Boundedly Rational Players Research Articles

Interdicting Attack Plans with Boundedly Rational Players and Multiple Attackers: An Adversarial Risk Analysis Approach

Cybersecurity planning supports the selection of and implementation of security controls in resource-constrained settings to manage risk. Doing so requires considering adaptive adversaries with different levels of strategic sophistication in modeling efforts to support risk management. However, most models in the literature only consider rational or nonstrategic adversaries. Therefore, we study how to inform defensive decision making to mitigate the risk from boundedly rational players, with a particular focus on making integrated, interdependent planning decisions. To achieve this goal, we introduce a modeling framework for selecting a portfolio of security mitigations that interdict adversarial attack plans that uses a structured approach for risk analysis. Our approach adapts adversarial risk analysis and cognitive hierarchy theory to consider a maximum-reliability path interdiction problem with a single defender and multiple attackers who have different goals and levels of strategic sophistication. Instead of enumerating all possible attacks and defenses, we introduce a solution technique based on integer programming and approximation algorithms to iteratively solve the defender’s and attackers’ problems. A case study illustrates the proposed models and provides insights into defensive planning. Funding: A. Peper and L. A. Albert were supported in part by the National Science Foundation [Grant 2000986].

On Markov Games Played by Bayesian and Boundedly-Rational Players

We present a new game-theoretic framework in which Bayesian players with bounded rationality engage in a Markov game and each has private but incomplete information regarding other players' types. Instead of utilizing Harsanyi's abstract types and a common prior, we construct intentional player types whose structure is explicit and induces a {\em finite-level} belief hierarchy. We characterize an equilibrium in this game and establish the conditions for existence of the equilibrium. The computation of finding such equilibria is formalized as a constraint satisfaction problem and its effectiveness is demonstrated on two cooperative domains.

From reasonable preferences, via argumentation, to logic

This article demonstrates that typical restrictions which are imposed in dialogical logic in order to recover first-order logical consequence from a fragment of natural language argumentation are also forthcoming from preference profiles of boundedly rational players, provided that these players instantiate a specific player type and compute partial strategies. We present two structural rules, which are formulated similarly to closure rules for tableaux proofs that restrict players' strategies to a mapping between games in extensive forms (i.e., game trees) and proof trees. Both rules are motivated from players' preferences and limitations; they can therefore be viewed as being player-self-imposable. First-order logical consequence is thus shown to result from playing a specific type of argumentation game. The alignment of such games with the normative model of the Pragma-dialectical theory of argumentation is positively evaluated. But explicit rules to guarantee that the argumentation game instantiates first-order logical consequence have now become gratuitous, since their normative content arises directly from players' preferences and limitations. A similar naturalization for non-classical logics is discussed.

Robust Adversarial Risk Analysis: A Level-k Approach

Adversarial risk analysis is an active and important area of decision analytic research. Both single-actor decision analysis and multiple-actor game theory have been applied to this problem, with game theoretic methods being particularly popular. Although game theory models do explicitly capture strategic interactions between attackers and defenders, two of the key assumptions—decision making based on subjective expected utility maximization and common knowledge of rationality—are known to be descriptively inaccurate in some situations. This paper addresses these shortcomings by proposing, formulating, and illustrating the application of robust optimization methodologies to a level-k game theory model for adversarial risk analysis. Level-k game theory provides a practical method for modeling bounded rationality. Robust optimization provides an alternative way to model the actions of conservative players facing “deep” uncertainties about their environment—uncertainties that are possible to bound but that are difficult or impossible to represent using probability distributions. Our approach thus combines level-k and robust optimization insights to provide a computationally tractable model of boundedly rational players who are faced with significant and difficult to quantify uncertainties.

Playing off-line games with bounded rationality

We study a two-person zero-sum game where players simultaneously choose sequences of actions, and the overall payoff is the average of a one-shot payoff over the joint sequence. We consider the maxmin value of the game played in pure strategies by boundedly rational players and model bounded rationality by introducing complexity limitations. First we define the complexity of a sequence by its smallest period (a nonperiodic sequence being of infinite complexity) and study the maxmin of the game where player 1 is restricted to strategies with complexity at most n and player 2 is restricted to strategies with complexity at most m . We study the asymptotics of this value and a complete characterization in the matching pennies case. We extend the analysis of matching pennies to strategies with bounded recall.

