Kalai, E., & Lehrer, E. (1993). Rational learning leads to Nash equilibrium. Econometrica, 61, 1019--1045.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... can only converge to a Nash equilibrium (if it converges at all) In addition, an important trend in modern game theory is the development of learning models, and there too, it has been shown that Nash equilibria result also from rational learning through repeated play among the same players [44]. A more general (and hence weaker) notion of stability is the existence of an ffl Nash equilibrium. Let the ffl best replies be S i (s Gammai ) fs i 2 S i (s Gammai ) u i (s i ; s Gammai ) u i (s i ; s Gammai ) Gamma ffl; 8s i 2 S i (s Gammai )g. An ffl Nash equilibrium is a ....

....(i.e. than they would at the efficient equilibrium) end up cancelling out each others advantages. These are very preliminary results and further work needs to be done. The appropriate analytical tools for this context can are in the branch of game theory that deals with learning and evolution [44, 23]. In conclusion, the auction game needs to be further understood in terms of user behaviour over time. The dual question is how to price resources over time, when the players do not play repeatedly, but rather only once. This arises naturally when considering connection oriented network ....

E. Kalai and E. Lehrer. Rational learning leads to Nash equilibrium. Econometrica, 61(5):1019--1045, September 1993.

....agents. The value of information associated with an action includes the information it provides about other agents s strategies, not just the environment model. Both of these changes require that an agent possess some model of the strategies of other agents, for which we adopt a Bayesian view [11]. Putting these together, we derive optimal exploration methods for (Bayesian) multiagent systems. After reviewing relevant background, and describing related work The existence of multiple equilibria can have a negative impact on known theoretical results for MARL [9, 10] in MARL, we ....

....the agents. With no means of breaking the symmetry, and the risk of incurring the penalty if they choose different optimal equilibria, the agents might in fact focus on the suboptimal equilibrium #a1,b1#. Learning models have become popular as a means tackling the equilibrium selection problem [16, 11, 7]. Assuming repeated play of some stage game, these methods require an agent to make some prediction about the play of others at the current stage based on the history of interactions, and play the current stage game using these predictions. One simple model is fictitious play [16] at each ....

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E. Kalai and E. Lehrer. Rational learning leads to Nash equilibrium. Econometrica, 61(5):1019--1045, 1993.

.... on the complexity on agents [47] In comparison, economic models of metadeliberation select a level of deliberation within a decisiontheoretic framework, based on the expected value of further deliberation [22, 54, 48] In contrast to the recent literature on bounded rational learning in games [32, 44], we assume in the portfolio selection problem that an agent s opponent (the market) plays the same strategy for all agent strategies. Prices do not depend on investment actions. Furthermore, there is no exploration versus exploitation problem, as occurs for example in the classic Multiarmed ....

Ehud Kalai and Ehud Lehrer. Rational learning leads to Nash equilibrium. Econometrica, 61(5):1019--1045, 1993.

....1994# deals with coordination and negotiation issues by giving pre computed solutions to speci#c problems. There has been much research reported on developing theoretical models in which learning plays an eminent role, especially in the area of adaptive dynamics of games #e.g. #Jordan 1992, Kalai Lehrer 1993##. However, to build autonomous agents that improve their negotiation competence based on learning from their interactions with other agents is still an emerging area. We are interested in developing autonomous agents capable of reasoning based on experience and improving their negotiation ....

....of beliefs about other agents. 4 Learning in Negotiation The importance of learning in negotiation has been recently recognized in the game research community as fundamental for understanding human behavior as well as for developing new solution concepts #Osborne Rubinstein 1994, Jordan 1992, Kalai Lehrer 1993#. Theoretical results #most of which are partial and preliminary#, however, are available only for the simplest game settings. Multi agent learning has also increasingly drawn research e#orts from Distributed AI community #e.g. #Mor, Goldman Rosenschein 1995, Sen Sekaran 10 1995##. In the ....

