arXiv subjects

Martin Hoefer

Publications and source records attributed to Martin Hoefer.

At least 19 recordsLinked to original sources

arXiv2026

Computing Tarski Fixed Points in Financial Networks

Modern financial networks are highly connected and result in complex interdependencies of the involved institutions. In the prominent Eisenberg-Noe model, a fundamental aspect is clearing -- to determine the amount of assets available to each financial institution in the presence of potential defaults and bankruptcy. A clearing state represents a fixed point that satisfies a set of natural axioms. Existence can be established (even in broad generalizations of the model) using Tarski's theorem. While a maximal fixed point can be computed in polynomial time, the complexity of computing other fixed points is open. In this paper, we provide an efficient algorithm to compute a minimal fixed point that runs in strongly polynomial time. It applies in a broad generalization of the Eisenberg-Noe model with any monotone, piecewise-linear payment functions and default costs. Moreover, in this scenario we provide a polynomial-time algorithm to compute a maximal fixed point. For networks without default costs, we can efficiently decide the existence of fixed points in a given range. We also study claims trading, a local network adjustment to improve clearing, when networks are evaluated with minimal clearing. We provide an efficient algorithm to decide existence of Pareto-improving trades and compute optimal ones if they exist.

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arXiv2025

Fare Zone Assignment on Trees

Designing fare systems for public transportation networks is a challenging task. A popular approach is to partition the network into fare zones (``zoning'') and fix journey prices depending on the number of traversed zones (``pricing''). In this paper, we focus on finding revenue-optimal solutions to the zoning problem for a given subadditive pricing function. We consider tree networks with $n$ vertices, since trees already pose non-trivial algorithmic challenges. Our main results are efficient algorithms that yield a simple $\mathcal{O}(\log n)$-approximation as well as a more involved $\mathcal{O}(\log n/\log \log n)$-approxi\-ma\-tion. We show that rooted instances, in which all demand arises at a single source, can be solved exactly. We further show APX-hardness for general instances on star graphs. For paths, we prove strong NP-hardness and outline a PTAS. Moreover, we show that computing an optimal solution is in FPT or XP for several natural problem parameters.

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arXiv2025

Persuading Agents in Opinion Formation Games

Prominent opinion formation models such as the one by Friedkin and Johnsen (FJ) concentrate on the effects of peer pressure on public opinions. In practice, opinion formation is also based on information about the state of the world and persuasion efforts. In this paper, we analyze an approach of Bayesian persuasion in the FJ model. There is an unknown state of the world that influences the preconceptions of n agents. A sender S can (partially) reveal information about the state to all agents. The agents update their preconceptions, and an equilibrium of public opinions emerges. We propose algorithms for the sender to reveal information in order to optimize various aspects of the emerging equilibrium. For many natural sender objectives, we show that there are simple optimal strategies. We then focus on a general class of range-based objectives with desired opinion ranges for each agent. We provide efficient algorithms in several cases, e.g., when the matrix of preconceptions in all states has constant rank, or when there is only a polynomial number of range combinations that lead to positive value for S. This generalizes, e.g., instances with a constant number of states and/or agents, or instances with a logarithmic number of ranges. In general, we show that subadditive range-based objectives allow a simple n-approximation, and even for additive ones, obtaining an $n^{1-c}$-approximation is NP-hard, for any constant $c > 0$.

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arXiv2025

Welfare and Beyond in Multi-Agent Contracts

A principal delegates a project to a team $S$ from a pool of $n$ agents. The project's value if all agents in $S$ exert costly effort is $f(S)$. To incentivize the agents to participate, the principal assigns each agent $i\in S$ a share $\rho_i\in [0,1]$ of the project's final value (i.e., designs $n$ linear contracts). The shares must be feasible -- their sum should not exceed $1$. It is well-understood how to design these contracts to maximize the principal's own expected utility, but what if the goal is to coordinate the agents toward maximizing social welfare? We initiate a systematic study of multi-agent contract design with objectives beyond principal's utility, including welfare maximization, for various classes of value functions $f$. Our exploration reveals an arguably surprising fact: If $f$ is up to XOS in the complement-free hierarchy of functions, then the optimal principal's utility is a constant-fraction of the optimal welfare. This is in stark contrast to the much larger welfare-utility gaps in auction design, and no longer holds above XOS in the hierarchy, where the gap can be unbounded. A constant bound on the welfare-utility gap immediately implies that existing algorithms for designing contracts with approximately-optimal principal's utility also guarantee approximately-optimal welfare. The downside of reducing welfare to utility is the loss of large constants. To obtain better guarantees, we develop polynomial-time algorithms directly for welfare, for different classes of value functions. These include a tight $2$-approximation to the optimal welfare for symmetric XOS functions. Finally, we extend our analysis beyond welfare to the project's value under general feasibility constraints. Our results immediately translate to budgeted welfare and utility.

