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on Microeconomics |
| By: | Francesc Dilmé |
| Abstract: | A Nash equilibrium of a game in extensive form is a sequential equilibrium in mixed strategies if it can be approximated through equilibria of close-by games with slightly perturbed payoffs and small-probability behavioral types. We show that sequential equilibria in mixed strategies are equivalent to (i) weakly sequential equilibria (Reny, 1992), (ii) normal-form perfect equilibria (Selten, 1975) in games with generic payoffs, and (iii) purifiable Nash equilibria (Harsanyi, 1973). A corollary of our results is that extensive-form perfect equilibria are normal-form perfect equilibria in games with generic payoffs. |
| Keywords: | Sequential equilibria, mixed strategies, purification |
| JEL: | C72 C73 |
| Date: | 2025–10 |
| URL: | https://d.repec.org/n?u=RePEc:bon:boncrc:crctr224_2025_703 |
| By: | Mariann Ollár; Antonio Penta |
| Abstract: | We study a framework for robust mechanism design that can accommodate various degrees of robustness with respect to agents' beliefs, which encompasses both the belief-free and Bayesian settings as special cases. For general belief restrictions, we characterize the set of incentive compatible direct mechanisms in general environments with interdependent values. Our main results, which we obtain based on a first-order approach, inform the design of transfers via 'belief-based' terms to attain incentive compatibility. In environments that satisfy a property of generalized independence, our results imply a robust version of revenue equivalence in non-Bayesian settings. Instead, under a notion of comovement between types and beliefs, which extends the idea of correlated information to non-Bayesian settings, we show that any allocation rule can be implemented, even if standard single-crossing and monotonicity conditions do not hold. Yet, unless the environment is Bayesian, information rents typically remain, and they decrease monotonically as the robustness requirements are weakened. |
| Keywords: | belif restrictions, incentive compatibility, interdependent values, moment conditions, robust mechanism design |
| JEL: | D62 D82 D83 |
| Date: | 2025–09 |
| URL: | https://d.repec.org/n?u=RePEc:bge:wpaper:1513 |
| By: | Francesc Dilmé; Aaron Kolb |
| Abstract: | We revisit the classic chain-store paradox by introducing a novel element: the arrival of exogenous, public signals about the incumbent’s private type over time. As the horizon lengthens, two opposing forces come into play. On one hand, standard reputational incentives grow stronger; on the other, the increasing availability of information makes it more difficult to sustain a reputation. We show that full deterrence can still emerge as the horizon grows arbitrarily long, though not always, and we provide a complete characterization of the conditions under which it arises. |
| Keywords: | Entry deterrence, reputation, chain-store parardox |
| JEL: | C72 C73 |
| Date: | 2025–10 |
| URL: | https://d.repec.org/n?u=RePEc:bon:boncrc:crctr224_2025_704 |
| By: | Dong, Xiaoge (Center for Mathematical Economics, Bielefeld University) |
| Abstract: | We study “willful ignorance” - choosing not to learn whether a task is illegal - in a lawmaker-principal-agent game and characterize the penalty policies that implement welfare-maximizing behavior. The model delivers an implementability frontier : which equilibrium behaviors can exist and be selected by penalties. With perfect inquiry, this frontier is aligned with the welfare ordering, so the lawmaker can make the welfare-maximizing behavior both exist and be preferred by all parties. With imperfect inquiry, noise breaks that alignment and produces two failures: inquiry that is socially desirable may be infeasible at any penalty, and inquiry that is socially undesirable may persist because it cannot be switched off. We compare harm-based, compliance-based, and dual-penalty rules: harm-based rules preserve control but tightens feasibility; compliance-based rules relax feasibility but sacrifices control; dual penalty rules recover both levers subject to simple bounds. The framework yields practical guidance for calibrating penalties to harm, inquiry accuracy, and inquiry costs. It also implies that ignorance cannot serve as a shield: the absence of knowing crime in equilibrium is driven by incentives rather than morality, making non-inquiry the true strategic margin of liability design |