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Kalai, E. & Lehrer, E. #1993#. Rational learning leads to nash equilibrium, Econometrica 61#5#: 1019#1045.

....1994# deals with coordination and negotiation issues by giving pre computed solutions to speci#c problems. There has been much research reported on developing theoretical models in which learning plays an eminent role, especially in the area of adaptive dynamics of games #e.g. #Jordan 1992; Kalai Lehrer 1993##. However, to build autonomous agents that improve their negotiation competence based on learning from their interactions with other agents is still an emerging area. We are interested in developing autonomous agents capable of reasoning based on experience and improving their negotiation ....

.... agents #the distribution P over # Learning in Negotiation The importance of learning in negotiation has been recently recognized in the game research community as fundamental for understanding human behavior as well as for developing new solution concepts #Osborne Rubinstein 1994; Jordan 1992; Kalai Lehrer 1993#. Theoretical results #most of which are partial and preliminary #, however, are available only for the simplest game settings. Multi agent learning has also increasingly drawn research e#orts from Distributed AI community #e.g. #Mor, Goldman, Rosenschein 1995; Sen Sekaran 1995##. In the ....

Kalai, E., and Lehrer, E. 1993. Rational learning leads to nash equilibrium. Econometrica 61#5#:1019# 1045.

....two type two action game with two different states in 0 Q . The actions are labeled In and Out; the types are labeled Timid (T) and Brave (B) the 0 Q states are labeled L and R. Both types get a payoff of 0 from Out. Payoffs from In are given in the table below. 8 Battigalli [1988] and Kalai and Lehrer [1993] defined similar concepts. This is the unitary version of self confirming equilibria because it supposes that there is a single # # # for each player i. The unitary version of the concept is appropriate here because of our assumption that there is a single agent in each player role; when we ....

Kalai, Ehud and Ehud Lehrer [1993] "Rational Learning Leads to Nash Equilibrium," Econometrica; 61(5), 1019-45.

....use of stopping times, but argue that their definition is equivalent to this simpler one . 6 Our specification allows Q measures to put different probabilities on tail events, which prevents the measures from merging as Blackwell and Dubins (1962) show will occur under absolute continuity. See Kalai and Lerner (1993) and Jackson, Kalai, and Smordoninsky (1999) for implications of absolute continuity for learning. 5 Since any random variable that is F t measurable is also F t measurable, it follows from this equation that EQ g t = E P g t q t : The only way that this equation and (6) can both hold for ....

....dt where f B t : t 0g is a Brownian motion. Thus we parameterize Q by the drift distortion fh t g and express the distorted state evolution equation: dx t = c t ; x t )dt oe(c t ; x t )dB t = c t ; x t )dt oe(c t ; x t ) h t dt d B t ) 10) 10 See Blackwell and Dubins (1962) Kalai and Lerner (1993), and Jackson, Kalai, and Smordoninsky (1999) for some of the ramifications of absolute continuity. 8 Since f B t : t 0g remains a Brownian motion under all of the probability models Q 2 Q, we use E to denote the expectation operator that integrates over Brownian motion specification for ....

Kalai, E. and E. Lerner (1993). Rational learning leads to nash equilibrium. Econometrica 61 (5), 1019--1045.

.... models of the structure but simplified models of others decisions (Fudenberg and Kreps (1993) Crawford (1995) Crawford and Broseta (1998) Ho, Camerer, and Weigelt (1998) Camerer and Ho (1998) and models whose cognitive requirements approach those of the deductive rationale for equilibrium (Kalai and Lehrer (1993), Stahl (1996) Even the least sophisticated among these learning models have a strong tendency to converge to equilibrium in many environments. 9 The theory must be common knowledge to make players decisions mutual knowledge because otherwise a player might doubt whether the others know that ....

Kalai, Ehud, and Ehud Lehrer (1993): "Rational Learning Leads to Nash Equilibrium," Econometrica, 61, 1019-1045.

