Units are the formal standards that convert observed strategic and market behavior into evidence. In Interaction, units specify how actions, responses, exchanges, and institutional effects are measured, stabilized, and rendered comparable across agents, settings, and time. Without units, interactional behavior remains anecdotal or narrative; with units, it becomes evaluable.
A unit in Interaction is not a label. It is a conventional mapping between a conceptual feature of strategic or market interaction and a repeatable observational, transactional, or recording procedure. This mapping is what allows observed interactions to enter the evidence layer.
Function of Units
Within the SAT framework, units in Interaction perform four irreducible functions:
- Stabilization — ensure repeated observations refer to the same kind of interactional behavior.
Units fix what counts as the same action, same response, same transaction, or same strategic move across agents, markets, and institutional contexts. - Comparability — allow interactions to be evaluated across agents, environments, and time.
Units make it possible to compare strategic behavior or market outcomes without assuming identical objectives, beliefs, or informational states. - Aggregation — enable individual interactions to be combined into patterns, distributions, networks, or market-level regularities.
This supports movement from bilateral or local interactions to system-level structure without collapsing strategic heterogeneity. - Transmission — preserve semantic meaning as interaction data move between observers, datasets, and analytical frameworks.
Units ensure that recorded actions, prices, bids, contracts, or responses retain their meaning across institutional and analytical boundaries.
If any function fails, evidence about interaction degrades into impression, narrative, or ungrounded interpretation.
Conceptual Boundaries
Units in Interaction must be distinguished from related concepts:
- Variables identify what interactional behavior is observed; units define how it is quantified.
- Scales order or space observed actions or outcomes; units anchor those observations to defined magnitudes or categories.
- Indicators / proxies approximate latent strategic states or market forces; units define the observable interaction itself.
- Dimensions describe abstract interaction attributes (price, quantity, time, information); units are the concrete instantiations used to record exchanges and responses.
Confusing these categories produces structural errors in interaction evidence, often leading to false conclusions about market behavior or strategic incentives.
Validity Conditions
For a unit to function within the Evidence Layer for Interaction, it must satisfy most of the following:
- Operational definability — tied to a repeatable transaction, action, rule application, or observation protocol.
- Stability — meaning does not drift across agents or institutions without explicit redefinition of the interaction environment.
- Resolution — sufficient granularity to distinguish meaningful differences in strategic or market behavior.
- Characterizable error — noise from timing, reporting, institutional frictions, or strategic misrepresentation can be bounded or described.
- Inter-observer consistency — different observers record compatible interaction outcomes under similar institutional conditions.
Variation across agents, strategies, and institutions is expected and informative, but unit definitions must remain fixed.
Structural Types of Units
Fundamental Units
Fundamental units in Interaction are the independently defined measurement standards that underpin all market, strategic, and mechanism-based observations. They are not created by interaction theory; they are imported into Interaction from physical, mathematical, or institutional systems and used directly to record timing, amounts, probabilities, and quantities that structure multi-agent exchange.
These units do not rely on combinations, normalization, ranking, counting, or rule activation to exist. If every other unit type were removed, they would remain meaningful. In interaction settings, they anchor observable phenomena such as prices posted, bids submitted, quantities exchanged, signals sent, and responses timed—without presuming equilibrium, strategy, or rationality.
Because all higher-order interaction units (derived, count, ordinal, index, threshold) are constructed from these primitives, any instability in a fundamental unit propagates upward. Within the SAT Evidence Layer, fundamental units therefore define the measurement substrate of interaction: the point at which strategic behavior becomes recordable without interpretation.
Derived Units
Derived units in Interaction are units that are constructed from combinations of more primitive units—such as time, monetary amounts, quantities, and probabilities—through arithmetic or formal operations. They do not exist independently: altering the definition or measurement of any underlying unit immediately alters the derived unit.
