Reasoning, games, action and rationality

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1 Reasoning, games, action and rationality August 11-15, ESSLLI 2008 Eric Pacuit Olivier Roy August 10, 2008 This document contains an extended outline of the first lecture including a bibliography. The idea of this reader is to provide a bird s eye view of the literature and to list the main examples, definitions and theorems we will discuss throughout the course. 1 General Introduction and Definitions 1.1 Introduction and Motivation Starting from the work of Ramsey [23], de Finetti [10], von Neumann and Morgenstern [28] and Savage [24], the formal analyses carried in decision and game theory have provided important insights for the theory of rational decision making. More recently, the epistemic program in game theory [13, 2, 8] has highlighted the importance of mutual expectations for the understanding of interactive rationality, that is for rational decision making in situation of social interaction. Game theory has inherited from decision theory its instrumental understanding of rationality. In both disciplines to choose rationally is to choose, in the light of one s expectations, the best means to achieve one s ends. Decision theory studies instrumental rationality in situations where one agent chooses among various actions on the basis of their expected consequences. Crucially, in decision theoretic scenario it is the agent s environment, or Nature, which determines the consequences of his actions. Game theory, on the other hand, is concerned with the interaction of many rational decision makers. Here the consequences of one agent s decision depend on the choices of all the agents involved in the situation. The expectations of an individual 1

2 are thus no more about a passive or external environment, but rather about the choices and expectations of other rational decision makers. Acknowledging this apparently small difference, one vs many agents, complicates the picture of instrumental rationality. In games the players expectations become interrelated: what one expects from his opponents depends on what one thinks the others expect from him, and what the others expect from a given player depends on what they think his expectations about them are. Dynamic epistemic logic [22, 5, 25, 27] provides here a fruitful environment to study such entangled expectations. It allows for an elegant analysis of information and information about information, that is of higher-order information. In this course we will study various foundational issues that arise from the epistemic outlook on games, and show how dynamic epistemic logic sheds new lights on them. We will, in other words, take a logical perspective on conceptual problems regarding the notion of rationality, expectations and choices in interactive situations. The kind of problems that we are interested in and the methods we draw from make the present course a contribution to contemporary formal epistemology [12, 14], while our emphasis on interaction is also relevant for social software [21]. 1.2 Decisions in uncertain environment Even though interactive situations are our main interest, it is instructive to start with an overview of how expectations determine choices in decisiontheoretic scenarios. Decision theory deals with two kinds of uncertainty: endogenous and exogenous. Situations of exogenous uncertainty are situations in which the results of the agents actions depend on random or non-deterministic occurrences in the environment. Exogenous uncertainty is thus the result of objective non-determinacy, in the world so to speak. Buying a lottery ticket is a typical example. Someone cannot simply choose to buy the winning ticket. All one can do is pick one and wait for the drawing to determine whether it is a winner. In models of decision making under exogenous uncertainly, the decision maker has to choose between different lotteries or simply probability distributions over a set of outcomes. Buying a lottery ticket, for instance, would then be represented as an action which gives the agent a certain probability of winning. Situations of endogenous uncertainty, on the other hand, are situations in which the outcome of the agent s decision depends on the actual state 2

3 of the world, about which he has only partial information. The following is a classical example, from Savage [24]. Imagine an agent who is making an omelet. He has already broken five eggs into the bowl, and he is about to break the sixth and last one. Whether this will result in a good omelet depends on the state of this last egg, which can be rotten or not. The agent, however, does not know whether the egg is rotten. In slightly more technical terms, he lacks information about the actual state of the egg, and this makes the outcome of his decision uncertain. Note the contrast with buying a lottery ticket. There is no chance or randomness involved here. The egg is rotten or it is not. What matters is that the agent does not know the state of the egg. Decision theoretic models, for example the one proposed by Anscombe and Aumann [1], usually represent such situations with a set of possible states. In the omelet example there would be two states, the egg is rotten and the egg is not rotten. The uncertainty over these states is represented either qualitatively, by an epistemic accessibility relation, or quantitatively, by a probability distribution. The most wide-spread expression of instrumental rationality in the face of uncertainty, whether exogenous or endogenous, is Bayesian rationality, also known as maximization of expected utility. A decision problem usually consists of a set of states, used to represent the agent s uncertainty, a set of outcomes, which have a certain value for the agent, and a set of actions which are functions from the set of states to the set of outcomes. This expresses the idea that the result (outcome) of an agent s action depends on the state of the world. The agent s uncertainty is usually represented by a probability distribution over the set of states. The expected value of an action is the sum of the values of all its possible outcomes at each state weighted by the probability of the latter. To be instrumentally rational, in the Bayesian sense, is to choose the action, or one of the actions, which maximizes this sum. In other words, the agent is instrumentally rational when he chooses an action from which he can expect the highest payoff. It is important to see that in such decision problems, the expectations of the agent are fixed by the probability distribution induced by the objective random event or by the (partial) information he has about the state of the world. These expectations are fixed in the sense that they do not depend on the agents preferences and possible choices. They only depend on Nature, the decision-making environment, which do not have preferences or form expectations about the decision maker. This is where decision-theoretic scenario differ from game-theoretic ones. For more references on decision theory, see Jeffrey [15], Myerson [19], Joyce 3

