Papers

We study whether model diversity survives selection into trading. In synthetic markets with a fixed mixture of three language-model families, news presentation changes their representation among submitted orders. At the announcement round, Qwen's share of submitted orders shifts by 48 percentage points in the financing event, with little change in net order counts. In the workforce-reduction event, Mistral's share shifts by 40 percentage points while net counts reverse sign. Homogeneous populations remove opposing flow when their active decisions share a direction. An analytical decomposition shows why selection can improve or worsen price accuracy even at unchanged aggregate demand sensitivity. The evidence concerns presentation bundles and submitted flow; cleaner replication and a known-value validation are specified prospectively.

The article reviews the robust design and analysis of auctions, focusing on the predominant paradigm in the computer science literature, which quantifies robustness by the worst-case, over environments, ratio between the robust auction's performance and that of an optimal auction for the environment. Applications include robust analysis of canonical auctions such as the first-price auction for the welfare objective, prior-robust auction design for the revenue objective, and prior-robust design of exchange mechanisms for the gains-from-trade objective. Finally, the revelation principle is not without loss in robust mechanism design. The potential loss can be quantified by the revelation gap, the fraction of the optimal robust guarantee that is lost when restricting from all mechanisms to revelation mechanisms.

Refreshment samples are the standard remedy for panel attrition, and the identification results behind them maintain that participation does not change measurement. We characterize what a refreshment sample identifies about panel conditioning, modelled as a deterministic monotone map at reinterview, when attrition is unrestricted. A candidate map is consistent with the data if and only if the retention-scaled distribution of the stayers' implied latent outcomes is setwise dominated by the refreshment distribution; every such map is rationalized by an explicit attrition process. Without attrition the map is identified on the latent-outcome support; with attrition, a density-ratio condition governs the identified set, and tail behaviour alone does not determine it. For an unrestricted map the survivors' mean effect has the familiar trimming bounds; for an item with all categories reported, the model reduces to a test of no conditioning. Within a cohort, other waves, dropout patterns and entry-wave items leave the set unchanged unless restrictions link selection across waves. Under explicit selection restrictions, survival matching, symmetric matching and entry-wave correction identify survivor effects. We derive their biases and give rank conditions under which refreshment schedules identify curvature in the conditioning path. A Japanese panel illustrates the results.

The increasing volatility of electricity prices driven by renewable energy integration, market shocks, and regulatory changes has reinforced the need for forecasting methods that go beyond point predictions and accurately describe the full conditional price distribution. This paper applies the Generalised Additive Models for Location, Scale and Shape (GAMLSS) framework to forecast Spanish day-ahead electricity prices using hourly data from 2020 to 2024. Alternative specifications based on Normal, Johnson's SU (JSU), and Sinh-Arcsinh (SHASH) distributions are considered, allowing the location, scale, and shape parameters to vary with market fundamentals, including electricity demand, renewable generation, seasonal effects, and regulatory and geopolitical risk factors. Forecasts are generated using a rolling-window approach and evaluated through the mean absolute error (MAE), pinball loss, and Diebold-Mariano tests. The results show that flexible distributional specifications improve forecasting performance relative to a naive benchmark and the standard normal specification. While SHASH and JSU specifications provide the lowest point forecasting errors, the hourly analysis reveals substantial intraday variation in relative performance across specifications. JSU specification with all four parameters driven by covariates achieves the best probabilistic forecasting performance, particularly in the tails of the distribution. Diebold-Mariano tests confirm the statistical significance of these improvements. These findings highlight the importance of modelling time-varying shape distributional parameters and demonstrate the value of GAMLSS models for forecasting and risk management in increasingly volatile electricity markets.

We develop the first empirically implementable selective-inference framework for difference-in-differences with controls selected by pre-treatment fit. We construct confidence intervals for the average treatment effect on the treated (ATT), incorporating that selection and estimating the unknown full covariance from the same observations. Counterexamples establish undercoverage from Roth's fixed-comparison pretest adjustment and direct Gaussian-score application of Andrews–Kitagawa–McCloskey to this selection, despite parallel trends for every candidate. Retaining the actual polynomial or piecewise-polynomial event, we establish uniform validity at perfect fit and population ties, endpoint equivalence with known-covariance inference, and unequal-sample guarantees. We accommodate staggered adoption, reused observations, and fitting-rule choice. In police-deployment and minimum-wage reanalyses, we find changes in significance and economically relevant effect ranges while retaining the selected comparison. For a smooth randomized test, we obtain higher-order accuracy through stabilized joint mean–covariance correction preserving normal-sampling baseline rejection probabilities. Researchers should disclose selection rules alongside ATT uncertainty.