A Comparative Study of Game Theoretic and Evolutionary Models of Bargaining for Software Agents

Most of the existing work in the study of bargaining behavior uses techniques from game theory. Game theoretic models for bargaining assume that players are perfectly rational and that this rationality is common knowledge. However, the perfect rationality assumption does not hold for real-life bargaining scenarios with humans as players, since results from experimental economics show that humans find their way to the best strategy through trial and error, and not typically by means of rational deliberation. Such players are said to be boundedly rational. In playing a game against an opponent with bounded rationality, the most effective strategy of a player is not the equilibrium strategy but the one that is the best reply to the opponent's strategy. The evolutionary model provides a means for studying the bargaining behaviour of boundedly rational players. This paper provides a comprehensive comparison of the game theoretic and evolutionary approaches to bargaining by examining their assumptions, goals, and limitations. We then study the implications of these differences from the perspective of the software agent developer.

Finitely Repeated Prisoners' Dilemma Games - Do Players use Backward or Forward Induction?

In the one-shot version of the Prisoners' Dilemma (PD) game, individuals pursue mutually destructive strategies (they both defect). The repeated PD examines whether interactions over time can induce players to adopt cooperative strategies. Cooperation is possible in the infinitely repeated PD. However, in the finitely repeated version, backward induction forces immediate mutual defection. Cooperation becomes possible if players are unsure of when the final round will be (Myerson 1991, Brams and Kilgour 2002), or when players are 'absent-minded' regarding the current round of play (Dilger 1998), since players will use forward, rather than backward, induction. In this paper, we develop a model in which cooperation may be sustainable in a finitely repeated PD game, even when players have perfect knowledge of the current round number and final round number. We analyse whether boundedly rational players use backward or forward induction in such a game. Players use forward induction if there are many rounds of play remaining, and the end round is a distant prospect. They only switch to backward induction at a critical round (when the end round is in sight). We believe that the insights of the model are so important for practical conflict scenarios, that they should be tested in an experimental game. In the final section, we suggest specifications for such a game.

Les justifications des notions d'équilibre de jeux

Résumé Les notions d’équilibre statiques ou dynamiques proposées en théorie des jeux peuvent être justifiées selon deux perspectives. Du point de vue épistémique, les équilibres résultent du seul raisonnement de joueurs hyper-intelligents qui ont une connaissance commune de la structure du jeu et de leurs rationalités respectives. Si l’équilibre rationalisable ou l’équilibre corrélé sont aisément justifiés, l’équilibre de Nash n’est obtenu que sous des conditions très drastiques ; quant à l’équilibre parfait, sa justification est très sensible aux hypothèses faites. Du point de vue évolutionniste, les équilibres résultent de la convergence d’un processus d’apprentissage ou d’évolution de joueurs en rationalité limitée, mais observant le déroulement passé du jeu. L’équilibre de Nash, du moins en stratégies pures, est souvent obtenu comme état asymptotique et certains de ses raffinements peuvent même être sélectionnés ; l’équilibre parfait est aussi justifié sous des conditions très étendues.

On the stability of monotone discrete selection dynamics with inertia

In this paper we study the learning behavior of a population of boundedly rational players who interact by repeatedly playing an evolutionary game. A simple imitation type learning rule for the agents is suggested and it is shown that the evolution of the population strategy is described by the discrete time replicator dynamics with inertia. Conditions are derived which guarantee the local stability of a Nash-equilibrium with respect to these dynamics and for quasi-strict ESS with support of three pure strategies the crucial level of inertia is derived which ensures stability and avoids overshooting. These results are illustrated by an example and generalized to the class of monotone selection dynamics.

Alternating-offer bargaining and common knowledge of rationality

This paper reconsiders Rubinstein's alternating-offer bargaining game with complete information. We define rationalizability and trembling- hand rationalizability (THR) for multi-stage games with observed actions. We show that rationalizability does not exclude perpetual disagreement or delay, but that THR implies a unique solution. Moreover, this unique solution is the unique subgame perfect equilibrium (SPE). Also, we reconsider an extension of Rubinstein's game where a smallest money unit is introduced: THR rules out the non-uniqueness of SPE in some particular case. Finally, we investigate the assumption of boundedly rational players. Perpetual disagreement is excluded, but not delay. Furthermore, we cannot use the asymmetric Nash bargaining solution as an approximation of the alternating-offer bargaining model once the players are boundedly rational ones.

