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on Game Theory |
| By: | Emiliano Catonini; Antonio Penta |
| Abstract: | Backward Induction is only defined for games with perfect information, but its logic is also invoked in many equilibrium concepts for games with imperfect or incomplete information. Yet, the meaning of ‘backward induction reasoning’ is unclear in these settings, and we lack a way to apply its simple logic to general games. We remedy this by introducing a solution concept, Backwards Rationalizability, that satisfies several properties normally ascribed to backward induction reasoning, foremost the possibility of being computed via a tractable backwards procedure. We also show that Backwards Rationalizability characterizes the robust predictions of a ‘perfect equilibrium’ notion that introduces the backward induction logic and nothing more into equilibrium analysis. We discuss a few applications, including a new version of peer-confirming equilibrium (Lipnowski and Sadler (2019)) that, thanks to Backwards Rationalizability, restores in dynamic games the natural comparative statics that the original concept only displays in static settings. |
| Keywords: | backward induction, backwards procedure, backwards rationalizability, Incomplete Information, interim perfect equilibrium, perfect bayesian equilibrium rationalizability, robustness to bounded rationality |
| JEL: | C72 C73 D82 |
| Date: | 2024–09 |
| URL: | https://d.repec.org/n?u=RePEc:bge:wpaper:1462 |
| By: | Daniele Condorelli; Massimiliano Furlan |
| Abstract: | We train two neural networks adversarially to play normal-form games. At each iteration, a row and column network take a new randomly generated game and output individual mixed strategies. The parameters of each network are independently updated via stochastic gradient descent to minimize expected regret given the opponent's strategy. Our simulations demonstrate that the joint behavior of the networks converges to strategies close to Nash equilibria in almost all games. For all $2 \times 2$ and in 80% of $3 \times 3$ games with multiple equilibria, the networks select the risk-dominant equilibrium. Our results show how Nash equilibrium emerges from learning across heterogeneous games. |
| Date: | 2024–09 |
| URL: | https://d.repec.org/n?u=RePEc:arx:papers:2409.15197 |
| By: | Mallozzi, Lina; Vidal-Puga, Juan |
| Abstract: | We consider cooperative games where the characteristic function is valued in the space of the fuzzy numbers. By using different fuzzy calculation methods to transform the game into a crisp cooperative one, we define and characterize an efficient extension of the Shapley value. This solution is a relevant member of a wider family of more general, fuzzy calculation method dependent extensions of the Shapley value. |
| Keywords: | Cooperative game; Shapley value; fuzzy set; efficiency |
| JEL: | C71 |
| Date: | 2024–09–21 |
| URL: | https://d.repec.org/n?u=RePEc:pra:mprapa:122168 |
| By: | Jasmina Karabegovic (University of Graz, Austria) |
| Abstract: | This paper introduces an explicit algorithm for computing perfect public equilibrium (PPE) payoffs in repeated games with imperfect public monitoring, public randomization, and discounting. The method adapts the established framework by Abreu, Pearce, & Stacchetti (1990) into a practical tool that balances theoretical accuracy with computational efficiency. The algorithm simplifies the complex task of identifying PPE payoff sets for any given discount factor δ. A stand-alone implementation of the algorithm can be accessed on GitHub. |
| Keywords: | Repeated games, imperfect public monitoring, computation. |
| JEL: | C63 C72 C73 |
| Date: | 2024–09 |
| URL: | https://d.repec.org/n?u=RePEc:grz:wpaper:2024-15 |
| By: | Yohan Pelosse (Humanities and Social Sciences, Swansea University) |
| Abstract: | Following Aumann (1976) and Brandenburger-Friedenberg-Keisler (2008), the standard epistemic approach of strategic games imposes a given type structure to the players with the implicit idea that it is an objectively well-defined ’ commonly known environment or context’. But imposing such ’contexts’ has some important implications for the analysis e.g., under RCBR, it turns strategic uncertainty into the exogenous uncertainty of a (subjective) correlating device in a correlated equilibrium (Brandenburger and Dekel, 1987). Crucially, for a given game, there exists an infinite set of such possible ’contexts’ – each assigning a particular subset of types/ belief restrictions to the description of the game ( Battigalli and Friedenberg, 2009). So, the choice of one vs.another type structure remains an important