We study an attribute-based model of stochastic choice in which attention to each attribute is limited, and characterize two choice rules within it. The Multiplicative Attention Rule (MAR) rewards an alternative only for the attributes on which it ranks first, combining attention to these multiplicatively across attributes; it captures choice deferral driven by the cognitive load of attending to `too many' attributes. The Additive Attention Rule (AAR) instead averages attention across attributes and credits an alternative by its full ordinal rank within the menu. This difference in aggregation is what separates the two rules behaviourally: AAR reproduces the Attraction Effect while holding the relevant attributes fixed across menus, something MAR can do only when the attribute set itself changes, though AAR, in turn, cannot accommodate the Compromise Effect. Both rules are characterized by axioms on observable choice data, and in both cases the underlying attention parameters and attribute-based preferences are uniquely identified from that data.

In global metropolitan areas, public transport benefits from bus systems. Bus design widely applies theoretical models, which typically assume static ridership. However, ridership randomness exists, and lack of attention to it might lead to wrong design decisions. Moving beyond static ridership, we extend the traditional single-line model with stochastic ridership. Further, to optimize the bus system under various ridership randomness, we introduce a hybrid model that combines conventional buses (CBs) and flexible buses (FBs). In both models, we apply a continuous approximation approach and find that: 1) Bus capacity increases with ridership randomness in the extended single-line model; 2) The hybrid model exhibits a binary state: either CBs serve the ridership with rejections under low ridership randomness, or CBs serve the majority while FBs serve the minority with rejections under high ridership randomness. Our findings establish a link between bus system design and ridership randomness, contributing to a more adaptive and efficient public transport framework.

Machine learning predicts outcomes well, but predictive accuracy does not ensure reliable counterfactual responses. We examine how to combine machine learning and theory for measurement and counterfactual analysis, using maritime CO2 emissions where physics provides a benchmark speed response. Matching hourly tracking data for dry bulk and container ships to annual fuel consumption reported under EU regulations, we compare engineering calculations, structural regressions, hybrid models, and machine learning. Out of sample, all estimated models predict within a few percent of reported totals, outperforming standard engineering calculations. Yet pure machine learning and unrestricted structural regression imply attenuated speed responses. Hybrids preserve the structural component's speed response by excluding speed-related inputs from the machine learning component. A cost-benefit analysis of speed reductions illustrates the policy stakes: an attenuated speed response can flip the sign of net benefits. Accurate aggregate predictions therefore cannot substitute for scrutiny of the restrictions determining counterfactual responses.

This paper examines the post-2010 expansion of the Hungarian construction sector as a case study of large-scale government intervention in a single market. A political economy framework conceptualises the intervention as two mutually reinforcing policies - expansionary demand (public procurement and housing subsidies) and preferential allocation towards politically connected firms - operating under inelastic supply through two mechanisms: market-level subsidy capitalisation and firm-level rent extraction. Using a structured descriptive case study design, the paper draws on Hungarian Central Statistical Office and Eurostat macroeconomic statistics, firm-level data on Hungary's largest companies (CrefoPort), policy documents, and a regional comparison to assess four propositions derived from the framework. The evidence supports all four: real construction output grew faster than in any regional peer except Romania; construction producer prices persistently outpaced the GDP deflator; ownership of the leading firms shifted towards politically connected domestic owners, whose profitability rose sharply while their exports remained negligible; labour productivity per hour worked returned by 2024 to roughly its 2000 level; and connected firms captured a rising share of sector profits, as the subsidies' value was partly eroded by the price increases they induced. Competing explanations do not jointly account for this configuration. The findings illustrate how combined demand stimulus and favouritism produce extensive growth without efficiency gains.

We study the measure of restrictiveness proposed in Fudenberg, Gao, and Liang (2026) to evaluate complexity of economic models using synthetic data. We show that Rademacher complexity is an affine transformation of a particular case of a consistent finite-sample estimate of restrictiveness. Our results show that restrictiveness inherits a cardinal interpretation as a bound on generalization error while avoiding the inability of limiting Rademacher complexity to distinguish between some falsifiable models.

