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...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...

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...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...

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...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...

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...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...

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...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...

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...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...

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...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...

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...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...

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...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 ...

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...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...

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...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...

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...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...

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...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...

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...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...

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... + 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...

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...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 ...

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...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...

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...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 ...

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...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...

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...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...

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...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...

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...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...

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...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...

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...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...

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...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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...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...

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...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...

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...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...

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...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...

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...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...

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...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-...

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...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...

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...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...

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...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...

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...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...

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...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...

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...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...

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... 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....

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...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...

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..., 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...

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...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...

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... 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...

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...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...

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...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...

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...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...