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## On the Implications of Lookahead Search in Game Playing (2012)

### Citations

5441 | Artificial Intelligence: A Modern Approach, 2nd edition - Russell, Norvig - 763 |

2526 | The foundations of statistics - Savage - 1954 |

2519 |
The Theory of Industrial Organization
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Citation Context ...rial Organisation: Cournot Competition Next we consider the classical game theoretic topic of duopolistic competition. Economists have considered a number of alternative models for market competition =-=[75]-=-, prominent amongst them is the Cournot model [17]. Our main result here is that the social surplus increases when firms are not myopic; surprisingly, social welfare is actually maximized when firms u... |

2458 | Judgement Under Uncertainty: Heuristics and Biases - Tversky, Kahneman - 1974 |

1435 |
A behavioral model of rational choice
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Citation Context ...ther than by the realism of its assumptions. On this scale rationality often (but not always) does very well. However, motivated by considerations of computational power and predictive ability, Simon =-=[69]-=- argued that “the task is to replace the global rationality of economic man with a kind of rational behaviour that is compatible with the access to information and the computational capacities that ar... |

1060 |
The Methodology of Positive Economics
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Citation Context ...ximisation. Whilst the non-existence of economic man is not in doubt, rationality remains a central assumption in economic thought. This is typically justified using an as if as expounded by Friedman =-=[26]-=-: whether people are actually rationality or not is unimportant provided their actions can be viewed in a way that is consistent with rational decision making - that is, provided agents act as if they... |

846 |
Heuristics: Intelligent Search Strategies for Computer Problem Solving
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(Show Context)
Citation Context ...duction Our goal here is not to prescribe how games should be played. Rather, we wish to analyse how games actually are played. To wit we consider the strategy of lookahead search, described by Pearl =-=[58]-=- in in his classical book on heuristic search as being used by “almost all game-playing programs”. To understand the lookahead method and the reasons for its ubiquity in practice, consider an agent tr... |

627 | Rational Choice and the Structure of the Environment - Simon - 1956 |

603 | Sources of power: How people make decisions. - Klein - 1998 |

541 |
Subjectivity and correlation in randomized strategies
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Citation Context ...rgence, and predictability of equilibria and game dynamics, various equilibrium concepts other than Nash equilibria have been studied in the economics literature. Among them are correlated equilibria =-=[2]-=-, stable equilibria [44], stochastic adjustment models [38], strategy subsets closed under rational behaviour (CURB set) [6], iterative elimination of dominated strategies, the set of undominated stra... |

525 | The evolution of conventions
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(Show Context)
Citation Context ...at is found in the search, this aspiration level may be changed. 5In fact, Simon sent his student George Baylor to help translate De Groot’s work into English. 6dynamics, noisy best-response dynamics =-=[20, 79, 51]-=-, where players occasionally make mistakes, and simultaneous Nash dynamics [7], where all players change their strategies simultaneously, are both well-studied. In many other models the effect of lear... |

366 | Maps of bounded rationality: Psychology for behavioral economics. - Kahneman - 2003 |

357 | Bounded Rationality in Macroeconomics. - Sargent - 1993 |

338 | Why bounded rationality - Conlisk - 1996 |

336 | A simple adaptive procedure leading to correlated equilibrium.
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Citation Context ...7], where all players change their strategies simultaneously, are both well-studied. In many other models the effect of learning algorithms [80] is examined, for example, regret minimisation dynamics =-=[25, 32, 33, 10, 8, 9, 19]-=- and fictitious play [11]. In most of these studies the most important factor is the stability of equilibria, and not measurements of the social value of equilibria. Furthermore, most of them are moti... |

323 | Position auctions - Varian - 2006 |

322 | On the Strategic Stability of Equilibria.
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(Show Context)
Citation Context ...ity of equilibria and game dynamics, various equilibrium concepts other than Nash equilibria have been studied in the economics literature. Among them are correlated equilibria [2], stable equilibria =-=[44]-=-, stochastic adjustment models [38], strategy subsets closed under rational behaviour (CURB set) [6], iterative elimination of dominated strategies, the set of undominated strategies, etc. Convergence... |