....of irrational or commitment types. Undiscounted repeated games of incomplete information have, however, been studied in some depth, especially in the zero sum case. 1 (A brief review of the relevant results is contained in Section 3. Some recent results exist for the discounted case, however. Kalai and Lehrer (1993) 1 and Jordan (1995) have established that play must converge to Nash play of the true game. Jordan (1995) has also proved the existence of an equilibrium for this class of games. Perfect Bayesian equilibria of such games must have a Markov property (Bergin (1989) McKelvey and Palfrey (1992, ....

Kalai, E., and E. Lehrer (1993): "Rational Learning Leads to Nash Equilibrium," Econometrica, 61, 1019--1045.

....make them indifferent. Our result extends this observation to the case where players are arbitrarily forward looking, and we show that it has important implications for prediction as well as for learning equilibrium. Of course, players can learn to predict under certain conditions. In particular, Kalai and Lehrer (1993) show that if the players initial beliefs about the others strategies are sufficiently correct to begin with, then with probability one they learn to predict more and more accurately as time goes on. Our result therefore implies that, in some games of incomplete information, no beliefs are ....

....by i s prior beliefs, and will be called i s forecasting function. Given any vector of forecasting functions f = f 1 , f 2 , f n ) it can be shown that there exists a set of prior beliefs such that the f i describe the period by period forecasts of players with these beliefs (see Kalai and Lehrer, 1993). 9 Consider the situation just after the players have been informed privately of their payoff functions u i . Because of the independence of the draws among players, no one knows anything he did not already know about the others payoffs. This has an implication for the forecasting functions. ....

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Kalai, Ehud, and Ehud Lehrer (1993): "Rational Learning Leads to Nash Equilibrium," Econometrica, 61, 1019-1045.

....we argue that if the players employ predictive learning algorithms, assuming rationality, play does not converge to Nash equilibrium in sfbp. Equivalently, if play converges to Nash equilibrium, then either play is not rational or play is not learned. In a seminal work by Kalai and Lehrer [10], sucient conditions are presented for predictivity speci cally, an absolute continuity assumption which suggests that convergence to Nash equilibrium is at least possible. Our negative results complement the work of Nachbar [12] and Foster and Young [6] who argue that the conditions sucient for ....

E. Kalai and E. Lehrer. Rational learning leads to Nash equilibrium. Econometrica, 61:1019-1045, 1993.

....so far, and always plays the best response to this model at each iteration (Owen, 1995) While it is known that the time averages of the strategies played form a Nash equilibrium, the strategies themselves do not converge to Nash, nor are the averaged payoffs to the players guaranteed to be Nash. Kalai and Lehrer (1993) proposed a Bayesian strategy for players in a repeated game that requires the players to have informed priors , and showed that under this condition play converges to a Nash equilibrium. A series of recent results has shown that the informed prior condition is actually quite restrictive, ....

Kalai E. and Lehrer E. (1993). Rational Learning leads to Nash Equilibrium. Econometrica.

.... can only converge to a Nash equilibrium (if it converges at all) In addition, an important trend in modern game theory is the development of learning models, and there too, it has been shown that Nash equilibria result also from rational learning through repeated play among the same players [13]. A more general (and hence weaker) notion of stability is the existence of an ffl Nash equilibrium. Let the ffl best replies be S ffl i (s Gammai ) fs i 2 S i (s Gammai ) u i (s i ; s Gammai ) u i (s 0 i ; s Gammai ) Gamma ffl; 8s 0 i 2 S i (s Gammai )g. An ffl Nash equilibrium ....

E. Kalai and E. Lehrer. Rational learning leads to Nash equilibrium. Econometrica, 61(5):1019--1045, September 1993.

....modeled in the theoretical literature on evolutionary behavior and learning dynamics. The standard assumption in this literature is that players do not attempt to influence other players future actions. Naive learning in this context can result in behavior that converges to a Nash equilibrium (Kalai and Lehrer, 1993, and Jordan, 1991) Friedman (1991) Mailath (1992) and Marimon and McGrattan (1995) survey theoretical work in this area. 1 For a discussion of this approach, see Kohlberg and Mertens (1986) and Cho and Kreps (1987) 2 A theoretical criterion for systematically ruling out a subset of Nash ....

