A Theoretical Examination of Practical Game Playing: Lookahead Search

Abstract

Abstract. Lookahead search is perhaps the most natural and widely used game playing strategy. Given the practical importance of the method, the aim of this paper is to provide a theoretical performance examination of lookahead search in a wide variety of applications.To determine a strategy play using lookahead search, each agent predicts multiple levels of possible re-actions to her move (via the use of a search tree), and then chooses the play that optimizes her future payoff accounting for these re-actions. There are several choices of optimization function the agents can choose, where the most appropriate choice of function will depend on the specifics of the actual game -we illustrate this in our examples. Furthermore, the type of search tree chosen by computationally-constrained agent can vary. We focus on the case where agents can evaluate only a bounded number, k, of moves into the future. That is, we use depth k search trees and call this approach k-lookahead search.We apply our method in five well-known settings: industrial organization (Cournot's model); AdWord auctions; congestion games; valid-utility games and basic-utility games; cost-sharing network design games. We consider two questions. First, what is the expected social quality of outcome when agents apply lookahead search? Second, what interactive behaviours can be exhibited when players use lookahead search?Myopic game playing (whose corresponding equilibria are Nash equilibria), where each player can only foresee the immediate effect of her own actions, is the special case of 1-lookahead search. Thus, for the first question, it is natural to ask whether social outcomes improve when players use more foresight than in myopic behaviour. The answer depends on the game played: (i) For the Cournot game, applying 2-lookahead leads to a 12.5% increase in output and a 5.5% increase in social surplus compared with myopic competition. Similar bounds arise as the length k of foresight increases.(ii) In AdWord auctions (or generalized second-price auctions), we show that 2-lookahead game playing results in outcomes that are always optimal to within a constant factor; in contrast, myopic game play can produce arbitrarily poor equilibrium outcomes.(ii) For congestion games, as with myopic game playing, lookahead search leads to constant factor qualitative guarantees. (iv) For basic-utility games, on the other hand, whilst myopic game playing always leads to constant factor approximations, additional foresight can lead to arbitrarily bad solutions! (v) In a simple Shapley network design game, qualitative guarantees improve with the length of foresight. Regarding the second question, a variety of interesting game playing characteristics also arise with lookahead search. Stackelberg leader-follower behaviours can be induced when the players have asymmetric computational power. For example, Stackelberg equilibria can be produced in the Cournot game. Lookahead search can also generate "uncoordinated" cooperative behaviour!

On lookahead equilibria in congestion games

“…This result significantly improves the previous upper bound of (1 + √ 5) 2 ≈ 10.47 given V. Bilò,A. Fanelli and L. Moscardelli 200 in Mirrokni et al (2012). All mentioned bounds hold either with or without consecutive moves.…”

“…In this work, following the study initiated by Mirrokni et al (2012), we focus on the settings in which each player has full knowledge of the strategies and costs of the other players, so that, based on such a knowledge, she can make predictions about the others' reactions to her move. We also assume that each player is an entity with limited computational abilities, thus she has the ability of making predictions only on the consequences of a fixed constant number of subsequent consecutive moves.…”

Risk Sensitivity of Price of Anarchy under Uncertainty

“…Our model of win-lose games was inspired by Immorlica et al [2011]. At a high level, the interplay of the decision making process of a single agent and the price of anarchy is related to the work on lookahead search in game playing by Mirrokni et al [2012]. Back to uncertain payoffs, stochastic games from game theory, first introduced by Shapley [1953], refer to dynamic games in which agents make probabilistic transitions (e.g., see [Neyman and Sorin 2003;Filar and Vrieze 1996]).…”

Informed production optimization in hydrocarbon reservoirs

“…For more information, see Birge and Louveaux (1997), Ruszczyski and Shapiro (2003), Georghiou et al (2011). In the formalism from decision and game theory, non-myopic decision rules are the ones where the decision makers look ahead and consider future information, opposed to the myopic approach in which the influence of current decisions on the future state of uncertainty (i.e., conditional posterior distributions) is ignored (Mirrokni et al 2012). The artificial intelligence community refers to this class of problems as partially observable Markov decision Barros et al (2016a) processes (POMDPs), related to applications where direct observations of the state of the uncertain processes are not available (Smallwood and Sondik 1973;Hauskrecht 2000).…”