}},$ it suffices to note that if i is in an intrinsic equilibrium σ = (σ ⁱ, σ ^{− i}), then a faithful representation of ${{\Gamma }^{i}_{G}}$ implies that the (co)-vectors of weights ω _{(w†, w†)} and ω _{(w, w)} must fulfill the equality,

$$\omega_{(w^{{\dag}},w)}(s^{i})\omega_{(w,w^{{\dag}})}(s^{i}) = \omega_{(w^{{\dag}},w^{{\dag}})}(s^{i}),$$

with ω _{(w†, w†)}(s ⁱ) = σ ⁱ(s ⁱ). Together with the condition that σ ⁱ is in the unit simplex, conditions (i) and (ii) imply that ${\sum }_{s^{i}\in \texttt {supp}(\sigma ^{i})}\omega _{(w^{{\dag }},w)}(s^{i}){\omega }_{(w,w^{{\dag }})}(s^{i}) = 1$ where ${\omega }_{(w,w^{{\dag }})}(s^{i}) = \overline {\omega }_{(w^{{\dag }},w)}(s^{i})$ is the complex conjugate of ω _{(w†, w)}(s ⁱ). From this last condition, we infer that ω _{(w, w†)} is in 𝕊 and ω _{(w†, w)} is its co-vector. This shows that in equilibrium, the intrinsic state space of i (at w) is in 𝕊. Finally, the fact that the set of vectors ω _{(w, w†)} lying in $\mathfrak {g}$ forms a one-dimensional subspace of 𝕊 follows easily from the Born rule. □

Proof of Theorem 1 (3)

(orthogonal projector)

The analysis of the choice of player i in terms of a probability measure over a set of projectors requires the following definitions.

First, we have to define a measure μ on the complex Hilbert space $\mathbb {H}=\mathbb {C}^{m_{i}}.$ Let $\mathcal {P}(\mathbb {H})$ be the set of orthogonal projectors of $\mathbb {H}.$ Consider a unit vector a _{s i} and the associated one dimensional projector ${\Pi }_{a_{s^{i}}}$. A measure is a mapping which assigns a non-negative real number $\mu ({\Pi }_{a_{s^{i}}})$ to each projector ${\Pi }_{a_{s^{i}}}$. Hence, we can identify an empirical model, $e^{i}(\cdot \left |\mathfrak {g}\right .)$, as a mapping $\mu :\mathcal {P}(\mathbb {H})\rightarrow \mathbb {R}$ by setting μ(π) = f(Im(π)). This mapping must be defined such that, if $\left \{a_{s^{i}}\right \}$ are mutually orthogonal, then the measure of ${\sum }_{s^{i}\in S^{i}}{\Pi }_{a_{s^{i}}}$ has to satisfy the (sub-)additivity property $\mu ({\sum }_{s^{i}\in S^{i}}{\Pi }_{a_{s^{i}}}) = {\sum }_{s^{i}\in S^{i}}\mu ({\Pi }_{a_{s^{i}}})$. Any such measure is determined by its values on the one dimensional projections such that:

• (1) If $\left \{{\Pi }_{a_{s^{i}}}\right \}$ is a family of pairwise orthogonal projectors i.e. ${\Pi }_{a_{s^{i}}}{\Pi }_{a_{s^{i'}}}=0$ for $s^{i}\neq s^{i^{\prime }}$ with ${\sum }_{s^{i}\in S^{i}}{\Pi }_{a_{s^{i}}}=I,$ then $\left \{\mu ({\Pi }_{a_{s^{i}}})\right \}$ is summable with sum μ(id).

• (2) If $\left \{{\Pi }_{a_{s^{i}}}\right \}$ is a family of pairwise orthogonal projectors i.e. ${\Pi }_{a_{s^{i}}}{\Pi }_{a_{s^{i^{\prime }}}}=0$ for $s^{i}\neq s^{i^{\prime }}$, then $\left \{\mu ({\Pi }_{a_{s^{i}}})\right \}$ is summable with $\sum \mu ({\Pi }_{a_{s^{i}}}) = \mu (\sum {\Pi }_{a_{s^{i}}}).$

In this case μ is a measure on $\mathcal {P}(\mathbb {H}).$

Next, we will use Gleason’s Theorem in order to establish a contradiction, namely that if we use a non canonical self-projective model, with a set of perspectives greater than two, then we cannot derive a well-defined probability measure, e ⁱ.

