Bram Driesen | The University Of Glasgow (original) (raw)

Papers by Bram Driesen

ABSTRACT Robinson (1951) showed that the learning process of Discrete Ficti-tious Play converges ... more ABSTRACT Robinson (1951) showed that the learning process of Discrete Ficti-tious Play converges to Nash equilibrium in two-player zero-sum games for any initial condition. In several earlier works, Brown (1949, 1951) makes some heuristic arguments for a similar convergence result for the case of Continuous Fictitious Play (CFP). The standard reference for a formal proof is Harris (1998); his argument requires several technical lemmas, and moreover, involves the advanced machinery of Lyapunov functions. In this note we present a simple alternative proof. In particular, we show that Brown's convergence result follows easily from a result obtained by Monderer et al. (1997).

International Journal of Game Theory

This paper provides a non-cooperative foundation for (asymmetric generalizations of) the continuo... more This paper provides a non-cooperative foundation for (asymmetric generalizations of) the continuous Raiffa solution. Specifically, we consider a continuous-time variation of the classic Ståhl-Rubinstein bargaining model, in which there is a finite deadline that ends the negotiations, and in which each player's opportunity to make proposals is governed by a player-specific Poisson process, in that the rejecter of a proposal becomes proposer at the first next arrival of her process. Under the assumption that future payoffs are not discounted, it is shown that the expected payoffs players realize in subgame perfect equilibrium converge to the continuous Raiffa solution outcome as the deadline tends to infinity. The weights reflecting the asymmetries among the players correspond to the Poisson arrival rates of their respective proposal processes.

Journal of Mathematical Economics, 2016

People interested in the research are advised to contact the author for the final version of the ... more People interested in the research are advised to contact the author for the final version of the publication, or visit the DOI to the publisher's website. • The final author version and the galley proof are versions of the publication after peer review. • The final published version features the final layout of the paper including the volume, issue and page numbers. Link to publication General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal. If the publication is distributed under the terms of Article 25fa of the Dutch Copyright Act, indicated by the "Taverne" license above, please follow below link for the End User Agreement:

Mathematical Social Sciences, 2016

h i g h l i g h t s • We consider Thomson and Lensberg's (1989) characterization of the Leximin b... more h i g h l i g h t s • We consider Thomson and Lensberg's (1989) characterization of the Leximin bargaining solution. • We remove Pareto Optimality from their axiom set. • The remaining axioms characterize a class of Truncated Leximin solutions. • These truncate agents' Leximin solution payoffs at a given utility level α. • We discuss efficiency-free characterizations of the Leximin solution.

We consider bargaining problems under the assumption that players are loss averse, i.e., experien... more We consider bargaining problems under the assumption that players are loss averse, i.e., experience disutility from obtaining an outcome lower than some reference point. We follow the approach of Shalev (2002) by imposing the self-supporting condition on an outcome: an outcome z in a bargaining problem is self-supporting under a given bargaining solution, whenever transforming the problem using outcome z as a reference point, yields a transformed problem in which the solution is z. We show that n-player bargaining problems have a unique self-supporting outcome under the Kalai-Smorodinsky solution. For all possible loss aversion coefficients we determine the bargaining solutions that give exactly these outcomes, and characterize them by the standard axioms of Scale Invariance, Individual Monotonicity, and Strong Individual Rationality, and a new axiom called Proportional Concession Invariance (PCI). A bargaining solution satisfies PCI if moving the utopia point in the direction of the solution outcome does not change this outcome.

Social Choice and Welfare, 2015

We reconsider the class of weighted Kalai-Smorodinsky solutions of Dubra (2001), and using method... more We reconsider the class of weighted Kalai-Smorodinsky solutions of Dubra (2001), and using methods of Imai (1983), extend their characterization to the domain of multilateral bargaining problems. Aside from standard axioms in the literature, this result involves a new property that weakens the axiom Bilateral Consistency (Lensberg, 1988), by making the notion of consistency dependent on how ideal values in a reduced problem change relative to the original problem.

In this article we define and characterize a class of asymmetric leximin solutions, that contains... more In this article we define and characterize a class of asymmetric leximin solutions, that contains both the symmetric leximin solution of Imai[5] and the two-person asymmetric Kalai-Smorodinsky solution of Dubra [3] as special cases. Solutions in this class combine three attractive features: they are defined on the entire domain of convex n-person bargaining problems, they generally yield Pareto efficient solution

Economics Letters, 2012

In this article we introduce a new axiom for bargaining solutions, named Proportional Concession ... more In this article we introduce a new axiom for bargaining solutions, named Proportional Concession Monotonicity (PCM), which imposes that no player benefit when all players collectively make proportional concessions with respect to their respective utopia values. We reconsider the leximin solution (Imai, 1983), and obtain an alternative characterization on the basis of PCM.