In markets, strategic settings, and mechanisms, derived units capture relationships created by interaction rather than raw observations. They express spreads between offers, values created by exchange, imbalances between demand and supply, outcomes of matching processes, and the timing and likelihood of agreement. These units are what make interaction analyzable as a system rather than a collection of isolated actions.
Because derived units combine multiple inputs, they inherit all assumptions and errors of their components. Mis-specified prices, quantities, probabilities, or time bases propagate directly into derived measures such as surplus, rates, and probabilities of trade. For this reason, derived units are informative but fragile: they require careful alignment of definitions, time windows, and reference sets.
Within the SAT Evidence Layer, derived units for Interaction occupy the core analytic space between fundamentals and higher-order constructions. They remain grounded in observable quantities, yet they are rich enough to describe exchange, coordination, congestion, and strategic timing without invoking equilibrium concepts or behavioral rules.
Count Units
Count units in Interaction are units that represent raw, discrete tallies of entities or events. They answer only the question “how many” and carry no information about value, success, role, or outcome. Each increment reflects the occurrence of an event or the presence of an entity, without interpretation.
In market and strategic settings, count units are used to enumerate who participated and what occurred, prior to any classification by outcome or rule. They provide the basic quantitative substrate for interaction data, supporting aggregation and comparison without embedding assumptions about behavior, incentives, or efficiency.
Count units are additive over identical entities or events and permit no transformations beyond summation. Their validity depends entirely on stable definitions of what constitutes an agent, an interaction, an offer, or an observation. If those definitions change, counts lose comparability immediately.
Within the SAT Evidence Layer, count units play a strictly descriptive role. They do not explain interaction outcomes or strategic behavior; they simply record occurrence. All higher-order measures in Interaction—such as rates, probabilities, surpluses, or thresholds—are constructed from these counts and inherit their definitional constraints.
Ordinal Units
Interaction does not yield ordinal units at the level of direct measurement.
In interactive settings, ordering is never observed as a standalone, order-only quantity taken at a single point. Instead, order is always realized through relations among multiple agents and through sequences of actions over time. Who moves first, who responds, which offer precedes another, or which bid dominates another are all properties of process, not isolated measurements.
Because of this, interactional order is inseparable from timing, comparison, and context. It is expressed through timestamps, elapsed durations, counts of events, and constructed comparisons rather than through primitive rank positions. What looks like “ranking” in interaction is always the result of multiple observations brought into relation, not a direct unit recorded in isolation.
The consequence is simple and substantive:
interaction expresses order through construction, not declaration. Ordinality appears downstream—as a property of sequences, exchanges, or aggregated comparisons—rather than as a basic unit of observation.
This is not a limitation; it reflects the fact that interaction is inherently relational and dynamic. Order exists, but it exists as an outcome of interaction, not as a unit captured independently.
Index Units
Index units in Interaction are baseline-relative measures that express how observed market or strategic outcomes compare to a defined reference state. They are not raw measurements of price, quantity, or activity; instead, they are normalized constructions whose values have meaning only with respect to the baseline embedded in their definition.
In interaction settings, index units are used to summarize complex relational outcomes—such as pricing behavior, participation patterns, and welfare results—into comparable measures. By anchoring observations to explicit benchmarks (for example, a reference price, an equal-share allocation, or a maximal-surplus outcome), index units allow interaction data to be compared across markets, periods, or institutional designs without assuming identical underlying conditions.
Because index units are constructed from other units, they inherit all assumptions and uncertainties of their components as well as of their baseline. Any change in the reference price, benchmark allocation, or feasible set immediately alters the index’s interpretation. As a result, index units are informative but context-dependent: they require the baseline to be stated alongside the value for the measure to be meaningful.
Within the Evidence Layer, index units occupy the comparative end of interaction measurement. They do not describe events directly, and they do not encode behavioral rules. Instead, they provide a standardized way to evaluate relative performance, dispersion, and efficiency in interactive economic systems.