Gleason’s Theorem [33]: In a Hilbert space of finite dimension d ≥ 3, any bounded measure on the orthogonal projectors of ℍ is such that π → tr(Aπ) where A is a linear Hermitian operator of ℍ.

In the particular case where the measure is assumed to be a probability measure, Gleason’s Theorem implies that,

$$\mu({\Pi}) = \texttt{tr}(\rho{\Pi}),$$

where ρ is a density operator. We have the following lemma. □

Lemma 4

Fix the empirical canonical Krikpean model for player i in G, $\mathcal {M}^{iO}_{G},$ where σ ⁱ has been determined as being true at perspective w ⁱⁱ . There exists an empirical distribution $e^{i}(w^{s^{i}}) = \texttt {tr}(\rho {\Pi }_{a_{s^{i}}})$ with $\rho ={\Pi }_{\omega _{{e}_{-i,i}}}$ if and only if we have the faithful representation with $\rho \in (\mathfrak {g},\wedge ).$

Proof

“Only if”. We have the following lemmata:lemmata Suppose an equilibrium σ of G has been determined as true in a self-projective model, $\mathcal {M}^{i}_{G}.$ If player i has more than two pure strategies i.e. m _i ≥ 3, then, any empirical distribution of $\mathcal {M}^{Oi}_{G}$ must be of the form, $e^{i}(w^{s^{i}}) = \texttt {tr}(\rho {\Pi }_{a_{s^{i}}})$ with ρ a self-adjoint endomorphism of trace 1 (a density matrix indeed). □

Proof

By Lemma 2, $e^{i}(w^{s^{i}}) = \mu ({\Pi }_{a_{s^{i}}}), \forall s^{i}$ with μ a bounded measure. By Gleason’s Theorem [33] we have that $\mu ({\Pi }_{a_{s^{i}}}) = \texttt {tr}(\rho {\Pi }_{a_{s^{i}}})$ with ρ a self-adjoint endomorphism. The necessity to have a trace 1 follows since we want a probability measure. □

The above results allow to conclude that the global intrinsic state of i is necessarily $\rho =h((w,w^{\dag }))\wedge h((w^{\dag },w))\in (\mathfrak {g},\wedge )$. The fact that h((w, w ^†)) ∧ h((w ^†, w)) is an orthogonal projector, ${\Pi }_{{\omega }_{(w,w^{\dag })}}= \left |{\omega }_{(w,w^{\dag })} \right \rangle \left \langle {\omega }_{(w,w^{\dag })} \right |,$ follows immediately from the fact that ω _{(w, w†)} is a unit vector by property (1) of Theorem 1 above.

Proof of Theorem 1 (4)

(Born rule)

By definition of an empirical distribution, we must have e ⁱ(w ^si) = σ ⁱ. If we have a faithful representation $\mathfrak {g},$ proposition 2 implies that $e^{i}(w^{s^{i}}) = \overline {h}((w^{\prime },w))(s^{i})h((w^{\prime },w))(s^{i}), \forall s^{i}$, which can be rewritten as

$$e^{i}(w^{s^{i}}) = \left|\omega_{(w,w^{{\dag}})}(s^{i})\right|^{2}.$$

□

Lemma 5 establishes the existence of an empirical distribution in the canonical model and states that we must have a faithful representation of the self-projective graph to have an empirical distribution. We now give the proof that an empirical distribution does not exist for self-projective models $\mathcal {M}^{i}_{G}$ with more than two perspectives, whenever m _i ≥ 3.

Before we embark in the proof of this result, we need some preliminary definitions.

Definition 8

Let $\mathcal {M}^{'i}_{G},$ and $\mathcal {M}^{i}_{G},$ be two self-projective models for G with $\mathcal {W}^{\prime }\subset \mathcal {W}$. If $A^{\sigma ^{i}}$ can be determined as being relatively true in both models, then we say that $\mathcal {M}^{i}_{G},$ is a reducible Krikpean model for G. When $\mathcal {M}^{i}_{G},$ is not reducible it is said to be irreducible.

We will refer to the irreducible self-projective model with $\mathcal {W}=\left \{{w^{i}_{i}},w^{i}_{-i}\right \}$ as the canonical self-projective model for G.

Lemma 5

In a reducible self-projective model for a game G, $\mathcal {M}^{i}_{G}$ , with m _i ≥ 3 if player i determines a rational strategy σ ⁱ as being relatively true i.e. a mixed NE profile (σ ⁱ) _{i ∈ N} , then $\mathcal {M}^{i}_{G}$ has no empirical distribution e ⁱ = σ ⁱ.