Artificial-intelligence weather prediction (AIWP) models have made it possible to produce high-quality tailored forecasts with limited computational resources. This advance has the potential to benefit hundreds of millions of farmers in low- and middle-income countries who lack access to forecasts of critical weather phenomena. However, it can be difficult for key stakeholders to evaluate forecast quality, risking a "race to the bottom" as cheap but low-quality forecasts crowd out forecasts that would benefit farmers. We propose a set of principles and protocols for evaluating agriculturally-relevant forecasts as a starting point for standards that would let forecasters credibly convey their forecasts' quality.

We study the anatomy of commodity risk by distinguishing micro, market-level, and economy-wide sources. We develop a two-stage "divide-and-conquer" framework that allows sensitivities to these risk sources to vary across commodities while treating economy-wide risk as latent. The first stage uses defactored instrumental-variable estimation to recover commodity-specific sensitivities to micro and market conditions. The second combines principal components with high-dimensional variable selection to identify an observable representation of macro-financial risk. We then construct Risk Intensity Indices (RIIs), which combine estimated sensitivities with prevailing risk conditions to quantify the relative importance of each risk source on a common scale. Market risk is the largest component on average, accounting for about two fifths of total risk intensity and more than half for energy commodities. Risk intensity is also highly concentrated across individual commodities: the top 20% account for approximately half of micro and market risk intensity, whereas macro risk is more broadly dispersed. The composition of risk varies substantially across sectors and over time, with market risk becoming particularly prominent during episodes of commodity-market stress. Micro and market RIIs also contain information about future volatility and absolute returns. These findings provide investors, risk managers, and policymakers with a diagnostic of where commodity risk is concentrated, which risk layers are most important, and how their importance changes over time. More broadly, our divide-and-conquer framework provides a flexible approach to decomposing layered risk in settings where common risk is latent.

In many statistical settings, the available data and maintained assumptions do not suffice to uniquely identify the model parameters of interest. In such cases, one can only identify sets which are guaranteed to contain the true parameters. These are often characterized through linear programs that optimize over models compatible with the observed data. These programs can be infinite-dimensional in the optimizer and the number of constraints. We provide a unified way to characterize and solve such optimization problems by phrasing them as optimal transport problems on path spaces. This allows us to regularize the problem with an entropy penalty, recasting it as a multi-marginal entropic optimal transport problem, which can be solved efficiently via Sinkhorn iterations. In addition, it allows us to establish convergence of the regularized value to the sharpest bound, derive consistency rates for a plug-in estimator, and obtain asymptotic distribution for approximate bounds. The method is general and accommodates settings ranging from instrumental variable models with continuous variables to welfare estimation in heterogeneous demand models. We verify the statistical and computational properties in simulations and provide an application to demand estimation.

Veterans face multiple hardships after they leave the military. Is food insecurity one of these hardships? This paper examines the long-term effects of military service on food insecurity using new data from the PSID. Military veteran-headed households experienced lower food insecurity than non-veteran-headed households, and participated less in federal nutrition assistance programs. We use the Vietnam-era lottery draft to identify causal effects, and find no evidence that military service adversely affects food insecurity. Our results suggest policies that assist currently enlisted service members, and veterans unlikely to head a traditional household, may most effectively tackle food insecurity among military veterans.

Many school choice mechanisms give students higher priority at schools they rank highly and aim to reward students' eagerness. This paper studies whether a centralized mechanism can incorporate such rank-dependent priorities while preserving truthful reporting. I consider a mechanism that jointly specifies how reported preferences modify schools' original priorities and how assignments are determined based on the modified priorities. I require the modification rule to satisfy rank monotonicity and the resulting matching to be stable with respect to the modified priorities. These two requirements jointly capture the policy objective of rewarding eagerness. My main result shows that any mechanism satisfying these requirements as well as strategy-proofness is outcome equivalent to the student-proposing Deferred Acceptance mechanism using the original priorities. Thus, if eagerness affects the outcome while the rank-dependent policy goal is maintained, the mechanism must be manipulable.

In a recent study, Ekici and Yenmez (2026) characterize top trading cycles (TTC) on the unrestricted domain by strategy-proofness, individual rationality, and pair rationality, uncovering a correspondence between the axiomatic foundations of TTC and deferred acceptance, matching theory's two canonical rules. We show that their characterization fails on the single-peaked domain, but it holds on the single-dipped domain even without strategy-proofness. They also show that pair rationality can be replaced in their characterization by two weaker conditions concerning agents' first and second choices. In contrast, this replacement fails on the single-dipped domain, even when strategy-proofness is imposed.

