A decision maker is asked to express her beliefs by assigning probabilities to certain possible s... more A decision maker is asked to express her beliefs by assigning probabilities to certain possible states. We focus on the relationship between her database and her beliefs. We show that if beliefs given a union of two databases are a convex combination of beliefs given each of the databases, the belief formation process follows a simple formula: beliefs are a similarity-weighted average of the beliefs induced by each past case.
A valuation for a player in a game in extensive form is an assignment of numeric values to the pl... more A valuation for a player in a game in extensive form is an assignment of numeric values to the players moves. The valuation reflects the desirability moves. We assume a myopic player, who chooses a move with the highest valuation. Valuations can also be revised, and hopefully improved, after each play of the game. Here, a very simple valuation revision is considered, in which the moves made in a play are assigned the payoff obtained in the play. We show that by adopting such a learning process a player who has a winning strategy in a win-lose game can almost surely guarantee a win in a repeated game. When a player has more than two payoffs, a more elaborate learning procedure is required. We consider one that associates with each move the average payoff in the rounds in which this move was made. When all players adopt this learning procedure, with some perturbations, then, with probability 1, strategies that are close to subgame perfect equilibrium are played after some time. A single player who adopts this procedure can guarantee only her individually rational payoff.
medRxiv (Cold Spring Harbor Laboratory), Aug 25, 2020
The overuse of antibiotics is exacerbating the antibiotic resistance crisis. Since this problem i... more The overuse of antibiotics is exacerbating the antibiotic resistance crisis. Since this problem is a classic common-goods dilemma, it naturally lends itself to a game-theoretic analysis. Hence, we designed a model wherein physicians weigh whether antibiotics should be prescribed, given that antibiotic usage depletes its future effectiveness. The physicians' decisions rely on the probability of a bacterial infection before definitive laboratory results are available. We show that the physicians' equilibrium decision-rule of antibiotic prescription is not socially optimal. However, we prove that discretizing the information provided to physicians can mitigate the gap between their equilibrium decisions and the social optimum of antibiotic prescription. Despite this problem's complexity, the effectiveness of the discretization solely depends on the distribution of available information. This is demonstrated on theoretic distributions and a clinical dataset. Our results provide a game-theory based guide for optimal output of current and future decision support systems of antibiotic prescription. .
The overuse of antibiotics is exacerbating the antibiotic resistance crisis. Since this problem i... more The overuse of antibiotics is exacerbating the antibiotic resistance crisis. Since this problem is a classic common-goods dilemma, it naturally lends itself to a game-theoretic analysis. Hence, we designed a model wherein physicians weigh whether antibiotics should be prescribed, given that antibiotic usage depletes its future effectiveness. The physicians' decisions rely on the probability of a bacterial infection before definitive laboratory results are available. We show that the physicians' equilibrium decision rule of antibiotic prescription is not socially optimal. However, we prove that discretizing the information provided to physicians can mitigate the gap between their equilibrium decisions and the social optimum of antibiotic prescription. Despite this problem's complexity, the effectiveness of the discretization solely depends on the type of information available to the physician to determine the nature of infection. This is demonstrated on theoretic distributions and a clinical dataset. Our results provide a game-theory based guide for optimal output of current and future decision support systems of antibiotic prescription.
Zeno's paradoxes of motion, which claim that moving from one point to another cannot be accomplis... more Zeno's paradoxes of motion, which claim that moving from one point to another cannot be accomplished in finite time, seem to be of serious concern when moving towards an agreement is concerned. Parkinson's Law of Triviality implies that such an agreement cannot be reached in finite time. By explicitly modeling dynamic processes of reaching interim agreements and using arguments similar to Zeno's, we show that if utilities are von Neumann-Morgenstern, then no such process can bring about an agreement in finite time in linear bargaining problems. To extend this result for all bargaining problems, we characterize a particular path illustrated by Raiffa (1953), and show that no agreement is reached along this path in finite time.