Proof

• Step 1. Every self-projective model of G, $\mathcal {M}^{i}_{G},$ with a set of m > 2 perspectives, is reducible. Consider a self-projective model of G, $\mathcal {M}^{i}_{G},$ with its associated weighted directed self-projective graph, ${{\Gamma }^{i}_{G}} = \left \langle \mathcal {W},\mathcal {E},{\Omega }\right \rangle$ where $\mathcal {W}=\left \{{w^{i}_{J}}:J\in M\right \}$ is an arbitrary class of subsets of W with a cardinality $\left |\mathcal {W}\right |>2.$ This induces a graph ${{\Gamma }^{i}_{G}}$ with m > 2 vertices. Theorem 1 asserts that relative truth σ ⁱ can indeed be determined in any such a model. Hence, this immediately shows that $\mathcal {M}^{i}_{G}$ is indeed reducible.

• Step 2. We prove that any $\mathcal {M}^{i}_{G}$ that is reducible has no empirical distribution e ⁱ. By step 1, any $\mathcal {M}^{i}_{G}$ that is reducible has a set of perspectives of cardinality greater than 2. Next, our aim is to apply Gleason’s Theorem [33] in order to show that e ⁱ cannot form a probability measure—hence an empirical distribution—when the set of vertices of ${{\Gamma }^{i}_{G}}$ is of cardinality greater than 2. Let π be projection operator π on a Hilbert space $\mathbb {H}.$ ³⁹ Expand the unit vector of weights, $\left |\omega _{({w^{i_{{K}}},w^{i_{{L}}}})}\right \rangle \in \mathbb {C}^{m_{i}}$ in the standard basis, $\left \{a_{s^{i}}\right \}_{s^{i}\in S^{i}}$, as $\left |\omega _{({w^{i_{{K}}},w^{i_{{L}}}})}\right \rangle ={\sum }_{s^{i}\in S^{i}}\left \langle a_{s^{i}}\left |\right . \omega _{({w^{i_{{K}}},w^{i_{{L}}}})}\right \rangle \left |a_{s^{i}}\right \rangle .$ Lemma 5 tells us that $e^{i}(w^{s^{i}}) = \mu ({\Pi }_{a^{s^{i}}}).$ Hence, the empirical distribution must be a function of ${\Pi }_{a_{s^{i}}}\omega _{({w^{i_{{K}}},w^{i_{{L}}}})},$ for $\omega _{({w^{i_{{K}}},w^{i_{{L}}}})}\in \mathbb {H},$ with $({w^{i_{{K}}},w^{i_{{L}}}})\in \mathcal {E}.$ Thus, there must exist a function f such that: $e^{i}(w^{s^{i}}) = f({\Pi }_{a_{s^{i}}}\omega _{({w^{i_{{K}}},w^{i_{{L}}}})}), \forall s^{i}.$ A direct application of Gleason’s Theorem allows to conclude that the unique empirical probability measure e ⁱ is of the form

  $$e^{i}(w^{s^{i}}) = \left\|{\Pi}_{a_{s^{i}}}\omega_{({w^{i_{{K}}},w^{i_{{L}}}})}\right\|^{2} \forall s^{i}.$$

  Next, we note that the Indivisibility axiom implies necessarily the Born rule in the canonical model, without using Gleason’s Theorem. From this, we will be able to show that the structure of the graph of any non canonical model does not induce the Born rule, which allows to conclude that such model does not induce a well-defined empirical distribution e ⁱ as a consequence of Gleason’s Theorem.

The axiom of Indivisibility implies that the intrinsic state of player i is $w_{0}\in E^{s^{i}}=K_{i_{i}\otimes i_{-i}}E^{s^{i}}$ iff we have ${\Pi }_{a_{s^{i}}}\left |\omega _{(w,w^{{\dag }})}\right \rangle$ for i _{− i} and ${\Pi }_{a_{s^{i}}}\left (\left \langle \omega _{(w,w^{{\dag }})}\right |\right )^{\dag }$ for i _i . In the canonical self-projective model, this immediately implies that, $e^{i}(w^{s^{i}}) = \left \|{\Pi }_{a_{s^{i}}}\left |\omega _{(w,w^{{\dag }})}\right \rangle \right \|^{2},$ where ∥⋅∥ is the norm derived from the inner product on $\mathbb {C}^{m_{i}}.$ This by definition is the Born rule. This of course also requires to have the faithful representation of the canonical graph given in Theorem 1. Hence, Gleason’s Theorem and the axiom of Indivisibility allows to conclude that the Born rule cannot be constructed from a self-projective graph with more than two perspectives. This can be seen by noting that if ${{\Gamma }^{i}_{G}}$ has more than two vertices, then the empirical distribution is