Kalai, Ehud and Ehud Lehrer, 1993, Rational learning leads to Nash equilibrium, Econometrica 61, 1019-1045.

....Economics The work on learning in economics falls roughly into two camps. The highrationality approach involves learning algorithms that aim to predict their opponents strategies, and then optimize with respect to those predictions. Prediction methods can be Bayesian (as in Kalai and Lehrer [32]) calibrated (as in Foster and Vohra [13] or consistent (as in Fudenberg and Levine [21, 20] just to name a few examples. Typically, the asymptotic play of highrationality learning is either a correlated or Nash equilibrium. Since these algorithms depend on knowledge of the underlying ....

E. Kalai and E. Lehrer. Rational learning leads to Nash equilibrium. Econometrica, 61:1019-1045, 1993.

....in order to induce player 2 to experiment on the same arm and extract information from 2 s subsequent moves. For this reason, it is not clear if direct communication among players leads to honest information sharing. 2 See, for example, Jordan [14, 15] and Nyarko [20] See also Kalai and Lehrer [16] for a discussion on Rothschild s [21] conclusion on a single person two armed bandit model. 4 in each trial) of arms X and Y , respectively. The requirement is that x and y be different by a fixed margin with probability one. The conclusion is valid even if players begin with some private ....

E. Kalai and E. Lehrer, Rational learning leads to Nash equilibrium, Econometrica 61 (1993), 1019-1045.

....The work on learning falls roughly into two camps. The high rationality approach involves learning algorithms which aim to predict the strategies of their opponents, and myopically optimize with respect to those predictions. The prediction methods can be Bayesian (as in Kalai and Lehrer [33]) calibrated (as in Foster and Vohra [18] or consistent (as in Fudenberg and Levine [22, 21] Typically the asymptotic play of such algorithms are either correlated or Nash equilibria. Since these algorithms depend on knowledge of the underlying structure of the game, they are not applicable ....

E. Kalai and E. Lehrer. Rational learning leads to Nash equilibrium. Econometrica, 61:1019--1045, 1993.

....1994) deals with coordination and negotiation issues by giving pre computed solutions to specific problems. There has been much research reported on developing theoretical models in which learning plays an eminent role, especially in the area of adaptive dynamics of games (e.g. Jordan 1992, Kalai Lehrer 1993)) However, to build autonomous agents that improve their negotiation competence based on learning from their interactions with other agents is still an emerging area. We are interested in developing autonomous agents capable of reasoning based on experience and improving their negotiation ....

....of beliefs about other agents. 4 Learning in Negotiation The importance of learning in negotiation has been recently recognized in the game research community as fundamental for understanding human behavior as well as for developing new solution concepts (Osborne Rubinstein 1994, Jordan 1992, Kalai Lehrer 1993). Theoretical results (most of which are partial and preliminary) however, are available only for the simplest game settings. Multi agent learning has also increasingly drawn research efforts from Distributed AI community (e.g. Mor, Goldman Rosenschein 1995, Sen Sekaran 1995) In the ....

[Article contains additional citation context not shown here]

Kalai, E. & Lehrer, E. (1993). Rational learning leads to nash equilibrium, Econometrica 61(5): 1019--1045.

....of myopic best response dynamics (Young, 1993; Kandori et al. 1993) A common assumption, that is used to justify myopic play, is that agents play a random matching game in a stationary environment, which ignores the possibility that other agents in the game might also adapt their strategies. Kalai and Lehrer (1993) and Milgrom and Roberts (1991) allow for more sophisticated learning. Multiagent systems research complements this literature with a more principled normative approach to choosing an appropriate level of complexity with which to model other agents in a game this can be somewhere between a full ....