The Journal of Applied Behavioral Science, 2010

Mathematical Social Sciences

We consider bargaining problems under the assumption that players are loss averse, i.e., experien... more We consider bargaining problems under the assumption that players are loss averse, i.e., experience disutility from obtaining an outcome lower than some reference point. We follow the approach of Shalev (2002) by imposing the self-supporting condition on an outcome: an outcome z in a bargaining problem is self-supporting under a given bargaining solution, whenever transforming the problem using outcome z as a reference point, yields a transformed problem in which the solution is z.We show that n-player bargaining problems have a unique self-supporting outcome under the Kalai–Smorodinsky solution. For all possible loss aversion coefficients we determine the bargaining solutions that give exactly these outcomes, and characterize them by the standard axioms of Scale Invariance, Individual Monotonicity, and Strong Individual Rationality, and a new axiom called Proportional Concession Invariance (PCI). A bargaining solution satisfies PCI if moving the utopia point in the direction of the...

This paper presents a general technique for comparing the concavity of different utility function... more This paper presents a general technique for comparing the concavity of different utility functions when probabilities need not be known. It generalizes: (a) Yaariʼs comparisons of risk aversion by not requiring identical beliefs; (b) Kreps and Porteusʼ information-timing preference by not requiring known probabilities; (c) Klibanoff, Marinacci, and Mukerjiʼs smooth ambiguity aversion by not using subjective probabilities (which are not

Theory and Decision, 2010

In this article three different types of loss aversion equilibria in bimatrix games are studied. ... more In this article three different types of loss aversion equilibria in bimatrix games are studied. Loss aversion equilibria are Nash equilibria of games where players are loss averse and where the reference points-points below which they consider payoffs to be lossesare endogenous to the equilibrium calculation. The first type is the fixed point loss aversion equilibrium, introduced in Shalev (2000; Int. J. Game Theory 29 under the name of 'myopic loss aversion equilibrium.' There, the players' reference points depend on the beliefs about their opponents' strategies. The second type, the maximin loss aversion equilibrium, differs from the fixed point loss aversion equilibrium in that the reference points are only based on the carriers of the strategies, not on the exact probabilities. In the third type, the safety level loss aversion equilibrium, the reference points depend on the values of the own payoff matrices. Finally, a comparative statics analysis is carried out of all three equilibrium concepts in 2 × 2 bimatrix games. It is established when a player benefits from his opponent falsely believing that he is loss averse.

Mathematical Social Sciences, 2011

We consider bargaining problems under the assumption that players are loss averse, i.e., experien... more We consider bargaining problems under the assumption that players are loss averse, i.e., experience disutility from obtaining an outcome lower than some reference point. We follow the approach of Shalev (2002) by imposing the self-supporting condition on an outcome: an outcome z in a bargaining problem is self-supporting under a given bargaining solution, whenever transforming the problem using outcome z as a reference point, yields a transformed problem in which the solution is z.

Mathematical Social Sciences, 2012

The Rubinstein alternating offers bargaining game is reconsidered under the assumption that each ... more The Rubinstein alternating offers bargaining game is reconsidered under the assumption that each player is loss averse and the associated reference point is equal to the highest turned down offer of the opponent in the past. This makes the payoffs and therefore potential equilibrium strategies dependent on the history of play. A subgame perfect equilibrium is constructed, in which the strategies depend on the history of play through the current reference points. It is shown that this equilibrium is unique under some assumptions that it shares with the equilibrium in the classical model: immediate acceptance of equilibrium offers, indifference between acceptance and rejection of such offers, and strategies depending only on the current reference points. It is also shown that in this equilibrium loss aversion is a disadvantage. Moreover, a relation with asymmetric Nash bargaining is established, where a player's bargaining power is negatively related to own loss aversion and positively to the opponent's loss aversion.

We consider bargaining games under the assumption that bargainers are loss averse, i.e. experienc... more We consider bargaining games under the assumption that bargainers are loss averse, i.e. experience disutility from obtaining an outcome lower than some reference point. We follow the approach of Shalev (2002) by imposing the self-supporting condition on a solution. Given a bargaining game, we say outcome z is self-supporting under a given bargaining solution, whenever transforming the game using outcome z as reference point, yields a transformed game in which the solution is z.We show that n-player bargaining games have a unique self-supporting outcome under the Kalai-Smorodinsky (KS) solution. We define a bargaining solution, giving exactly this outcome, and characterize it by the standard axioms of Scale Invariance [SI], Individual Monotonicity [IM], and Strong Individual Rationality [SIR], and a novel axiom called Proportional Concession Invariance [PCI].A bargaining solution satisfies PCI if moving the utopia point in the direction of the solution outcome, does not change this o...