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arXiv2025

Fractional Claims Trades and Donations in Financial Networks

Exploring measures to improve financial networks and mitigate systemic risks is an ongoing challenge. We study claims trading, a notion defined in Chapter 11 of the U.S. Bankruptcy Code. For a bank $v$ in distress and a trading partner $w$, the latter is taking over some claims of $v$ and in return giving liquidity to $v$. The idea is to rescue $v$ (or mitigate contagion effects from $v$'s insolvency). We focus on the impact of trading claims fractionally, when $v$ and $w$ can agree to trade only part of a claim. In addition, we study donations, in which $w$ only provides liquidity to $v$. They can be seen as special claims trades. When trading a single claim or making a single donation in networks without default cost, we show that it is impossible to strictly improve the assets of both banks $v$ and $w$. Since the goal is to rescue $v$ in distress, we study creditor-positive trades, in which $v$ improves and $w$ remains indifferent. We show that an optimal creditor-positive trade that maximizes the assets of $v$ can be computed in polynomial time. It also yields a (weak) Pareto-improvement for all banks in the entire network. In networks with default cost, we obtain a trade in polynomial time that weakly Pareto-improves all assets over the ones resulting from the optimal creditor-positive trade. We generalize these results to trading multiple claims for which $v$ is the creditor. Instead, when trading claims with a common debtor $u$, we obtain NP-hardness results for computing trades in networks with default cost that maximize the assets of the creditors and Pareto-improve the assets in the network. Similar results apply when $w$ donates to multiple banks in networks with default costs. For networks without default cost, we give an efficient algorithm to compute optimal donations to multiple banks.

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arXiv2024

Designing Exploration Contracts

We study a natural application of contract design in the context of sequential exploration problems. In our principal-agent setting, a search task is delegated to an agent. The agent performs a sequential exploration of $n$ boxes, suffers the exploration cost for each inspected box, and selects the content (called the prize) of one inspected box as outcome. Agent and principal obtain an individual value based on the selected prize. To influence the search, the principal a-priori designs a contract with a non-negative payment to the agent for each potential prize. The goal of the principal is to maximize her expected reward, i.e., value minus payment. Interestingly, this natural contract scenario shares close relations with the Pandora's Box problem. We show how to compute optimal contracts for the principal in several scenarios. A popular and important subclass is that of linear contracts, and we show how to compute optimal linear contracts in polynomial time. For general contracts, we obtain optimal contracts under the standard assumption that the agent suffers cost but obtains value only from the transfers by the principal. More generally, for general contracts with non-zero agent values for outcomes we show how to compute an optimal contract in two cases: (1) when each box has only one prize with non-zero value for principal and agent, (2) for i.i.d. boxes with a single prize with positive value for the principal.

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arXiv2024

Algorithms for Claims Trading

The recent banking crisis has again emphasized the importance of understanding and mitigating systemic risk in financial networks. In this paper, we study a market-driven approach to rescue a bank in distress based on the idea of claims trading, a notion defined in Chapter 11 of the U.S. Bankruptcy Code. We formalize the idea in the context of financial networks by Eisenberg and Noe. For two given banks v and w, we consider the operation that w takes over some claims of v and in return gives liquidity to v to ultimately rescue v. We study the structural properties and computational complexity of decision and optimization problems for several variants of claims trading. When trading incoming edges of v, we show that there is no trade in which both banks v and w strictly improve their assets. We therefore consider creditor-positive trades, in which v profits strictly and w remains indifferent. For a given set C of incoming edges of v, we provide an efficient algorithm to compute payments by w that result in maximal assets of v. When the set C must also be chosen, the problem becomes weakly NP-hard. Our main result here is a bicriteria FPTAS to compute an approximate trade. The approximate trade results in nearly the optimal amount of assets of v in any exact trade. Our results extend to the case in which banks use general monotone payment functions and the emerging clearing state can be computed efficiently. In contrast, for trading outgoing edges of v, the goal is to maximize the increase in assets for the creditors of v. Notably, for these results the characteristics of the payment functions of the banks are essential. For payments ranking creditors one by one, we show NP-hardness of approximation within a factor polynomial in the network size, when the set of claims C is part of the input or not. Instead, for proportional payments, our results indicate more favorable conditions.