4 [16] and Bradley [7]. 1.3 Games - General Overview The key ingredients of a game-theoretic scenario are similar to those in decision theory. A number of agents have to choose an action, and the combination of these choices, one for each agent, determines an outcome which each agent values differently. Two main models are used in game theory: extensive and strategic representations. Extensive representations are often pictured as trees, as in Figure 1. The order of nodes in the tree represents the sequential Ann Hi (2, 2) Bob Hi Lo (0, 0) Lo Ann hi (0, 0) lo (1, 1) Figure 1: A simple game in extensive form structure of the game. In this example Bob plays first, choosing Hi or Lo, and Ann plays second, choosing between Hi or Lo if Bob played Hi, and hi or lo if Bob played Lo. A strategy for a player i, in an extensive games is a function which specifies an action to take at each nodes in which player i has to make a decision. It is, so to speak a plan for all eventualities. We will introduce the formal details about extensive games in the notes for Lecture 5. For now we will focus on games in strategic form, which abstract away from the sequential structure of decisions. The players are thought to choose once a strategy for the whole game. At this point it is worthwhile to get more formal. Definition 1.1 (Strategic games) A strategic game G is a tuple I, S i, v i such that : I is a finite set of agents. 4

5 S i is a finite set of actions or strategies for i. A strategy profile σ Π i I S i is a vector of strategies, one for each agent in I. The strategy s i which i plays in the profile σ is noted σ i. v i : Π i I S i R is an utility function that assigns to every strategy profile σ Π i I S i the utility valuation of that profile for agent i. In what follow we will often reason with the partial order i induced in the obvious way on the set of strategy profiles : σ i σ iff v i (σ) v i (σ ). Games in strategic forms are often represented as matrices, as in Table 1. This game is a two-players game. The row player, which we will call Ann, can A B a 1, 1 0, 0 b 0, 0 1, 1 Table 1: A simple game in strategic form choose between two actions or strategies, a and b. Bob, the column player, has also two strategies, A and B. In this game the payoffs are identical for both players. If they coordinate, either on aa or bb they get 1. Otherwise they get 0. This game, often called a coordination game, is of particular interest to understand the difference between game- and decision-theoretic situations. Let us ask ourself, what is the rational thing for Ann to choose? Intuitively, it all depends on what she expects Bob to choose. If she thinks Bob will choose A, then it is rational for her to choose a, and symmetrically if she thinks Bob will choose B. However, Ann also knows that, just like herself, Bob will make his decision on the basis of what he expects she will choose. This means that what Ann expects Bob to choose depends also on what she thinks Bob expects her to choose. If Ann thinks Bob expects her to choose a, than under the assumption that he is rational she will expect him to choose A, and thus being herself rational she will choose a. This chain of reasoning involving higher order expectations, expectations about expectations, does not stop here, though. What Ann thinks Bob expects her to choose depends on what she thinks Bob thinks she expects him to choose. This expectation of Ann about what Bob thinks she thinks of him should in turn be based on still another layer of mutual expectations, and so on ad infinitum. The same holds for Bob. 5

