---
title: Invitation-Promoted Monotonicity (IP-MON)
url: https://www.emergentmind.com/topics/invitation-promoted-monotonicity-ip-mon
type: topic
---

# Invitation-Promoted Monotonicity (IP-MON)

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 [2507.14472]. 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 [2507.14472]. 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 [1710.04246] [2011.09049] [2502.06278].

## 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 \(N(s)\) [2507.14472].

In the multi-unit unit-demand model, bidder \(i\) has private type
\[
\theta_i=(v_i,r_i),
\]
where \(v_i\in\mathbb{R}_{\ge 0}\) is her valuation and \(r_i\subseteq N(i)\) is the set she actually invites. A direct-revelation mechanism is \(\mathcal M=(f,p)\), with allocation rule \(f=(f_i)\) and payment rule \(p=(p_i)\), where
\[
f_i:\Theta\to\{0,1\},\qquad p_i:\Theta\to\mathbb{R}.
\]
In a \(k\)-unit auction with unit-demand bidders, feasibility requires \(\sum_{i\in N} f_i(\theta)\le k\). Utility is quasi-linear:
\[
u_i(\theta;\mathcal M)=f_i(\theta)\cdot v_i-p_i(\theta).
\]

The strategic requirement is stronger than in canonical auctions. Each bidder reports both a valuation \(v_i'\) and an invitation set \(r_i'\subseteq N(i)\). Strategyproofness in this environment requires truthfulness in both dimensions: reporting \(v_i'=v_i\) and inviting all neighbors \(r_i'=r_i\) is a dominant strategy. This is the paper’s DSIC notion. Individual rationality is stated as
\[
u_i((v_i,r_i'),\theta'_{-i};\mathcal M)\ge 0
\]
for all \(\theta'\), all \(i\), and any invitation report \(r_i'\subseteq r_i\); strategyproofness requires
\[
u_i((v_i,r_i),\theta'_{-i};\mathcal M)\ge u_i((v_i',r_i'),\theta'_{-i};\mathcal M)
\]
for all \(\theta'\) and all \(i\) [2507.14472].

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 \(\theta'_{-i}\) and an invitation set \(r_i'\), whenever
\[
f_i((v_i',r_i'),\theta'_{-i})=1,
\]
then
\[
f_i((v_i'',r_i'),\theta'_{-i})=1
\]
for all \(v_i''\ge v_i'\). The associated critical winning bid is
\[
v_i^*(r_i')=\inf\{v_i'\ge 0:f_i((v_i',r_i'),\theta'_{-i})=1\}.
\]
This is the bid threshold induced by a fixed invitation report [2507.14472].

IP-MON is defined through a partial order on types. For \(\theta_i^1=(v_i^1,r_i^1)\) and \(\theta_i^2=(v_i^2,r_i^2)\), the invitation-promoted order is
\[
\theta_i^1 \succeq_{\mathcal P} \theta_i^2
\quad\text{if}\quad
v_i^1\ge v_i^2\ \text{and}\ r_i^2\subseteq r_i^1.
\]
Inviting more neighbors is therefore “better” in the order. The allocation rule \(f\) satisfies IP-MON if, for fixed \(\theta_{-i}\),
\[
f_i(\theta_i,\theta_{-i})=1 \Rightarrow f_i(\theta_i',\theta_{-i})=1
\quad\text{for all } \theta_i'\succeq_{\mathcal P}\theta_i.
\]
Equivalently, keeping others fixed, increasing \(v_i\) and expanding \(r_i\) weakly preserves winning [2507.14472].

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 [2507.14472].

Under IP-MON, the critical winning threshold is monotone in the invitation set in the DSIC-compatible direction:
\[
r_i^2\subseteq r_i^1 \Rightarrow v_i^*(r_i^1)\le v_i^*(r_i^2).
\]
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 [2507.14472].

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 [2507.14472].

## 3. Network implementability and payment characterization

The payment side is expressed through a decomposition
\[
p_i(\theta)=f_i(\theta)\cdot \tilde p_i(\theta)+(1-f_i(\theta))\cdot \bar p_i(\theta),
\]
where \(\tilde p_i\) is the winner payment and \(\bar p_i\) the loser payment. A DSIC characterization cited in the paper states that DSIC holds if and only if four conditions are met: \(f\) is value-monotone; \(\tilde p_i\) and \(\bar p_i\) are bid-independent and invitational-monotone; \(\tilde p_i(r_i)-\bar p_i(r_i)=v_i^*(r_i)\); and \(\bar p_i(\emptyset)\le 0\), which is used for IR [2507.14472].

The main implementability theorem for IP-MON is direct: every IP-MON allocation rule \(f\) is network-implementable. That is, there exists a payment rule \(p\) such that \((f,p)\) 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 [2507.14472].

The revenue-maximizing payment rule under IP-MON has the simplest possible form. For each bidder \(i\),
\[
\tilde p_i(\theta)=v_i^*(r_i),\qquad \bar p_i(\theta)=0.
\]
Equivalently,
\[
p_i(\theta)=f_i(\theta)\cdot v_i^*(r_i).
\]
Winners pay their critical values, dependent on their own invitation sets, and losers pay \(0\). 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 [2507.14472].

