Equal-Marginal-Value Rule

• Equal-Marginal-Value Rule is a fair division method that compares agents’ realized utility surpluses over their means to assign resources.
• It reconciles egalitarian guarantees with marginal productivity by achieving a balance between equal sharing and incentive-based allocation.
• The rule is applicable in distinct settings, including random object division for two agents and transferable utility games, adjusting fairness benchmarks accordingly.

The Equal-Marginal-Value Rule denotes, in the materials considered here, two related allocation constructions defined in distinct environments. In the fair division of a random object, it is the rule that assigns the realized object to the agent with the larger realized utility relative to that agent’s mean utility, namely the agent maximizing $u_i(x)-\mu_i$; this rule is presented as a special two-agent case of the Top-Heavy family in Bogomolnaia–Moulin–Sandomirskiy (Bogomolnaia et al., 2019). In cooperative games with transferable utility, the Equal–Marginal–Value Rule $\phi_i^{EMV}(v;k)$, also called the $k$-solidarity Egalitarian–Marginal Value, uses equal sharing in sufficiently small coalitions and marginal contribution in sufficiently large coalitions, with limiting cases given by the Shapley value and the Equal-Division rule (Choudhury et al., 2022). A plausible implication is that the shared label “Equal-Marginal-Value” identifies a common design objective: reconciling egalitarian guarantees with sensitivity to realized or marginal productivity.

1. Two formal environments

The two uses of the term differ in primitives, observables, and fairness benchmarks.

Setting	Primitive objects	EMV rule
Random object division	$N=\{1,2\}$, random object $X$, realized utilities $u_i(x)$, known means $\mu_i$	Assign to $i^*(x)=\arg\max_i(u_i(x)-\mu_i)$
Transferable utility game	Coalition game $v$, player set $N=\{1,\dots,n\}$, switch size $\phi_i^{EMV}(v;k)$0, random permutation $\phi_i^{EMV}(v;k)$1	Equal sharing up to size $\phi_i^{EMV}(v;k)$2, marginal contribution thereafter

In the random-object model, the manager observes only the realized utilities $\phi_i^{EMV}(v;k)$3 and the means $\phi_i^{EMV}(v;k)$4, while “no other feature of the distribution of $\phi_i^{EMV}(v;k)$5 is used” (Bogomolnaia et al., 2019). In the cooperative-game model, the rule is parameterized by an integer $\phi_i^{EMV}(v;k)$6 and is defined by averaging a player’s coalitional contribution across all permutations of entry into the grand coalition (Choudhury et al., 2022).

The two frameworks also differ in the status of fairness. In the random-object setting, the central notions are Fair Share in expectation and ex ante no-envy, alongside explicit recognition that ex post Pareto-optimality and ex post envy-freeness may fail (Bogomolnaia et al., 2019). In the cooperative-game setting, the central concerns are efficiency, symmetry, linearity or its substitutes, and implementation in subgame perfect Nash equilibrium on zero-monotonic games (Choudhury et al., 2022).

2. Equal-Marginal-Value in the division of a random object

The random-object model is stated for two agents, $\phi_i^{EMV}(v;k)$7, and a random object $\phi_i^{EMV}(v;k)$8, each with probability $\phi_i^{EMV}(v;k)$9 (Bogomolnaia et al., 2019). Each agent $k$0 has a random cardinal utility $k$1, with known mean

$k$2

Upon realization $k$3, the manager observes $k$4 and $k$5 and must allocate the single object, possibly randomly, between the two agents.

The rule is defined after normalizing utilities by their means:

$k$6

The entire object is then assigned to the agent whose “marginal gain above expectation” is larger. Equivalently,

$k$7

Ties can be broken by a fair coin-flip (Bogomolnaia et al., 2019). The rule is summarized in the formulation: “Give $k$8 to the agent who likes it more relative to his mean.”

The motivating example is the case in which one agent likes oranges much more than apples while the other likes apples much more than oranges. Equal division is fair for each realization, but an agreement that awards the realized fruit to the agent who likes it more gives a higher expected utility to each agent and remains fair “in the average sense” (Bogomolnaia et al., 2019). The formal contribution is to recover a rule with this flavor under the strong informational restriction that only means, not full distributions, are known to the manager.