Threshold Units
Threshold units in Interaction represent rule-defined categorical cutoffs that determine whether an action, offer, or event is permitted, accepted, or constrained. They convert continuous or complex inputs—such as prices, budgets, eligibility criteria, or time—into binary states where only crossing the cutoff matters, not the distance from it.
In interactive settings, these units encode the activation of institutional or procedural rules that shape what can happen next: whether a bid qualifies, an offer meets minimum requirements, a participant is eligible, or a deadline has been met. They do not measure value, preference, or intensity; they record compliance with a rule.
Threshold units allow no arithmetic or ordinal manipulation beyond classification. Treating them as magnitudes or rankings violates their evidentiary role. Their validity rests entirely on the clarity and stability of the underlying rule: when the rule changes, the threshold unit’s meaning changes immediately.
Within the Evidence Layer, threshold units mark the points where constraints enter interaction. They delineate feasibility, admissibility, and timing boundaries, enabling interaction dynamics to be analyzed without embedding assumptions about behavior or outcomes beyond the rule itself.
SAT – Evidence – Units – Interaction (Markets, Strategy & Mechanisms)
| Unit | Unit Type | Measurement Structure | Reference Dependence | Transformability | Temporal Behavior | Error Profile | Aggregation Compatibility | Interpretive Load | Boundary Conditions |
|---|---|---|---|---|---|---|---|---|---|
| Second | Fundamental | Continuous, additive, linear | Absolute (physical standard) | Differences, ratios, scaling | Time-indexed; instantaneous or cumulative | Instrument resolution, clock drift | Fully aggregatable | Minimal (purely quantitative) | Breaks at relativistic / non-operational regimes |
| Millisecond | Fundamental | Continuous, additive, linear (instrument-bounded) | Absolute (physical standard) | Differences, ratios, scaling | Time-indexed; instantaneous | Resolution limits, precision noise | Fully aggregatable | Minimal | Invalid below measurement resolution |
| Currency Unit (e.g., dollar) | Fundamental | Continuous, additive, linear | Institutional absolute (legal definition) | Differences, ratios; regime-consistent conversions | Time-indexed; stock or flow | Reporting error, redenomination risk | Aggregatable within same currency regime | Moderate (institutional meaning) | Invalid across regimes or redenomination |
| Currency Subunit (e.g., cent) | Fundamental | Discrete, additive, linear | Institutional absolute | Differences, ratios | Time-indexed | Rounding, discretization error | Aggregatable within same currency | Moderate | Breaks under rounding rules or reform |
| Probability (0–1) | Fundamental | Continuous, bounded, nonlinear at extremes | Axiomatic absolute | Differences, ratios (not additive) | Timeless per presentation | Elicitation/misunderstanding error | Aggregatable only under identical definitions | Moderate | Invalid if probabilities are subjective/inconsistent |
| Percentage | Fundamental | Continuous, bounded, normalized | Relative (representation of probability) | Differences, ratios | Timeless | Scaling/representation error | Same as probability | Moderate | Invalid if baseline representation shifts |
| Quantity Unit (homogeneous goods) | Fundamental | Discrete or continuous, additive | Absolute (unit definition) | Differences, ratios (unit-consistent) | Timeless per transaction | Counting/measurement error | Aggregatable for identical units | Low | Invalid if unit definition changes |
| Distance (e.g., meter) | Fundamental | Continuous, additive, linear | Absolute (physical standard) | Differences, ratios, scaling | Timeless per measure | Instrument resolution error | Fully aggregatable | Minimal | Breaks outside valid physical regimes |
| Price difference | Derived | Continuous, additive, linear | Relative (currency baseline) | Differences, ratios | Timeless per comparison | Quotation/recording error | Aggregatable within same currency regime | Moderate | Invalid across currency regimes |
| Bid–ask spread | Derived | Continuous, additive, linear | Relative (bid and ask quotes) | Differences, ratios | Time-indexed | Quotation timing and reporting error | Aggregatable under identical market definitions | Moderate | Invalid if bid/ask definitions change |