An advisor knows who would benefit from using a resource but cannot assign its use. Individuals have private assessments and may ignore advice. We study how their choices can respect a fixed usage limit. In our model, whenever coordination is possible, eliciting private assessments makes some harmful use unavoidable. Policies that maximize welfare, or even minimize harm, also forgo beneficial uses that direct assignment would permit. In those environments, observing assessments instead allows advice to achieve as much beneficial use as direct assignment, without harmful use. The loss therefore reflects the incentive cost of eliciting private beliefs while leaving use voluntary.

How does an inherited cross-type wage gap enter Markov-perfect discretionary monetary policy when transfers are passive? In a two-agent New Keynesian model with sticky prices and type-specific own-lag wage adjustment, the gap changes implementable allocations and the second-order welfare loss. A positive lower bound establishes its value relevance; explicit rank conditions characterize when current price inflation, wage inflation, and the output gap fail to determine the implementing nominal rate. An illustrative parameterization satisfies these conditions, although the additional state explains little nominal-rate variance after conditioning on all three aggregate variables. Welfare comparisons with fixed rules are driven mainly by aggregate wage-inflation stabilization and do not isolate the value of distributional information. Unrestricted targeted transfers separate aggregate allocation from the legacy wage-gap transition. In the CES-consistent distribution block, optimal smoothing eliminates consumption dispersion, improves on immediate wage-gap elimination, and coincides under discretion and date-0 commitment.

This paper develops a framework for estimation and inference on the volumes of sets that are projections of critical function sets, focusing particularly on the convex body beneath the optimal receiver operating characteristic (ROC) surface. Specifically, we propose a volume calculation method that first uses an Aumann expectation representation and then applies Minkowski mixed volumes. Using this framework, we show that the population volume under the ROC surface (VUS) is proportional to the expectation of a symmetric U-statistic kernel. We then propose a double/debiased machine learning estimator of the VUS, derive its asymptotic properties, and develop an inference procedure. Further applications of this framework include an analysis of the feasible error set across pre-defined groups and a natural generalization of the Gini coefficient for measuring inequality.

Imputation-based causal estimation is typically viewed as relying exclusively on an outcome model, in contrast to augmented inverse-probability weighting, whose consistency is protected by fitting two nuisance models. This paper argues that this view can be misleading by highlighting a hidden dual weighting structure in least-squares sieve regression imputation. Although only outcome regressions are explicitly fitted, the resulting imputation estimator admits an exact weighting representation whose induced weights balance every function in the sieve space and the corresponding population weighting functions are the $L^2$ projections of the inverse propensity scores onto the same sieve space. This projection structure yields an implicit form of double robustness and, under standard sieve approximation and growth conditions, asymptotic linearity with the efficient influence function. Thus, weighting, covariate balance, double robustness, and semiparametric efficiency can all emerge from imputation alone through the geometry of least-squares projection.

Global games select a unique cutoff when a player's incentive moves monotonically with private information. We show that this logic can fail when a privately informed participant can take a costly, simultaneous, and unobserved action that defends the status quo. In a currency-attack model, the defender acts only at intermediate signals. Along the traders' cutoff equation, this interval response creates an endogenous defense-loss term that vanishes at the endpoints of an activation component but can dominate the decreasing benchmark payoff margin in between. We establish a primitive region in which defense is inactive and uniqueness survives, and a finite-CDF overshoot condition under which two active-defense cutoff roots coexist with the inactive benchmark root. Because marginal indifference need not imply global incentives, we also give sufficient conditions for a genuine monotone cutoff best response. Finally, outward-rounded interval arithmetic proves that a proper-prior parameter vector has exactly three symmetric monotone-cutoff equilibria and that this topology persists locally. The result isolates a payoff-based boundary of global-game selection: hidden intervention can restore multiplicity without public signalling or a perturbation of the information hierarchy.

