Invitation-Promoted Monotonicity (IP-MON)

• IP-MON is a principle defining monotonicity in network auctions by ensuring that increased valuations and more invitations maintain or improve a bidder’s chance to win.
• It addresses limitations in traditional value-monotonicity by integrating invitation effects, thus supporting DSIC and individual rationality in complex strategic settings.
• The paper establishes tractable payment rules that build on critical threshold computations, offering a revenue-maximizing mechanism for network-implementable allocation rules.

Searching arXiv for the cited papers and closely related work to ground the article. Invitation-Promoted Monotonicity (IP-MON) is a monotonicity principle for mechanism design on social networks in which bidders are strategic not only about valuations but also about whom they invite into the market. In the formulation introduced for network auctions, IP-MON requires that, with other reports fixed, a bidder who wins under a type $\theta_i=(v_i,r_i)$ must continue to win after increasing her valuation and expanding her invitation set; it is therefore the invitation-dimension analogue of value-monotonicity in Myerson-style single-parameter design (Guo et al., 19 Jul 2025). The concept was introduced together with Invitation-Depressed Monotonicity (ID-MON) to characterize network-implementable allocation rules, explain why several earlier multi-unit diffusion auctions fail strategyproofness, and support revenue-maximizing payment design in both multi-unit and single-minded combinatorial settings (Guo et al., 19 Jul 2025). The same label, or a closely analogous one, also appears in approval-based multi-winner voting, invitation-based cooperative games, and adaptive clinching auctions, but with different formal objects and objectives (Sánchez-Fernández et al., 2017, Zhang et al., 2020, Sato, 10 Feb 2025).

1. Network-auction setting and strategic environment

The canonical IP-MON setting is a social network $G=(V,E)$ with vertex set $V=N\cup\{s\}$, where $s$ is the seller and $N$ is the set of potential bidders. For each bidder $i\in N$, the neighbor set is $N(i)=\{j\in N:(i,j)\in E\}$. Each bidder can invite a subset of her neighbors; the invitation component is denoted $r_i\subseteq N(i)$, and the diffusion process expands the set of informed bidders beyond the seller’s neighbors $N(s)$ (Guo et al., 19 Jul 2025).

In the multi-unit unit-demand model, bidder $i$ has private type

$G=(V,E)$0

where $G=(V,E)$1 is her valuation and $G=(V,E)$2 is the set she actually invites. A direct-revelation mechanism is $G=(V,E)$3, with allocation rule $G=(V,E)$4 and payment rule $G=(V,E)$5, where

$G=(V,E)$6

In a $G=(V,E)$7-unit auction with unit-demand bidders, feasibility requires $G=(V,E)$8. Utility is quasi-linear: $G=(V,E)$9

The strategic requirement is stronger than in canonical auctions. Each bidder reports both a valuation $V=N\cup\{s\}$0 and an invitation set $V=N\cup\{s\}$1. Strategyproofness in this environment requires truthfulness in both dimensions: reporting $V=N\cup\{s\}$2 and inviting all neighbors $V=N\cup\{s\}$3 is a dominant strategy. This is the paper’s DSIC notion. Individual rationality is stated as

$V=N\cup\{s\}$4

for all $V=N\cup\{s\}$5, all $V=N\cup\{s\}$6, and any invitation report $V=N\cup\{s\}$7; strategyproofness requires

$V=N\cup\{s\}$8

for all $V=N\cup\{s\}$9 and all $s$0 (Guo et al., 19 Jul 2025).

This model isolates the network-specific difficulty: invitations alter market participation and hence the competitive environment. A bidder can manipulate not only her announced value but also the set of rivals and downstream invitees who become active. That feature is precisely what makes value-monotonicity alone insufficient.

2. Formal definition of IP-MON

The network-auction paper first defines value-monotonicity in the usual threshold sense: fixing $s$1 and an invitation set $s$2, whenever

$s$3

then

$s$4

for all $s$5. The associated critical winning bid is

$s$6

This is the bid threshold induced by a fixed invitation report (Guo et al., 19 Jul 2025).