cs.GT↗

arXiv2023

Information Design for Congestion Games with Unknown Demand

We study a novel approach to information design in the standard traffic model of network congestion games. It captures the natural condition that the demand is unknown to the users of the network. A principal (e.g., a mobility service) commits to a signaling strategy, observes the realized demand and sends a (public) signal to agents (i.e., users of the network). Based on the induced belief about the demand, the users then form an equilibrium. We consider the algorithmic goal of the principal: Compute a signaling scheme that minimizes the expected total cost of the induced equilibrium. We concentrate on single-commodity networks and affine cost functions, for which we obtain the following results. First, we devise a fully polynomial-time approximation scheme (FPTAS) for the case that the demand can only take two values. It relies on several structural properties of the cost of the induced equilibrium as a function of the updated belief about the distribution of demands. We show that this function is piecewise linear for any number of demands, and monotonic for two demands. Second, we give a complete characterization of the graph structures for which it is optimal to fully reveal the information about the realized demand. This signaling scheme turns out to be optimal for all cost functions and probability distributions over demands if and only if the graph is series-parallel. Third, we propose an algorithm that computes the optimal signaling scheme for any number of demands whose time complexity is polynomial in the number of supports that occur in a Wardrop equilibrium for some demand. Finally, we conduct a computational study that tests this algorithm on real-world instances.

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arXiv2023

Threshold Testing and Semi-Online Prophet Inequalities

We study threshold testing, an elementary probing model with the goal to choose a large value out of $n$ i.i.d. random variables. An algorithm can test each variable $X_i$ once for some threshold $t_i$, and the test returns binary feedback whether $X_i \ge t_i$ or not. Thresholds can be chosen adaptively or non-adaptively by the algorithm. Given the results for the tests of each variable, we then select the variable with highest conditional expectation. We compare the expected value obtained by the testing algorithm with expected maximum of the variables. Threshold testing is a semi-online variant of the gambler's problem and prophet inequalities. Indeed, the optimal performance of non-adaptive algorithms for threshold testing is governed by the standard i.i.d. prophet inequality of approximately $0.745+o(1)$ as $n \to \infty$. We show how adaptive algorithms can significantly improve upon this ratio. Our adaptive testing strategy guarantees a competitive ratio of at least $0.869-o(1)$. Moreover, we show that there are distributions that admit only a constant ratio $c < 1$, even when $n \to \infty$. Finally, when each box can be tested multiple times (with $n$ tests in total), we design an algorithm that achieves a ratio of $1-o(1)$.

cs.DS↗

arXiv2023

Dynamic Debt Swapping in Financial Networks

A debt swap is an elementary edge swap in a directed, weighted graph, where two edges with the same weight swap their targets. Debt swaps are a natural and appealing operation in financial networks, in which nodes are banks and edges represent debt contracts. They can improve the clearing payments and the stability of these networks. However, their algorithmic properties are not well-understood. We analyze the computational complexity of debt swapping. Our main interest lies in semi-positive swaps, in which no creditor strictly suffers and at least one strictly profits. These swaps lead to a Pareto-improvement in the entire network. We consider network optimization via sequences of v-improving debt swaps from which a given bank v strictly profits. For ranking-based clearing, we show that every sequence of semi-positive v-improving swaps has polynomial length. In contrast, for arbitrary v-improving swaps, the problem of reaching a network configuration that allows no further swaps is PLS-complete. In global optimization, the goal is to maximize the utility of a given bank $v$ by performing a sequence of debt swaps in the network. This problem is NP-hard to approximate for multiple types of swaps. Moreover, we study reachability problems -- deciding if a sequence of swaps exists between given initial and final networks. We design a polynomial-time algorithm to decide this question for arbitrary swaps and derive hardness results for several other types of swaps. Many of our results can be extended to networks with arbitrary monotone clearing.