316 |
Recherches sur les Principes Mathématiques de la Théorie des Richesses
- Cournot
- 1838
(Show Context)
Citation Context ...sider the classical game theoretic topic of duopolistic competition. Economists have considered a number of alternative models for market competition [75], prominent amongst them is the Cournot model =-=[17]-=-. Our main result here is that the social surplus increases when firms are not myopic; surprisingly, social welfare is actually maximized when firms use 2-lookahead. The Cournot model assumes players ... |

261 |
Unraveling in guessing games: an experimental study
- Nagel
- 1995
(Show Context)
Citation Context ...icial intelligence community [55]; for examples in effective diagnostics and real-time planning see [40] and [63]. Lookahead search is also related to the sequential thinking framework in game theory =-=[52, 73]-=-. However, compared to these works and the research carried out by the two schools above, our focus is more theoretical and less experimental and psychological. Specifically, we desire quantitative pe... |

258 | Rational fools: A critique of the behavioral foundations of economic theory. - Sen - 1990 |

249 | On players’ models of other players: Theory and experimental evidence. - Stahl, Wilson - 1995 |

226 |
Iterative Solutions of Games by Fictitious Play,
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(Show Context)
Citation Context ...imultaneously, are both well-studied. In many other models the effect of learning algorithms [80] is examined, for example, regret minimisation dynamics [25, 32, 33, 10, 8, 9, 19] and fictitious play =-=[11]-=-. In most of these studies the most important factor is the stability of equilibria, and not measurements of the social value of equilibria. Furthermore, most of them are motivated by theoretical game... |

208 |
Programming a computer for playing chess,”
- Shannon
- 1950
(Show Context)
Citation Context ...that makes lookahead search such a natural method, especially for humans and for dynamic (or repeated) games.2 Interestingly, the lookahead method was formally proposed as long ago as 1950 by Shannon =-=[68]-=-, who considered it a practical way for machines to tackle complex problems that require “general principles, something of the nature of judgement, and considerable trial and error, rather than a stri... |

185 |
Experimental evidence on players’ models of other players
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- 1994
(Show Context)
Citation Context ...icial intelligence community [55]; for examples in effective diagnostics and real-time planning see [40] and [63]. Lookahead search is also related to the sequential thinking framework in game theory =-=[52, 73]-=-. However, compared to these works and the research carried out by the two schools above, our focus is more theoretical and less experimental and psychological. Specifically, we desire quantitative pe... |

179 | The Sciences of the Artificial, 3rd edition, - Simon - 1996 |

169 | The complexity of pure Nash equilibria. In:
- Fabrikant, Papadimitriou, et al.
- 2004
(Show Context)
Citation Context ...f them capture lookahead dynamics. In another line of work, convergence of best-response dynamics to (approximate) equilibria and the complexity of game dynamics and sink equilibria have been studied =-=[22, 1, 14, 72, 21, 49]-=-, but our paper does not focus on these types of dynamics or convergence to equilibria. Motivated by concerns of stability, convergence, and predictability of equilibria and game dynamics, various equ... |

140 | The price of routing unsplittable flow.
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(Show Context)
Citation Context ... + 12E(f ′′|f) where f ′′ is as above. Theorem 4.1. In the average-case 2-lookahead leaf model, the coordination ratio for an equilibrium is at most (1 + √ 5)2. Proof. This proof adapts the result in =-=[3]-=- to our setting. Let f be any flow at a lookahead equilibrium and f∗ be an optimal flow. Suppose player i is taking path Pj in flow f and path P ∗j in flow f ∗. Let J(e) be the set of players using ed... |

138 |
Strategic Learning and Its Limits.
- Young
- 2004
(Show Context)
Citation Context ...occasionally make mistakes, and simultaneous Nash dynamics [7], where all players change their strategies simultaneously, are both well-studied. In many other models the effect of learning algorithms =-=[80]-=- is examined, for example, regret minimisation dynamics [25, 32, 33, 10, 8, 9, 19] and fictitious play [11]. In most of these studies the most important factor is the stability of equilibria, and not ... |

124 | Nash equilibria in competitive societies, with applications to facility location, traffic routing and auctions, - Vetta - 2002 |