6 Observe the contrast with decision theoretic scenarios. Here the decision maker s expectations only depends on his uncertainty about the state of his environment, but there is no higher-order expectations here. What the agent thinks is the actual state of the world, or what are his chances to get a certain outcome does not depend on what the environment expects him to choose. In short, mutual expectations (knowledge and beliefs) are at the very heart of interactive reasoning in game theory. Some authors have raised objection to the idea there is a genuine divide between (single-agent) decision and (many-agent) game-theoretic scenarios, and that the seconds are ultimately reducible to the firsts (see e.g.[17]). We will return briefly to this objection in Lecture 3. We will see that gametheoretic rationality can indeed be phrased in decision-theoretic or Bayesian terms but that, however, equilibrium solutions concepts in game theory seem to crucially involve higher-order expectations, a phenomenon which is intrinsically interactive. For more on strategic games: the seminal work of von Neumann and Morgenstern [28], and more recent textbooks: Myerson [19], Osborne and Rubinstein [20]. 1.4 Game Models An epistemic model of a given game G is a structure that represents what the agents might know, believe and prefer in diverse scenarios or game playing situations [9]. Such situations are ex interim [3]: they are situations where the agents have made their decision about which action to take, but might still be uncertain about the decisions of others. In comparison, an ex ante situation is a case where no decision has been made yet, and an ex post situation is when the choices of all players are openly disclosed. For the main part of this course we will consider ex interim situations, except while discussing correlated equilibrium, because they allow for a straightforward assessment of the agent s rationality given their expectations, i.e. partial information, about the choices of others. Two main types of models have been used in the literature to represent game-playing situations: type spaces [13] and the so-called Aumannor Kripke-structures [11, 6]. Type spaces and Kripke structures differ in the way they represent the (partial) information available to all agents. The first represent it quantitatively, using probability distributions, while the second represent it qualitatively, using partitions and/or plausibility orderings. In the game-theoretic literature, these two modeling paradigms have given 6

7 rise to different styles of epistemic analysis [8]. The probabilistic nature of type spaces have naturally led towards belief -based characterizations. Aumann or Kripke structures, on the other hand, have mostly provided knowledge-based characterizations. Recent work [18, 4], however, have extended the qualitative analysis to deal with various epistemic attitudes, including (conditional) beliefs and various strength of knowledge. In what follows we take advantage of these advances, and provide a belief-based analysis both in terms of type spaces and qualitative plausibility models Qualitative models Definition 1.2 (Epistemic Plausibility Model [4]) An epistemic plausibility model M of the game G is a tuple W, f, { i, i } i I such that: W is a set of states. Then f : W Π i I S i is a strategy function that assigns to each w W a strategy profile. From convenience we write σ(w) for the σ = f(w) and σ i (w) for the i th component of this profile. i is an epistemic accessibility equivalence relation such that if w i w then σ i (w) = σ i (w ). We write [w] i for {w : w i w }. i is a reflexive and transitive plausibility ordering on W such that if w i w then w i w. This relation is say to be locally connected when, for all w and w, w [w] i, either w i w or w i w. Each state w of a epistemic model represents a possible play of the game. The strategy that each player chooses in that play is given by the function f. Observe that there needs not be a one to one correspondence between the possible states and the set of profile. There can be many states in which the same profile is chosen, and there can be profiles which are chosen in no state of the model. The information of a player i at a state is given by the relations i and i. The relation i gives, at a given state w, all the states w that an agent i considers possible. Among all these states, however, there might be states that i considers more or less plausible than others, and the relation i is here to represents this fact. If w i w we thus say that i considers w at least as plausible as w. The condition that if w i w then w i w makes sure that 7

8 whenever a state w is at least as plausible as another w then w is considered possible at w. In the ex interim stage represented by a game playing situation, each agent has hard information [26], i.e. is certain and not mistaken about his own choice and information, as well as about the structure of the game: his uncertainty concerns what the others know, believe and choose. The constraint on the epistemic accessibility relation i are intended to capture this fact. By virtue of it being an equivalence relation, i makes i s hard information fully introspective and veridical; the agents know what they know and what they don t know, and what they know is in fact the case. To see this, let an event E be a set of states, i.e. a subset of W. We say that E occurs at w just in case w E. A player is said to know at a state w that a certain event E occurs whenever this event occurs all states that he considers possible, i.e. if [w] i E. A player thus knows that a certain event occurs when he does not consider possible that it might not occur. Anticipating our formal language (next lecture), let us write K i E for the set of states where player i knows that E occurs, E for the complement of E in W and for the empty set of states, the impossible event. With this in hand it is a standard correspondence exercise [6] to show that i is transitive if and only if K i (E) K i K i (E), that it is symmetric if and only if K i (E) K i ( K i (E)) and that it is reflexive if and only if K i (E) E. The first two conditions respectively express i s positive and negative introspection about is own information: whenever he knows something he knows that he knows it (K i (E) K i K i (E)), and whenever he does not know something he knows that as well ( K i (E) K i ( K i (E))). The third condition, K i (E) E, expresses the fact that i s hard information is veridical. Observe that this implies that agents never have inconsistent information, that is we never have [w] i =. Finally, the ex interim character of a game-playing situation is embodied by the condition that σ i (w) = σ i (w ) whenever w i w, which enforces that at each state the agents know which choices they are making, and do not consider possible that they are actually choosing something else. Being uncertain about the information and choices of others does not preclude that some facts might be more plausible than others, something which the plausibility ordering i is intended to capture. This ordering can be seen as the qualitative counterpart of the probability weights in type structures, which we will introduce shortly, and just like in these structures one can use it to define beliefs and conditional beliefs. At state w, agent i is said to believe that event E occurs, that is w B i E, whenever E occurs in all states that he considers the most plausible, that is in all w in the 8