Kalai, E., and Lehrer, E. (1993). "Rational learning leads to Nash equilibrium," Econometrica 61, 1019--1045.

....1994) deals with coordination and negotiation issues by giving pre computed solutions to specific problems. There has been much research reported on developing theoretical models in which learning plays an eminent role, especially in the area of adaptive dynamics of games (e.g. Jordan 1992; Kalai Lehrer 1993)) However, to build autonomous agents that improve their negotiation competence based on learning from their interactions with other agents is still an emerging area. We are interested in developing autonomous agents capable of reasoning based on experience and improving their negotiation ....

.... P over Omega Gamma Learning in Negotiation The importance of learning in negotiation has been recently recognized in the game research community as fundamental for understanding human behavior as well as for developing new solution concepts (Osborne Rubinstein 1994; Jordan 1992; Kalai Lehrer 1993). Theoretical results (most of which are partial and preliminary) however, are available only for the simplest game settings. Multi agent learning has also increasingly drawn research efforts from Distributed AI community (e.g. Mor, Goldman, Rosenschein 1995; Sen Sekaran 1995) In the ....

Kalai, E., and Lehrer, E. 1993. Rational learning leads to nash equilibrium. Econometrica 61(5):1019-- 1045.

....with the game situation. Learning techniques have been well studied in game theory, not only for coordination in cooperative games, but also for the more general problem of equilibrium selection [12, 5] Models applied to this problem include fictitious play [13] and Bayesian best response methods [8, 19, 4] (evolutionary models have also attracted a great deal of attention [1, 11] These have especially nice behavior in coordination problems [19] However, these models tend to assume that each agent can observe the exact action performed by all others at each interaction. Such action observable ....

.... a convention restricts (or forces) consideration to a subset of feasible or optimal joint actions (such as the convention of driving on the right hand side of the street) Finally, coordinated action choice might be learned through repeated play of the game, either with the same agents [4, 8, 10] or a random selection of similar agents [1, 15, 11, 19] We focus here on learning models in which agents repeatedly interact with the same set of players in state games. In this section, we assume that each agent can observe the actions of the others at each interaction. Intuitively, each agent ....

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Ehud Kalai and Ehud Lehrer. Rational learning leads to nash equilibrium. Econometrica, 61(5):1019--1045, 1993.

....their opponents, a Nash equilibrium is a point at which (if it is ever reached) this iteration would naturally stop. A strong trend in recent game theoretic research is the development of models where Nash equilibrium play results from rational learning through repeated play among the same players [KL93, AS95]. 3.1 Specifications We now present some properties we would like the auction game described above to have. In other words, these are the specifications for the mechanism design. The specifications come from fundamental characteristics in the absence of which the validity of this approach to ....

Ehud Kalai and Ehud Lehrer, "Rational learning leads to nash equilibrium," Econometrica, vol. 61, pp. 1019--1045, September 1993.

....learners (ILs) apply Q learning in the classic sense, ignoringthe existence of other agents. Joint action learners (JALs) in contrast, learn the value of their own actions in conjunction with those of other agents via integration of RL with equilibrium (or coordination) learning methods [24, 5, 6, 9]. We then briefly consider the importance of exploitive exploration strategies and examine, through a series of examples, how game structure and exploration strategies influence the dynamics of the learning process and the convergence to equilibrium. We show that both JALs and ILs will converge to ....

.... instance, communication between agents might be admitted [22] or one could impose conventions or rules that restrict behavior so as to ensure coordination [18] Here we entertain the suggestion that coordinated action choice might be learned through repeated play of the game with the same agents [5, 6, 9, 11]. Repeated play with a random selection of similar agents from a large population has also been the object of considerable study [17, 10, 24] we will see that interesting issues emerge. One especially simple, yet often effective, learning model for achieving coordination is fictitious play ....