114 | Calibrated learning and correlated equilibrium.
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Citation Context ...7], where all players change their strategies simultaneously, are both well-studied. In many other models the effect of learning algorithms [80] is examined, for example, regret minimisation dynamics =-=[25, 32, 33, 10, 8, 9, 19]-=- and fictitious play [11]. In most of these studies the most important factor is the stability of equilibria, and not measurements of the social value of equilibria. Furthermore, most of them are moti... |

83 | Sink equilibria and convergence.
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- 2005
(Show Context)
Citation Context ...lookahead equilibria can be combined with other techniques to bound the coordination ratio for k-lookahead dynamics. We show how to do this for congestions games in Section 4; see also Goemans et al. =-=[30]-=- for several examples with respect to 1-lookahead dynamics. Consequently, for both simplicity and brevity, most of the results we give here concern the coordination ratio for lookahead equilibria. We ... |

77 | Adaptive heuristics.
- Hart
- 2005
(Show Context)
Citation Context ...7], where all players change their strategies simultaneously, are both well-studied. In many other models the effect of learning algorithms [80] is examined, for example, regret minimisation dynamics =-=[25, 32, 33, 10, 8, 9, 19]-=- and fictitious play [11]. In most of these studies the most important factor is the stability of equilibria, and not measurements of the social value of equilibria. Furthermore, most of them are moti... |

73 | A.: Convergence to approximate Nash equilibria in congestion games.
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(Show Context)
Citation Context ...f them capture lookahead dynamics. In another line of work, convergence of best-response dynamics to (approximate) equilibria and the complexity of game dynamics and sink equilibria have been studied =-=[22, 1, 14, 72, 21, 49]-=-, but our paper does not focus on these types of dynamics or convergence to equilibria. Motivated by concerns of stability, convergence, and predictability of equilibria and game dynamics, various equ... |

70 | Greedy bidding strategies for keyword auctions.
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Citation Context ...her bidders corresponds to a range of bid values that will result in the same outcome from i’s perspective. Among these set of bid values, we focus on a specific bid value bi, called the balanced bid =-=[13]-=-. The balanced bid bi is a best-response bid that is as high as possible such that player i cannot be harmed by a player with a better slot undercutting him, i.e. bidding just below him. It is easy to... |

70 | Market sharing games applied to content distribution in ad hoc networks,” - Goemans, Li, et al. - 2006 |

66 | Internal consistency of choice. - Sen - 1993 |

61 | On the impact of combinatorial structure on congestion games.
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Citation Context ...f them capture lookahead dynamics. In another line of work, convergence of best-response dynamics to (approximate) equilibria and the complexity of game dynamics and sink equilibria have been studied =-=[22, 1, 14, 72, 21, 49]-=-, but our paper does not focus on these types of dynamics or convergence to equilibria. Motivated by concerns of stability, convergence, and predictability of equilibria and game dynamics, various equ... |

59 | Regret minimization and the price of total anarchy
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Citation Context ...of game dynamics to approximately optimal solutions [50, 30] and to sink equilibria [30, 21]. Numerous articles study the convergence rate of best-response dynamics to approximately optimal solutions =-=[15, 23, 4, 9]-=-. For example, polynomial-time bounds has been proven for the speed of convergence to approximately optimal solutions for approximate Nash dynamics in a large class of potential games [4], and for lea... |

58 | Routing without regret: on convergence to Nash equilibria of regret-minimizing algorithms in routing games.
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Citation Context |

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48 | A.: Convergence issues in competitive games.
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Citation Context ...re quantitative performance guarantees for our heuristics. Our research is also related to works on the price of anarchy in a game, and convergence of game dynamics to approximately optimal solutions =-=[50, 30]-=- and to sink equilibria [30, 21]. Numerous articles study the convergence rate of best-response dynamics to approximately optimal solutions [15, 23, 4, 9]. For example, polynomial-time bounds has been... |

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Citation Context ...is student George Baylor to help translate De Groot’s work into English. 6dynamics, noisy best-response dynamics [20, 79, 51], where players occasionally make mistakes, and simultaneous Nash dynamics =-=[7]-=-, where all players change their strategies simultaneously, are both well-studied. In many other models the effect of learning algorithms [80] is examined, for example, regret minimisation dynamics [2... |

