Distribution through Repeated Market with Buying Rights

TL;DR

The paper investigates fair resource distribution under scarcity by embedding buying rights into a repeated hybrid market. It formalizes a two-stage round structure where central rights are allocated and tradable, analyzes long-horizon equilibria under Greedy strategies, and derives an upper bound showing that asymptotically the expected frustration is at most half of the free-market value. Through both theory and experiments, it demonstrates that the buying-right mechanism reduces inequity while maintaining price stability close to the free-market benchmark, even under time-varying supply. The findings highlight the potential of tradable entitlement mechanisms to improve fairness in decentralized distribution—relevant to policy design, emissions markets, and welfare provisioning—while suggesting directions for refining truthfulness and dynamic dynamics in future work.

Abstract

Resource distribution is a fundamental problem in economic and policy design, particularly when demand and supply are not naturally aligned. Without regulation, wealthier individuals may monopolize this resource, leaving the needs of others unsatisfied. While centralized distribution can ensure fairer division, it can struggle to manage logistics efficiently, and adapt to changing conditions, often leading to shortages, surpluses, and bureaucratic inefficiencies. Building on previous research on market-based redistribution, we examine a repeated hybrid market that incorporates buying rights. These rights, distributed iteratively by a central authority (for instance, as digital tokens), are intended to enhance fairness in the system - a unit of right is required to acquire a unit of the resource, but the rights themselves can also be traded alongside the resource in the market. We analyze how this regulatory mechanism influences the distribution of the scarce resource in the hybrid market over time. Unlike past works that relied on empirical methods, we explore the exact analytical properties of a system in which traders optimize over multiple rounds. We identify its market equilibrium, which is a natural generalization of the free market equilibrium, and show that it is coalition-proof. To assess the fairness in the system, we use the concept of frustration, which measures the gap between the resources a buyer is entitled to through their buying rights and what they actually obtain through trading. Our main theoretical result shows that using buying rights reduces the frustration by at least half compared to the free market. Empirical evaluations further support our findings, suggesting the system performs well even beyond the theoretically studied assumptions.

Figures (6)

• Figure 1: An illustration of a Repeated Market consisting of a series of Markets indexed by $\tau$. The distribution function $\phi$ is followed by the Trading phase, which is done through the market mechanism $\mu$. The Money and Good obtained by traders are transferred to the next Market via the transition function $\rho$. The "+" operation is used to aggregate the utility $u_T^\tau$ of traders $T$.
• Figure 2: Comparison of the Repeated Market with our hybrid mechanism involving trading Right (denoted "fairness", see Definition \ref{['def: informal market mechanism']}) under the Greedy equilibrium strategy, and the free market under its equilibrium. The two rows show two scenarios, which differ only by scaling the Claim five-times smaller. The horizontally-left figures show the Claim $D_b$ (in blue) of each of the three buyers. Colors used for the right distribution mechanisms $\phi$ (i.e. violet and red) show the corresponding amount of Right $R_b$ each buyer receives next to their Claim. The horizontally-centered figures show the evolution of the expected frustration $\mathcal{E}_f^\tau$. The horizontally-right figures show the evolution of the price of both Good and Right in each scenario.
• Figure 3: Comparison of the Repeated Market with our hybrid mechanism involving trading Right, denoted "fairness", as a function of the number of buyers. To generate parameters of the Repeated Market, we sample the Claim $D_b$ and income $m_b$ of buyers from the Dirichlet distribution. To generate a scenario, we randomly order the buyers. We than set the mean Claim of a buyer inversely proportional to her position. The income is set in the same way, but with respect to the inverse ordering. Figure \ref{['fig: experiments constant supply']} uses the same scheme, without applying the Dirichlet noise. Similar to Figure \ref{['fig: experiments constant supply']}, the mean Claim is $|B|$-times smaller in the second row. The horizontally-left, resp. horizontally-right figures show an estimate of the asymptotic expected frustration, resp. price of both Good and Right. The shaded regions show standard errors. For each data-point, we sample 10 environments and perform $\mathcal{T} = 10|B|$ steps.
• Figure 4: Repeated Market with time-variable supply of Good when our hybrid mechanism is used, compared to the free market. The left column shows the total volume of Good entering the system $\sum_{s\in S}g_s^\tau$ as a function of the Market number $\tau$. This volume is held constant in the previous experiments. The centered column shows the evolution of the expected frustration $\mathcal{E}_f^\tau$ as a function of the Market number $\tau$. The right column shows the evolution of the price of both Good and Right in the Repeated Market. In the first row, the supply follows a cosine wave. The second and third row show linear dependence. In the fourth row, the supply follows the logistic curve.
• Figure 5: Repeated Market with time-variable supply of Good when our hybrid mechanism is used, compared to the free market. The left column shows the total volume of Good entering the system $\sum_{s\in S}g_s^\tau$ as a function of the Market number $\tau$. This volume is held constant in the previous experiments. The centered column shows the evolution of the expected frustration $\mathcal{E}_f^\tau$ as a function of the Market number $\tau$. The right column shows the evolution of the price of both Good and Right in the Repeated Market. The first row shows a situation where the supply of Good abruptly changes in one timestamp. The second row depict a simple model of the bullwhip effect in supply chain lee1997bullwhip. Finally, the third row shows the Hubbert peak supply curve hubbert1956nuclearcalvo2017assessing. In all cases, the price in our hybrid market closely follows the free market price. Moreover, the expected frustration is about half of the free-market value, despite the lack of theoretical guarantees of Theorem \ref{['thm:nonmyopic:poa']}.

Theorems & Definitions (13)

• Remark 1
• definition 1
• definition 2
• definition 3: market mechanism
• definition 4
• definition 5
• definition 6
• Remark 2: Correctness of Definition \ref{['def: greedy strategy']}
• Remark 3: To Intentional Storing
• Remark 4: To the Money in the Next Market