$$e^{i}\left(w^{s^{i}}\right) :=\mu\left({\Pi}_{a_{s^{i}}}\right),~\mathrm{ such\; that\;} e^{i}\left(w^{s^{i}}\right) \neq \left\|{\Pi}_{a_{s^{i}}}\omega_{({w^{i_{{K}}},w^{i_{{L}}}})}\right\|^{2}.$$

This immediately implies that the graph must have a unique pair of weights and hence $\mathcal {M}^{i}_{G}$ must be canonical i.e. irreducible. If not, either the indivisibility axiom is not met, or ${{\Gamma }^{i}_{G}}$ is not well-defined i.e. it does not assign some weights at every edge, which entails that h does not form an isomorphism. From this we conclude that e ⁱ never forms a well-defined probability measure for any reducible self-projective models.

That a canonical model with a faithful representation induces necessarily an empirical distribution is shown in the proof of Theorem 1 (3) below. □

Since any reducible model has no empirical distribution, the rest of the appendix is devoted to the study of canonical self-projective models. Henceforth it is convenient to set w ⁱⁱ := w ^† and w ⁱ⁻ⁱ := w.

Proof of Corollary 1

By Theorem 1, any global equilibrium intrinsic state of player i is given by the density matrix $\left |{\omega }_{(w^{i_{-i}},w^{i_{i}})}\right \rangle \left \langle {\omega }_{(w^{i_{-i}},w^{i_{i}})}\right |.$ This operator can be rewritten as,

$$\begin{array}{@{}rcl@{}} &&\sum\limits_{s^{'i}\in \texttt{supp}(\sigma^{i})}\left|{\omega}_{\left(w^{i_{-i}},w^{i_{i}}\right)}\left({s^{'i}}\right)\right|^{2} \left.\right|\left.a_{s^{'i}} \right\rangle\left\langle a_{s^{'i}} \right.\left.\right|\\&&+\sum\limits_{s^{i}\in \texttt{supp}(\sigma^{i}):s^{i}\neq s^{'i}}{\omega}_{\left(w^{i_{-i}},w^{i_{i}}\right)}({s^{i}}){\omega}_{\left(w^{i_{-i}},w^{i_{i}}\right)}\left({s^{'i}}\right)\left.\right|\left.a_{s^{i}} \right\rangle\left\langle a_{s^{'i}} \right.\left.\right|. \end{array} $$

By the classical indifference condition, σ ⁱ is an equilibrium strategy for player i if and only if very s ⁱ ∈ supp(σ ⁱ) is a best reply. On the other hand, a faithful representation implies that in the empirical model, the event,

$$\left\|\Box^{i_{i}}A^{s^{i}}\wedge\Box^{i_{-i}}A^{s^{'i}}\right\|^{\mathcal{M}^{i}_{G}},$$

has a non-zero (complex) off-diagonal weight given by,

$${\omega}_{\left(w^{i_{i}},w^{i_{-i}}\right)}({s^{i}}){\omega}_{\left(w^{i_{-i}},w^{i_{i}}\right)}\left({s^{'i}}\right)\neq 0,$$

for all pair of pure actions, s ⁱ, s ^′i ∈ supp(σ ⁱ). This proves the existence of off-diagonal terms. □

Proof of Corollary 2

By Theorem 1, any mixed strategy $\sigma ^{i}_{\omega }$ gets identified by an orthogonal projector, π _ω , with complex off-diagonal terms. Hence, if player i is in a pure intrinsic state π _ω with a probability p _ω , we have an empirical distribution of the form, $e^{i}={\sum }_{\omega } p_{\omega }\sigma ^{i}_{\omega }.$

Necessity

By contradiction. Suppose that the global intrinsic state of player i cannot be expressed by ${\sum }_{\omega } p_{\omega }{\Pi }_{\omega }.$ This means that player i does not determine $\sigma ^{i}_{\omega }$ as a rational strategy with a probability p _ω . By definition, this implies that $e^{i}\neq {\sum }_{\omega } p_{\omega }\sigma ^{i}_{\omega }.$ □