open question. Here, we follow the suggestion in Brandenburger and Friedenberg (2010) and drop the standard assumption that a type structure (or ’context’ ) is exogenously assigned to the players. Our analysis treats the canonical case where each player is on a par with an analyst facing endogenous uncertainty with no underlying exogenous signals determining the players’ types i.e., no initial restricted subset of beliefs can be taken to be ’known’ by all the players. We show the existence of a ’meta- self-referring epistemicmodel’ where the notion of ’self-belief/knowledge’ arises as an ’intertwined/meta’ version of the ’self-evident events’ (Monderer and Samet [1989]): The subset of types at which the analyst believes possible a subset of states of the world–generating a certain subset of belief hierarchies for the players– must be those that agree with the subset of states at which the agent believes these types. In this self-referringmodel each ’context’–belief system– is an inherently subjective structure that always belongs to the mind of a single agent: An observer assigns subjectively a specific belief systemto the players if and only if he ’self-believes/knows’ that all the agents have the samemodel as his. Our results show that every such subjective player-specific type structure corresponds to a player-independent type structure whose types/beliefs are intertwined i.e., non-separable. In our central result we obtain that any notion of ’context’ of a game –wherein players are analysts allowed to choose their type-structures– is equivalent to the subjective specific type-structure of a meta-observer/analyst. These results suggest that the presence of endogenous uncertainty–wherein no exogenous ex ante stage is assumed– is incompatible with the existence of a ’player independent belief system’. |
| Date: | 2024–10–09 |
| URL: | https://d.repec.org/n?u=RePEc:swn:wpaper:2024-07 |
| By: | Roberto Pinheiro |
| Abstract: | In this paper, we present a model of repeated first-price private value auctions in which the bidders have access to a cheap talk communication mechanism. In this framework, messages allow bidders to transmit their preference rankings over the goods to be auctioned, similar to Pesendorfer (2000). We show that collusion through this static mechanism not only dominates the static bid rotation mechanism presented by McAfee and McMillan (1992), but it is also not strictly dominated by the dynamic bid rotation mechanism presented by Aoyagi (2003). However, we show that asymptotic efficiency of collusion through increasing the number of ordered goods, presented by Pesendorfer (2000), demands patience rates to asymptotically approach one, making collusion increasingly more difficult to sustain. Finally, we study mechanisms through which the auctioneer may try to break bidders' collusion. |
| Keywords: | collusion; auctions; cheap talk communication; repeated games |
| JEL: | D44 C72 L41 |
| Date: | 2024–10–07 |
| URL: | https://d.repec.org/n?u=RePEc:fip:fedcwq:98920 |
| By: | Pradeep Dubey; Siddhartha Sahi; Guanyang Wang |
| Abstract: | We give examples of situations -- stochastic production, military tactics, corporate merger -- where it is beneficial to concentrate risk rather than to diversify it, that is, to put all eggs in one basket. Our examples admit a dual interpretation: as optimal strategies of a single player (the `principal') or, alternatively, as dominant strategies in a non-cooperative game with multiple players (the `agents'). The key mathematical result can be formulated in terms of a convolution structure on the set of increasing functions on a Boolean lattice (the lattice of subsets of a finite set). This generalizes the well-known Harris inequality from statistical physics and discrete mathematics; we give a simple self-contained proof of this result, and prove a further generalization based on the game-theoretic approach. |
| Date: | 2024–03 |
| URL: | https://d.repec.org/n?u=RePEc:arx:papers:2403.15957 |
| By: | Tsuritani, Ryosuke |
| Abstract: | In a vertical market, the price of the final good is high if a seller has strong bargaining power. Thus, a policy that strengthens the bargaining power of sub-suppliers may be desirable from a fairness perspective while undesirable from an efficiency perspective. We consider a vertical market with one sub-supplier, focal supplier, and manufacturer. The focal supplier purchases inputs from the sub-supplier and sells its products to the manufacturer. Suppliers' selling prices are determined through Nash bargaining. We find that although suppliers' vertical separation induces triple-markup inefficiency in vertical relations, if the focal supplier has weak bargaining power over the manufacturer or strong bargaining power over the sub-supplier, the suppliers have the incentive to remain separated. This is because suppliers' vertical separation may be a price-increasing commitment and transfer the bargaining surplus from the manufacturer to the suppliers. Therefore, a policy that strengthens the bargaining power of sub-suppliers may also be justified from an efficiency perspective because it may encourage vertical integration. |
