Invitation-Depressed Monotonicity (ID-MON)

• Invitation-Depressed Monotonicity is a monotone allocation principle in network auctions that guarantees a bidder’s win is preserved when valuation increases and invitations decrease.
• It formalizes the competitive effect where inviting more neighbors weakly reduces winning chances by expanding the informed market, linking critical bids to strategyproof payments.
• The framework under ID-MON enables revenue-maximizing mechanisms for multi-unit and combinatorial auctions with computationally efficient payment computation via binary search.

Searching arXiv for the specified papers and closely related work on Invitation-Depressed Monotonicity. Search query: (Guo et al., 19 Jul 2025) Invitation-Depressed Monotonicity (ID-MON) is a monotone allocation principle for auctions over social networks in which each bidder’s type has both a valuation component and an invitation component. In the formulation introduced in "Strategyproofness and Monotone Allocation of Auction in Social Networks" (Guo et al., 19 Jul 2025), ID-MON requires that if a bidder wins under a given type, then the bidder must also win after increasing valuation and reducing invitations. The principle formalizes the competitive effect of invitations: inviting more neighbors expands the informed market and can weakly decrease the inviter’s chance of winning. Within that framework, ID-MON is one of two identified categories of monotone allocation rules on networks, together with Invitation-Promoted Monotonicity (IP-MON), and it is used to connect allocation monotonicity, critical bids, payments, and strategyproofness in network auctions (Guo et al., 19 Jul 2025).

1. Formal model and definition

In the social-network auction model, the market is represented by a graph $G = (N \cup \{s\}, E)$, where $N$ is the set of potential bidders, $s$ is the seller node, and $E$ is the set of edges. Each bidder $i \in N$ has neighbor set $N(i) = \{j \in N : (i,j) \in E\}$. Information diffuses by invitations: the seller invites her neighbors, invited bidders may invite their neighbors, and so on. Each bidder has a two-dimensional private type $\theta_i = (v_i, r_i)$, where $v_i \ge 0$ is the valuation and $r_i \subseteq N(i)$ is the set of neighbors she can invite. The reported type is $\theta_i' = (v_i', r_i')$ with $N$0 and $N$1. The mechanism is $N$2, where the allocation rule $N$3 satisfies feasibility and each $N$4 (Guo et al., 19 Jul 2025).

The allocation environments covered by the framework are single-parameter environments: single-item, $N$5-unit with unit demand, or single-minded combinatorial. Agents have quasi-linear utility

$N$6

Strategyproofness in this setting means both truthful valuation reporting and inviting all neighbors are dominant.

For bidder $N$7 and two types $N$8 and $N$9, the invitation-depressed partial order is defined by

$s$0

This order treats inviting fewer neighbors as higher in the order because fewer invitations imply less competition. An allocation rule $s$1 is ID-MON if, fixing $s$2, whenever $s$3, then for every $s$4 one also has

$s$5

Thus, winning must be preserved under higher valuation and a smaller invitation set (Guo et al., 19 Jul 2025).

The contrast class is IP-MON, based on the opposite order in which inviting more neighbors raises allocation priority. The two categories encompass existing allocation rules of network auctions as specific instances. Classical efficient top-$s$6 allocation extended to the network setting is ID-MON, whereas mechanisms such as DNA-MU-Refined are IP-MON. If an allocation satisfies both ID-MON and IP-MON, invitations become irrelevant for allocation, producing an invitation-independent or degenerated case (Guo et al., 19 Jul 2025).

2. Critical bids, payment monotonicity, and implementability

The framework extends value-monotonicity by incorporating monotonicity over invitations. Value-monotonicity requires that, fixing invitations, a bidder who wins at a bid $s$7 must also win at any higher bid $s$8. Payments are decomposed into winning and losing components:

$s$9

The components are bid-independent if changing the reported valuation does not change $E$0 or $E$1, and payments are invitational-monotone if inviting more neighbors weakly reduces both the winning and losing payment components (Guo et al., 19 Jul 2025).

For fixed $E$2 and invitation action $E$3, the critical winning bid is

$E$4

The network-auction characterization quoted in the paper states that a network single-item auction mechanism is IR and SP if and only if four conditions hold: $E$5 is value-monotone; $E$6 and $E$7 are bid-independent and invitational-monotone; $E$8; and $E$9 (Guo et al., 19 Jul 2025).

Within this characterization, ID-MON yields a precise comparative statics result for thresholds. If $i \in N$0 is ID-MON and $i \in N$1, then

$i \in N$2

Inviting more neighbors therefore weakly increases the minimum winning bid required. This is the threshold version of the competitive effect embedded in the definition of ID-MON (Guo et al., 19 Jul 2025).

The central implementability theorem states that every ID-MON allocation is network-implementable. More generally, for a value-monotone allocation, implementability follows from the existence of functions

$i \in N$3

such that

$i \in N$4

and both $i \in N$5 and $i \in N$6 are minimized at the full invitation set. Under ID-MON, $i \in N$7 and $i \in N$8 can be chosen as non-increasing functions of $i \in N$9, which makes invitational-monotonicity compatible with strategyproofness (Guo et al., 19 Jul 2025).