39 | Convergence and approximation in potential games. In:
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Citation Context ...of game dynamics to approximately optimal solutions [50, 30] and to sink equilibria [30, 21]. Numerous articles study the convergence rate of best-response dynamics to approximately optimal solutions =-=[15, 23, 4, 9]-=-. For example, polynomial-time bounds has been proven for the speed of convergence to approximately optimal solutions for approximate Nash dynamics in a large class of potential games [4], and for lea... |

38 | An investigation of the causes of pathology in games - Nau - 1982 |

37 | Online linear optimization and adaptive routing. - Awerbuch, Kleinberg - 2008 |

35 | Information processing and bounded rationality: A survey - Lipman - 1995 |

33 | The complexity of game dynamics: BGP oscillations, sink equilibria, and beyond
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Citation Context ...antees for our heuristics. Our research is also related to works on the price of anarchy in a game, and convergence of game dynamics to approximately optimal solutions [50, 30] and to sink equilibria =-=[30, 21]-=-. Numerous articles study the convergence rate of best-response dynamics to approximately optimal solutions [15, 23, 4, 9]. For example, polynomial-time bounds has been proven for the speed of converg... |

32 | Convergence to equilibrium in local interaction games.
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Citation Context ...at is found in the search, this aspiration level may be changed. 5In fact, Simon sent his student George Baylor to help translate De Groot’s work into English. 6dynamics, noisy best-response dynamics =-=[20, 79, 51]-=-, where players occasionally make mistakes, and simultaneous Nash dynamics [7], where all players change their strategies simultaneously, are both well-studied. In many other models the effect of lear... |

32 | Inapproximability of pure Nash equilibria. In:
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Citation Context |

30 | Pure and bayes-nash price of anarchy for generalized second price auctions.
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Citation Context ...ead equilibrium if no player has a lookahead improving move. It is known that the social welfare of Nash equilibria for myopic game playing can be arbitrarily bad [13] unless we disallow over-bidding =-=[46]-=-. Here, we prove the advantage of additional foresight by showing that 2-lookahead equilibria have much better social welfare. In particular, we show that all such equilibria are optimal in the worst-... |

29 | Multiplicative updates outperform generic no-regret learning in congestion games. - Kleinberg, Piliouras, et al. - 2009 |

20 | Eds., Bounded Rationality —-The Adaptive Toolbox. - Gigerenzer, Selten - 2002 |

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20 |
Modeling Bounded Rationality
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Citation Context ...ing systems. The label bounded rationality is currently used in a number of disparate areas some of which actually go against the main thrust of Simon’s original ideas; see Selten [64] and Rubenstein =-=[59]-=- for some discussion on this point. Two schools of thought developed by psychologists, experimental economists, and behavioural economists are, however, well worth mentioning here. First, the Heuristi... |

18 | Developing expertise in decision making”, - Klein - 1997 |

18 |
Decision quality as a function of search depth on game trees
- Nau
- 1983
(Show Context)
Citation Context ...ts and by the expertise of the player), and there is a decision rule (backwards induction). The value of lookahead search in decision-making has been examined by the artificial intelligence community =-=[55]-=-; for examples in effective diagnostics and real-time planning see [40] and [63]. Lookahead search is also related to the sequential thinking framework in game theory [52, 73]. However, compared to th... |

16 |
Strategy Subsets Closed Under
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Citation Context ...n studied in the economics literature. Among them are correlated equilibria [2], stable equilibria [44], stochastic adjustment models [38], strategy subsets closed under rational behaviour (CURB set) =-=[6]-=-, iterative elimination of dominated strategies, the set of undominated strategies, etc. Convergence and strategic stability of equilibria in evolutionary game theory is also an important subject of s... |

15 | Slovic P, Tversky A, eds. Judgements Under Uncertainty: Heuristic and Biases - Kahneman - 1982 |

