Holoscopic Economics II: Holoscopicity in Games [z2gfmk]

For any set \(X\), the tuple-to-multiset projection is the map \[q_X:X^n\to \operatorname {Multiset}_n(X)\] defined by \[q_X\left (\langle x_1,\ldots ,x_n\rangle \right )=⦃x_1,\ldots ,x_n⦄.\] The projection forgets the positions in a tuple while preserving its elements and their multiplicities. The subscript records the set from which those elements are drawn.

For any set \(X\) and tuple \(\boldsymbol {x}\in X^n\), the permutation orbit of \(\boldsymbol {x}\) in \(X^n\) is \[\operatorname {Orb}_X(\boldsymbol {x})=q_X^{-1}\left (\left \{q_X(\boldsymbol {x})\right \}\right ).\] Equivalently, \(\operatorname {Orb}_X(\boldsymbol {x})\) is the set of all distinct reorderings of \(\boldsymbol {x}\). When entries repeat, permutations that produce the same tuple contribute only one element to the orbit.

To separate participant position from individual state, compare the actual profile with every profile obtained by reordering its entries. These state profiles form \(\operatorname {Orb}_\Omega (\boldsymbol {\omega })\). We write an arbitrary member of this orbit as \(\boldsymbol {\eta }=\langle \eta _1,\ldots ,\eta _n\rangle \). Moving within the orbit changes which participant position carries each state but introduces no state and removes none. It is therefore the smallest collection of state profiles needed to compare every exchange of the states in the actual profile.

Let \[\mathcal {P}=\left \{\langle \theta ,s\rangle \;\middle |\;\theta \in \Omega \text { and }s\in S_\theta \right \}\] be the set of valid state–strategy pairs. The labeled outcome domain over the actual and counterfactual state profiles is \[L_{\boldsymbol {\omega }}=\left \{\langle \langle \eta _1,s_1\rangle ,\ldots ,\langle \eta _n,s_n\rangle \rangle \in \mathcal {P}^n\;\middle |\;\langle \eta _1,\ldots ,\eta _n\rangle \in \operatorname {Orb}_\Omega (\boldsymbol {\omega })\right \}.\]

An element \(\ell \in L_{\boldsymbol {\omega }}\) records both which state–strategy pairs occur and the participant positions in which they occur. Define the corresponding anonymous outcome domain as the image \[M_{\boldsymbol {\omega }}:=q_{\mathcal P}\left (L_{\boldsymbol {\omega }}\right )\subseteq \operatorname {Multiset}_n(\mathcal P).\] Restricting the tuple-to-multiset projection on \(\mathcal P^n\) to \(L_{\boldsymbol {\omega }}\) with that codomain gives the forgetting map \[q_{\boldsymbol {\omega }}:=\left .q_{\mathcal P}\right |_{L_{\boldsymbol {\omega }}}:L_{\boldsymbol {\omega }}\to M_{\boldsymbol {\omega }}.\]

The map \(q_{\boldsymbol {\omega }}\) forgets only the positions of state–strategy pairs; it never separates a state from its strategy, and by construction it is onto \(M_{\boldsymbol {\omega }}\). The domain \(L_{\boldsymbol {\omega }}\) is closed under permutations of pairs; consequently, for every \(\ell \in L_{\boldsymbol {\omega }}\), the orbit \(\operatorname {Orb}_{\mathcal P}(\ell )\) lies in \(L_{\boldsymbol {\omega }}\) and is exactly the fiber of \(q_{\boldsymbol {\omega }}\) containing \(\ell \).

Part II writes \(M_{\boldsymbol {\omega }}\) rather than \(M\) to keep the state profile explicit while several state profiles are in view. Technically, for every \(\boldsymbol {\eta }\in \operatorname {Orb}_\Omega (\boldsymbol {\omega })\), applying the same construction with \(\boldsymbol {\eta }\) gives \(L_{\boldsymbol {\eta }}=L_{\boldsymbol {\omega }}\), and hence \(M_{\boldsymbol {\eta }}=M_{\boldsymbol {\omega }}=M\), where \(M\) is the anonymous outcome domain from Part I.

