Multiplicative Kantian Equilibrium (MKE)

• Multiplicative Kantian Equilibrium (MKE) is a concept where players assess if a common proportional rescaling of all actions improves payoffs, leading to the condition ∇u(x)·x = 0.
• MKE differs from Nash equilibrium by evaluating universal deviations along the current status-quo ray, thereby embedding efficiency and fairness into the outcome selection.
• The approach impacts practical applications in public goods and cooperation, with extensions addressing how coordinate shifts and scaling affect Pareto efficiency and strategic behavior.

Multiplicative Kantian Equilibrium (MKE) is a Kantian equilibrium concept in which each player evaluates an action profile by asking whether a common proportional rescaling of all actions would improve her payoff. In an $n$-player game with action profile $x=(x_1,\dots,x_n)$ and utility $u_i(x)$, player $i$ considers the one-dimensional problem $\max_{t>0} u_i(tx)$, where $tx$ denotes the profile obtained by multiplying every action by the same factor $t$. A profile $x^\star$ is an MKE if $t=1$ is optimal for every player; under differentiability and interiority, this yields the first-order condition $\nabla u_i(x)\cdot x=0$ for all $x=(x_1,\dots,x_n)$0 (Kordonis, 2016). In the broader “variation function” formalism, MKE is the multiplicative specialization $x=(x_1,\dots,x_n)$1 on numerical action spaces $x=(x_1,\dots,x_n)$2 (Istrate, 2021).

1. Definition, deviation protocol, and status-quo dependence

The defining feature of MKE is the deviation protocol. Nash equilibrium evaluates unilateral deviations, holding other players’ actions fixed. MKE instead evaluates universal proportional deviations: a player asks what would happen if everyone changed action in the same multiplicative proportion. Formally, with profile $x=(x_1,\dots,x_n)$3, player $x=(x_1,\dots,x_n)$4 studies the ray $x=(x_1,\dots,x_n)$5 and requires that no such common scaling improve $x=(x_1,\dots,x_n)$6. In interior differentiable environments, the first-order condition at $x=(x_1,\dots,x_n)$7 is

$x=(x_1,\dots,x_n)$8

while boundary cases require the corresponding one-sided inequality along feasible directions (Kordonis, 2016).

This formulation is explicitly status-quo dependent. The ray along which proportional scaling occurs is determined by the current or previous action profile, so the relevant counterfactual is not “what if everyone chose some common alternative action,” but “what if everyone scaled the present profile by the same factor” (Kordonis, 2016). That dependence distinguishes MKE from replacement-style Kantian models, in which agents compare a diagonal profile against all other diagonal profiles, and from additive variants, in which universalization is modeled by $x=(x_1,\dots,x_n)$9 rather than $u_i(x)$0 (Istrate, 2021).

The extended abstract on moral and other-regarding agents places MKE within a general template of Kantian equilibria. There, a pure Kantian equilibrium is a profile $u_i(x)$1 such that each player weakly prefers the current profile to the profile produced when every action is transformed by the same variation function $u_i(x)$2. MKE is the case $u_i(x)$3, with $u_i(x)$4 and $u_i(x)$5 (Istrate, 2021). This makes MKE a special case of a broader family of morally motivated but non-Nashian solution concepts.

A recurrent misconception is that MKE is merely Nash equilibrium with altruistic or utilitarian preferences. That characterization is inaccurate. The defining change is not primarily in the utility function but in the admissible counterfactual deviation: MKE keeps material payoffs $u_i(x)$6 and replaces unilateral deviations by proportional co-moves of the full profile (Istrate, 2021). This suggests that MKE should be understood as a change in optimization protocol rather than a direct internalization of others’ utility.

2. First-order conditions, efficiency, and Pareto structure

In symmetric environments, MKE is closely tied to Pareto efficiency. Under the Roemer-style scaling condition described in the partial Kantian cooperation model, and under mild conditions, the resulting Kantian profile lies on the Pareto frontier (Kordonis, 2016). A more explicit characterization is provided in symmetric two-player games satisfying Assumption A: if both players are Kantian and $u_i(x)$7, the interior MKE condition becomes

$u_i(x)$8

Hence the efficient interior MKE is $u_i(x)$9, where $i$0 is the unique maximizer of the diagonal objective, while $i$1 remains an inefficient Kantian candidate (Sloev et al., 1 May 2026).

For canonical public-good specifications $i$2, the efficient symmetric Kantian action solves

$i$3

whereas the Nash action solves

$i$4

The additional factor $i$5 in the Kantian condition is the formal expression of the universalization step: the player optimizes as if an incremental change in own action were replicated by all agents (Sloev et al., 1 May 2026). In standard social dilemmas this yields $i$6.