ABSTRACT Robinson (1951) showed that the learning process of Discrete Ficti-tious Play converges ... more ABSTRACT Robinson (1951) showed that the learning process of Discrete Ficti-tious Play converges to Nash equilibrium in two-player zero-sum games for any initial condition. In several earlier works, Brown (1949, 1951) makes some heuristic arguments for a similar convergence result for the case of Continuous Fictitious Play (CFP). The standard reference for a formal proof is Harris (1998); his argument requires several technical lemmas, and moreover, involves the advanced machinery of Lyapunov functions. In this note we present a simple alternative proof. In particular, we show that Brown's convergence result follows easily from a result obtained by Monderer et al. (1997).

International Journal of Game Theory

This paper provides a non-cooperative foundation for (asymmetric generalizations of) the continuo... more This paper provides a non-cooperative foundation for (asymmetric generalizations of) the continuous Raiffa solution. Specifically, we consider a continuous-time variation of the classic Ståhl-Rubinstein bargaining model, in which there is a finite deadline that ends the negotiations, and in which each player's opportunity to make proposals is governed by a player-specific Poisson process, in that the rejecter of a proposal becomes proposer at the first next arrival of her process. Under the assumption that future payoffs are not discounted, it is shown that the expected payoffs players realize in subgame perfect equilibrium converge to the continuous Raiffa solution outcome as the deadline tends to infinity. The weights reflecting the asymmetries among the players correspond to the Poisson arrival rates of their respective proposal processes.

Journal of Mathematical Economics, 2016

People interested in the research are advised to contact the author for the final version of the ... more People interested in the research are advised to contact the author for the final version of the publication, or visit the DOI to the publisher's website. • The final author version and the galley proof are versions of the publication after peer review. • The final published version features the final layout of the paper including the volume, issue and page numbers. Link to publication General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal. If the publication is distributed under the terms of Article 25fa of the Dutch Copyright Act, indicated by the "Taverne" license above, please follow below link for the End User Agreement:

Mathematical Social Sciences, 2016

h i g h l i g h t s • We consider Thomson and Lensberg's (1989) characterization of the Leximin b... more h i g h l i g h t s • We consider Thomson and Lensberg's (1989) characterization of the Leximin bargaining solution. • We remove Pareto Optimality from their axiom set. • The remaining axioms characterize a class of Truncated Leximin solutions. • These truncate agents' Leximin solution payoffs at a given utility level α. • We discuss efficiency-free characterizations of the Leximin solution.

We consider bargaining problems under the assumption that players are loss averse, i.e., experien... more We consider bargaining problems under the assumption that players are loss averse, i.e., experience disutility from obtaining an outcome lower than some reference point. We follow the approach of Shalev (2002) by imposing the self-supporting condition on an outcome: an outcome z in a bargaining problem is self-supporting under a given bargaining solution, whenever transforming the problem using outcome z as a reference point, yields a transformed problem in which the solution is z. We show that n-player bargaining problems have a unique self-supporting outcome under the Kalai-Smorodinsky solution. For all possible loss aversion coefficients we determine the bargaining solutions that give exactly these outcomes, and characterize them by the standard axioms of Scale Invariance, Individual Monotonicity, and Strong Individual Rationality, and a new axiom called Proportional Concession Invariance (PCI). A bargaining solution satisfies PCI if moving the utopia point in the direction of the solution outcome does not change this outcome.

Social Choice and Welfare, 2015

We reconsider the class of weighted Kalai-Smorodinsky solutions of Dubra (2001), and using method... more We reconsider the class of weighted Kalai-Smorodinsky solutions of Dubra (2001), and using methods of Imai (1983), extend their characterization to the domain of multilateral bargaining problems. Aside from standard axioms in the literature, this result involves a new property that weakens the axiom Bilateral Consistency (Lensberg, 1988), by making the notion of consistency dependent on how ideal values in a reduced problem change relative to the original problem.

In this article we define and characterize a class of asymmetric leximin solutions, that contains... more In this article we define and characterize a class of asymmetric leximin solutions, that contains both the symmetric leximin solution of Imai[5] and the two-person asymmetric Kalai-Smorodinsky solution of Dubra [3] as special cases. Solutions in this class combine three attractive features: they are defined on the entire domain of convex n-person bargaining problems, they generally yield Pareto efficient solution

Economics Letters, 2012

In this article we introduce a new axiom for bargaining solutions, named Proportional Concession ... more In this article we introduce a new axiom for bargaining solutions, named Proportional Concession Monotonicity (PCM), which imposes that no player benefit when all players collectively make proportional concessions with respect to their respective utopia values. We reconsider the leximin solution (Imai, 1983), and obtain an alternative characterization on the basis of PCM.