Game theoretic models of learning which are based on the strategic form of the game cannot explai... more Game theoretic models of learning which are based on the strategic form of the game cannot explain learning in games with large extensive form. We study learning in such games by using valuation of moves. A valuation for a player is a numeric assessment of her moves that purports to reflect their desirability. We consider a myopic player, who chooses moves with the highest valuation. Each time the game is played, the player revises her valuation by assigning the payoff obtained in the play to each of the moves she has made. We show for a repeated win-lose game that if the player has a winning strategy in the stage game, there is almost surely a time after which she always wins. When a player has more than two payoffs, a more elaborate learning procedure is required. We consider one that associates with each move the average payoff in the rounds in which this move was made. When all players adopt this learning procedure, with some perturbations, then, with probability 1 there is a time after which strategies that are close to subgame perfect equilibrium are played. A single player who adopts this procedure can guarantee only her individually rational payoff.
Nonwmmetric Shapley values for coalitional form games with transferable utility are studied. The ... more Nonwmmetric Shapley values for coalitional form games with transferable utility are studied. The nonsymmetries are modeled through nonsymmetric weight systems defined on the players of the games. It is shown axiomatically that two families of solutions of this type are possible. These families are strongly related to each other through the duality relationship on games. While the first family lends itself to applications of nonsymmetric revenue sharing problems the second family is suitable for applications of cost allocation problems. The intersection of these two families consists essentially of the symmetric Shapley value. These families are also characterized by a prObabilistic arrival time to the game approach. It is also demonstrated that lack of symmetries may arise naturally when players in a game represent nonequal size constituencies.
ABSTRACT Nonatomic vector measures are shown to be open maps from the σ- field on which they are ... more ABSTRACT Nonatomic vector measures are shown to be open maps from the σ- field on which they are defined to their range, where the σ-field is equipped with the pseudometric of the symmetric difference with rspect to a given scalar measure.
A learning process is belief affirming if the difference between a player's expected payoff in th... more A learning process is belief affirming if the difference between a player's expected payoff in the next period, and the average of his or her past payoffs converges to zero. We show that every smooth discrete fictitious play and every continuous fictitious play is belief affirming. We also provide conditions under which general averaging processes are belief affirming. Journal of Economic Literature Classification Numbers: C72, C73, D83. 1997 Academic Press Consider players engaged in the repeated play of a finite game in strategic form. In each period, each player, on the basis of the history of past moves, forms a belief about the next joint move of the other players. She then chooses an action that is a myopic pure best reply, that is, an article no.
A decision maker is asked to express her beliefs by assigning probabilities to certain possible s... more A decision maker is asked to express her beliefs by assigning probabilities to certain possible states. We focus on the relationship between her database and her beliefs. We show that if beliefs given a union of two databases are a convex combination of beliefs given each of the databases, the belief formation process follows a simple formula: beliefs are a similarity-weighted average of the beliefs induced by each past case.
Theoretical Aspects of Rationality and Knowledge, Jul 8, 2001
Game theoretic models of learning which are based on the strategic form of the game cannot explai... more Game theoretic models of learning which are based on the strategic form of the game cannot explain learning in games with large extensive form. We study learning in such games by using valuation of moves. A valuation for a player is a numeric assessment of her moves that purports to reflect their desirability. We consider a myopic player, who chooses moves with the highest valuation. Each time the game is played, the player revises her valuation by assigning the payoff obtained in the play to each of the moves she has made. We show for a repeated win-lose game that if the player has a winning strategy in the stage game, there is almost surely a time after which she always wins. When a player has more than two payoffs, a more elaborate learning procedure is required. We consider one that associates with each move the average payoff in the rounds in which this move was made. When all players adopt this learning procedure, with some perturbations, then, with probability 1 there is a time after which strategies that are close to subgame perfect equilibrium are played. A single player who adopts this procedure can guarantee only her individually rational payoff.