An explicit continuous-action illustration appears in the 2021 extended abstract. In the two-player symmetric public-effort game

$i$7

the diagonal payoff is $i$8. Under multiplicative variation, the first-order condition at $i$9 yields $\max_{t>0} u_i(tx)$0, and the second-order condition is strictly negative, so the unique diagonal MKE is $\max_{t>0} u_i(tx)$1. The corresponding Nash equilibrium is $\max_{t>0} u_i(tx)$2 (Istrate, 2021). In this example, MKE selects a more cooperative outcome than Nash while remaining entirely within individual optimization over material payoffs.

A major development in later work is the realization result for the entire interior Pareto frontier. If one introduces a coordinate shift $\max_{t>0} u_i(tx)$3 and defines net actions relative to that origin, then the MKE condition becomes

$\max_{t>0} u_i(tx)$4

Under continuous differentiability, concavity, strict concavity in own action, unidirectional externalities, and interiority of the target point, every interior Pareto-efficient allocation $\max_{t>0} u_i(tx)$5 can be implemented as an MKE under some nonnegative shift $\max_{t>0} u_i(tx)$6 lying on the common tangent line to the players’ indifference surfaces at $\max_{t>0} u_i(tx)$7 (Sloev et al., 19 May 2026). The paper interprets this as an efficiency–fairness separation: Pareto efficiency comes from Kantian optimization of net actions, while the selected point on the frontier is determined by the chosen origin $\max_{t>0} u_i(tx)$8.

3. Embedding within broader Kantian and partial-cooperation frameworks

The continuum-player model of partial Kantian cooperation provides a larger framework in which MKE appears as a special case. In that model, each agent of type $\max_{t>0} u_i(tx)$9 imagines a virtual group described by a sub-probability measure $tx$0, and chooses a strategy mapping $tx$1 for that group as if all imagined group members were bound to follow the same strategy. The resulting equilibrium notion is the $tx$2-Kant-Nash equilibrium, or, with partial observability through a reduced position variable $tx$3, the $tx$4-Kant-Nash equilibrium (Kordonis, 2016).

Within that framework, MKE differs in the object of deviation. In $tx$5-Kant-Nash equilibrium, the deviation is a change in a strategy function $tx$6; in MKE, the deviation is a scalar $tx$7 that rescales the current action profile (Kordonis, 2016). The former is group-wise strategy symmetrization by type, whereas the latter is proportional motion along a status-quo ray. The two notions coincide only under restrictive choices: the virtual group must be the entire population, players must be risk-neutral, weights must be uniform, and the action representation must make “the same strategy” equivalent to proportional scaling (Kordonis, 2016).

The paper gives an explicit construction of this coincidence. Let a previous action profile $tx$8 define the status quo, and reparameterize decisions by $tx$9. With $t$0, $t$1, uniform weights, and $t$2 selecting the status-quo component, the $t$3-Kant-Nash equilibrium coincides with Roemer’s multiplicative Kantian equilibrium: each player chooses a common multiplicative factor along the status-quo ray (Kordonis, 2016). This embedding is conceptually important because it shows that MKE is not an isolated construction but a specialization of a more general family of ethical game-theoretic models.

The same continuum framework also locates MKE relative to other classical solution concepts. Setting $t$4 and $t$5 recovers mean-field Nash. Setting $t$6 and $t$7 yields the Bentham–Harsanyi utilitarian solution. Taking $t$8 and $t$9 recovers Rawls’ difference principle, while $x^\star$0 yields a best-off representative criterion (Kordonis, 2016). In this taxonomy, full multiplicative Kantian cooperation is one ethically structured case within a continuum between purely self-regarding and fully group-regarding optimization.

The fishing examples in the same paper illustrate how partial Kantian cooperation interpolates between Nash and full Kantian outcomes. When the imagined group is a fraction $x^\star$1 of the population, the equilibrium action in the simplest symmetric fishing specification is

$x^\star$2

so $x^\star$3 yields Nash and $x^\star$4 yields full Kantian cooperation (Kordonis, 2016). A plausible implication is that MKE should be viewed less as a single equilibrium point than as an endpoint of a broader family of Kantian-response protocols indexed by perceived group scope.

4. Strategic non-equivalence and endogenous scaling

Recent work emphasizes that MKE is not strategically equivalent under monotone relabelings of the strategy space. If $x^\star$5 is a strictly monotone change of coordinates, a Kantian optimizing in $x^\star$6-space evaluates proportional deviations $x^\star$7, which do not generally correspond to proportional deviations in $x^\star$8-space. In the two-player case, the natural-coordinate Kantian first-order condition is

$x^\star$9

whereas the rescaled condition becomes

$t=1$0

These coincide for all states only if $t=1$1 is proportional to $t=1$2, as with power transformations, and generally fail under transformations such as $t=1$3 (Sloev et al., 1 May 2026).