| Keywords: | Vertical market; Vertical integration; Three-tier supply chain; Bargaining; Subcontracting Act |
| JEL: | D42 L23 L40 |
| Date: | 2024–09–15 |
| URL: | https://d.repec.org/n?u=RePEc:pra:mprapa:122071 |
| By: | Andrzej Baranski; Sumit Goel |
| Abstract: | We study the classical contest design problem of allocating a budget across different prizes to maximize effort in a finite type-space environment. For any contest, we characterize the unique symmetric equilibrium. In this equilibrium, different agent types mix over contiguous intervals so that more efficient agents always exert greater effort than less efficient agents. We then solve for the expected equilibrium effort, and identify conditions under which the effect of increasing competition under linear costs is informative about the effect of such a transformation under general costs. As a result, we find that the winner-takes-all contest is optimal under linear and concave costs. Lastly, we obtain an equilibrium convergence result for the continuum type-space, and since the finite type-space encompasses the complete information environment as a special case, our analysis offers a unified approach to studying contests in these classical environments. |
| Date: | 2024–10 |
| URL: | https://d.repec.org/n?u=RePEc:arx:papers:2410.04970 |
| By: | Gioacchino Fazio; Stefano Fricano; Claudio Pirrone |
| Abstract: | Immersive technologies such as Metaverse, AR, and VR are at a crossroads, with many actors pondering their adoption and potential sectors interested in integration. The cultural and tourism industries are particularly impacted, facing significant pressure to make decisions that could shape their future landscapes. Stakeholders' perceptions play a crucial role in this process, influencing the speed and extent of technology adoption. As immersive technologies promise to revolutionize experiences, stakeholders in these fields weigh the benefits and challenges of embracing such innovations. The current choices will likely determine the trajectory of cultural preservation and tourism enhancement, potentially transforming how we engage with history, art, and travel. Starting from a decomposition of stakeholders' perceptions into principal components using Q-methodology, this article employs an evolutionary game model to attempt to map possible scenarios and highlight potential decision-making trajectories. The proposed approach highlights how evolutionary dynamics lead to identifying a dominant long-term strategy that emerges from the complex system of coexistence among various stakeholders. |
| Date: | 2024–08 |
| URL: | https://d.repec.org/n?u=RePEc:arx:papers:2409.06720 |
| By: | Neil A. Chriss |
| Abstract: | This paper develops a mathematical framework for building a position in a stock over a fixed period of time while in competition with one or more other traders doing the same thing. We develop a game-theoretic framework that takes place in the space of trading strategies where action sets are trading strategies and traders try to devise best-response strategies to their adversaries. In this setup trading is guided by a desire to minimize the total cost of trading arising from a mixture of temporary and permanent market impact caused by the aggregate level of trading including the trader and the competition. We describe a notion of equilibrium strategies, show that they exist and provide closed-form solutions. |
| Date: | 2024–09 |
| URL: | https://d.repec.org/n?u=RePEc:arx:papers:2409.03586 |
| By: | Vincent Meisner; Pascal Pillath |
| Abstract: | We design the profit-maximizing mechanism to sell an excludable and non-rival good with network effects. Buyers have heterogeneous private values that depend on how many others also consume the good. In optimum, an endogenous number of the highest types shares consumption, and we provide an algorithm that implements this allocation in dominant strategies. We apply our insights to digital content creation, and we are able to rationalize features seen in monetization schemes in this industry such as voluntary contributions, community subsidies, and exclusivity bids. |