9 set min i [w] i = {w : w w for all w [w] i }. This local or statedependent definition of beliefs is conditional on i s hard information at a state. Indeed, the states used to assess i s beliefs are those which he considers possible at that state. Conditional beliefs are defined similarly: i is said to believe conditional on learning that F at state w that event E occurs, which we note w B F i E, whenever E occurs in all F -states that he considers the most plausible, that is in all w in the set min i [w] i F. The reader can check that beliefs and conditional beliefs are introspective, both in terms of beliefs and knowledge: if an agent believes something he not only believes that he believes it, he knows that as well. In fact, it is generally the case that everything which is known is also believed, but not the other way around. Finally, beliefs, unlike knowledge, can be mistaken. For most of this course we will work with locally connected plausibility orderings, that is orderings which are not necessarily fully connected, but which are connected for every equivalence class [w] i. On these models the relation i is superfluous, as it becomes equivalent to the connected components of i, or the union of this relation and its converse. We introduced it separately to facilitate the understanding. Much more about these models, and the relations that they induce between knowledge and various forms and strengths of beliefs can be found in [4] Quantitative models A state w in a epistemic plausibility model M for a game G encapsulates the two main ingredients of game playing situation: the strategy choice σ i (w) and the information [w] i of each player. A type structure for a given strategic game G encapsulates, somewhat differently, the same features. Definition 1.3 (Type Structure) A type structure T is a pair T i, λ i such that : T i is a finite set of types. λ i : T i (Π j i S j Π j i T j ) is a function which gives, for each type in T i of i a probability distribution on the set of possible combinations of strategy choices and types (σ i, t i ) of the other players. 9

10 Types and their associated image under λ i encode the players (probabilistic) information about the others information, which boils down for them of being of a certain type, and strategy choices. In type structures events, beliefs and knowledge are relative to types. An event E for a type t i is a set of pairs (σ j, t j ), i.e. a set of strategy choice and types for all the other players. For convenience we use λ i (t i )(E) to denote the sum of the probabilities that λ i (t i ) assigns to the elements of E. Type t i of player i is said to believe the event E whenever λ i (t i )(E) = 1. Conditional beliefs are computed in the standard way: type t i believes that E given F whenever: λ i (t i )(E F ) = 1 λ i (t i )(F ) A state in a type structure is a tuple (σ, t) where σ is a strategy profile and t is type profile, a combination of type, one for each player. We will consider that an agent i knows that an event E occurs at a state (σ, t) whenever his type t i believes that E occurs, λ i (t i ) assigns non-zero probability to (σ i, t i ) and (σ i, t i ) is in E. In other words, when he non-mistakingly believes that E occurs at (σ, t). Uncertainty in type structure bears on the strategy choices and information of others, even more so than in epistemic plausibility models, as the function λ i does not assigns probabilistic beliefs to events related to i s own strategy choice and information. Issues related to knowledge of one s choice and type are somehow bypassed, as long as they concern what one believes about himself. What one thinks the other thinks about himself is, on the other and, clearly specified by a given type: unlike higher-order information about oneself, higher-order information concerning the others is built-in type structures. Let B i (E) = {(σ j, t j ) : t i believes thate} be the event for j that i believes that E. Agent j believes that i believes that E simply when λ j (t j )(B i E) = 1. One thus can thus go on and compute any (finite) level of such higher-order information, which together form what is often called the agent s belief hierarchy : starting from his beliefs about the other s strategy choices, the ground facts, and building up to higher and higher order of beliefs about the other s beliefs. We invite the reader to check that this belief hierarchy can also be recovered from a state in an epistemic plausibility model. 10