On the dynamics of word of mouth learningwith and without anticipations

We analyze the learning behavior of two populations engaged in playing a “battle of thesexes” game. The boundedly rational players change their strategy with some positive probabilityif they learn, via direct communication with other players, about a strategy whichcurrently has a higher payoff than their own. In games with no risk‐dominant equilibrium,this learning rule leads to convergence towards one of the pure strategies' coordinationequilibria if the initial population distributions are asymmetric. For symmetric initialpopulation distributions, depending on the players' propensity to adopt new strategies, convergencetowards the mixed strategies' equilibrium or periodic and complex behavior mightoccur. The introduction of anticipations leads to the emergence of stable fixed points of thelearning process, which are no Nash equilibria, via a fold and a transcritical bifurcation. Ifone equilibrium is risk dominant, this equilibrium has a larger basin of attraction than theother coordination state for both the dynamics with and without anticipations. However, theintroduction of anticipations increases the basin of attraction of the risk‐dominated equilibrium.

Is step-j thinking an arbitrary modelling restriction or a fact of human nature?

In `Boundedly Rational Rule Learning in a Guessing Game,' Games and Economic Behavior, 16 (1996), we combined Nagel's (1995) model of boundedly rational players with a `law of effect' learning model, and the synthesis outperformed alternative theories when confronting Nagel's data. In that model, there were four boundedly rational behavioral rules (step-j, j=0, 1, 2, 3), each corresponding to an integer level of depth of reasoning. It is legitimate to ask for a justification of the restriction to these `integer' rules. Why is it not reasonable to suppose that some player believes that 50% of the population is step-0, and 50% is step-1, and so himself adopts something like a step-1.5 rule? This paper constructs a tractable model with potentially infinitely many non-integer rules and conducts comparison tests. The main conclusion is that allowing for non-integer rules does not help to explain the data. Therefore, by Occam's Razor, the integer rule model is preferred. These results suggest that step-j thinking is a fact of human nature rather that an arbitrary modelling restriction.

Skill, complexity and strategic behavior

Boundedly rational players with differing skill are affected differently by complexity. Thus players have preferences for complexity that depend on their own skill and the skill of their opponents. Such preferences are likely to be important determinants of players' choices. To illustrate, this paper analyzes a model where differently skilled players sort themselves across two-person win-draw-loss games of differing complexity, are randomly paired against opponents who have made similar choices and then play the games they have chosen.

Boundedly Rational Rule Learning in a Guessing Game

We combine Nagel's “step-k” model of boundedly rational players with a “law of effect” learning model. Players begin with a disposition to use one of the step-krules of behavior, and over time the players learn how the available rules perform and switch to better performing rules. We offer an econometric specification of this dynamic process and fit it to Nagel's experimental data. We find that the rule of learning model vastly outperforms other nested and nonnested learning models. We find strong evidence for diverse dispositions and reject the Bayesian rule-learning model.Journal of Economic LiteratureClassification Numbers: C70, C52, D83.

Conventions, local interaction, and automata networks

This paper deals with the evolution of conventions in a finite population of boundedly rational players. We apply methods from the theory of automata networks to analyze the effects of different interaction structures. It is shown that the size of the reference group for each player has an important impact on the evolutionary stability of a particular convention, and that the interaction structure crucially determines the distribution of conventions in the population.

Finite automata play repeated prisoner's dilemma with information processing costs

Abstract We study the finitely repeated Prisoner's Dilemma game. Our players are modeled as finite automata. A population of boundedly rational players compete in a ‘survival of the fittest’ evolution contest simulated using Holland's genetic algorithm. Starting from a hostile population which plays defection frequently, our simulation results show that players converge to play cooperatively. This emergence of cooperative behavior breaks down when we penalize a complex strategy based on the size of the machine. On the other hand, a penalty cost that increases with the frequency an automaton switches states will not hurt the development of cooperative behavior.