| Average transaction price | Derived | Continuous, additive, linear | Relative (set of transaction prices) | Differences, ratios | Time-indexed | Sampling and averaging error | Aggregatable with consistent transaction sets | Moderate | Invalid if transaction set changes |
| Price dispersion (from prices) | Derived | Continuous, non-additive, nonlinear | Relative (price distribution) | Differences only (statistic-dependent) | Time-indexed | Sampling and distributional error | Limited aggregation (requires homogeneity) | Moderate | Invalid if price distribution shifts |
| Excess demand | Derived | Continuous, additive, linear | Relative (demand − supply) | Differences | Time-indexed | Measurement and timing error | Aggregatable under consistent definitions | Moderate | Invalid if demand/supply definitions change |
| Excess supply | Derived | Continuous, additive, linear | Relative (supply − demand) | Differences | Time-indexed | Measurement and timing error | Aggregatable under consistent definitions | Moderate | Invalid if demand/supply definitions change |
| Transaction value (price × quantity) | Derived | Continuous, additive, linear | Relative (currency × quantity) | Differences, ratios | Time-indexed | Propagated price/quantity error | Aggregatable within same units | Moderate | Invalid if unit definitions change |
| Net payoff (payoff − cost) | Derived | Continuous, additive, linear | Relative (payoff and cost baselines) | Differences, ratios | Timeless per outcome | Component mismeasurement error | Aggregatable under consistent baselines | Moderate | Invalid if baseline redefined |
| Profit (revenue − cost) | Derived | Continuous, additive, linear | Relative (revenue and cost definitions) | Differences, ratios | Time-indexed or per transaction | Accounting and classification error | Aggregatable within same accounting rules | High | Invalid if accounting regime changes |
| Loss (cost − revenue) | Derived | Continuous, additive, linear | Relative (cost and revenue definitions) | Differences, ratios | Time-indexed or per transaction | Accounting and classification error | Aggregatable within same accounting rules | High | Invalid if accounting regime changes |
| Net revenue (revenue − offsets) | Derived | Continuous, additive, linear | Relative (revenue and offset definitions) | Differences, ratios | Time-indexed | Accounting and classification error | Aggregatable under identical accounting rules | High | Invalid if offset definitions change |
| Expected payoff | Derived | Continuous, additive, linear | Relative (probability-weighted payoffs) | Differences, ratios | Timeless per option | Probability and payoff specification error | Aggregatable under identical probability spaces | Moderate | Invalid if probabilities or payoff definitions change |
| Expected profit | Derived | Continuous, additive, linear | Relative (probability-weighted profit) | Differences, ratios | Timeless per option | Probability and accounting error | Aggregatable under identical rules | High | Invalid if profit definition changes |
| Expected loss | Derived | Continuous, additive, linear | Relative (probability-weighted loss) | Differences, ratios | Timeless per option | Probability and loss framing error | Aggregatable under identical definitions | High | Invalid if loss baseline shifts |
| Total surplus (buyer + seller) | Derived | Continuous, additive, linear | Relative (buyer and seller surplus definitions) | Differences, ratios | Timeless per transaction | Component mismeasurement error | Aggregatable under consistent definitions | High | Invalid if surplus components redefined |
| Buyer surplus (value − price) | Derived | Continuous, additive, linear | Relative (value and price definitions) | Differences, ratios | Timeless per transaction | Valuation and price error | Aggregatable under consistent valuation rules | High | Invalid if value definition changes |
| Seller surplus (price − cost) | Derived | Continuous, additive, linear | Relative (price and cost definitions) | Differences, ratios | Timeless per transaction | Cost accounting error | Aggregatable under consistent cost rules | High | Invalid if cost definition changes |
| Waiting time to transaction | Derived | Continuous, additive, linear | Absolute (time difference) | Differences, ratios | Time-indexed, cumulative | Timing and censoring error | Aggregatable under consistent observation windows | Low | Invalid if start/end definitions change |