tion for the existence of a common prior in this case is a straightforward result of the separati... more tion for the existence of a common prior in this case is a straightforward result of the separation theorem. For f g R , denote by Ef the element of R defined by i Z .Z. Z. Ef vstvf. ii 172 0899-8256r98 $25.00 Copyright Q 1998 by Academic Press All rights of reproduction in any form reserved. Z. CLAIM. When there are two agents n s 2, then there exists a common prior, iff there is no f in R , such that E f ) 0 )Ef. 12 Proof. There is no common prior iff P and P can be strongly 12 separated, that is, iff there are g g R and c g R, such that xg)c) 1 xg, for each x g P , and x g P . Subtracting c from all the coordinates 21122 of g yields a vector f , such that xf)0)xffor each x g P , and 12 11 xgP. However, these inequalities hold iff they hold for the extreme 22 points of P and P , i.e., iff Ef)0)Ef. B 12 1 2 We now show that a generalization of this claim to any number of agents is the immediate result of a generalization of the separation theorem that gives a necess
We thank the following people for pointing out errors and improvements: Lorand Ambrus-Lakatos, Pi... more We thank the following people for pointing out errors and improvements: Lorand Ambrus-Lakatos, Pierpaolo Battigalli, Jean-Pierre Benôıt, Jim Bergin, Richard Boylan, Boudewijn P. de Bruin, Joel Cohen, Juan Dubra, Peter Forsyth, Haruo Imai, Christopher Kah, Karthik Kalyanaraman, Nicolas Klein, Fuhito Kojima, Vijay Krishna, Hui Li, Tsen Lim, Kin Chung Lo, Salvatore Modica, Robert Murphy, Yasuyuki Noguchi, Marc Pauly, Bezalel Peleg, Daniel Probst, Phil Reny, Al Roth, Dov Samet, Giora Slutzki, Lutz Veldman, Shmuel Zamir.
Ceva’s theorem, which concerns triangles, is a central result of post-Euclidean plane geometry. T... more Ceva’s theorem, which concerns triangles, is a central result of post-Euclidean plane geometry. The three-dimensional generalization of a triangle is a tetrahedron, and the n-dimensional generalization of these is an n-simplex. We extend Ceva’s theorem to n-simplices and in doing so illustrate the considerations and choices that can be made in generalizing from plane geometry to high-dimensional geometries.
A decision maker is asked to express her beliefs by assigning probabilities to certain possible s... more A decision maker is asked to express her beliefs by assigning probabilities to certain possible states. We focus on the relationship between her database and her beliefs. We show that if beliefs given a union of two databases are a convex combination of beliefs given each of the databases, the belief formation process follows a simple formula: beliefs are a similarity-weighted average of the beliefs induced by each past case.
A valuation for a player in a game in extensive form is an assignment of numeric values to the pl... more A valuation for a player in a game in extensive form is an assignment of numeric values to the players moves. The valuation reflects the desirability moves. We assume a myopic player, who chooses a move with the highest valuation. Valuations can also be revised, and hopefully improved, after each play of the game. Here, a very simple valuation revision is considered, in which the moves made in a play are assigned the payoff obtained in the play. We show that by adopting such a learning process a player who has a winning strategy in a win-lose game can almost surely guarantee a win in a repeated game. When a player has more than two payoffs, a more elaborate learning procedure is required. We consider one that associates with each move the average payoff in the rounds in which this move was made. When all players adopt this learning procedure, with some perturbations, then, with probability 1, strategies that are close to subgame perfect equilibrium are played after some time. A single player who adopts this procedure can guarantee only her individually rational payoff.
medRxiv (Cold Spring Harbor Laboratory), Aug 25, 2020
The overuse of antibiotics is exacerbating the antibiotic resistance crisis. Since this problem i... more The overuse of antibiotics is exacerbating the antibiotic resistance crisis. Since this problem is a classic common-goods dilemma, it naturally lends itself to a game-theoretic analysis. Hence, we designed a model wherein physicians weigh whether antibiotics should be prescribed, given that antibiotic usage depletes its future effectiveness. The physicians' decisions rely on the probability of a bacterial infection before definitive laboratory results are available. We show that the physicians' equilibrium decision-rule of antibiotic prescription is not socially optimal. However, we prove that discretizing the information provided to physicians can mitigate the gap between their equilibrium decisions and the social optimum of antibiotic prescription. Despite this problem's complexity, the effectiveness of the discretization solely depends on the distribution of available information. This is demonstrated on theoretic distributions and a clinical dataset. Our results provide a game-theory based guide for optimal output of current and future decision support systems of antibiotic prescription. .