A state-permutation game family is a tuple \[\mathfrak {G}=\langle N,\Omega ,\boldsymbol {\omega },S,\mathbf {u}\rangle ,\] where \(N\) is the participant set, \(\Omega \) is the individual-state space, \(\boldsymbol {\omega }\) is the actual state profile, \(S\) assigns each state its nonempty set of available strategies, and \(\mathbf {u}=(u_i)_{i\in N}\) is the utility family for all participants, with \[u_i:L_{\boldsymbol {\omega }}\to \mathbb {R}\qquad (i\in N).\]

For each state profile \(\boldsymbol {\eta }\in \operatorname {Orb}_\Omega (\boldsymbol {\omega })\), restricting the labeled utilities to outcomes with that profile defines the normal-form game \[G_{\boldsymbol {\eta }}=\left \langle N,\bigl (S_{\eta _i}\bigr )_{i\in N},\bigl (u_i^{\boldsymbol {\eta }}\bigr )_{i\in N}\right \rangle ,\] where \[u_i^{\boldsymbol {\eta }}(s_1,\ldots ,s_n)=u_i\left (\langle \langle \eta _1,s_1\rangle ,\ldots ,\langle \eta _n,s_n\rangle \rangle \right ).\]

Thus the family contains one ordinary game for every placement of the fixed multiset of states. The actual-profile game \(G_{\boldsymbol {\omega }}\) models the situation under analysis; the remaining games assign those states counterfactually to participant positions. Only the utilities in \(G_{\boldsymbol {\omega }}\) determine its best responses and Nash equilibria. The counterfactual values characterize the utilities across positions but do not alter the actual-profile game.

A labeled utility \(u_i:L_{\boldsymbol {\omega }}\to \mathbb {R}\) is holoscopic when its value does not depend on the participant positions forgotten by \(q_{\boldsymbol {\omega }}\). Equivalently, it factors through that map: \[u_i\text { is holoscopic}\quad \Longleftrightarrow \quad \exists v:M_{\boldsymbol {\omega }}\to \mathbb {R}\text { such that }u_i=v\circ q_{\boldsymbol {\omega }}.\]

Because \(q_{\boldsymbol {\omega }}\) is onto, the factor is unique whenever it exists; for a holoscopic \(u_i\), denote it by \(v_i:M_{\boldsymbol {\omega }}\to \mathbb {R}\). The same condition can be stated without introducing the factor: \(u_i\) is holoscopic exactly when it is constant on \(\operatorname {Orb}_{\mathcal P}(\ell )\) for every \(\ell \in L_{\boldsymbol {\omega }}\).

A labeled utility is meroscopic when it is not holoscopic—that is, when changing participant positions while holding the anonymous outcome fixed changes its value. Holoscopicity is therefore a property derived from each utility function, not membership in a predetermined set of “holoscopic participants.” A single game family may contain both holoscopic and meroscopic utilities.

If every \(u_i\) is holoscopic, the game at the actual profile \(\boldsymbol {\omega }\) agrees with the corresponding holoscopic game from Part I. Indeed, \[\begin {aligned}u_i^{\boldsymbol {\omega }}(s_1,\ldots ,s_n)&=u_i\left (\langle \langle \omega _1,s_1\rangle ,\ldots ,\langle \omega _n,s_n\rangle \rangle \right )\\&=v_i\left (q_{\boldsymbol {\omega }}\left (\langle \langle \omega _1,s_1\rangle ,\ldots ,\langle \omega _n,s_n\rangle \rangle \right )\right )\\&=v_i\left (⦃\langle \omega _1,s_1\rangle ,\ldots ,\langle \omega _n,s_n\rangle ⦄\right ),\end {aligned}\] which is precisely the utility formula from Part I.

Suppose \(L_{\boldsymbol {\omega }}\) is nonempty, and choose a metric \(d\) on the functions \(L_{\boldsymbol {\omega }}\to \mathbb {R}\). Let \[\mathcal H_{\boldsymbol {\omega }}=\left \{v\circ q_{\boldsymbol {\omega }}\;\middle |\;v:M_{\boldsymbol {\omega }}\to \mathbb {R}\right \}\] be the class of holoscopic utilities on the labeled domain. The absolute meroscopic bias of a labeled utility \(u:L_{\boldsymbol {\omega }}\to \mathbb {R}\) is its distance from this class: \[\operatorname {Bias}_d(u)=\inf _{h\in \mathcal H_{\boldsymbol {\omega }}}d(u,h).\]

This definition does not require \(L_{\boldsymbol {\omega }}\) to be finite. On an infinite domain, however, constructing an integral metric such as an \(L^2\) distance requires a measure and suitable integrability conditions. Those choices belong to the application rather than to the definition of holoscopicity or meroscopic bias.

Absolute bias alone does not say how large position dependence is relative to the utility’s overall variation. To make that comparison, contrast the distance from holoscopic utilities with the distance from constant utilities. Let \[\mathcal C=\left \{k:L_{\boldsymbol {\omega }}\to \mathbb {R}\;\middle |\;k\text { is constant}\right \}.\] The utility spread under \(d\) is the distance from \(u\) to this class, \[\operatorname {Spread}_d(u)=\inf _{k\in \mathcal C}d(u,k),\] and the meroscopic bias ratio is \[\operatorname {BiasRatio}_d(u)=\begin {cases}\operatorname {Bias}_d(u)/\operatorname {Spread}_d(u)&\operatorname {Spread}_d(u)>0,\\0&\operatorname {Spread}_d(u)=0.\end {cases}\]

Every constant utility is holoscopic, so \(\mathcal C\subseteq \mathcal H_{\boldsymbol {\omega }}\). It follows that \(\operatorname {Bias}_d(u)\leq \operatorname {Spread}_d(u)\) and hence \(0\leq \operatorname {BiasRatio}_d(u)\leq 1\) for any metric \(d\). No additional property of the metric is needed for this bound.