| Time to agreement | Derived | Continuous, additive, linear | Absolute (time difference) | Differences, ratios | Time-indexed | Timing and recording error | Aggregatable under consistent protocols | Low | Invalid if agreement definition changes |
| Time between offers | Derived | Continuous, additive, linear | Absolute (time difference) | Differences, ratios | Time-indexed | Timing and sequencing error | Aggregatable under consistent sequencing | Low | Invalid if offer definition changes |
| Arrival rate (events ÷ time) | Derived | Continuous, non-additive, nonlinear (rate) | Relative (event definition and time base) | Ratios only | Time-indexed | Event classification and timing error | Aggregatable only under identical time bases | Moderate | Invalid if event definition or time window changes |
| Matching rate (matches ÷ time) | Derived | Continuous, non-additive, nonlinear (rate) | Relative (matching rule and time base) | Ratios only | Time-indexed | Matching definition and timing error | Aggregatable under identical matching protocols | Moderate | Invalid if matching criteria or time base changes |
| Probability of trade (trades ÷ opportunities) | Derived | Continuous, bounded, nonlinear | Relative (opportunity definition) | Differences, ratios | Timeless per measurement window | Sampling and opportunity definition error | Aggregatable under identical opportunity sets | Moderate | Invalid if opportunity set changes |
| Probability of acceptance (acceptances ÷ offers) | Derived | Continuous, bounded, nonlinear | Relative (offer definition) | Differences, ratios | Timeless per measurement window | Offer classification error | Aggregatable under identical offer definitions | Moderate | Invalid if offer definition changes |
| Probability of agreement (agreements ÷ interactions) | Derived | Continuous, bounded, nonlinear | Relative (interaction definition) | Differences, ratios | Timeless per measurement window | Interaction classification error | Aggregatable under identical interaction definitions | Moderate | Invalid if interaction definition changes |
| Number of agents | Count | Discrete, additive | Contextual (agent definition) | Addition only | Timeless per dataset | Identification/deduplication error | Fully aggregatable | Minimal | Invalid if agent identity definition changes |
| Number of interactions | Count | Discrete, additive | Contextual (interaction definition) | Addition only | Time-indexed | Event classification error | Fully aggregatable under same definition | Minimal | Invalid if interaction definition changes |
| Number of interaction periods | Count | Discrete, additive | Contextual (period definition) | Addition only | Time-indexed | Period boundary error | Fully aggregatable | Minimal | Invalid if periodization changes |
| Number of offers | Count | Discrete, additive | Contextual (offer definition) | Addition only | Time-indexed | Recording/coding error | Fully aggregatable | Minimal | Invalid if offer definition changes |
| Number of bids | Count | Discrete, additive | Contextual (bid definition) | Addition only | Time-indexed | Recording/coding error | Fully aggregatable | Minimal | Invalid if bid definition changes |
| Number of asks | Count | Discrete, additive | Contextual (ask definition) | Addition only | Time-indexed | Recording/coding error | Fully aggregatable | Minimal | Invalid if ask definition changes |
| Number of counteroffers | Count | Discrete, additive | Contextual (counteroffer definition) | Addition only | Time-indexed | Recording/coding error | Fully aggregatable | Minimal | Invalid if counteroffer definition changes |
| Number of observations | Count | Discrete, additive | Contextual (observation rule) | Addition only | Time-indexed | Missing/duplicate records | Fully aggregatable | Minimal | Invalid if observation protocol changes |
| Price index relative to reference price | Index | Continuous, normalized, dimensionless | Explicit reference price | Differences, ratios (baseline-preserving) | Time-indexed | Price measurement & normalization error | Aggregatable only with identical reference | Moderate | Invalid if reference price changes |