This lack of invariance sharply contrasts with Nash equilibrium. Under a strictly monotone one-to-one relabeling, Nash best responses map bijectively between coordinate systems, so Nash equilibria are strategically equivalent. MKE lacks that property because proportional co-movement is representation-dependent (Sloev et al., 1 May 2026). The result is not a minor technicality: it implies that measurement scale, or more generally the subjective description of strategy space, can materially change Kantian best responses and equilibrium outcomes.

A parallel formulation uses a common monotone scale $t=1$4, chosen before play. The corresponding local elasticity

$t=1$5

modifies the Kantian first-order condition to

$t=1$6

The chosen scale therefore acts as a commitment device: material payoffs and feasible actions remain unchanged, but the Kantian stationarity condition acquires endogenous elasticity weights (Sloev et al., 20 Jun 2026).

This machinery yields a sharp substitutes–complements dichotomy. Under strategic substitutes, for every $t=1$7 there exists an increasing $t=1$8 scale such that the stationary transformed Nash–Kantian outcome gives the Kantian player at least $t=1$9, but the Nash benchmark itself is not attained by any interior stationary outcome under a common monotone scale (Sloev et al., 20 Jun 2026). Scaling is therefore defensive: it removes the loss associated with naive Kantian behavior without overtaking the Nash payoff benchmark. Under strategic complements, by contrast, there exists an increasing $\nabla u_i(x)\cdot x=0$0 scale such that the stationary outcome coincides with the Stackelberg leader solution and gives the Kantian player a payoff strictly above the Nash benchmark (Sloev et al., 20 Jun 2026). In that regime, scaling is offensive and functions as a form of strategic leadership.

The same line of research shows that with player-specific scales the Kantian first-order condition can be aligned pointwise with the Stackelberg condition along the entire reaction curve if and only if the reaction slope is positive. Under strategic substitutes, exact alignment would require a negative elasticity, violating monotonicity (Sloev et al., 20 Jun 2026). This establishes a universality–flexibility trade-off: a common scale preserves the universality of the Kantian deviation protocol, while player-specific scales expand implementability but weaken that universality.

5. Distributional conflict, inequality, and coordination barriers

Although MKE often restores efficiency in social dilemmas, later work shows that voluntary adoption under inequality can be difficult. In a two-agent public-good model with endowments $\nabla u_i(x)\cdot x=0$1, contributions $\nabla u_i(x)\cdot x=0$2, private consumptions $\nabla u_i(x)\cdot x=0$3, public good $\nabla u_i(x)\cdot x=0$4, and preferences

$\nabla u_i(x)\cdot x=0$5

Nash equilibrium yields $\nabla u_i(x)\cdot x=0$6 in the interior case and thus underprovides the public good relative to the MKE level $\nabla u_i(x)\cdot x=0$7 (Sloev et al., 20 Jun 2026).

Under contribution-space MKE, proportional scaling applies to contributions. The equilibrium is uniquely

$\nabla u_i(x)\cdot x=0$8

so MKE fixes equal sacrifice ratios across agents (Sloev et al., 20 Jun 2026). Under consumption-space MKE, proportional scaling applies instead to private consumption, and the equilibrium condition becomes

$\nabla u_i(x)\cdot x=0$9

which yields a continuum of Pareto-efficient outcomes rather than a unique point (Sloev et al., 20 Jun 2026). Both parametrizations satisfy the Samuelson efficiency condition, but they have different distributive consequences.

Three barriers then arise. First, entry into Kantian cooperation need not be a Pareto improvement under inequality. The poorer agent prefers contribution-space MKE to Nash if and only if

$x=(x_1,\dots,x_n)$00

while the richer agent always prefers contribution-space MKE (Sloev et al., 20 Jun 2026). Second, parametrization itself becomes distributive. When symmetric consumption-space MKE is feasible, the poor prefers the consumption parametrization and the rich prefers the contribution parametrization, because the two scales allocate private consumption differently (Sloev et al., 20 Jun 2026). Third, consumption-space MKE creates an additional coordination problem: the efficient set is a one-parameter family of points on the Pareto frontier, and utilities move in opposite directions along that frontier (Sloev et al., 20 Jun 2026).

