Holoscopic Economics I: Holoscopic Games [6yq1u5]

In a conventional normal-form game, participant \(i\) evaluates a labeled strategy profile \(\langle s_1,\ldots ,s_n\rangle \) through a utility function, obtaining the value \(u_i(s_1,\ldots ,s_n)\).

The position of each strategy in this tuple identifies the participant who chose it. A utility may therefore treat the first, second, or any other position specially. If exchanging the perspectives of two participants is not supposed to change an evaluation, one way to guarantee this invariance is to make those position labels unavailable to the utility.

A natural way to eliminate position labels is to use multisets, in which the same element may occur more than once but no position is distinguished. We will use the notation \(⦃\ldots ⦄\) to denote a multiset. For example, \(⦃x, x, y⦄ = ⦃x, y, x⦄\), since both contain two copies of \(x\) and one copy of \(y\), whereas \(⦃x, x, y⦄ \neq ⦃x, y⦄\), since the multiplicity of \(x\) differs. The first attempt is therefore to replace the labeled tuple \(\langle s_1,\ldots ,s_n\rangle \) with an anonymous multiset \(⦃s_1,\ldots ,s_n⦄\). Reordering its elements does not change the multiset, so no position is distinguished as “oneself” or as any other particular participant.

A multiset of strategies alone, however, forgets too much. The same action can have a different significance when chosen under different needs, knowledge, abilities, education, assets, social connections, legal status, or other relevant circumstances. We represent those circumstances by an individual state \(\omega _i\) and keep each chosen strategy \(s_i\) attached to the state under which it is available. The resulting anonymous outcome is \[⦃\langle \omega _1,s_1\rangle ,\ldots ,\langle \omega _n,s_n\rangle ⦄.\]

The multiset of state–strategy pairs forgets who occupies each participant position, but it preserves both the circumstances that occur and the strategy chosen under each circumstance. The distinction matters: without states, the model could express simple equality of actions but not equity between people in different situations. Exchanging perspectives rearranges state–strategy pairs; it never detaches a strategy from its state or changes either component.

Which information belongs in a state is therefore a substantive modeling choice. In a game about allocating money, current assets may be relevant while food preferences may be irrelevant; another game may require hunger, mobility, education, or access to information. Omitting a relevant circumstance can make an apparently impartial evaluation merely insensitive. Conversely, encoding a participant’s identity itself as part of the state restores the same identity distinctions that the anonymous representation was meant to remove.

Holoscopicity concerns only evaluation. Participant \(i\) still controls only the strategy chosen at position \(i\); the utility function simply evaluates the resulting collection of state–strategy pairs without distinguishing which participant position carries each pair. Nor does holoscopicity require all participants to share one value system. Different participants may assign different utilities to the same anonymous outcome.

Consider two participants whose circumstances differ. A already has \(100\) monetary units, B already has \(150\), and an additional \(50\) units are available. Either participant may take it. How can we model this choice as a holoscopic game?

• \(N = \{1, 2\}\)
• \(\Omega = \{\text {already has }100, \text {already has }150\}\)
• \(\boldsymbol {\omega } = \langle \text {already has }100, \text {already has }150\rangle \)
• \(S_{\text {already has }100} = S_{\text {already has }150} = \{\text {take }50, \text {do not take}\}\)

Suppose both participants evaluate the anonymous outcome according to the following table. The row and column labels identify participants by their current states because the utility values follow states and actions rather than participant identities.

Holoscopic preferences do not eliminate disagreement or coordination problems. The definition removes distinguished participant positions from each utility’s input; it does not force different utility functions to rank anonymous outcomes alike. When participants’ rankings do align, however, cooperative outcomes may arise from their independent best responses without an explicit coalition or binding agreement.

Some coordination problems may be resolved through focal points without communication. Suppose all participants have the same state, a strategy \(s\) is salient, and the anonymous outcome \(⦃\langle \omega ,s\rangle , \langle \omega ,s\rangle , \ldots , \langle \omega ,s\rangle ⦄\) is among the best outcomes. It is then likely to be a focal point. For example, if six equally hungry people regard an equal division of a pizza as one of the best outcomes, “everyone takes one sixth” may be salient even without communication.

Public signals offer another coordination mechanism. Signals such as coin flips, dice rolls, and traffic lights can allow participants to coordinate on correlated equilibria.

Existing behavior can itself provide a coordinating force. In real life, changing an existing pattern of play usually incurs communication costs, creating rational inertia. Communication costs can therefore allow several existing patterns of play to remain practically stable.

Evolutionary game theory offers tools for studying how behavior changes over time. A more detailed treatment could introduce population dynamics and update rules to study which equilibria are attractive in the long run, including the formation of some focal points. Such an extension may help explain how fair outcomes become evolutionarily favored when preferences are holoscopic.

Holoscopic games differ from conventional games in what their utility functions are allowed to distinguish. Conventional games permit each participant’s utility to treat that participant’s own position differently. Even anonymous games and aggregative games usually treat a participant’s own strategy separately from the distribution or aggregate of the others’ strategies. A utility that actually depends on one or more privileged positions may be described as meroscopic.

Participant states differ from Bayesian types. The states defined here are fixed inputs to the game, and this article does not model uncertainty about them. By contrast, “types” in a Bayesian game encode information that shapes participants’ beliefs under uncertainty. Extensions could introduce incomplete information and information-acquisition costs; a Bayesian extension is one possibility.

Holoscopic preferences differ from inequity aversion. Models of inequity aversion encode a preference against unequal outcomes within participant-centered utility functions; holoscopic games instead change the domain of evaluation by removing the distinguished participant position. Holoscopicity does not by itself favor equality or fairness.

By removing position labels, holoscopicity may shift some disputes from competition between identity groups toward disagreement over which anonymous outcome should be pursued. This could support deliberation organized around proposed outcomes rather than allegiance to “our side,” but the formal model alone establishes neither that psychological shift nor its political effects. These remain empirical hypotheses, not consequences of the definition.

A separate open question is the psychological interpretation of holoscopic preferences. Holoscopicity provides a language for studying impartial or conscience-like preferences without taking self-interest as the only primitive. A stronger hypothesis suggested by the motivating observations is that the underlying decision system is inherently holoscopic and self-interested evaluation arises when self-related cognitive processes modify its operation, rather than that the underlying system is inherently selfish and only regulated by conscience and moral rules. The game-theoretic definition in this article does not decide between these psychological accounts.