IP-MON is defined through a partial order on types. For $s$7 and $s$8, the invitation-promoted order is

$s$9

Inviting more neighbors is therefore “better” in the order. The allocation rule $N$0 satisfies IP-MON if, for fixed $N$1,

$N$2

Equivalently, keeping others fixed, increasing $N$3 and expanding $N$4 weakly preserves winning (Guo et al., 19 Jul 2025).

For contrast, the same paper defines Invitation-Depressed Monotonicity (ID-MON), in which shrinking the invitation set is the monotone direction. The two notions partition the ways an allocation can react to invitations. The paper states that they encompass all existing allocation rules of network auctions as specific instances (Guo et al., 19 Jul 2025).

Under IP-MON, the critical winning threshold is monotone in the invitation set in the DSIC-compatible direction: $N$5 Thus, inviting more neighbors does not increase the bidder’s winning threshold. This threshold monotonicity is the key bridge from allocation monotonicity to payment design (Guo et al., 19 Jul 2025).

A central misconception addressed by the paper is that Myerson’s value-monotonicity should suffice. In network auctions it does not. Since types are two-dimensional, payment design must align incentives over invitations as well as bids. Earlier multi-unit extensions can therefore be value-monotone and still fail DSIC, because some agents benefit from inviting fewer neighbors. The paper identifies DNA-MU as such a case and uses IP-MON to explain why its refined version DNA-MU-R succeeds (Guo et al., 19 Jul 2025).

3. Network implementability and payment characterization

The payment side is expressed through a decomposition

$N$6

where $N$7 is the winner payment and $N$8 the loser payment. A DSIC characterization cited in the paper states that DSIC holds if and only if four conditions are met: $N$9 is value-monotone; $i\in N$0 and $i\in N$1 are bid-independent and invitational-monotone; $i\in N$2; and $i\in N$3, which is used for IR (Guo et al., 19 Jul 2025).

The main implementability theorem for IP-MON is direct: every IP-MON allocation rule $i\in N$4 is network-implementable. That is, there exists a payment rule $i\in N$5 such that $i\in N$6 is IR and DSIC. The mechanism-design significance is that invitation-aware monotonicity in the allocation dimension is sufficient to recover a full truthful mechanism, not merely a monotone ranking rule (Guo et al., 19 Jul 2025).

The revenue-maximizing payment rule under IP-MON has the simplest possible form. For each bidder $i\in N$7,

$i\in N$8

Equivalently,

$i\in N$9

Winners pay their critical values, dependent on their own invitation sets, and losers pay $N(i)=\{j\in N:(i,j)\in E\}$0. The paper proves that this mechanism is IR and DSIC and, among all IR+DSIC payment rules compatible with a fixed IP-MON allocation, maximizes the seller’s revenue (Guo et al., 19 Jul 2025).

The invitation incentive is then transparent. If a bidder is already a winner, inviting more neighbors weakly preserves winning by IP-MON and weakly decreases the threshold $N(i)=\{j\in N:(i,j)\in E\}$1, so the bidder pays no more than before. If a bidder is a loser, the payment remains $N(i)=\{j\in N:(i,j)\in E\}$2. Hence withholding invitations cannot improve utility. This is the network-auction form of “invite all neighbors” as a dominant strategy (Guo et al., 19 Jul 2025).

The threshold payments are also computationally tractable. If the IP-MON allocation rule $N(i)=\{j\in N:(i,j)\in E\}$3 runs in time $N(i)=\{j\in N:(i,j)\in E\}$4, then the revenue-maximizing payment profile can be computed in

$N(i)=\{j\in N:(i,j)\in E\}$5

where $N(i)=\{j\in N:(i,j)\in E\}$6. The stated method is binary search on $N(i)=\{j\in N:(i,j)\in E\}$7 while running the allocation rule with $N(i)=\{j\in N:(i,j)\in E\}$8’s invitation set fixed. The paper also notes that DSIC requires consistent tie-breaking in $N(i)=\{j\in N:(i,j)\in E\}$9, such as BFS order or predetermined priorities, so that thresholds are well defined (Guo et al., 19 Jul 2025).