3. Revenue-maximizing payments under ID-MON

For a fixed ID-MON allocation, the framework gives an explicit revenue-maximizing payment rule. For each bidder $N(i) = \{j \in N : (i,j) \in E\}$0,

$N(i) = \{j \in N : (i,j) \in E\}$1

and the payment vector is

$N(i) = \{j \in N : (i,j) \in E\}$2

The resulting mechanism $N(i) = \{j \in N : (i,j) \in E\}$3 is IR and SP, and it revenue-dominates every other IR and SP mechanism with the same allocation rule (Guo et al., 19 Jul 2025).

The economic interpretation is asymmetric between winners and losers. Winners pay the no-invitation critical bid $N(i) = \{j \in N : (i,j) \in E\}$4. Losers receive the transfer

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

which is weakly negative in the sign convention of the paper. Because ID-MON makes $N(i) = \{j \in N : (i,j) \in E\}$6 non-decreasing in the invitation set, inviting more neighbors weakly decreases the losing payment, possibly implying a larger subsidy. This compensates for the fact that more invitations increase competition and can reduce winning probability (Guo et al., 19 Jul 2025).

The payment rule is constructive rather than existential. If an ID-MON allocation rule runs in time $N(i) = \{j \in N : (i,j) \in E\}$7, then the revenue-maximizing payment can be computed in

$N(i) = \{j \in N : (i,j) \in E\}$8

by binary search for $N(i) = \{j \in N : (i,j) \in E\}$9 and $\theta_i = (v_i, r_i)$0 for each bidder. This gives a computationally feasible route from a monotone allocation oracle to a strategyproof mechanism with maximal revenue among IR and SP implementations of that allocation (Guo et al., 19 Jul 2025).

4. Applications in multi-unit and combinatorial network auctions

The paper uses ID-MON to reorganize multi-unit and combinatorial network-auction design. It states that pioneering research in multi-unit network auctions with single-unit demand fails to be strategyproof, and attributes the failure to the absence of a general allocation principle connecting invitations, payments, and SP. In the ID-MON reformulation, the standard efficient allocation that selects the top-$\theta_i = (v_i, r_i)$1 bids among informed bidders is ID-MON: increasing a bidder’s value preserves winning, and reducing invitations cannot introduce new competitors that displace that bidder from the top-$\theta_i = (v_i, r_i)$2 (Guo et al., 19 Jul 2025).

This leads to the VCG-Revenue-Maximizing mechanism (VCG-RM) for $\theta_i = (v_i, r_i)$3-unit auctions with unit-demand bidders. Its construction is: allocate items to the top-$\theta_i = (v_i, r_i)$4 bids among informed agents; construct the invitational-domination tree (IDT); and charge winner $\theta_i = (v_i, r_i)$5 the threshold $\theta_i = (v_i, r_i)$6, while loser $\theta_i = (v_i, r_i)$7 is assigned the payment $\theta_i = (v_i, r_i)$8. The stated properties are EF, IR, SP, and

$\theta_i = (v_i, r_i)$9

The paper also notes that extending VCG to networks with efficient top-$v_i \ge 0$0 allocation may incur deficits, whereas the ID-MON revenue-maximizing payment rule remedies revenue performance (Guo et al., 19 Jul 2025).

For single-minded combinatorial bidders, the efficient allocation problem is NP-hard. The paper extends the classical $v_i \ge 0$1-approximation allocation to the network setting via ID-MON and obtains the mechanism Net-$v_i \ge 0$2-APM, which is stated to be $v_i \ge 0$3-EF, IR, SP, but not WBB. The mechanism first applies the classical $v_i \ge 0$4-approximation priority rule over informed bidders, which is ID-MON, and then applies the ID-MON revenue-maximizing payment rule (Guo et al., 19 Jul 2025).

The taxonomy also explains earlier failures. DNA-MU fails SP because its payment violates invitational-monotonicity, so bidders can benefit by inviting fewer neighbors. DNA-MU-Refined instead achieves IP-MON allocation with revenue-maximizing payments, restoring SP and WBB. The boundary is therefore not monotonicity alone, but compatibility between the direction of monotonicity in allocation and the corresponding payment monotonicity (Guo et al., 19 Jul 2025).

5. Relation to Myerson’s lemma and the efficiency–budget boundary

The classical reference point is Myerson’s monotone-allocation paradigm for single-parameter auctions, where strategyproofness is characterized by value-monotone allocations and critical payments. In network auctions, bidders choose both values and invitation actions, so one-dimensional value-monotonicity is insufficient. The network-auction characterization supplies conditions on value-monotonicity, payment bid-independence, invitational-monotonicity, and critical-bid identities, but ID-MON provides the missing allocation principle in the invitation dimension (Guo et al., 19 Jul 2025).