The overuse of antibiotics is exacerbating the antibiotic resistance crisis. Since this problem i... more The overuse of antibiotics is exacerbating the antibiotic resistance crisis. Since this problem is a classic common-goods dilemma, it naturally lends itself to a game-theoretic analysis. Hence, we designed a model wherein physicians weigh whether antibiotics should be prescribed, given that antibiotic usage depletes its future effectiveness. The physicians' decisions rely on the probability of a bacterial infection before definitive laboratory results are available. We show that the physicians' equilibrium decision rule of antibiotic prescription is not socially optimal. However, we prove that discretizing the information provided to physicians can mitigate the gap between their equilibrium decisions and the social optimum of antibiotic prescription. Despite this problem's complexity, the effectiveness of the discretization solely depends on the type of information available to the physician to determine the nature of infection. This is demonstrated on theoretic distributions and a clinical dataset. Our results provide a game-theory based guide for optimal output of current and future decision support systems of antibiotic prescription.
Zeno's paradoxes of motion, which claim that moving from one point to another cannot be accomplis... more Zeno's paradoxes of motion, which claim that moving from one point to another cannot be accomplished in finite time, seem to be of serious concern when moving towards an agreement is concerned. Parkinson's Law of Triviality implies that such an agreement cannot be reached in finite time. By explicitly modeling dynamic processes of reaching interim agreements and using arguments similar to Zeno's, we show that if utilities are von Neumann-Morgenstern, then no such process can bring about an agreement in finite time in linear bargaining problems. To extend this result for all bargaining problems, we characterize a particular path illustrated by Raiffa (1953), and show that no agreement is reached along this path in finite time.
Game theoretic models of learning which are based on the strategic form of the game cannot explai... more Game theoretic models of learning which are based on the strategic form of the game cannot explain learning in games with large extensive form. We study learning in such games by using valuation of moves. A valuation for a player is a numeric assessment of her moves that purports to reflect their desirability. We consider a myopic player, who chooses moves with the highest valuation. Each time the game is played, the player revises her valuation by assigning the payoff obtained in the play to each of the moves she has made. We show for a repeated win-lose game that if the player has a winning strategy in the stage game, there is almost surely a time after which she always wins. When a player has more than two payoffs, a more elaborate learning procedure is required. We consider one that associates with each move the average payoff in the rounds in which this move was made. When all players adopt this learning procedure, with some perturbations, then, with probability 1 there is a time after which strategies that are close to subgame perfect equilibrium are played. A single player who adopts this procedure can guarantee only her individually rational payoff.
Nonwmmetric Shapley values for coalitional form games with transferable utility are studied. The ... more Nonwmmetric Shapley values for coalitional form games with transferable utility are studied. The nonsymmetries are modeled through nonsymmetric weight systems defined on the players of the games. It is shown axiomatically that two families of solutions of this type are possible. These families are strongly related to each other through the duality relationship on games. While the first family lends itself to applications of nonsymmetric revenue sharing problems the second family is suitable for applications of cost allocation problems. The intersection of these two families consists essentially of the symmetric Shapley value. These families are also characterized by a prObabilistic arrival time to the game approach. It is also demonstrated that lack of symmetries may arise naturally when players in a game represent nonequal size constituencies.
ABSTRACT Nonatomic vector measures are shown to be open maps from the σ- field on which they are ... more ABSTRACT Nonatomic vector measures are shown to be open maps from the σ- field on which they are defined to their range, where the σ-field is equipped with the pseudometric of the symmetric difference with rspect to a given scalar measure.