| Keywords: | willful ignorance, ostrich instruction, law and economics, asymmetric information. |
| Date: | 2025–09–26 |
| URL: | https://d.repec.org/n?u=RePEc:bie:wpaper:752 |
| By: | Heng-fu Zou |
| Abstract: | We analyze a continuum of risk-neutral agents working under a risk-neutral principal. Each agent's output depends on hidden effort and random shocks, while the principal observes both individual outcomes and their cross-sectional average. Agents value consumption linearly but face quadratic effort costs, with all parties discounting at a common rate. We derive the optimal contract in closed form. It consists of a fixed salary plus a relative-performance component that rewards an agent's outcome compared to the group average. This design preserves incentives, since no individual can influence the average, while filtering out common risks and transitory fluctuations. In the unique symmetric equilibrium, all agents exert constant efficient effort, and the fixed salary adjusts to ensure participation. Because of risk neutrality, the contract is independent of the level of randomness. |
| Keywords: | principal-agent problem; mean field games; contract theory; relative performance evaluation; optimal incentives; symmetric equilibrium; risk neutrality |
| Date: | 2025–09–12 |
| URL: | https://d.repec.org/n?u=RePEc:cuf:wpaper:789 |
| By: | Kevin He (University of Pennsylvania); Jonathan Libgober (University of Southern California) |
| Abstract: | We extend the indirect evolutionary approach to the selection of (possibly misspecified) models. Agents with different models match in pairs to play a stage game, where models define feasible beliefs about game parameters and about others’ strategies. In equilibrium, each agent adopts the feasible belief that best fits their data and plays optimally given their beliefs. We define the stability of the resident model by comparing its equilibrium payoff with that of the entrant model, and provide conditions under which the correctly specified resident model can only be destabilized by misspecified entrant models that contain multiple feasible beliefs (that is, entrant models that permit inference). We also show that entrants may do well in their matches against the residents only when the entrant population is large, due to the endogeneity of misspecified beliefs. Applications include the selection of demand-elasticity misperception in Cournot duopoly and the emergence of analogy-based reasoning in centipede games. |
| Keywords: | misspecified Bayesian learning, endogenous misspecifications, evolutionary stability, analogy classes |
| Date: | 2025–09–19 |
| URL: | https://d.repec.org/n?u=RePEc:pen:papers:25-020 |
| By: | Francesc Dilmé |
| Abstract: | An equilibrium outcome of a game in extensive form is fully self-justifiable if it is supported by justifiable equilibria (McLennan, 1985) regardless of the order in which actions implausible under the given outcome are excluded. We show that the set of fully self-justifiable outcomes is non-empty and contains the set of sequentially stable outcomes (Dilmé, 2024). In signaling games, fully self-justifiable outcomes pass all the selection criteria in Cho and Kreps (1987). Full self-justifiability allows for the systematic use of the logic of selection criteria in signaling games to select equilibria in any finite extensive form game. |
| Keywords: | Justfiable equilibira, sequentially stable outcomes |
| JEL: | C72 C73 |
| Date: | 2025–10 |
| URL: | https://d.repec.org/n?u=RePEc:bon:boncrc:crctr224_2025_702 |
| By: | Krishna Dasaratha (Boston University); Kevin He (University of Pennsylvania) |
| Abstract: | We study learning on social media with an equilibrium model of users interacting with shared news stories. Rational users arrive sequentially, observe an original story (i.e., a private signal) and a sample of predecessors’ stories in a news feed, then decide which stories to share. The observed sample of stories is jointly determined by predecessors’ sharing behavior and the sampling algorithm generating news feeds. We focus on how often this algorithm selects more viral (i.e., widely shared) stories. Showing users viral stories can increase information aggregation, but it can also generate steady states where most shared stories are wrong. These misleading steady states self-perpetuate, as users who observe wrong stories develop wrong beliefs, and thus rationally continue to share them. Finally, we describe several consequences for platform design and robustness. |