These distributional findings sharpen an earlier concern from the finite-game literature: Kantian equilibria can suffer from a high price of miscoordination when multiple Kantian actions exist. In symmetric diagonal games under replacement variation, the price of miscoordination can be as high as $x=(x_1,\dots,x_n)$01, and the corresponding mixed Kantian equilibrium problem is NP-hard in two-player symmetric games (Istrate, 2021). Those results are not MKE-specific, since the paper’s complexity and miscoordination analysis concern replacement variation rather than multiplicative variation. Still, they caution against conflating the moral appeal of Kantian reasoning with automatic implementability.

The inequality paper suggests a broader lesson. MKE can select efficient outcomes, but when agents differ in endowments or positions, the relevant questions are not exhausted by efficiency. Which parametrization is used, and which point on the induced efficient set is selected, become independent political or institutional choices (Sloev et al., 20 Jun 2026). This resonates with the coordinate-shift result that the entire interior Pareto frontier can be realized as MKE under suitable origins (Sloev et al., 19 May 2026).

6. Stability, mixed populations, and current research frontier

A major recent development concerns the stability of Kantian optimization in environments where Kantian and Nash types coexist. In a two-player symmetric game, if a Kantian player can publicly adopt the affine rescaling $x=(x_1,\dots,x_n)$02, where $x=(x_1,\dots,x_n)$03 is the symmetric Nash action, then there exists a symmetric Kantian–Nasher equilibrium in which the Kantian receives the Nash payoff and the Nasher cannot free-ride on naive Kantian behavior (Sloev et al., 1 May 2026). In the quadratic example $x=(x_1,\dots,x_n)$04, original units allow the Nasher to obtain $x=(x_1,\dots,x_n)$05 against the Kantian’s $x=(x_1,\dots,x_n)$06, whereas the affine shift yields the equilibrium $x=(x_1,\dots,x_n)$07 with both players receiving $x=(x_1,\dots,x_n)$08 (Sloev et al., 1 May 2026).

That same framework extends to endogenous type choice. In an $x=(x_1,\dots,x_n)$09-player environment where agents first choose type $x=(x_1,\dots,x_n)$10 and then interact pairwise, the all-Kantian outcome is a subgame-perfect Nash equilibrium under the specified rescaling and equilibrium-selection assumptions. Kantian–Kantian pairs coordinate on the efficient MKE and receive $x=(x_1,\dots,x_n)$11, Nash–Nash pairs receive $x=(x_1,\dots,x_n)$12, and Kantian–Nasher pairs receive $x=(x_1,\dots,x_n)$13 under the neutralizing rescaling (Sloev et al., 1 May 2026). In the evolutionary version, with observable types and pairwise matching, expected payoffs are

$x=(x_1,\dots,x_n)$14

so Kantian optimization is an evolutionarily stable strategy, whereas Nash optimization is not (Sloev et al., 1 May 2026).

These results do not remove the main limitations of the concept. MKE presumes a numerically meaningful action space on which multiplicative scaling is well defined; this is straightforward on $x=(x_1,\dots,x_n)$15 but becomes problematic in finite action spaces or in domains with no natural multiplicative geometry (Istrate, 2021). Existence, uniqueness, and algorithmic theory are also uneven across the literature. The 2021 extended abstract proves existence and tractability only for replacement-based Kantian equilibria in certain symmetric finite games, and explicitly states that it does not develop separate existence, uniqueness, or complexity theorems for MKE (Istrate, 2021). By contrast, later continuous-action analyses obtain local or global characterizations under differentiability, concavity, quasi-concavity along proportional rays, and interiority assumptions (Sloev et al., 1 May 2026, Sloev et al., 19 May 2026).

Another technical frontier concerns auxiliary variables that may become negative after strategic recentering. To preserve the interpretation of “similar change by all,” the stability paper introduces an extended MKE definition for the case in which transformed coordinates have opposite signs: if $x=(x_1,\dots,x_n)$16, the relevant co-move is $x=(x_1,\dots,x_n)$17 rather than a common $x=(x_1,\dots,x_n)$18-multiplication (Sloev et al., 1 May 2026). This suggests that the multiplicative intuition can be preserved beyond the strictly nonnegative orthant, but only at the cost of a more delicate definition.

Taken together, the current literature presents MKE as a family of representation-sensitive Kantian fixed points rather than a single canonical equilibrium. In its classical form, MKE asks whether $x=(x_1,\dots,x_n)$19 is optimal along the ray $x=(x_1,\dots,x_n)$20. In its modern extensions, the effective ray may be shifted, rescaled, or reparameterized, and those choices can determine not only whether cooperation is stable but also which efficient outcome is selected (Kordonis, 2016, Sloev et al., 20 Jun 2026). A plausible implication is that future work on MKE will focus less on proving efficiency in abstract social dilemmas and more on specifying the institutions, conventions, or commitment devices that determine the relevant geometry of proportional deviation.