E. Kalai and E. Lehrer. Rational learning leads to Nash equilibrium. Econometrica, 61(5):1019--1045, 1993.

.... can only converge to a Nash equilibrium (if it converges at all) In addition, the dominant trend in modern game theory is the development of learning models, and there too, it has been shown that Nash equilibria result also from rational learning through repeated play among the same players [3]. Thus, we take the existence of a Nash equilibrium as the definition stability. Define the partial ordering on S to be the usual component wise relation: x = x 1 ; x 2 ) y = y 1 ; y 2 ) if x 1 x 2 and y 1 y 2 . Let x y = x 1 y 1 ; x 2 y 2 ) 0 20 40 60 80 100 0 5 10 15 0 100 200 300 ....

E. Kalai and E. Lehrer. Rational learning leads to nash equilibrium. Econometrica, 61(5):1019--1045, September 1993.

....1995) deals with coordination and negotiation issues by giving pre computed solutions to specific problems. There has been much research reported on developing theoretical models in which learning plays an eminent role, especially in the area of adaptive dynamics of games (e.g. Jordan 1992; Kalai Lehrer 1993)) However, to build autonomous agents that improve their negotiation competence based on learning from their interactions with other agents is still an emerging area. Learning in negotiation is closely coupled with the issue of how to model the overall negotiation process, i.e. what negotiation ....

....community as fundamental for understanding human behavior as well as for developing new solution concepts (Osborne Rubinstein 1994; Harsanyi Selten 1972) In (Jordan 1992) the author studied the impact of Bayesian learning processes for finite strategy normal form games. Kalai and Lehrer (Kalai Lehrer 1993) analyzed infinitely repeated games in which players as subjective utility maximizers learn to predict opponents future strategies. These theoretical results, however, are available only for the simplest game settings and valid only under very restrictive assumptions such as only a subset of ....

Kalai, E., and Lehrer, E. 1993. Rational learning leads to nash equilibrium. Econometrica 61(5):1019-- 1045.

.... makers include: a) the design of conventions or social laws that restrict agents to selecting coordinated actions [9, 15] b) allowing communication among agents before action selection [16] and (c) the use of learning methods, whereby agents learn to coordinate through repeated interaction [5, 6, 8, 11]. Unfortunately, none of these approaches explicitly considers the impact of coordination problems in the context of larger sequential decision problems. If the agents run the risk of miscoordination at a certain state in a decision problem, how should this impact their policy decisions at other ....

E. Kalai and E. Lehrer. Rational learning leads to Nash equilibrium. Econometrica, 61(5):1019--1045, 1993.

....understanding of rationality: players choose strategies which maximize their immediate payoffs, given their beliefs. Rational learning is generally accepted to mean Bayesian updating, namely the updating of prior beliefs in light of new information to obtain posterior beliefs. Kalai and Lehrer [5] prove that repeated play of strategic form games among two rational players (in the two fold sense described above) converges to approximate Nash equilibrium (specifically, ffl Nash equilibrium) provided that players initial belief sets contain a grain of truth: i.e. players assign positive ....

....Figure 2) Let the set of neutral conventional strategies be fH ; T ; randomize (50 : 50)g, where H is heads always and T is tails always. Assume grain of truth; in particular, let both players assign positive probability to all conventional strategies. By results in Kalai and Lehrer [5], conventional prediction holds. In particular, if player 2 plays H , then player 1 eventually learns that player 2 is playing H . However, conventional optimization fails since none of the conventional strategies have the flexibility to adjust to learned information. We might try adding ....

Ehud Kalai and Ehud Lehrer. Rational learning leads to nash equilibrium. Econometrica, 61:1019--1045, 1993.

....one, it also has a couple of additional drawbacks in terms of interpretability and applicability. Section 2 presents the model of [AS95] and then summarizes their main results. Section 3 discusses the results and compares them with those of the traditional approach, using Kalai and Lehrer [KL93] as the benchmark for the latter. 2 Summary of Anderlini and Sabourian 2.1 Model The model consists of N populations of players participating in an N player normal form game. A population i has a fixed strategy space and payoff function. Each population contains a (possibly infinite) number of ....