The Journal of Applied Behavioral Science, 2010

Mathematical Social Sciences

We consider bargaining problems under the assumption that players are loss averse, i.e., experien... more We consider bargaining problems under the assumption that players are loss averse, i.e., experience disutility from obtaining an outcome lower than some reference point. We follow the approach of Shalev (2002) by imposing the self-supporting condition on an outcome: an outcome z in a bargaining problem is self-supporting under a given bargaining solution, whenever transforming the problem using outcome z as a reference point, yields a transformed problem in which the solution is z.We show that n-player bargaining problems have a unique self-supporting outcome under the Kalai–Smorodinsky solution. For all possible loss aversion coefficients we determine the bargaining solutions that give exactly these outcomes, and characterize them by the standard axioms of Scale Invariance, Individual Monotonicity, and Strong Individual Rationality, and a new axiom called Proportional Concession Invariance (PCI). A bargaining solution satisfies PCI if moving the utopia point in the direction of the...

This paper presents a general technique for comparing the concavity of different utility function... more This paper presents a general technique for comparing the concavity of different utility functions when probabilities need not be known. It generalizes: (a) Yaariʼs comparisons of risk aversion by not requiring identical beliefs; (b) Kreps and Porteusʼ information-timing preference by not requiring known probabilities; (c) Klibanoff, Marinacci, and Mukerjiʼs smooth ambiguity aversion by not using subjective probabilities (which are not

Theory and Decision, 2010

In this article three different types of loss aversion equilibria in bimatrix games are studied. ... more In this article three different types of loss aversion equilibria in bimatrix games are studied. Loss aversion equilibria are Nash equilibria of games where players are loss averse and where the reference points-points below which they consider payoffs to be lossesare endogenous to the equilibrium calculation. The first type is the fixed point loss aversion equilibrium, introduced in Shalev (2000; Int. J. Game Theory 29 under the name of 'myopic loss aversion equilibrium.' There, the players' reference points depend on the beliefs about their opponents' strategies. The second type, the maximin loss aversion equilibrium, differs from the fixed point loss aversion equilibrium in that the reference points are only based on the carriers of the strategies, not on the exact probabilities. In the third type, the safety level loss aversion equilibrium, the reference points depend on the values of the own payoff matrices. Finally, a comparative statics analysis is carried out of all three equilibrium concepts in 2 × 2 bimatrix games. It is established when a player benefits from his opponent falsely believing that he is loss averse.

Mathematical Social Sciences, 2011

We consider bargaining problems under the assumption that players are loss averse, i.e., experien... more We consider bargaining problems under the assumption that players are loss averse, i.e., experience disutility from obtaining an outcome lower than some reference point. We follow the approach of Shalev (2002) by imposing the self-supporting condition on an outcome: an outcome z in a bargaining problem is self-supporting under a given bargaining solution, whenever transforming the problem using outcome z as a reference point, yields a transformed problem in which the solution is z.

Mathematical Social Sciences, 2012

The Rubinstein alternating offers bargaining game is reconsidered under the assumption that each ... more The Rubinstein alternating offers bargaining game is reconsidered under the assumption that each player is loss averse and the associated reference point is equal to the highest turned down offer of the opponent in the past. This makes the payoffs and therefore potential equilibrium strategies dependent on the history of play. A subgame perfect equilibrium is constructed, in which the strategies depend on the history of play through the current reference points. It is shown that this equilibrium is unique under some assumptions that it shares with the equilibrium in the classical model: immediate acceptance of equilibrium offers, indifference between acceptance and rejection of such offers, and strategies depending only on the current reference points. It is also shown that in this equilibrium loss aversion is a disadvantage. Moreover, a relation with asymmetric Nash bargaining is established, where a player's bargaining power is negatively related to own loss aversion and positively to the opponent's loss aversion.

We consider bargaining games under the assumption that bargainers are loss averse, i.e. experienc... more We consider bargaining games under the assumption that bargainers are loss averse, i.e. experience disutility from obtaining an outcome lower than some reference point. We follow the approach of Shalev (2002) by imposing the self-supporting condition on a solution. Given a bargaining game, we say outcome z is self-supporting under a given bargaining solution, whenever transforming the game using outcome z as reference point, yields a transformed game in which the solution is z.We show that n-player bargaining games have a unique self-supporting outcome under the Kalai-Smorodinsky (KS) solution. We define a bargaining solution, giving exactly this outcome, and characterize it by the standard axioms of Scale Invariance [SI], Individual Monotonicity [IM], and Strong Individual Rationality [SIR], and a novel axiom called Proportional Concession Invariance [PCI].A bargaining solution satisfies PCI if moving the utopia point in the direction of the solution outcome, does not change this o...