| Transaction price index relative to benchmark price | Index | Continuous, normalized, dimensionless | Explicit benchmark price | Differences, ratios | Time-indexed | Sampling & benchmark error | Aggregatable under same benchmark | Moderate | Invalid if benchmark definition changes |
| Bid–ask spread index relative to baseline spread | Index | Continuous, normalized | Explicit baseline spread | Differences, ratios | Time-indexed | Quotation & timing error | Aggregatable under same spread definition | Moderate | Invalid if spread definition shifts |
| Price dispersion index relative to reference distribution | Index | Continuous, composite, non-additive | Explicit reference distribution | Differences only (statistic-dependent) | Time-indexed | Distributional & sampling error | Limited aggregation (homogeneous sets only) | Moderate | Invalid if distributional baseline changes |
| Price volatility index relative to baseline period | Index | Continuous, composite | Explicit baseline period | Differences only | Time-indexed | Windowing & estimation error | Limited aggregation | Moderate | Invalid if baseline period changes |
| Market share index relative to equal-share baseline | Index | Continuous, normalized, bounded | Equal-share reference | Differences, ratios | Time-indexed | Classification & denominator error | Aggregatable under identical market definition | Moderate | Invalid if agent set changes |
| Interaction share index relative to fixed interaction baseline | Index | Continuous, normalized, bounded | Fixed interaction baseline | Differences, ratios | Time-indexed | Counting & baseline error | Aggregatable under fixed baseline | Moderate | Invalid if baseline shifts |
| Trade share index relative to feasible trade set | Index | Continuous, normalized, bounded | Explicit feasible-trade baseline | Differences, ratios | Time-indexed | Feasibility mis-specification | Limited aggregation | Moderate | Invalid if feasibility definition changes |
| Total surplus index relative to benchmark allocation | Index | Continuous, normalized | Explicit benchmark allocation | Differences, ratios | Timeless per allocation | Component valuation error | Aggregatable under identical benchmark | High | Invalid if benchmark allocation changes |
| Realized surplus index relative to maximal feasible surplus | Index | Continuous, normalized | Explicit maximal-surplus baseline | Differences, ratios | Timeless per allocation | Estimation & feasibility error | Limited aggregation | High | Invalid if feasibility frontier changes |
| Budget constraint satisfied / violated | Threshold | Binary, non-metrical | Explicit budget rule (price ≤ budget) | None beyond classification | Timeless per interaction | Price/budget measurement error | Aggregatable as counts under identical rules | Low | Invalid if budget definition or currency changes |
| Eligibility threshold met / not met | Threshold | Binary, non-metrical | Explicit eligibility criteria | None beyond classification | Timeless per interaction | Classification and rule-application error | Aggregatable under identical eligibility rules | Low | Invalid if eligibility criteria change |
| Bid meets reserve price / does not meet | Threshold | Binary, non-metrical | Explicit reserve price rule | None beyond classification | Time-indexed (per bid) | Bid recording error | Aggregatable under identical reserve rules | Low | Invalid if reserve price definition changes |
| Ask meets minimum price / does not meet | Threshold | Binary, non-metrical | Explicit minimum price rule | None beyond classification | Time-indexed (per ask) | Ask recording error | Aggregatable under identical minimum rules | Low | Invalid if minimum price definition changes |
| Offer meets minimum requirement / does not meet | Threshold | Binary, non-metrical | Explicit offer requirement | None beyond classification | Time-indexed (per offer) | Requirement misclassification | Aggregatable under identical requirements | Low | Invalid if requirement definition changes |
| Deadline met / missed | Threshold | Binary, non-metrical | Explicit deadline cutoff | None beyond classification | Time-indexed | Timing and timestamp error | Aggregatable under identical deadlines | Low | Invalid if deadline definition shifts |
| Time limit exceeded / not exceeded | Threshold | Binary, non-metrical | Explicit time-limit rule | None beyond classification | Time-indexed | Timing measurement error | Aggregatable under identical time limits | Low | Invalid if time-limit definition changes |