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 \(v_i^*(r_i)\), so the bidder pays no more than before. If a bidder is a loser, the payment remains \(0\). Hence withholding invitations cannot improve utility. This is the network-auction form of “invite all neighbors” as a dominant strategy [2507.14472].

The threshold payments are also computationally tractable. If the IP-MON allocation rule \(f\) runs in time \(T\), then the revenue-maximizing payment profile can be computed in
\[
O(|N|\cdot T\cdot \log V_{\max}),
\]
where \(V_{\max}=\max_{i\in N} v_i\). The stated method is binary search on \(v_i\) while running the allocation rule with \(i\)’s invitation set fixed. The paper also notes that DSIC requires consistent tie-breaking in \(f\), such as BFS order or predetermined priorities, so that thresholds are well defined [2507.14472].

## 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 \(v_i^*(r_i)\), losers pay \(0\) |
| 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 \(i\) in a BFS order and awards \(i\) if
\[
v_i \ge v^k(N\setminus T_i),
\]
where \(T_i\) is the invitational-domination subtree rooted at \(i\), that is, the set of bidders dominated by \(i\). With winner payments \(v_i^*(r_i)\) and loser payments \(0\), the paper proves that DNA-MU-R is IP-MON, revenue-maximizing, IR, DSIC, and weakly budget balanced [2507.14472].

The six-agent example in the paper makes the invitation effect explicit. The seller \(s\) is connected to \(A\) and \(B\); \(B\) invites \(C\) and \(F\); \(C\) invites \(D\); and \(D\) can invite \(H\). Valuations are
\[
v_A=4,\ v_B=1,\ v_C=4,\ v_D=7,\ v_F=6,\ v_H=5,
\]
with \(k=3\). Under DNA-MU-R, when \(D\) invites \(H\), the winners are \(\{B,F,C\}\), with payments \(A(0), B(0), F(4), C(1), D(0), H(0)\). When \(D\) does not invite \(H\), the winners are \(\{A,B,F\}\), with payments \(A(4), B(0), F(4), C(0), D(0), H(0)\). 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 \(0\) [2507.14472].

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 [2507.14472].

## 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 \(k\) heterogeneous items \(\mathcal K\), each bidder \(i\) has a publicly known favorite bundle \(S_i^*\subseteq \mathcal K\), and the private component is the value \(v_i\) for that bundle. The paper states that the classic efficient allocation is NP-hard and that a \(\sqrt{k}\)-approximation exists in the canonical setting [2507.14472].

For the network setting, the IP-MON mechanism is NSA. It generalizes the ranking criterion
\[
\frac{v_i}{\sqrt{|S_i^*|}}
\]
using the invitational-domination set \(T_i\) and a BFS priority order. For bidder \(i\) in BFS order, let \(N_{-T_i}=(N\setminus T_i)\cup\{i\}\). If \(i\) is the \(\arg\max\) of \(v_j/\sqrt{|S_j^*|}\) over \(j\in N_{-T_i}\), and the bundle \(S_i^*\) is disjoint from the union of bundles already allocated to the current winner set \(W\), then \(i\) is selected. With the IP-MON revenue-maximizing payment rule, NSA is IR, DSIC, and WBB [2507.14472].

The parallel ID-MON mechanism is Net-\(\sqrt{k}\)-APM. It applies the \(\sqrt{k}\)-approximation allocation globally, ignoring \(T_i\), and uses the ID-MON revenue-maximizing payment:
\[
\text{winners pay } v_i^*(\emptyset),\qquad
\text{losers pay } v_i^*(\emptyset)-v_i^*(r_i).
\]
The mechanism is \(\sqrt{k}\)-efficient, IR, and DSIC, but not necessarily WBB [2507.14472].

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 [2507.14472]. 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 \(f\) is polynomial-time [2507.14472].

Open directions are stated in broad terms. The paper points to extensions beyond deterministic \(0\)-\(1\) allocations, Bayesian truthfulness on networks, and broader domains including budgets, complementarities, and dynamic networks [2507.14472]. 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 \(G\) of candidates already present in a winning committee. Strong SMWPI requires that all candidates in \(G\) remain in some, or in the universal form all, winning committees after the population increase; weak SMWPI requires that at least one candidate in \(G\) 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 [1710.04246]. 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 \(v\) 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 \(\phi_i\) weakly increases after adding an invitation edge. The weighted permission Shapley value
\[
\phi_i^{\mathrm{perm},\omega}(v,\rho)=\phi_i^\omega(v^\rho),\qquad v^\rho(S)=v(\alpha(S)),
\]
is shown to satisfy payoff IP-MON under monotone \(v\) and proper weights \(\omega(i)=f(d_i)\) that depend only on distance to the initial set [2011.09049]. 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 \(B_i=\beta\), 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 [2502.06278]. 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 [2507.14472].

Source: https://www.emergentmind.com/topics/invitation-promoted-monotonicity-ip-mon