Pub Date : 2023-11-20DOI: 10.1007/s00182-023-00881-0
Arvind Srinivasan, Burton Simon
The Lowest Unique Positive Integer game, a.k.a. Limbo, is among the simplest games that can be played by any number of players and has a nontrivial strategic component. Players independently pick positive integers, and the winner is the player that picks the smallest number nobody else picks. The Nash equilibrium for this game is a mixed strategy, ((p(1),p(2),ldots )), where p(k) is the probability you pick k. A recursion for the Nash equilibrium has been previously worked out in the case where the number of players is Poisson distributed, an assumption that can be justified when there is a large pool of potential players. Here, we summarize previous results and prove that as the (expected) number of players, n, goes to infinity, a properly scaled version of the Nash equilibrium random variable converges in distribution to a Unif(0, 1) random variable. The result implies that for large n, players should choose a number uniformly between 1 and (phi _n sim O(n/ln (n))). Convergence to the uniform is rather slow, so we also investigate a continuous analog of the Nash equilibrium using a differential equation derived from the recursion. The resulting approximation is unexpectedly accurate and is interesting in its own right. Studying the differential equation yields some useful analytical results, including a precise expression for (phi _n), and efficient ways to sample from the continuous approximation.
{"title":"Exact asymptotics and continuous approximations for the Lowest Unique Positive Integer game","authors":"Arvind Srinivasan, Burton Simon","doi":"10.1007/s00182-023-00881-0","DOIUrl":"https://doi.org/10.1007/s00182-023-00881-0","url":null,"abstract":"<p>The Lowest Unique Positive Integer game, a.k.a. Limbo, is among the simplest games that can be played by any number of players and has a nontrivial strategic component. Players independently pick positive integers, and the winner is the player that picks the smallest number nobody else picks. The Nash equilibrium for this game is a mixed strategy, <span>((p(1),p(2),ldots ))</span>, where <i>p</i>(<i>k</i>) is the probability you pick <i>k</i>. A recursion for the Nash equilibrium has been previously worked out in the case where the number of players is Poisson distributed, an assumption that can be justified when there is a large pool of potential players. Here, we summarize previous results and prove that as the (expected) number of players, <i>n</i>, goes to infinity, a properly scaled version of the Nash equilibrium random variable converges in distribution to a Unif(0, 1) random variable. The result implies that for large <i>n</i>, players should choose a number uniformly between 1 and <span>(phi _n sim O(n/ln (n)))</span>. Convergence to the uniform is rather slow, so we also investigate a continuous analog of the Nash equilibrium using a differential equation derived from the recursion. The resulting approximation is unexpectedly accurate and is interesting in its own right. Studying the differential equation yields some useful analytical results, including a precise expression for <span>(phi _n)</span>, and efficient ways to sample from the continuous approximation.</p>","PeriodicalId":14155,"journal":{"name":"International Journal of Game Theory","volume":"7 4","pages":""},"PeriodicalIF":0.6,"publicationDate":"2023-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138494998","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"经济学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-11-07DOI: 10.1007/s00182-023-00877-w
Estela Sánchez-Rodríguez, Miguel Ángel Mirás Calvo, Carmen Quinteiro Sandomingo, Iago Núñez Lugilde
{"title":"Coalition-weighted Shapley values","authors":"Estela Sánchez-Rodríguez, Miguel Ángel Mirás Calvo, Carmen Quinteiro Sandomingo, Iago Núñez Lugilde","doi":"10.1007/s00182-023-00877-w","DOIUrl":"https://doi.org/10.1007/s00182-023-00877-w","url":null,"abstract":"","PeriodicalId":14155,"journal":{"name":"International Journal of Game Theory","volume":"34 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135480166","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"经济学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Winning strategies of (i, j) multimove games on Thai, Japanese, and Chinese chess games","authors":"Tanayot Prapaithrakul, Saharath Sanguanpong, Nantapath Trakultraipruk","doi":"10.1007/s00182-023-00880-1","DOIUrl":"https://doi.org/10.1007/s00182-023-00880-1","url":null,"abstract":"","PeriodicalId":14155,"journal":{"name":"International Journal of Game Theory","volume":"238 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135476225","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"经济学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-11DOI: 10.1007/s00182-023-00873-0
Quan Li, Kang Rong
{"title":"Full disclosure in competitive Bayesian persuasion","authors":"Quan Li, Kang Rong","doi":"10.1007/s00182-023-00873-0","DOIUrl":"https://doi.org/10.1007/s00182-023-00873-0","url":null,"abstract":"","PeriodicalId":14155,"journal":{"name":"International Journal of Game Theory","volume":"28 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136062477","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"经济学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-11DOI: 10.1007/s00182-023-00876-x
Peter Borm, Hans Peters
{"title":"Professor Stef Tijs (1937–2023)","authors":"Peter Borm, Hans Peters","doi":"10.1007/s00182-023-00876-x","DOIUrl":"https://doi.org/10.1007/s00182-023-00876-x","url":null,"abstract":"","PeriodicalId":14155,"journal":{"name":"International Journal of Game Theory","volume":"122 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136208787","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"经济学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-11DOI: 10.1007/s00182-023-00874-z
Francesc Dilmé
Abstract This paper provides a new characterization of belief consistency in extensive games. We show that all consistent assessments are supported by sequences of strategy profiles with the property that all actions with vanishing probability are played according to power functions of the sequence index. The result makes it simpler to prove or disprove that a given assessment is consistent, facilitating the use of sequential equilibria.