Organizations often learn about competence only from project-level success or failure, even when a project contains several complementary tasks. We compare task architectures by the Blackwell order. An expert of unknown fixed competence can perform all tasks in one bundled project, one task in a project completed by an outside technology, or the same number of tasks across separate projects. Bundling dominates a single narrow assignment below an outside-reliability threshold and is otherwise incomparable with it. Holding the expert's workload fixed strictly lowers this threshold but does not overturn the result: bundling still dominates when the outside technology is sufficiently unreliable, separate projects dominate only when that technology is perfect, and the experiments are otherwise incomparable. We derive the thresholds for any number of tasks and show that the fixed-workload threshold decreases to zero as task scope grows. Explicit posterior-variance formulas measure the cost of coarse aggregation for particular decisions. Finally, in the two-task case, occasional stage-level audits expand the bundling-dominance region according to an exact frontier.

Xiao, Qiu, and Huang (AAMAS 2023) and independently Lonc (personal communication) asked whether indivisible chores located at the vertices of a tree can always be allocated to $n$ agents in connected bundles so that the cost of every agent is at most its connected maximin share; for goods, this is a theorem of Bouveret, Cechlárová, Elkind, Igarashi, and Peters. We answer the question affirmatively, even for monotone costs. The answer follows from a combinatorial theorem: if $\mathcal P_1,\ldots,\mathcal P_k$ are partitions of the vertex set of a finite tree, each into at most $k$ connected parts, then the vertex set can be split into disjoint sets $B_1,\ldots,B_k$, some possibly empty, such that each nonempty $B_i$ is connected and contained in a part of $\mathcal P_i$. Equivalently, if each of $k$ colours occurs on at most $k-1$ edges of a tree, then the vertices can be partitioned into connected sets labelled by distinct colours, none containing an edge of its own colour; in particular, one can choose for every colour a component of the forest obtained by deleting that colour so that the chosen components cover the tree. The bound is already best possible for paths, and for additive costs, the theorem is equivalent to the fair-division statement. The proof reduces the problem to inward partitions of oriented trees, which we obtain from the colourful KKM theorem on a simplex of edge weights, using a leaf-elimination labelling that remains compatible when weights vanish. We also give an algorithm running in time $k^{O(k)}$ plus polynomial time.

This paper develops a dynamic production-network model in which monetary non-neutrality emerges under fully flexible prices. Firms finance intermediate-input purchases from current nominal balances, and intermediate goods arrive with heterogeneous lead times across suppliers. A monetary shock then has two distinct effects. The first is miscoordination: because production uses dated input bundles assembled under past nominal conditions, the shock disturbs the composition of usable inputs (within and across firms) and thereby generates a deadweight loss. The second is reallocation: because the incidence of the shock is tilted toward smaller firms, which are disproportionately downstream, the shock redistributes resources along the supply chain. More specifically, positive shocks push resources downstream, negative shocks pull them upstream. This means that under a negative shock, both the miscoordination and the reallocation effect reduce downstream output and therefore GDP. Under a positive shock, reallocation raises GDP while miscoordination reduces it. The money-output relation is therefore sign-asymmetric: negative shocks generate sharp contractions, while positive shocks generate at most mild expansions, and sometimes none. Computational experiments on a reconstructed US production network generate output declines after a monetary contraction that are about three times as large as the increases after a monetary expansion.

Human verification depends on expertise that must be maintained before it is needed. This paper links reliance on automated checks, investment in human checking ability, and performance during an interruption. Better checking lowers the error reduction gained from an extra unit of human skill while the checker works. It can therefore reduce the incentive to preserve independent expertise, even when it lowers the best achievable expected cost of maintenance and errors. In an illustration, a more informative checker raises detection while it works from \hoPowWorkA{} to \hoPowWorkC{} percent, but the organization then keeps no routine practice and detects \hoPowOnsetC{} percent of errors when the checker first fails, against \hoPowOnsetA{} percent with a less informative checker. A detection requirement therefore concerns both current capability and its survival until new training becomes effective. Evidence from colonoscopy and aviation documents weaker unaided performance under routine automation, without isolating the mechanism. The framework connects a detection target to an explicit reserve of expertise and a training pipeline. Standard detection tests estimate each quantity and reveal automation bias and silent checker failures. The framework distinguishes the requirement from minimizing expected loss and proposes a longitudinal test.