....as to what will happen in an actual game in terms of the sequence of plays. They become useful only when, with the additional assumption of monotonic growth, they lead to Theorem 6. 3. 1 Interpretation of the model In the probabilistic learning models, typically (see e.g. Kalai and Lehrer [KL93]) the model has a fixed set of players with fixed learning strategies. Morevoer, their learning is assumed to be rational, i.e. purely by Bayesian updating of a set of a priori beliefs on opponents strategies. The updating takes place on the information from past play, specifically on the sample ....

[Article contains additional citation context not shown here]

Ehud Kalai and Ehud Lehrer, "Rational learning leads to nash equilibrium, " Econometrica, vol. 61, pp. 1019--1045, September 1993.

....learners (ILs) apply Q learning in the classic sense, ignoringthe existence of other agents. Joint action learners (JALs) in contrast, learn the value of their own actions in conjunction with those of other agents via integration of RL with equilibrium (or coordination) learning methods [24, 6, 5, 10]. We also examine the influence of partial observability on JALs, and how game structure and exploration strategies influence the dynamics of the learning process and the convergence to equilibrium. We conclude by mentioning several problems that promise to make the integration of RL with ....

.... communication between agents might be admitted [22, 23] or one could impose conventions or rules that restrict behavior so as to ensure coordination [12, 19] Here we entertain the suggestion that coordinated action choice might be learned through repeated play of the game with the same agents [5, 6, 10, 13]. Repeated play with a random selection of similar agents from a large population has also been the object of considerable study [1, 18, 11, 24] One especially simple, yet often effective, learning model for achieving coordination is fictitious play [4, 5] Each agent i keeps a count C j a j ....

Ehud Kalai and Ehud Lehrer. Rational learning leads to Nash equilibrium. Econometrica, 61(5):1019--1045, 1993.

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E. Kalai and E. Lehrer, Rational learning leads to Nash equilibrium, Econometrica 61 (1993), 1019#1045.

....to, one shot implementation even in problems that admit intertemporal solutions such as the growth model or sequential candidate choice. 2) If agents also do not learn or if they are extremely impatient, then we are constrained to Bayesian implementation. 3) If agents do learn, the results of Kalai and Lehrer (1993) and others tell us they will eventually play a Nash equilibrium in the true environment. So even if the planner does not learn, if the planner is patient enough then Nash implementation may be possible. 4) Surprisingly, we will show below that if the planner learns and is more patient than the ....

....the most general conditions would only cloud the issues by turning this into a learning, rather than an implementation, paper. We note, however, that the inference used in our simple mechanism is a special case of Bayesian learning, and we refer the interested reader to papers of Jordan (1991) and Kalai and Lehrer (1993) for more powerful generalizations. 5 2. Patient Implementation by Dominant Strategies The Environment A, with generic elements a,b, denotes a set of social alternatives. N, with generic elements i,j, denotes a finite set of economic agents. Q i , with generic elements q i , denotes a finite ....

Kalai, Ehud and E. Lehrer. 1993. Rational Learning Leads to Nash Equilibrium, Econometrica, Vol. 61, No. 5 1019-1045.

....to, one shot implementation even in problems that admit intertemporal solutions such as the growth model or sequential candidate choice. 2) If agents also do not learn or if they are extremely impatient, then we are constrained to Bayesian implementation. 3) If agents do learn, the results of Kalai and Lehrer (1993) and others tell us they will eventually play a Nash equilibrium in the true environment. So even if the planner does not learn, if the planner is patient enough then Nash implementation may be possible. 4) Somewhat surprisingly, we will show below that if the planner learns and is more patient ....

....the most general conditions would only cloud the issues by turning this into a learning, rather than an implementation, paper. We note, however, that the inference used in our simple mechanism is a special case of Bayesian learning, and we refer the interested reader to papers of Jordan (1991) and Kalai and Lehrer (1993) for more powerful generalizations. 5 2. Patient Implementation by Dominant Strategies The Environment A, with generic elements a,b, denotes a set of social alternatives. N, with generic elements i,j, denotes a finite set of economic agents. Q i , with generic elements q i , denotes a finite ....