{"title":"A characterization of consistent assessments using power sequences of strategy profiles","authors":"Francesc Dilmé","doi":"10.1007/s00182-023-00874-z","DOIUrl":"https://doi.org/10.1007/s00182-023-00874-z","url":null,"abstract":"Abstract This paper provides a new characterization of belief consistency in extensive games. We show that all consistent assessments are supported by sequences of strategy profiles with the property that all actions with vanishing probability are played according to power functions of the sequence index. The result makes it simpler to prove or disprove that a given assessment is consistent, facilitating the use of sequential equilibria.","PeriodicalId":14155,"journal":{"name":"International Journal of Game Theory","volume":"7 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136062439","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"经济学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-10DOI: 10.1007/s00182-023-00872-1
Yoshio Kamijo
{"title":"A note on the risk dominance of the Nash demand game","authors":"Yoshio Kamijo","doi":"10.1007/s00182-023-00872-1","DOIUrl":"https://doi.org/10.1007/s00182-023-00872-1","url":null,"abstract":"","PeriodicalId":14155,"journal":{"name":"International Journal of Game Theory","volume":"3 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136295450","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"经济学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-10DOI: 10.1007/s00182-023-00875-y
Endre Boros, Paolo Giulio Franciosa, Vladimir Gurvich, Michael Vyalyi
Abstract We prove that a deterministic n -person shortest path game has a Nash equlibrium in pure and stationary strategies if it is edge-symmetric (that is ( u , v ) is a move whenever ( v , u ) is, apart from moves entering terminal vertices) and the length of every move is positive for each player. Both conditions are essential, though it remains an open problem whether there exists a NE-free 2-person non-edge-symmetric game with positive lengths. We provide examples for NE-free 2-person edge-symmetric games that are not positive. We also consider the special case of terminal games (shortest path games in which only terminal moves have nonzero length, possibly negative) and prove that edge-symmetric n -person terminal games always have Nash equilibria in pure and stationary strategies. Furthermore, we prove that an edge-symmetric 2-person terminal game has a uniform (subgame perfect) Nash equilibrium, provided any infinite play is worse than any of the terminals for both players.
{"title":"Deterministic n-person shortest path and terminal games on symmetric digraphs have Nash equilibria in pure stationary strategies","authors":"Endre Boros, Paolo Giulio Franciosa, Vladimir Gurvich, Michael Vyalyi","doi":"10.1007/s00182-023-00875-y","DOIUrl":"https://doi.org/10.1007/s00182-023-00875-y","url":null,"abstract":"Abstract We prove that a deterministic n -person shortest path game has a Nash equlibrium in pure and stationary strategies if it is edge-symmetric (that is ( u , v ) is a move whenever ( v , u ) is, apart from moves entering terminal vertices) and the length of every move is positive for each player. Both conditions are essential, though it remains an open problem whether there exists a NE-free 2-person non-edge-symmetric game with positive lengths. We provide examples for NE-free 2-person edge-symmetric games that are not positive. We also consider the special case of terminal games (shortest path games in which only terminal moves have nonzero length, possibly negative) and prove that edge-symmetric n -person terminal games always have Nash equilibria in pure and stationary strategies. Furthermore, we prove that an edge-symmetric 2-person terminal game has a uniform (subgame perfect) Nash equilibrium, provided any infinite play is worse than any of the terminals for both players.","PeriodicalId":14155,"journal":{"name":"International Journal of Game Theory","volume":"74 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136295452","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"经济学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-09-07DOI: 10.1007/s00182-023-00869-w
Yu-Sung Tu, Wei-Torng Juang
{"title":"Evolution of preferences in multiple populations","authors":"Yu-Sung Tu, Wei-Torng Juang","doi":"10.1007/s00182-023-00869-w","DOIUrl":"https://doi.org/10.1007/s00182-023-00869-w","url":null,"abstract":"","PeriodicalId":14155,"journal":{"name":"International Journal of Game Theory","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2023-09-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45982337","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"经济学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-08-31DOI: 10.1007/s00182-023-00871-2
Igal Milchtaich
{"title":"Best-response equilibrium: an equilibrium in finitely additive mixed strategies","authors":"Igal Milchtaich","doi":"10.1007/s00182-023-00871-2","DOIUrl":"https://doi.org/10.1007/s00182-023-00871-2","url":null,"abstract":"","PeriodicalId":14155,"journal":{"name":"International Journal of Game Theory","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2023-08-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49256713","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"经济学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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