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arXiv2022

Best of Both Worlds: Agents with Entitlements

Fair division of indivisible goods is a central challenge in artificial intelligence. For many prominent fairness criteria including envy-freeness (EF) or proportionality (PROP), no allocations satisfying these criteria might exist. Two popular remedies to this problem are randomization or relaxation of fairness concepts. A timely research direction is to combine the advantages of both, commonly referred to as Best of Both Worlds (BoBW). We consider fair division with entitlements, which allows to adjust notions of fairness to heterogeneous priorities among agents. This is an important generalization to standard fair division models and is not well-understood in terms of BoBW results. Our main result is a lottery for additive valuations and different entitlements that is ex-ante weighted envy-free (WEF), as well as ex-post weighted proportional up to one good (WPROP1) and weighted transfer envy-free up to one good (WEF(1,1)). It can be computed in strongly polynomial time. We show that this result is tight - ex-ante WEF is incompatible with any stronger ex-post WEF relaxation. In addition, we extend BoBW results on group fairness to entitlements and explore generalizations of our results to instances with more expressive valuation functions.

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arXiv2022

Maximizing Nash Social Welfare in 2-Value Instances: Delineating Tractability

We study the problem of allocating a set of indivisible goods among a set of agents with \emph{2-value additive valuations}. In this setting, each good is valued either $1$ or $p/q$, for some fixed co-prime numbers $p,q\in \mathbb{N}$ such that $1\leq q < p$. Our goal is to find an allocation maximizing the \emph{Nash social welfare} (\NSW), i.e., the geometric mean of the valuations of the agents. In this work, we give a complete characterization of polynomial-time tractability of \NSW\ maximization that solely depends on the values of $q$. We start by providing a rather simple polynomial-time algorithm to find a maximum \NSW\ allocation when the valuation functions are \emph{integral}, that is, $q=1$. We then exploit more involved techniques to get an algorithm producing a maximum \NSW\ allocation for the \emph{half-integral} case, that is, $q=2$. Finally, we show it is \classNP-hard to compute an allocation with maximum \NSW\ whenever $q\geq3$.

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arXiv2022

Equilibria and Convergence in Fire Sale Games

The complex interactions between algorithmic trading agents can have a severe influence on the functioning of our economy, as witnessed by recent banking crises and trading anomalies. A common phenomenon in these situations are fire sales, a contagious process of asset sales that trigger further sales. We study the existence and structure of equilibria in a game-theoretic model of fire sales. We prove that for a wide parameter range (e.g., convex price impact functions), equilibria exist and form a complete lattice. This is contrasted with a non-existence result for concave price impact functions. Moreover, we study the convergence of best-response dynamics towards equilibria when they exist. In general, best-response dynamics may cycle. However, in many settings they are guaranteed to converge to the socially optimal equilibrium when starting from a natural initial state. Moreover, we discuss a simplified variant of the dynamics that is less informationally demanding and converges to the same equilibria. We compare the dynamics in terms of convergence speed.

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arXiv2022

Seniorities and Minimal Clearing in Financial Network Games

Financial network games model payment incentives in the context of networked liabilities. In this paper, we advance the understanding of incentives in financial networks in two important directions: minimal clearing (arising, e.g., as a result of sequential execution of payments) and seniorities (i.e., priorities over debt contracts). We distinguish between priorities that are chosen endogenously or exogenously. For endogenous priorities and standard (maximal) clearing, the games exhibit a coalitional form of weak acyclicity. A strong equilibrium exists and can be reached after a polynomial number of deviations. Moreover, there is a strong equilibrium that is optimal for a wide variety of social welfare functions. In contrast, for minimal clearing there are games in which no optimal strategy profile exists, even for standard utilitarian social welfare. Perhaps surprisingly, a strong equilibrium still exists and, for a wide range of strategies, can be reached after a polynomial number of deviations. In contrast, for exogenous priorities, equilibria can be absent and equilibrium existence is NP-hard to decide, for both minimal and maximal clearing.

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arXiv2022

Public Signals in Network Congestion Games

We consider a largely untapped potential for the improvement of traffic networks that is rooted in the inherent uncertainty of travel times. Travel times are subject to stochastic uncertainty resulting from various parameters such as weather condition, occurrences of road works, or traffic accidents. Large mobility services have an informational advantage over single network users as they are able to learn traffic conditions from data. A benevolent mobility service may use this informational advantage in order to steer the traffic equilibrium into a favorable direction. The resulting optimization problem is a task commonly referred to as signaling or Bayesian persuasion. Previous work has shown that the underlying signaling problem can be NP-hard to approximate within any non-trivial bounds, even for affine cost functions with stochastic offsets. In contrast, we show that in this case, the signaling problem is easy for many networks. We tightly characterize the class of single-commodity networks, in which full information revelation is always an optimal signaling strategy. Moreover, we construct a reduction from optimal signaling to computing an optimal collection of support vectors for the Wardrop equilibrium. For two states, this insight can be used to compute an optimal signaling scheme. The algorithm runs in polynomial time whenever the number of different supports resulting from any signal distribution is bounded to a polynomial in the input size. Using a cell decomposition technique, we extend the approach to a polynomial-time algorithm for multi-commodity parallel link networks with a constant number of commodities, even when we have a constant number of different states of nature.