Holoscopicity always implies zero bias, and with a technical closure condition on \(\mathcal H_{\boldsymbol {\omega }}\), the reverse holds. By the point-of-closure characterization, \(\operatorname {Bias}_d(u)=0\) if and only if \(u\) belongs to the \(d\)-closure of \(\mathcal H_{\boldsymbol {\omega }}\). We therefore recover exact holoscopicity when \(\mathcal H_{\boldsymbol {\omega }}\) is a closed subset of the metric space determined by \(d\).

When utilities are considered equivalent up to positive affine transformations, the meroscopic bias ratio should not depend on which representation is chosen. This invariance holds if, for all functions \(F,G\), \[d(aF+b,aG+b)=a\cdot d(F,G)\qquad (a>0,\ b\in \mathbb {R}).\] Applying the same transformation multiplies both meroscopic bias and utility spread by \(a\), leaving their ratio unchanged. A metric that does not satisfy this condition can still define the ratio, but its numerical value may depend on the chosen representation.

Consider B’s eight-value utility from the allocation example. Here \(L_{\boldsymbol {\omega }}\) has eight elements, so we use the uniform root mean square distance \[d_{\mathrm {RMS}}(F,G)=\sqrt {\frac {1}{8}\sum _{\ell \in L_{\boldsymbol {\omega }}}\left (F(\ell )-G(\ell )\right )^2}.\] On this finite domain, the class of holoscopic utilities is a closed subset of the resulting metric space, so zero RMS bias is equivalent to holoscopicity. The metric also satisfies \(d_{\mathrm {RMS}}(aF+b,aG+b)=a\,d_{\mathrm {RMS}}(F,G)\) for \(a>0\), so its bias ratio is invariant under positive affine transformations.

Under this metric, the nearest holoscopic utility is obtained by replacing the values within each permutation orbit by their arithmetic mean. Written back on the labeled domain, this orbitwise symmetrization is \[\operatorname {Sym}(u)(\ell )=\frac {1}{\left |\operatorname {Orb}_{\mathcal P}(\ell )\right |}\sum _{\ell '\in \operatorname {Orb}_{\mathcal P}(\ell )}u(\ell ').\]

Each outcome orbit in this example has two labeled representatives, one at each state profile. The function \(\operatorname {Sym}(u_B)\) assigns their arithmetic mean to both representatives. For example, when both participants take the \(50\), B assigns the two representatives utilities \(-100\) and \(-75\), so their common symmetrized value is \((-100-75)/2=-87.5\). The remaining entries are calculated in the same way:

The eight absolute deviations from these four orbit means are four copies of \(12.5\), two copies of \(37.5\), and two copies of \(50\). Their root mean square is B’s meroscopic bias: \[\begin {aligned}\operatorname {Bias}_{d_{\mathrm {RMS}}}(u_B)&=\sqrt {\frac {4(12.5)^2+2(37.5)^2+2(50)^2}{8}}\\&\approx 32.48.\end {aligned}\]

Under root mean square distance, the closest constant utility takes the arithmetic mean of the utility values. Let \(\bar {u}_B\) be this function. For every \(\ell \in L_{\boldsymbol {\omega }}\), \[\bar {u}_B(\ell )=\frac {-100-75+75-25+25+100+50+25}{8}=9.375,\] so \[\operatorname {Spread}_{d_{\mathrm {RMS}}}(u_B)=d_{\mathrm {RMS}}(u_B,\bar {u}_B)\approx 66.07.\]

The general ratio therefore becomes \[\operatorname {BiasRatio}_{d_{\mathrm {RMS}}}(u_B)\approx \frac {32.48}{66.07}\approx 0.49.\] Thus B’s meroscopic bias is about \(49\) percent of the root mean square distance from \(u_B\) to the closest constant utility.

The value \(0.49\) summarizes the magnitude of B’s position dependence relative to B’s overall utility spread. Because the ratio is a scalar, it does not indicate which positional distinctions increase or decrease utility. This loss of direction is intentional: meroscopic bias measures position dependence, including dependence on which participant position is one’s own; it is not a measure of selfishness in the ordinary sense.

The value also depends on B’s utilities at the exchanged state profile, which are not determined by the four utilities in the actual-profile game. Different counterfactual extensions can therefore preserve the same best responses and Nash equilibria in that game while producing different holoscopicity classifications and meroscopic bias values.