A central auxiliary quantity is

$k$9

Because $N=\{1,2\}$0, the rule compares each agent’s realized “surprise” relative to the agent’s own baseline. The paper’s intuitive comment is that subtracting $N=\{1,2\}$1 compares “surprise” above or below average, and that allocating to the larger surprise treats equal surprises equally (Bogomolnaia et al., 2019).

3. Fairness, welfare, and limitations in the random-object model

Fair Share is defined as the requirement that each agent’s expected utility be at least half of the agent’s mean, $N=\{1,2\}$2 (Bogomolnaia et al., 2019). Under EMV,

$N=\{1,2\}$3

Using $N=\{1,2\}$4, the proof rewrites the realized utility of the winning agent as

$N=\{1,2\}$5

Since, by symmetry, $N=\{1,2\}$6 and $N=\{1,2\}$7, it follows that

$N=\{1,2\}$8

Hence EMV satisfies Fair Share (Bogomolnaia et al., 2019).

The welfare comparison is made against two benchmarks. Equal-Split (ES) gives the object to each agent with probability $N=\{1,2\}$9, so $X$0. The full-information optimum (OPT) chooses in each state the utilitarian allocation $X$1, yielding expected total welfare $X$2 (Bogomolnaia et al., 2019). Defining

$X$3

the paper states

$X$4

It is then shown that $X$5, strictly whenever $X$6 and $X$7 are not almost surely equal, while

$X$8

with strict inequality if the means differ or if sometimes $X$9 but $u_i(x)$0 (Bogomolnaia et al., 2019).

The paper also studies the competitive ratio

$u_i(x)$1

For EMV, a short calculation is said to show

$u_i(x)$2

and “in fact one can pin it down more tightly as $u_i(x)$3 in the two-agent case” (Bogomolnaia et al., 2019). This places EMV in the same range as the best rule even though the manager lacks full information about utility distributions.

The fairness and efficiency properties are deliberately ex ante rather than ex post. EMV is not ex post Pareto-optimal because it may allocate the object to a lower-utility agent when that agent’s gain over the agent’s own mean is larger. Ex ante envy-freeness does hold:

$u_i(x)$4

but ex post envy may arise because one agent receives the whole object and the other receives zero (Bogomolnaia et al., 2019). A recurrent source of confusion is therefore whether fairness is being asserted state by state or in expectation; in this model, the key guarantees are explicitly ex ante.

The same logic extends to $u_i(x)$5 agents. Writing again $u_i(x)$6, the $u_i(x)$7-agent EMV rule awards the object to

$u_i(x)$8

The stated conclusions are: Fair Share ex ante, $u_i(x)$9; ex ante no-envy among all $\mu_i$0 agents; improvement over naive Equal-Split; and lower welfare than the full-information rule that picks $\mu_i$1 (Bogomolnaia et al., 2019).

4. Equal–Marginal–Value in transferable utility games

In cooperative games with transferable utilities, the Equal–Marginal–Value Rule is defined on a game $\mu_i$2 with player set $\mu_i$3 and a switch parameter $\mu_i$4 (Choudhury et al., 2022). Players are imagined to enter one by one according to a random permutation $\mu_i$5, all permutations being equally likely. The notation is

$\mu_i$6

The construction imposes two regimes. In small coalitions of size up to $\mu_i$7, each player “accepts” an egalitarian share of the coalition’s worth. In larger coalitions, each player insists on the player’s marginal contribution. The coalitional contribution of player $\mu_i$8 at position $\mu_i$9 is

$i^*(x)=\arg\max_i(u_i(x)-\mu_i)$0

The $i^*(x)=\arg\max_i(u_i(x)-\mu_i)$1-EMV value is the expectation of these contributions over all $i^*(x)=\arg\max_i(u_i(x)-\mu_i)$2 permutations:

$i^*(x)=\arg\max_i(u_i(x)-\mu_i)$3

This is the formal definition given in the paper (Choudhury et al., 2022).