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arXiv2022

Delegated Online Search

In a delegation problem, a principal P with commitment power tries to pick one out of $n$ options. Each option is drawn independently from a known distribution. Instead of inspecting the options herself, P delegates the information acquisition to a rational and self-interested agent A. After inspection, A proposes one of the options, and P can accept or reject. Delegation is a classic setting in economic information design with many prominent applications, but the computational problems are only poorly understood. In this paper, we study a natural online variant of delegation, in which the agent searches through the options in an online fashion. For each option, he has to irrevocably decide if he wants to propose the current option or discard it, before seeing information on the next option(s). How can we design algorithms for P that approximate the utility of her best option in hindsight? We show that in general P can obtain a $\Theta(1/n)$-approximation and extend this result to ratios of $\Theta(k/n)$ in case (1) A has a lookahead of $k$ rounds, or (2) A can propose up to $k$ different options. We provide fine-grained bounds independent of $n$ based on two parameters. If the ratio of maximum and minimum utility for A is bounded by a factor $\alpha$, we obtain an $\Omega(\log \log \alpha / \log \alpha)$-approximation algorithm, and we show that this is best possible. Additionally, if P cannot distinguish options with the same value for herself, we show that ratios polynomial in $1/\alpha$ cannot be avoided. If the utilities of P and A for each option are related by a factor $\beta$, we obtain an $\Omega(1/ \log \beta)$-approximation, where $O(\log \log \beta/ \log \beta)$ is best possible.

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arXiv2022

Asynchronous Opinion Dynamics in Social Networks

Opinion spreading in a society decides the fate of elections, the success of products, and the impact of political or social movements. The model by Hegselmann and Krause is a well-known theoretical model to study such opinion formation processes in social networks. In contrast to many other theoretical models, it does not converge towards a situation where all agents agree on the same opinion. Instead, it assumes that people find an opinion reasonable if and only if it is close to their own. The system converges towards a stable situation where agents sharing the same opinion form a cluster, and agents in different clusters do not \mbox{influence each other.} We focus on the social variant of the Hegselmann-Krause model where agents are connected by a social network and their opinions evolve in an iterative process. When activated, an agent adopts the average of the opinions of its neighbors having a similar opinion. By this, the set of influencing neighbors of an agent may change over time. To the best of our knowledge, social Hegselmann-Krause systems with asynchronous opinion updates have only been studied with the complete graph as social network. We show that such opinion dynamics with random agent activation are guaranteed to converge for any social network. We provide an upper bound of $\mathcal{O}(n|E|^2 (\varepsilon/\delta)^2)$ on the expected number of opinion updates until convergence, where $|E|$ is the number of edges of the social network. For the complete social network we show a bound of $\mathcal{O}(n^3(n^2 + (\varepsilon/\delta)^2))$ that represents a major improvement over the previously best upper bound of $\mathcal{O}(n^9 (\varepsilon/\delta)^2)$. Our bounds are complemented by simulations that indicate asymptotically matching lower bounds.

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arXiv2021

Maximizing Nash Social Welfare in 2-Value Instances

We consider the problem of maximizing the Nash social welfare when allocating a set $\mathcal{G}$ of indivisible goods to a set $\mathcal{N}$ of agents. We study instances, in which all agents have 2-value additive valuations: The value of every agent $i \in \mathcal{N}$ for every good $j \in \mathcal{G}$ is $v_{ij} \in \{p,q\}$, for $p,q \in \mathbb{N}$, $p \le q$. Maybe surprisingly, we design an algorithm to compute an optimal allocation in polynomial time if $p$ divides $q$, i.e., when $p=1$ and $q \in \mathbb{N}$ after appropriate scaling. The problem is \classNP-hard whenever $p$ and $q$ are coprime and $p \ge 3$. In terms of approximation, we present positive and negative results for general $p$ and $q$. We show that our algorithm obtains an approximation ratio of at most 1.0345. Moreover, we prove that the problem is \classAPX-hard, with a lower bound of $1.000015$ achieved at $p/q = 4/5$.

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