A learning process is belief affirming if the difference between a player's expected payoff in th... more A learning process is belief affirming if the difference between a player's expected payoff in the next period, and the average of his or her past payoffs converges to zero. We show that every smooth discrete fictitious play and every continuous fictitious play is belief affirming. We also provide conditions under which general averaging processes are belief affirming. Journal of Economic Literature Classification Numbers: C72, C73, D83. 1997 Academic Press Consider players engaged in the repeated play of a finite game in strategic form. In each period, each player, on the basis of the history of past moves, forms a belief about the next joint move of the other players. She then chooses an action that is a myopic pure best reply, that is, an article no.
A decision maker is asked to express her beliefs by assigning probabilities to certain possible s... more A decision maker is asked to express her beliefs by assigning probabilities to certain possible states. We focus on the relationship between her database and her beliefs. We show that if beliefs given a union of two databases are a convex combination of beliefs given each of the databases, the belief formation process follows a simple formula: beliefs are a similarity-weighted average of the beliefs induced by each past case.
Theoretical Aspects of Rationality and Knowledge, Jul 8, 2001
Game theoretic models of learning which are based on the strategic form of the game cannot explai... more Game theoretic models of learning which are based on the strategic form of the game cannot explain learning in games with large extensive form. We study learning in such games by using valuation of moves. A valuation for a player is a numeric assessment of her moves that purports to reflect their desirability. We consider a myopic player, who chooses moves with the highest valuation. Each time the game is played, the player revises her valuation by assigning the payoff obtained in the play to each of the moves she has made. We show for a repeated win-lose game that if the player has a winning strategy in the stage game, there is almost surely a time after which she always wins. When a player has more than two payoffs, a more elaborate learning procedure is required. We consider one that associates with each move the average payoff in the rounds in which this move was made. When all players adopt this learning procedure, with some perturbations, then, with probability 1 there is a time after which strategies that are close to subgame perfect equilibrium are played. A single player who adopts this procedure can guarantee only her individually rational payoff.
tion for the existence of a common prior in this case is a straightforward result of the separati... more tion for the existence of a common prior in this case is a straightforward result of the separation theorem. For f g R , denote by Ef the element of R defined by i Z .Z. Z. Ef vstvf. ii 172 0899-8256r98 $25.00 Copyright Q 1998 by Academic Press All rights of reproduction in any form reserved. Z. CLAIM. When there are two agents n s 2, then there exists a common prior, iff there is no f in R , such that E f ) 0 )Ef. 12 Proof. There is no common prior iff P and P can be strongly 12 separated, that is, iff there are g g R and c g R, such that xg)c) 1 xg, for each x g P , and x g P . Subtracting c from all the coordinates 21122 of g yields a vector f , such that xf)0)xffor each x g P , and 12 11 xgP. However, these inequalities hold iff they hold for the extreme 22 points of P and P , i.e., iff Ef)0)Ef. B 12 1 2 We now show that a generalization of this claim to any number of agents is the immediate result of a generalization of the separation theorem that gives a necess
We thank the following people for pointing out errors and improvements: Lorand Ambrus-Lakatos, Pi... more We thank the following people for pointing out errors and improvements: Lorand Ambrus-Lakatos, Pierpaolo Battigalli, Jean-Pierre Benôıt, Jim Bergin, Richard Boylan, Boudewijn P. de Bruin, Joel Cohen, Juan Dubra, Peter Forsyth, Haruo Imai, Christopher Kah, Karthik Kalyanaraman, Nicolas Klein, Fuhito Kojima, Vijay Krishna, Hui Li, Tsen Lim, Kin Chung Lo, Salvatore Modica, Robert Murphy, Yasuyuki Noguchi, Marc Pauly, Bezalel Peleg, Daniel Probst, Phil Reny, Al Roth, Dov Samet, Giora Slutzki, Lutz Veldman, Shmuel Zamir.
Ceva’s theorem, which concerns triangles, is a central result of post-Euclidean plane geometry. T... more Ceva’s theorem, which concerns triangles, is a central result of post-Euclidean plane geometry. The three-dimensional generalization of a triangle is a tetrahedron, and the n-dimensional generalization of these is an n-simplex. We extend Ceva’s theorem to n-simplices and in doing so illustrate the considerations and choices that can be made in generalizing from plane geometry to high-dimensional geometries.
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