| Keywords: | Mechanism design, non-rival goods, club goods, network effects, digital content, creator economy |
| JEL: | D82 |
| Date: | 2024–09–20 |
| URL: | https://d.repec.org/n?u=RePEc:bdp:dpaper:0049 |
| By: | Yang Sun; Wei Zhao; Junjie Zhou |
| Abstract: | We study dynamic network formation from a centralized perspective. In each period, the social planner builds a single link to connect previously unlinked pairs. The social planner is forward-looking, with instantaneous utility monotonic in the aggregate number of walks of various lengths. We show that, forming a nested split graph at each period is optimal, regardless of the discount function. When the social planner is sufficiently myopic, it is optimal to form a quasi-complete graph at each period, which is unique up to permutation. This finding provides a micro-foundation for the quasi-complete graph, as it is formed under a greedy policy. We also investigate the robustness of these findings under non-linear best response functions and weighted networks. |
| Date: | 2024–09 |
| URL: | https://d.repec.org/n?u=RePEc:arx:papers:2409.14136 |
| By: | Neil A. Chriss |
| Abstract: | This paper extends the optimal-trading framework developed in arXiv:2409.03586v1 to compute optimal strategies with {\em real-world constraints}. The aim of the current paper, as with the previous, is to study trading in the context of {\em multi-player non-cooperative games}. While the former paper relies on methods from the calculus of variations and optimal strategies arise as the solution of partial differential equations, the current paper demonstrates that the entire framework may be re-framed as a quadratic programming problem and cast in this light constraints are readily incorporated into the calculation of optimal strategies. An added benefit is that two-trader equilibria may be calculated as the end-points of a dynamic process of traders forming repeated adjustments to each other's strategy. |
| Date: | 2024–09 |
| URL: | https://d.repec.org/n?u=RePEc:arx:papers:2409.15459 |
| By: | Thành Nguyen (Purdue University); Rakesh Vohra (University of Pennsylvania) |
| Abstract: | Any mapping from states to signals induces a distribution over possible posteriors. For decision-making purposes, one is interested in parameters of those posteriors, such as their quartiles or whether they first-order stochastically dominate some given distribution. We show that a generalization of Gale’s demand theorem can be used to characterize which distributions over possible posteriors with the requisite properties can be generated by some mapping from states to signals. |
| Keywords: | Bayesian Persuasion, Information Design, Polymatroids |
| Date: | 2024–08–01 |
| URL: | https://d.repec.org/n?u=RePEc:pen:papers:24-025 |
| By: | Michele Fabi (Telecom Paris, CREST, IP Paris) |
| Abstract: | We analyze the effect of block propagation latency on the performance and design of Nakamoto-style blockchains. Miners strategically choose block capacity, balancing the risk of invalidation from forking with transaction fee income. The model identifies a unique and symmetric Nash equilibrium block capacity, which increases with the ratio of block production time to transmission delay and decreases with the ratio of coinbase reward to transaction fee rate. We endogenize blockchain growth and derive the Fokker-Planck equation for pending mempool data. The results reveal a tradeoff between efficiency (low transaction load) and security (high miner participation). Reducing the coinbase reward while raising transaction fees improves efficiency but may weaken security. We also discuss testable implications and extend the model to include uncle block rewards and discrete latency. |
| Keywords: | Blockchain design, Nakamoto consensus, forks, coinbase, stochastic storage |
| JEL: | C62 C63 G10 G14 |
| Date: | 2024–09–15 |
| URL: | https://d.repec.org/n?u=RePEc:crs:wpaper:2024-10 |
| By: | S. Nageeb Ali; Maximilian Mihm; Lucas Siga |
| Abstract: | This paper offers a strategic rationale for zero-sum thinking in elections. We show that asymmetric information and distributional considerations together make voters wary of policies supported by others. This force impels a majority of voters to support policies contrary to their preferences and information. Our analysis identifies and interprets a form of "adverse correlation" that is necessary and sufficient for zero-sum thinking to prevail in equilibrium. |
| Date: | 2024–09 |
| URL: | https://d.repec.org/n?u=RePEc:arx:papers:2409.15946 |
| By: | David K Levine; Cesar Martinelli; Nicole Stoelinga |
| Date: | 2024–10–14 |
| URL: | https://d.repec.org/n?u=RePEc:cla:levarc:11694000000000199 |