| Keywords: | social learning, selective equilibrium sharing, social media, platform design, endogenous virality |
| Date: | 2025–07–31 |
| URL: | https://d.repec.org/n?u=RePEc:pen:papers:25-021 |
| By: | Heng-fu Zou |
| Abstract: | This paper develops a continuous-time model of contracting between a principal and many agents under moral hazard, linking dynamic contract theory to the mean field game(MFG) framework. Outputs depend on effort, competition, and both idiosyncratic and common shocks. The principal designs contracts using filtered signals that benchmark agents against peers. We show that the first-order approach remains valid: effort is linear in exposure with slope determined by the informativeness of fltered outputs. The Hamilton-Jacobi-Bellman equation implies that optimal incentives depend jointly on marginal revenue, signal variance, and intertemporal insurance. The optimal filter is the generalized least squares residual, which asymptotically eliminates common shocks as group size grows. In a linear-quadratic specialization, equilibrium exists, is unique, and is globally stable, with the uniqueness result coinciding with the Lasry-Lions MFG fixed-point condition. The framework unifies tournaments, RPE, and dynamic contracts, and explains empirical puzzles in executive pay, sales teams, and finance. |
| Keywords: | Dynamic contracts; Principal-agent problem; Multi-agent incentives; Relative performance evaluation; Tournaments; Mean field games; Hamilton-Jacobi-Bellman equation; Fokker-Planck dynamics; Incentives and insurance; Executive compensation; Organizational design |
| Date: | 2025–09–10 |
| URL: | https://d.repec.org/n?u=RePEc:cuf:wpaper:790 |
| By: | Max Riegel |
| Abstract: | I study how consumer preferences for one-stop-shopping affect the resource allocation of multiproduct firms. In many markets, consumers prefer to buy multiple products at a single seller. At the same time, multiproduct firms have to split their resources between their products, for instance by allocating R&D budgets or shelf space. One-stop-shoppers treat firms’ products as if offered as bundles. If there are only one-stop-shoppers, firms therefore cannot gain from a comparative advantage over a particular product. I show that firms become more inclined to specialize in different products and gain market power, as the share of one-stop-shoppers decreases. Thereby, industry profits rise, potentially to the detriment of consumers. |
| Keywords: | one-stop-shopping, specialization, multiproduct competition |
| JEL: | D43 L13 L81 |
| Date: | 2025–09 |
| URL: | https://d.repec.org/n?u=RePEc:bon:boncrc:crctr224_2025_701 |
| By: | Ganglmair, Bernhard; Klix, Julian; Shin, Dongsoo |
| Abstract: | Many business relationships begin with informal interactions and later transition to formal contracts. Using a repeated-games model with a finite horizon, we show that this hybrid-contracting approach can both prolong cooperation (intensive margin) and enable it across a broader range of settings (extensive margin). We model the contract as a "smooth-landing contract" that limits actions only near the end of the relationship. We show that this flexible design supports early cooperation and outperforms rigid contracts. Our findings are robust to changes in contracting costs and timing, with optimal contract length balancing profitability and implementability. |
| Keywords: | cooperation, hybrid contracting, relational contracts, repeated games, strategic alliances |
| JEL: | C73 D86 K12 L14 |
| Date: | 2025 |
| URL: | https://d.repec.org/n?u=RePEc:zbw:zewdip:327114 |
| By: | Heng-fu Zou |
| Abstract: | This paper develops a dynamic theory of contracting with one principal and many agents, extending the promised-utility framework of Sannikov (2008) to environments with competition, correlated outputs, and relative performance evaluation (RPE). We show that the first-order approach remains valid when the principal filters performance through generalized least squares (GLS) residuals of peer outputs, which minimize variance while preserving incentive slopes. The principal's Hamilton- Jacobi-Bellman equation links optimal effort to three forces: marginal revenue, signal informativeness, and intertemporal insurance. In a linear-quadratic environment, equilibrium effort is uniquely determined by a contraction mapping, resolving the multiplicity of equilibria found in static models such as Mookherjee (1984). We further unify Lazear-Rosen (1981) tournaments and linear RPE contracts as diffusion-driven special cases - Gaussian, CIR, and lognormal - within a mean field game system. The framework rationalizes empirical puzzles in executive pay, sales contests, and financial benchmarking, and generates new testable predictions about incentives and organizational scale. |
