Robust Equilibria in Shared Resource Allocation via Strengthening Border's Theorem

TL;DR

The paper addresses non-monetary, repeated allocation of a single shared resource among multiple agents and develops a Budgeted Robust Border (BRB) mechanism that achieves both a Bayes-Nash equilibrium and strong robustness guarantees. By linking online allocation with Border's theorem and strengthening the border polytope, the authors design a simple α_i-threshold bidding strategy that attains a Nash-equilibrium utility of at least λ_NASH = 1 − ∏_{j}(1 − α_j) times each agent's ideal utility, and a robustness of at least λ_ROB ≥ 1/2 against collusion. The construction uses a two-layer approach: budget-regulated bidding (all-pay-like budgeting) and probabilistic allocation with carefully chosen interim allocation probabilities p_i^S, supported by a novel flow-network argument and Schur-convexity analysis. They also provide computationally efficient implementations using a multiplicative-weights framework to compute allocation probabilities online, enabling practical deployment with near-optimal equilibrium and robustness properties. Overall, the work advances non-monetary mechanism design by delivering a practical mechanism with provable equilibrium and robustness guarantees, under minimal distributional assumptions, and introduces a technical strengthening of Border's theorem that may benefit broader implementation problems.

Abstract

We consider repeated allocation of a shared resource via a non-monetary mechanism, wherein a single item must be allocated to one of multiple agents in each round. We assume that each agent has i.i.d. values for the item across rounds, and additive utilities. Past work on this problem has proposed mechanisms where agents can get one of two kinds of guarantees: $(i)$ (approximate) Bayes-Nash equilibria via linkage-based mechanisms which need extensive knowledge of the value distributions, and $(ii)$ simple distribution-agnostic mechanisms with robust utility guarantees for each individual agent, which are worse than the Nash outcome, but hold irrespective of how others behave (including possibly collusive behavior). Recent work has hinted at barriers to achieving both simultaneously. Our work however establishes this is not the case, by proposing the first mechanism in which each agent has a natural strategy that is both a Bayes-Nash equilibrium and also comes with strong robust guarantees for individual agent utilities. Our mechanism comes out of a surprising connection between the online shared resource allocation problem and implementation theory, and uses a surprising strengthening of Border's theorem. In particular, we show that establishing robust equilibria in this setting reduces to showing that a particular subset of the Border polytope is non-empty. We establish this via a novel joint Schur-convexity argument. This strengthening of Border's criterion for obtaining a stronger conclusion is of independent technical interest, as it may prove useful in other settings.

Figures (5)

• Figure 4.1: Flow network that can be used to prove \ref{['thm:alpha_border']}. We let $S'$ be the random set of bidding agents where each agent $i$ lies in $S'$ independently with probability $\alpha_i$. Then, there are three kind of edges: edges whose flow corresponds to the probability of observing a specific $S'$ (left), edges whose flow corresponds to how we randomly allocate the item condition on observing a specific $S'$ (middle), and edges whose flow represent the probability that a specific agent gets the item (right). There is a flow of value $\Pr(S'\neq\emptyset)$ if and only if there exist allocation probabilities $p_i^S$ inducing the interim allocation probabilities $p_i$. In other words, the flows $p_i^S\Pr(S' = S)$ in the middle transform the probabilities that agents in a certain set $S$ bid to an agent $i\in S$ being allocated. We obtain the conditions in \ref{['thm:alpha_border']} by analyzing every minimum-cut of this network.
• Figure 5.1: Flow network for the proof of \ref{['lem:specific_border_modification']}, similar to the one in \ref{['fig:introductory_border_flow_network']} used in \ref{['thm:alpha_border']}. However, the red edges $(u_{\{i,j\}}, v_j)$ have explicit capacities as opposed to infinite capacity to enforce the additional bounds $\bar{p}_j^{\{i,j\}}\leq \frac{1+\alpha_i}{2}$.
• Figure 5.2: Plot of $f(X)$, the maximum value of the left-hand side of \ref{['eq:modifiedmaximizationobjective']} for a fixed size $m$ of $I$. It gets super close to $1$, and we prove that it never exceeds $1$.
• Figure D .1: Flow network that can be used to prove \ref{['thm:beta_border']}. We let $S'$ be the random set of bidding agents where each agent $i$ lies in $S'$ independently with probability $\beta_i$. Then, there are three kind of edges: edges whose flow corresponds to the probability of observing a specific $S'$ (left), edges whose flow corresponds to how we randomly allocate the item condition on observing a specific $S'$ (middle), and edges whose flow represent the probability that a specific agent gets the item (right). There is a flow of value $\Pr(S'\neq\emptyset)$ if and only if there exist allocation probabilities $p_i^S$ inducing the interim allocation probabilities $p_i$. In other words, the flows $p_i^S\Pr(S' = S)$ in the middle transform the probabilities that agents in a certain set $S$ bid to an agent $i\in S$ being allocated. We obtain the conditions in \ref{['thm:beta_border']} by analyzing every minimum-cut of this network.
• Figure D .2: Flow network that is used in the proof of \ref{['thm:border_arbitrary_upper_bound']}. The flow network is the same as \ref{['fig:beta_border_flow_network']} except the middle edges $(u_S, v_i)$ have capacity $\bar{p}_i^S\Pr(S'=S)$ to enforce the upper bounds on the allocation probabilities.

Theorems & Definitions (69)

• Definition 2.1: Ideal Utility
• Lemma 2.2
• Definition 3.1: $\beta$-aggressive strategy
• Definition 3.2: $\varepsilon$-approximate Nash equilibrium
• Proposition 3.3
• Definition 3.4: Interim allocation probability
• Theorem 3.5
• Theorem 3.6
• Lemma 4.1
• Theorem 4.2