Oversight changes the evidence it relies on. We ask when randomized audits and scoring align AI agents that can conceal misconduct and alter records. Stronger auditing makes undeterred violations better hidden. Because the provider writes the agent's objective, sanctions need not stop at forfeiture, and rare audits deter every type of agent if evidence survives concealment and audit draws cannot be learned in advance. When evidence can be erased, deterrence must come from lower gains from violation, such as credit for stopping, or from costlier or fewer ways to conceal. These conditions identify what failed when agents in OpenAI's cybersecurity evaluations compromised parts of Hugging Face's infrastructure in July 2026.

Network reconstruction from partial information is usually performed at the same resolution level at which constraints are observable. This becomes problematic when only coarse-grained information is available, while the relevant process occurs at a finer scale. Here we employ the multiscale model of weighted networks introduced in a companion paper and turn it into a probabilistic framework for reconstructing weighted networks across arbitrary aggregation levels. The model is built to preserve its functional form under coarse-graining, so that global parameters calibrated on an observable aggregate layer can be transferred to finer layers without refitting. We test the method on two empirical systems with different aggregation mechanisms. In the International Trade Network, countries are aggregated into geographic macro-regions and the observed coarse-grained layer is used to infer the underlying country-level network. In the Dutch production network, sectoral flows are reconstructed across the hierarchical industrial classification, using coarser sectoral layers to infer finer ones. In both geographical and sectoral settings, we benchmark our genuinely multiscale reconstruction method against a state-of-the-art weighted reconstruction model calibrated directly at the target resolution, and therefore using additional information available at the same (finer) scale at which performance is evaluated. Remarkably, despite this informational disadvantage, our method recovers the fine-scale binary structure with high accuracy, improving precision, specificity, accuracy and maximum degree-error diagnostics, while remaining nearly equivalent in sensitivity.

Firms follow changing economic opportunities, but rivalry changes their response. We develop ESCAPE, a rational-share game of distribution-valued positioning with heterogeneous capability costs, establish a unique equilibrium, and derive an exact reallocation restriction separating opportunity and crowding contributions. Competition need not push firms apart: rational firms can enter increasingly crowded regions, because the full reallocation gain can be positive even when its crowding component is negative. Rivalry instead changes how differently firms respond to the same opportunity shift. In the benchmark many-firm economy, the relative-response component accounts for 90.5% of the weighted squared composition-response contrast between strategic and independent firms; it is numerically zero under identical capabilities. The contrast remains distinguishable under corpus-scale measurement. Corporate news distributions document persistent but changing peer structure: optimized peer reconstructions retain 86.5% of their aggregate validation advantage one year later even as absolute distance to both frozen reconstructions increases. Peer structure is persistent without being a fixed destination, so current similarity can remain informative without fixing firms' future relative positions.

We study estimation and inference for a partially identified parameter whose identified set depends on a first-stage nuisance parameter that must itself be estimated. Combining the criterion-function approach with the theory of Neyman-orthogonal moments that underlies double/debiased machine learning, we propose a two-step procedure: the point-identified nuisance is estimated by flexible machine-learning methods, and the set-identified target is recovered as a level set of a sample criterion built from orthogonal moment inequalities with cross-fitting. When the contour level is bounded, we show that the resulting set estimator converges in Hausdorff distance at the parametric rate of the infeasible criterion built on the true nuisance. We further develop a subsampling procedure that delivers asymptotically valid coverage, provided the product of the first-stage estimation errors is $o(N^{-1/2})$. We illustrate the method on bounds for the wage distribution and the interquantile range under selection into employment and on the gender wage gap with an interval-censored wage. The empirical application studies the gender wage gap using the March supplement of the 2015 Current Population Survey.

We create a set of 6 axioms relating to strategic voting and study them on single transferable vote (STV), Meek STV, Bloc, Chamberlin-Courant, Monroe, and k-Borda. For Chamberlin-Courant, Monroe, and k-Borda, we implement “optimistic” and “pessimistic” models for handing incomplete ballots, thus turning each of the three into two separate voting rules. We determine which voting rules satisfy each axiom. We find that under Chamberlin-Courant and k-Borda, the optimistic models encourage burying, while the pessimistic models encourage truncation strategies, indicating that the handling of partial ballots has tremendous implications for strategic voting. We define an axiom called no skipped payment that differentiates similar rules, as Meek STV and Chamberlin-Courant satisfy it, while STV and Monroe fail. Each voting rule was found to fail favorite betrayal, revealing that none of them are completely immune to strategic voting. In general, STV variants are relatively resistant to strategic voting while Monroe is susceptible to a wide range of strategies.