4. Mechanism classes, corrected designs, and examples

The paper uses IP-MON and ID-MON as a classification scheme for known network-auction mechanisms.

Mechanism	Allocation class	Property noted
DNA-MU-R	IP-MON	IR, DSIC, WBB; winners pay $r_i\subseteq N(i)$0, losers pay $r_i\subseteq N(i)$1
MUDAN	IP-MON	Listed as an IP-MON mechanism
SNCA	IP-MON	Certain budgeted network mechanisms
VCG on networks	ID-MON	Listed as an ID-MON mechanism
VCG-RM	ID-MON	Revenue-maximizing payment for efficient allocation
LDM-Tree	ID-MON	Listed as an ID-MON mechanism

DNA-MU-R is the refined multi-unit mechanism used to repair the failure of DNA-MU. Its allocation checks bidder $r_i\subseteq N(i)$2 in a BFS order and awards $r_i\subseteq N(i)$3 if

$r_i\subseteq N(i)$4

where $r_i\subseteq N(i)$5 is the invitational-domination subtree rooted at $r_i\subseteq N(i)$6, that is, the set of bidders dominated by $r_i\subseteq N(i)$7. With winner payments $r_i\subseteq N(i)$8 and loser payments $r_i\subseteq N(i)$9, the paper proves that DNA-MU-R is IP-MON, revenue-maximizing, IR, DSIC, and weakly budget balanced (Guo et al., 19 Jul 2025).

The six-agent example in the paper makes the invitation effect explicit. The seller $N(s)$0 is connected to $N(s)$1 and $N(s)$2; $N(s)$3 invites $N(s)$4 and $N(s)$5; $N(s)$6 invites $N(s)$7; and $N(s)$8 can invite $N(s)$9. Valuations are

$i$0

with $i$1. Under DNA-MU-R, when $i$2 invites $i$3, the winners are $i$4, with payments $i$5. When $i$6 does not invite $i$7, the winners are $i$8, with payments $i$9. The paper uses this example to show that inviting fewer neighbors is not profitable: winners pay threshold values that weakly decrease with invitations, while losers pay $G=(V,E)$00 (Guo et al., 19 Jul 2025).

The paper also identifies a degenerate intersection of IP-MON and ID-MON: mechanisms whose allocations satisfy both monotonicities and whose revenue-maximizing payments coincide. In that case utilities become independent of invitations. This boundary case clarifies that the two monotonicity classes are not merely opposites; they can overlap in a way that suppresses invitation effects entirely (Guo et al., 19 Jul 2025).

5. Combinatorial extension, assumptions, and limitations

The same framework extends from multi-unit unit-demand auctions to combinatorial auctions with single-minded bidders. In this model the seller has $G=(V,E)$01 heterogeneous items $G=(V,E)$02, each bidder $G=(V,E)$03 has a publicly known favorite bundle $G=(V,E)$04, and the private component is the value $G=(V,E)$05 for that bundle. The paper states that the classic efficient allocation is NP-hard and that a $G=(V,E)$06-approximation exists in the canonical setting (Guo et al., 19 Jul 2025).

For the network setting, the IP-MON mechanism is NSA. It generalizes the ranking criterion

$G=(V,E)$07

using the invitational-domination set $G=(V,E)$08 and a BFS priority order. For bidder $G=(V,E)$09 in BFS order, let $G=(V,E)$10. If $G=(V,E)$11 is the $G=(V,E)$12 of $G=(V,E)$13 over $G=(V,E)$14, and the bundle $G=(V,E)$15 is disjoint from the union of bundles already allocated to the current winner set $G=(V,E)$16, then $G=(V,E)$17 is selected. With the IP-MON revenue-maximizing payment rule, NSA is IR, DSIC, and WBB (Guo et al., 19 Jul 2025).