| Keywords: | Dynamic contracts; Principal-agent problem; Multi-agent incentives; Relative performance evaluation; Tournaments; Mean field games; Hamilton-Jacobi-Bellman equation; Fokker-Planck dynamics; Incentives and insurance; Executive compensation; Organizational design |
| Date: | 2025–09–19 |
| URL: | https://d.repec.org/n?u=RePEc:cuf:wpaper:791 |
| By: | Anna Bogomolnaia (University of Glasgow, CES - Centre d'économie de la Sorbonne - UP1 - Université Paris 1 Panthéon-Sorbonne - CNRS - Centre National de la Recherche Scientifique); Hervé Moulin (University of Glasgow, Higher School of Economics [Saint-Pétersbourg]) |
| Abstract: | We divide efficiently a pile of indivisible goods in common property, using cash transfers to ensure fairness among agents with utility linear in money. We compare three cognitively feasible and privacy preserving division rules in terms of the guarantees (worst case utility) they offer to the participants. In the first version of Divide & Choose to n agents, they bid for the role of Divider then everyone bids on the shares of the Divider's partition. In the second version each agent announces a partition and they all bid to select the most efficient one. In the Bid & Sell rule the agents bid for the role of Seller: with two agents the smallest bid defines the Seller who then charges any price constrained only by her winning bid. Both rules reward subadditive utilities and penalise superadditive ones, and B&S more so than both D&C-s. B&S is also better placed to collect a larger share of the surplus when agents play safe. |
| Keywords: | Bid and Sell, Divide &, Choose, worst case, guarantees, safe play |
| Date: | 2025 |
| URL: | https://d.repec.org/n?u=RePEc:hal:journl:hal-05139641 |
| By: | Anna Bogomolnaia (University of Glasgow, CES - Centre d'économie de la Sorbonne - UP1 - Université Paris 1 Panthéon-Sorbonne - CNRS - Centre National de la Recherche Scientifique); Hervé Moulin (University of Glasgow, Higher School of Economics [Saint-Pétersbourg]) |
| Abstract: | We divide efficiently a pile of indivisible goods in common property, using cash transfers to ensure fairness among agents with utility linear in money. We compare three cognitively feasible and privacy preserving division rules in terms of the guarantees (worst case utility) they offer to the participants. In the first version of Divide & Choose to n agents, they bid for the role of Divider then everyone bids on the shares of the Divider's partition. In the second version each agent announces a partition and they all bid to select the most efficient one. In the Bid & Sell rule the agents bid for the role of Seller: with two agents the smallest bid defines the Seller who then charges any price constrained only by her winning bid. Both rules reward subadditive utilities and penalise superadditive ones, and B&S more so than both D&C-s. B&S is also better placed to collect a larger share of the surplus when agents play safe. |
| Keywords: | Bid and Sell, Divide &, Choose, worst case, guarantees, safe play |
| Date: | 2025 |
| URL: | https://d.repec.org/n?u=RePEc:hal:cesptp:hal-05139641 |
| By: | Kevin He (University of Pennsylvania); Jonathan Libgober (University of Southern California) |
| Abstract: | We study misperceptions of private-signal correlation in an incomplete-information Cournot duopoly game. Exaggerating the correlation between players’ demand signals is beneficial when agents hold flexible beliefs about price elasticity, but harmful when their beliefs are dogmatically correct. For agents with flexible beliefs who learn elasticity by observing prices, correlation misperceptions indirectly distort behavior through elasticity misinference. If agents know the true elasticity, this learning channel is eliminated. Correlation misperceptions have opposite direct and indirect effects on behavior, so the presence of elasticity inference can reverse an error’s viability. |
| Date: | 2025–08–27 |
| URL: | https://d.repec.org/n?u=RePEc:pen:papers:25-018 |
| By: | Bouvard, Matthieu; Jullien, Bruno; Martimort, David |
| Date: | 2025 |
| URL: | https://d.repec.org/n?u=RePEc:tse:wpaper:130945 |