After regrouping by coalition sizes, the rule has the closed form

$i^*(x)=\arg\max_i(u_i(x)-\mu_i)$4

The first term is the “egalitarian block on small coalitions $i^*(x)=\arg\max_i(u_i(x)-\mu_i)$5,” while the second is the “marginal block on larger coalitions $i^*(x)=\arg\max_i(u_i(x)-\mu_i)$6” (Choudhury et al., 2022).

The limiting cases are explicit. If $i^*(x)=\arg\max_i(u_i(x)-\mu_i)$7, the expression gives exactly the Shapley formula. If $i^*(x)=\arg\max_i(u_i(x)-\mu_i)$8, the second sum is empty and the first sum implies

$i^*(x)=\arg\max_i(u_i(x)-\mu_i)$9

which is the Equal-Division rule (Choudhury et al., 2022). The rule therefore interpolates between the two extremes identified in the abstract: Shapley as “an extreme case of marginalism” and Equal Division as “an extreme case of egalitarianism.”

A further representation uses the “$v$0-coefficient” form of Driessen–Radzik. There is a unique family $v$1 with $v$2, $v$3 for $v$4, and $v$5 for $v$6 such that

$v$7

In this representation, $v$8 “turns on” marginalism exactly when $v$9 (Choudhury et al., 2022).

5. Axiomatic characterizations and strategic implementation

The cooperative-game paper provides four equivalent axiomatic routes to the same value (Choudhury et al., 2022). The first uses Efficiency, Symmetry, Linearity, and the $N=\{1,\dots,n\}$0-Nullifying Null Player Property (k-NNPP). These are stated as follows: Efficiency requires $N=\{1,\dots,n\}$1; Symmetry requires equal payoffs for interchangeable players; Linearity requires $N=\{1,\dots,n\}$2; and k-NNPP requires $N=\{1,\dots,n\}$3 when player $N=\{1,\dots,n\}$4 “nullifies” all coalitions of size $N=\{1,\dots,n\}$5 and is thereafter a standard null player. Theorem 1 states that a linear, efficient, symmetric value satisfying k-NNPP is unique and coincides with $N=\{1,\dots,n\}$6 (Choudhury et al., 2022).

The second route replaces linearity with Coalitional $N=\{1,\dots,n\}$7-Strategic Equivalence (k-CSE). The axiom says that whenever $N=\{1,\dots,n\}$8 is a $N=\{1,\dots,n\}$9-nullifying null player in $\phi_i^{EMV}(v;k)$00, adding $\phi_i^{EMV}(v;k)$01 to any $\phi_i^{EMV}(v;k)$02 does not change $\phi_i^{EMV}(v;k)$03, that is,

$\phi_i^{EMV}(v;k)$04

Theorem 2 states that Efficiency, Symmetry, and k-CSE uniquely determine $\phi_i^{EMV}(v;k)$05 (Choudhury et al., 2022).

The third route uses $\phi_i^{EMV}(v;k)$06-Partial Monotonicity (k-PMon). This requires $\phi_i^{EMV}(v;k)$07 whenever, for all $\phi_i^{EMV}(v;k)$08 of size at least $\phi_i^{EMV}(v;k)$09,

$\phi_i^{EMV}(v;k)$10

and for all such $\phi_i^{EMV}(v;k)$11 of size less than $\phi_i^{EMV}(v;k)$12,

$\phi_i^{EMV}(v;k)$13

Theorem 3 states that Efficiency, Symmetry, and k-PMon imply $\phi_i^{EMV}(v;k)$14 (Choudhury et al., 2022).

The fourth route uses Efficiency, k-NNPP, and Fairness in the sense of van den Brink. For any symmetric pair $\phi_i^{EMV}(v;k)$15 and any increment game $\phi_i^{EMV}(v;k)$16,

$\phi_i^{EMV}(v;k)$17

The authors show that this fairness axiom is equivalent to Differential Marginality (DM), so Fairness can be replaced by DM. Theorem 4 states that Efficiency, k-NNPP, and Fairness uniquely determine $\phi_i^{EMV}(v;k)$18 (Choudhury et al., 2022).