Kalai, Ehud and E. Lehrer. 1993. "Rational Learning Leads to Nash Equilibrium," Econometrica, Vol. 61, No. 5 1019-1045.

....for later generations. To demonstrate this conclusion in the generality stated above, this paper combines purification ideas of Harsanyi [12] with results from the recent literature on rational learning (relating to the merging of measures as shown by Blackwell and Dubins [2] and Kalai and Lehrer [18]) Recent criticisms of the Kalai and Lehrer approach center on the assumption that the prior assigns positive probability 5 to single vectors of types (see Jordan [16] Nachbar 4 An unjustified false reputation refers to a false reputation in a homogeneous population. 5 Similar criticisms ....

....t jU i is a mixed strategy composed of j i;h t jv i and a second strategy (j ijC with C being the complement of v i ) 16 which assigns strictly positive probability to j i;h t jv i under the non isolation condition. A result about merging (e.g. Theorem 3 in Kalai and Lehrer [18]) is then sufficient to guarantee that Part (i) of the definition of ffi subjective equilibrium is satisfied after a sufficiently long random time T. Proof of Theorem 1: Fix fl 0 and v such that D( v OE ) 0 for all OE 0. By Lemma 1, find ffi and such that for any ffi ffi and ....

Kalai, E. and E. Lehrer [1993a], "Rational Learning Leads to Nash Equilibrium," Econometrica, Vol. 61, pp. 1019--1045.

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Kalai, E., & Lehrer, E. (1993). Rational learning leads to Nash equilibrium. Econometrica, 61, 1019--1045.

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Kalai, E., & Lehrer, E. (1993). Rational learning leads to Nash equilibrium. Econometrica, 61, 1019--1045.

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Ehud Kalai and Ehud Lehrer. Rational learning leads to Nash equilibrium. Econometrica, 61(5):1019-1045, 1993.

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E. Kalai and E. Lehrer. Rational learning leads to nash equilibrium. Econometrica, 61(5):1019--1045, 1993.

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Ehud Kalai and Ehud Lehrer. Rational learning leads to nash equilibrium. Econometrica, 61(5):1019--1045, 1993.

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E. Kalai and E. Lehrer. Rational learning leads to nash equilibrium. Econometrica, 61(5):1019--1045, September 1993.

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E. Kalai and E. Lehrer. Rational learning leads to nash equilibrium, 1993.

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Ehud Kalai and Ehud Lehrer. Rational learning leads to nash equilibrium. Econometrica, 61(5):1019--1045, 1993.

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E. Kalai and E. Lehrer, Rational learning leads to Nash equilibrium, Econometrica 61 (1993), 1019#1045.

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E. Kalai and E. Lehrer. Rational learning leads to Nash equilibrium. Econometrica, 61:1019--1045, 1993.

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E. Kalai and E. Lehrer. Rational Learning Leads to Nash Equilibrium. Econometrica, 61(5):1019{ 1045, September 1993. 36

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E. Kalai and E. Lehrer. Rational learning leads to Nash equilibrium. Econometrica, 61:1019-1045, 1993.

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E. Kalai and E. Lehrer. Rational learning leads to Nash equilibrium. Econometrica, 61(5):1019--1045, 1993.

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Ehud Kalai and Ehud Lehrer. Rational learning leads to Nash equilibrium. Econometrica, 61(5):1019-1045, 1993.

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E. Kalai and E. Lehrer, Rational learning leads to Nash equilibrium, Econometrica 61, No. 5 (1993), 1019#1047.

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Ehud Kalai and Ehud Lehrer. Rational learning leads to Nash equilibrium. Econometrica, 61(5):1019--1045, September 1993.

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Ehud Kalai and Ehud Lehrer. Rational learning leads to Nash equilibrium. Econometrica, 61:1019--1045, 1993.

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