15 |
One step lookahead is pretty good
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Citation Context ...ackwards induction). The value of lookahead search in decision-making has been examined by the artificial intelligence community [55]; for examples in effective diagnostics and real-time planning see =-=[40]-=- and [63]. Lookahead search is also related to the sequential thinking framework in game theory [52, 73]. However, compared to these works and the research carried out by the two schools above, our fo... |

13 |
On the convergence of regret minimization dynamics in concave games
- Even-dar, Mansour, et al.
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Citation Context |

13 |
The speed of convergence in congestion games under best-response dynamics.
- Fanelli, Flammini, et al.
- 2012
(Show Context)
Citation Context ...of game dynamics to approximately optimal solutions [50, 30] and to sink equilibria [30, 21]. Numerous articles study the convergence rate of best-response dynamics to approximately optimal solutions =-=[15, 23, 4, 9]-=-. For example, polynomial-time bounds has been proven for the speed of convergence to approximately optimal solutions for approximate Nash dynamics in a large class of potential games [4], and for lea... |

13 |
What is bounded rationality?”, in Bounded Rationality: the Adaptive Toolbox, G. Gigerenzer and R. Selten (eds
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Citation Context ... information processing systems. The label bounded rationality is currently used in a number of disparate areas some of which actually go against the main thrust of Simon’s original ideas; see Selten =-=[64]-=- and Rubenstein [59] for some discussion on this point. Two schools of thought developed by psychologists, experimental economists, and behavioural economists are, however, well worth mentioning here.... |

9 |
Forward looking Nash equilibrium for keyword auction,
- Bu, Deng, et al.
- 2008
(Show Context)
Citation Context ...a are constantfactor approximate solutions in the average-case lookahead model. 2.1. Worst-Case Lookahead. Our proof for the worst-case lookahead model can be seen as a generalisation of the proof of =-=[12]-=- for a slightly different model. We start by proving a useful lemma in this context. Lemma 2.1. Consider the worst-case lookahead model with the leaf model. Label the players so that player i is in sl... |

6 |
long run equilibria in games
- Kandori, Mailath, et al.
- 1993
(Show Context)
Citation Context ..., various equilibrium concepts other than Nash equilibria have been studied in the economics literature. Among them are correlated equilibria [2], stable equilibria [44], stochastic adjustment models =-=[38]-=-, strategy subsets closed under rational behaviour (CURB set) [6], iterative elimination of dominated strategies, the set of undominated strategies, etc. Convergence and strategic stability of equilib... |

5 | Bounding rationality by discounting time - Fortnow, Santhanam - 2010 |

5 | A.: On the complexity of nash dynamics and sink equilibria
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Citation Context |

5 | Boundedly Rational Qualitative Reasoning on Comparative Statics - Selten - 2004 |

4 | How to diagnose with very little information - Kleer, Raiman - 1993 |

4 | Discrete strategies in keyword auctions and their inefficiency for locally aware bidders.
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(Show Context)
Citation Context ... vi. To see this add dummy slots with ct = 0 if t > T . The player who wins the top slot should also bid truthfully under balanced bidding. Balanced bidding is the most commonly used bidding strategy =-=[13, 48]-=-. For some intuition behind this, note that balanced bidding has several desirable properties. For a competitive firm, bidding high obviously increases the chance of obtaining a good slot. Within a sl... |

2 |
Fast convergence to nearly-optimal solutions in potential games
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- 2008
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Citation Context |

2 |
regret minimization and correlated equilibria
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Citation Context |

2 | Internet advertising and the generalised second-price auction: selling billions of dollars worth of keywords - Edelman, Ostrovsky, et al. - 2007 |

2 | Thought and Choice in Chess, 2nd Edition - Groot - 1978 |

2 | The simulation heursitic”, in Judgement under Uncertainty: Heuristics and Biases - Kahneman, Tversky - 1982 |

2 | Real-time A* Search with Depth-k Lookahead
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Citation Context ...induction). The value of lookahead search in decision-making has been examined by the artificial intelligence community [55]; for examples in effective diagnostics and real-time planning see [40] and =-=[63]-=-. Lookahead search is also related to the sequential thinking framework in game theory [52, 73]. However, compared to these works and the research carried out by the two schools above, our focus is mo... |