The same paper also gives a strategic implementation. The mechanism is described as a sequential “bidding $\phi_i^{EMV}(v;k)$19 randomization $\phi_i^{EMV}(v;k)$20 take-it-or-leave-it bargaining” mechanism in the spirit of A.-Mas-Colell–Remila–Solal (Choudhury et al., 2022). The stages are summarized as follows. In Stage 1, all players simultaneously bid for which $\phi_i^{EMV}(v;k)$21 will be used, and the highest total bids determine a set of eligible $\phi_i^{EMV}(v;k)$22’s. In Stage 2, players also bid on the permutation $\phi_i^{EMV}(v;k)$23, and the highest bids select eligible permutations. In Stage 3, a pair $\phi_i^{EMV}(v;k)$24 is chosen uniformly at random among the winners, after which the player in position $\phi_i^{EMV}(v;k)$25 makes a take-it-or-leave-it offer concerning the remaining worth; if not unanimously accepted by the trailing $\phi_i^{EMV}(v;k)$26 players, the last proposer is out, takes zero, and the game continues. In Stage 4, total payoff equals the amount earned in the bargaining subgame plus a rebate or tax reflecting the bids, designed so that truthful bidding on $\phi_i^{EMV}(v;k)$27 and $\phi_i^{EMV}(v;k)$28 is a weakly dominant subgame-perfect strategy and expected payoff under truthful play is exactly $\phi_i^{EMV}(v;k)$29. Proposition 7 states that, on the domain of zero-monotonic games, the mechanism implements $\phi_i^{EMV}(v;k)$30 in unique SPNE (Choudhury et al., 2022).

6. Example, limiting behavior, and comparative interpretation

A 4-player example in the cooperative-game framework makes the interpolation explicit (Choudhury et al., 2022). Let $\phi_i^{EMV}(v;k)$31 and define $\phi_i^{EMV}(v;k)$32 by

$\phi_i^{EMV}(v;k)$33

$\phi_i^{EMV}(v;k)$34

$\phi_i^{EMV}(v;k)$35

For this game, the Shapley value is approximately

$\phi_i^{EMV}(v;k)$36

while Equal Division gives each player $\phi_i^{EMV}(v;k)$37. For $\phi_i^{EMV}(v;k)$38, the formula becomes

$\phi_i^{EMV}(v;k)$39

and the resulting allocation is approximately

$\phi_i^{EMV}(v;k)$40

As $\phi_i^{EMV}(v;k)$41 increases from $\phi_i^{EMV}(v;k)$42, one “continuously moves from the Shapley vector $\phi_i^{EMV}(v;k)$43 toward the egalitarian $\phi_i^{EMV}(v;k)$44” (Choudhury et al., 2022).

In the random-object setting, the limiting comparison is not indexed by a parameter $\phi_i^{EMV}(v;k)$45 but by benchmark rules. Equal-Split gives each agent exactly $\phi_i^{EMV}(v;k)$46 in expectation, whereas the full-information optimum chooses the statewise utilitarian allocation and reaches $\phi_i^{EMV}(v;k)$47 (Bogomolnaia et al., 2019). EMV lies between these: it strictly improves on Equal-Split whenever the realized utilities are not almost surely equal, but remains weakly below the full-information optimum.

A plausible interpretation is that the two literatures attach the same name to a common structural compromise. In the random-object model, the compromise is between equal probabilistic treatment and realized utility gains measured relative to means. In the transferable-utility model, the compromise is between equal sharing in coalitions up to size $\phi_i^{EMV}(v;k)$48 and marginal sharing once coalition size exceeds $\phi_i^{EMV}(v;k)$49. The technical objects differ—mean-normalized surprises in one case, coalition-size–dependent contribution formulas in the other—but both constructions are designed so that egalitarian guarantees remain operative while marginalist considerations are not suppressed.

Another plausible implication is that the phrase “equal-marginal-value” should not be read as a single universally fixed formula. In the materials considered here, it instead names a family resemblance across models: fairness is encoded through equal treatment at a baseline, while efficiency or productivity enters through above-baseline comparisons or marginal increments.