The reduction to the classical case is exact when invitations disappear. If all bidders are directly connected to the seller, the network model collapses to a standard auction, and ID-MON collapses to value-monotonicity. In general networks, the additional partial order over $v_i \ge 0$5 yields the threshold inequality

$v_i \ge 0$6

which is the condition exploited by the revenue-maximizing payment formula (Guo et al., 19 Jul 2025).

ID-MON does not eliminate the usual impossibility frontiers. The paper states that under IR and SP constraints there exists an instance where no IP-MON allocation is EF, and that mechanisms under ID-MON may fail WBB. The trade-off is summarized directly: ID-MON can accommodate EF but may require subsidies, whereas IP-MON tends to be WBB but can sacrifice efficiency. The degenerated intersection of the two classes is also explicitly characterized: if an allocation is both ID-MON and IP-MON and payments follow both revenue-maximizing formulas, then for any $v_i \ge 0$7 and any $v_i \ge 0$8,

$v_i \ge 0$9

so invitations have no effect on utility (Guo et al., 19 Jul 2025).

A remaining open question is whether, given any value-monotone allocation, a computationally tractable SP payment always exists. The paper also identifies Bayesian truthfulness, randomized allocations, and richer multi-dimensional type spaces as further directions (Guo et al., 19 Jul 2025).

6. Related monotonicity concepts in other domains

Although the literal term ID-MON is native to network-auction mechanism design, closely related invitation-depressed directions of monotonicity appear in several adjacent literatures.

Domain	Native property	Relation to ID-MON
Cake cutting	Population-monotonicity	Inviting more agents weakly lowers incumbents’ utility
Incomplete-information games	Weakly increasing differences	Greater choice-parameter beliefs induce greater optimal choices
Approval-based voting	SMWPI	New supportive voters should not displace supported winners
Preference learning	Monotonicity of scores/probabilities	Favoring $r_i \subseteq N(i)$0 can still reduce $r_i \subseteq N(i)$1
Coalition formation	DIC	Inviting more players weakly increases the inviter’s payoff

In cake cutting, population-monotonicity requires that when the set of agents grows from $r_i \subseteq N(i)$2 to $r_i \subseteq N(i)$3, every incumbent weakly loses utility:

$r_i \subseteq N(i)$4

The Nash-optimal rule, equivalently the Strong Competitive Equilibrium from Equal Incomes (SCEEI), is stated to satisfy both resource-monotonicity and population-monotonicity; in the supplied comparative exposition, the upward population-monotone direction is identified with ID-MON (Segal-Halevi et al., 2015).

In games with incomplete information, the comparative exposition on rationalizability and monotonicity uses the label for monotone comparative statics driven by weakly increasing differences. There, greater own parameter and greater composite choice-parameter beliefs imply greater optimal choices:

$r_i \subseteq N(i)$5

The same monotonicity governs iterative rationalizability, with extremal surviving choices generated by extremal beliefs (Sloun, 26 Jan 2025).

In approval-based multi-winner voting, the corresponding native notion is support monotonicity with population increase (SMWPI). Adding a new voter who approves only candidates in a set $r_i \subseteq N(i)$6 is the voting analogue of an invitation. Strong SMWPI requires that all candidates in $r_i \subseteq N(i)$7 remain in some winning committee after the voter enters; weak SMWPI requires that at least one member of $r_i \subseteq N(i)$8 remain. Counting rules satisfy strong SMWPI, whereas Monroe fails even weak SMWPI in the cited counterexample, and strong SMWPI is incompatible with Perfect Representation (Sánchez-Fernández et al., 2017).

In comparison-based preference learning for AI alignment, the term is used in the supplied exposition for the failure mode in which strengthening a preference signal $r_i \subseteq N(i)$9 can still reduce either the model’s score for $\theta_i' = (v_i', r_i')$0 or the probability $\theta_i' = (v_i', r_i')$1. The paper proves local pairwise monotonicity for the difference $\theta_i' = (v_i', r_i')$2 under DPO-, GPO-, and GBT-type objectives, but also shows that individual-score and individual-probability monotonicity can fail because of parameter-space coupling, softmax normalization, and regularization toward a reference policy (Bareilles et al., 10 Jun 2025).

In coalition formation with invitations, the closest analogue is inviter monotonicity. The weighted permission Shapley value satisfies Diffusion Incentive Compatibility (DIC), which means that when a player expands invitations, the inviter’s payoff weakly increases under monotone $\theta_i' = (v_i', r_i')$3, acyclic invitation graphs, and suitable weights. The same exposition also states that stronger global and coalition forms of monotonicity fail in general, even though inviter monotonicity holds (Zhang et al., 2020).

Across these literatures, the common structure is monotone response to enlarged participation or strengthened support, but the object that is depressed or promoted differs: winning probability in network auctions, utility in cake cutting, best responses in incomplete-information games, committee membership in voting, individual generation probability in preference learning, and inviter payoff in coalition formation. The network-auction definition in (Guo et al., 19 Jul 2025) is the most explicit formalization of ID-MON as a partial order over types combining valuation and invitations.