The parallel ID-MON mechanism is Net-$G=(V,E)$18-APM. It applies the $G=(V,E)$19-approximation allocation globally, ignoring $G=(V,E)$20, and uses the ID-MON revenue-maximizing payment: $G=(V,E)$21 The mechanism is $G=(V,E)$22-efficient, IR, and DSIC, but not necessarily WBB (Guo et al., 19 Jul 2025).

The assumptions behind these results are explicit: single-parameter valuations, independent private values, deterministic allocations, a known network graph, invitations that are observable and enforceable as part of the reported type, no collusion, quasi-linear utilities, and a seller-chosen tie-breaking rule such as BFS order (Guo et al., 19 Jul 2025). The limitations are equally explicit. IP-MON allocations need not be efficient; the paper states that there exist instances where no IP-MON allocation achieves efficiency. Efficient allocations are ID-MON, but revenue-maximizing payments for them may fail WBB, as in VCG on networks with deficits. Threshold computation may require repeated allocation runs, although the overall computation remains polynomial-time when $G=(V,E)$23 is polynomial-time (Guo et al., 19 Jul 2025).

Open directions are stated in broad terms. The paper points to extensions beyond deterministic $G=(V,E)$24-$G=(V,E)$25 allocations, Bayesian truthfulness on networks, and broader domains including budgets, complementarities, and dynamic networks (Guo et al., 19 Jul 2025). A plausible implication is that IP-MON is a foundational rather than terminal characterization: it resolves the single-minded obstacle identified by the paper, but not the full space of networked mechanism design.

6. Related uses of the term in adjacent literatures

Outside network-auction design, the same acronym is used for formally different monotonicity notions. In approval-based multi-winner voting, the corresponding concept is support monotonicity with population increase (SMWPI). There, a new voter enters and approves exactly a subset $G=(V,E)$26 of candidates already present in a winning committee. Strong SMWPI requires that all candidates in $G=(V,E)$27 remain in some, or in the universal form all, winning committees after the population increase; weak SMWPI requires that at least one candidate in $G=(V,E)$28 remain. The paper shows that AV, SAV, CC, PAV, and MAV satisfy strong SMWPI; SeqPAV, seq-Phragmén, and max-Phragmén satisfy only weak SMWPI; Monroe fails even weak SMWPI (Sánchez-Fernández et al., 2017). This use of “IP-MON” concerns committee stability under new support rather than bidder incentives or payment thresholds.

In invitation-based cooperative games, IP-MON is split into a value version and a payoff version. With a monotone characteristic function $G=(V,E)$29 and a permission structure over a DAG, value IP-MON says that adding invited and permitted players cannot reduce coalition value; payoff IP-MON says that an inviter’s allocation $G=(V,E)$30 weakly increases after adding an invitation edge. The weighted permission Shapley value

$G=(V,E)$31

is shown to satisfy payoff IP-MON under monotone $G=(V,E)$32 and proper weights $G=(V,E)$33 that depend only on distance to the initial set (Zhang et al., 2020). Here the object of monotonicity is payoff allocation in cooperative surplus sharing, not allocation feasibility in auctions.

In adaptive clinching auctions, IP-MON refers to monotonicity of aggregate objectives with respect to bidder entry. For symmetric budgets $G=(V,E)$34, adding bidders offline or online weakly increases both liquid welfare and revenue under the adaptive clinching auction; for asymmetric budgets, counterexamples show that these monotonicity properties can fail (Sato, 10 Feb 2025). In that literature, IP-MON is therefore a property of auction outcomes as functions of the participant set, rather than a monotonicity condition on an individual bidder’s own type.

These usages are not equivalent axioms. What they share is a common directional intuition: additional support, invitations, or participants should not worsen the relevant outcome. In the network-auction literature, however, IP-MON has a particularly sharp technical role because it yields an implementability theorem, a critical-threshold payment rule, and a tractable revenue-maximization result (Guo et al., 19 Jul 2025).