A crucial assumption underlying any game-theoretic analysis is that there is common knowledge that all the players are rational. Rationality here is understood in the decision-theoretic sense: The players' choices are optimal according to some choice rule (such as maximizing subjective expected utility). Recent work in epistemic game theory has focused on developing sophisticated mathematical models to study the implications of assuming that all the players are rational and this is commonly known (or commonly believed). 1 However, if common knowledge of rationality is to have an \explanatory" role in the analysis of a game-theoretic situation, then it is not enough to simply assume that it has obtained in an informational context of a game. It is also important to describe how the players were able to arrive at this crucial state of information. 2 There is now a growing body of literature that analyzes games in terms of the \process of deliberation" that leads the players to select their component of a rational outcome (see [17] for a discussion). There is a second reason why it is important to develop formal models of the players' process of deliberation in game situations. A number of researchers have questioned the usefulness of models that make explicit assumptions about the players' higher-order beliefs in game situations. In the end, we are interested only in what (rational) players are going to do. This, in turn, depends only on what the players believe the other players are going to do. A player's belief about what her opponents are thinking is relevant only because they shape the players' rst-order beliefs about what her opponents are going to do. Kadane and Larkey explain the issue nicely: \It is true that a subjective Bayesian will have an opinion not only on his opponent's behavior, but also on his opponent's belief about his own behavior, his opponent's belief about his belief about his opponent's behavior, etc. (He also has opinions about the phase of the moon, tomorrow's weather and the winner of the next Superbowl). However, in a single-play game, all aspects of his opinion except his opinion about his 1 See, for example, [20, 9], and the references therein, for a survey of this literature. 2 This general point about common knowledge was already appreciated by David Lewis when he rst formulated his notion of common knowledge [14]. See [6] for an illuminating discussion and a reconstruction of Lewis' notion of common knowledge, with applications to game theory.
opponent's behavior are irrelevant, and can be ignored in the analysis by
integrating them out of the joint opinion." as [12, pg. 239, my emphasis]
A theory of rational decision making in game situations need not require
that a player considers all of her higher-order beliefs in her decision-making
process. The assumption is only that the players recognize that their opponents
are \actively reasoning" agents. Precisely \how much" higher-order information
should be taken into account in such a situation is a very interesting, open
question (cf. [
In this paper, I will compare and contrast di erent models of rational deliberation in games. The goal is not only to develop a comprehensive comparison of the di erent frameworks, but also to show how explicitly representing the players' process of deliberation can shed some light on the role that higher-order information plays in the analysis of game situations. The main challenge is to nd the right balance between descriptive accuracy and normative relevance. While this is true for all theories of individual decision making and reasoning, focusing on game situations raises a number of compelling issues. Indeed, Robert Aumann and Jacques Dreze [1, pg. 81] note that: \[T]he fundamental insight of game theory [is] that a rational player must take into account that the players reason about each other in deciding how to play".
The di erent models of rational deliberation in games that I will discuss in
this paper include:
1. John Harsanyi's \tracing procedure": The goal of the tracing procedure is
to identify a unique Nash equilibrium in any nite strategic game. Harsanyi
thought of the tracing procedure as \a mathematical formalization of the
process by which rational players coordinate their choices of strategies" [
Although the details of these models of deliberation in games are di erent,
they share a common line of thought: The rational outcomes of a game are arrived
at through a process in which each player settles on an optimal choice given her
evolving beliefs about her own choices and the choices of her opponents. The
frameworks mentioned above di er in how they represent the players' \state of
indecision" during the deliberation process:
1. Harsanyi does not explicitly represent the players' uncertainty about their
strategy choices during his tracing procedure. Instead, the idea is to analyze
a continuum of games starting from the original game and a common prior3,
in which the payo s are replaced by expected payo s.
2. Skyrms' model represents the players' beliefs during deliberation as a
probability measure on i's set of strategies. The interpretation is that it is the
mixed strategy that the players would adopt if deliberation ended at the
current stage.
3. Cubitt and Sugden assume that the players categorize their strategies Si in
a game as a pair of sets (Si+; Si ) where elements of Si+ are the \admissible"
actions, elements of Si are deemed \irrational", and strategies such that
s 62 Si+ [ Si (if any) are not (yet) categorized.
4. van Benthem and colleagues use relational models, familiar from the
computer science and philosophy literature (see [
Each of the above models are intended to be a \snapshot" of the players' beliefs about their own choices and their opponents' choices during the deliberational process. The second aspect of the models of deliberation in games are the dynamical rules that are intended to describe how the players' beliefs evolve during deliberation. At each stage of the deliberation, the players determine which of their available strategies are \optimal" and which ones they ought to avoid. Typically, it is assumed that the players are guided by some decision-theoretic choice rule, such as maximizing expected utility or avoiding dominated strategies. Using the information about the players' own choices and what they expect their opponents to do, the current snapshot of the players' beliefs is transformed according to some xed dynamical rule. Di erent types of transformations represent how con dent the players are in their assessment of which of the available choices are \rational". The picture to keep in mind is:
What should I do? M0 initial model !'1
What should I do?
What should I do? 3 Harsanyi in collaboration with Reinhard Selten o er an algorithm for constructing a common prior given the structure of the game.
Deliberation concludes when the players reach a xed point in the above process. The goal is not to develop a formal account of the players' practical reasoning in game situations. Rather, it is to describe deliberation in terms of a sequence of belief changes about what the players are doing or what their opponents may be thinking. The central question is: What are the update mechanisms that match di erent game-theoretic analyses?
The di erent frameworks mentioned above can be compared and contrasted in terms of the speci cs of the models: How are the players' beliefs represented during the deliberational process and what are the properties of the operations that are used to transform the players' beliefs? The general conclusion is that each of the frameworks o er a di erent perspective on strategic reasoning in games. These perspectives are not competing; rather, they highlight di erent aspects of what it means to reason strategically in interactive situations.
A second dimension along which the di erent models of deliberation can be compared is in terms of the outcomes that can be reached by the deliberational processes and what are the properties of the paths that lead to these outcomes. For instance, the outcomes of both Harsanyi's tracing procedure and Skryms's model of dynamic deliberation are qualitatively similar: Both procedures lead players to choose their component of a Nash equilibrium. However, in Skyrms's model, the rate of convergence depends on the players' initial beliefs in an interesting way; and this, in turn, suggests a more re ned analysis of Nash equilibrium [21, pgs. 154 - 158].
Finally, moving away from the mathematical properties of the di erent frameworks, it is also important to focus on the underlying motivations. Here, the tracing procedure can be singled out as it is focused on identifying a unique rational solution to every strategic game. The other frameworks are more \personalistic": The rational outcomes of a game depends not only on the structure of the game, but also on the players' initial beliefs, which dynamical rule is being used by the players (in general, di erent players may be using di erent rules), and what exactly is commonly known about the process of deliberation.
I will conclude the paper by carefully comparing the above general approach
to modeling strategic reasoning in interactive situations with a recent
contribution from the cognitive science literature. In [