11 1.4.3 Models for games: example Take the game pictured in Table 1. We are going to build a model of this game where both Bob and Ann are certain each other s strategy choice, where Bob is also certain about Ann s information, but where Ann is uncertain about what Bob believes about her. Let us first build a type structure. We will encode the fact that Bob is certain about Ann s information by assigning her only one type, i.e. T Ann = {t Ann }. We could of course assign more types to Ann and set λ Bob such that the probability of (a, t Ann ) and (b, t Ann ) sums up to 1, but for now this would unnecessarily complicates things. For Ann to be uncertain about Bob s information means that he has more than one type, say T Bob = {t Bob, u Bob }. The function λ Ann gives Ann s partial beliefs about Bob s types and strategies. An insightful way to specify these functions is by using matrices, as in Tables 2 and 3. λ Ann (t Ann ) A B u Bob 1/2 0 T Bob 1/2 0 Table 2: Ann s beliefs about Bob λ Bob (t Bob ) a b u Ann 1 0 λ Bob (u Bob ) a b u Ann 0 1 Table 3: Bob s beliefs about Ann The rows in Table 2 are Bob s types, the columns are his possible strategies. The values in the cells represent Ann s degree of belief in Bob playing a strategy while being of a certain type. In this case Ann is sure that Bob will play A but that she is uncertain about what he believes. Ann s certainty about Bob s strategy choice boils down to put all the probability weight to the column where Bob plays A. This column sums up to one. One the other hand, Ann s uncertainty about Bob s information is represented by the fact that she assigns 1/2 to him being of type u b, and 1/2 of him being of type t b. In the first case, the matrix on the left of Table 3, Bob is certain that she will play a. In the second, the matrix on the right of Table 3, he is certain that she will play b. So, even though Ann is uncertain about Bob s information, she is sure that Bob is certain about what she will do. What she does not know is what Bob is convinced of. 11

12 The same situation can be transposed in an epistemic plausibility model, as in Figure 2 and 3. The eight states can be seen as two copies of the strategy profile set, one for each possible type of Bob or, to avoid speaking directly of types, one for each possible information Bob might have. The dashed arrows in Figure 2 represent Bob s relation Bob and the solid ones the relation Ann. All these should be seen as transitive. The italic-blue aa ab aa ab ba bb ba bb Figure 2: The plausibility relations for the epistemic model of the game in Table 1. states are those where Bob is of type t Bob, while the red states are those where he is of type u Bob. Figure 3 shows the partitions induced by the equivalence relations Ann and Bob, taken simply as the connected component of Ann and Bob, respectively. This figures also exhibits the sets max i [w] i : the gray areas are the sets for Ann, and Bob s set, which are all singletons, correspond to the states marked with an asterisk. That at type t Bob Bob is convinced of Ann s strategy choice is represented by the fact that, at all states where he is of that type, he considers possible only states where she plays a, and vice-versa at type u Bob. Taking a Ann to be the set of states where Ann plays a, one can indeed check that at all states w where Bob is of type t Bob, we have that w is in B Bob a Ann. The same goes for Ann: taking a similar notation, in every state w we have that w B Ann A Bob. She is, however, uncertain about Bob s information: at state ba, where Bob believes that she will play b, she considers it possible that Bob thinks she 12

13 aa* ab* aa ab ba bb ba* bb* Figure 3: The partitions induced by the relations i and the sets max i [w] i for epistemic model of the game in Table 1. will play a. Observe that at state ba both have correct beliefs about each other s choices. But this need not to be the case. At aa, for example, Bob believes that Ann plays b, while in fact she is playing a. States in type structures thus encode essentially the same two ingredients as states in epistemic plausibility models: the players strategy choices and their information. The difference is that in epistemic plausibility models this information is represented qualitatively, by the players plausibility orderings, while in type structures it is represented probabilistically, by the value of λ i at each player s type. With this in hand, we are ready to look at what it means to maximizing one s payoffs given one s information in game. 1.5 Rationality Rationality in games is a natural extension to the idea of Bayesian or instrumental rationality from decision theory: maximizing one s payoffs given one s expectation. In games the expectations are about the choices of other rational agents, who also form expectations about each other s choices. Model for games are precisely designed to cope with such higher-order expectations. Definition 1.4 (Expected Value in type structure) The expected value 13

14 for player i of playing strategy s i given that he is of type t i is defined as follows. EV ti (s i ) = λ i (t i )(σ i, t i)v i (s i, σ i) t i σ i This definition of expected payoff in type structure is close to the decisiontheoretic idea of expected value, see e.g. [24] and [19]. The payoff that player i gets from choosing a strategy s i depends on what the other players choose. Player i has some expectations or information relative to these choices, which are encoded by his type t i. Hence, the expected value of playing s i for type t i is calculated by taking into account, first, all possible outcomes of the game where i plays s i, that is all v i (s i, σ i). The payoff at each of these outcome is weighted according to t i s expectations, that is according to the probability that λ i (t i ) assigns to their occurrence in combinations with the various types t i of the other players. A player s strategy s i is rational, given that he is of type t i, simply when it maximizes expected value. In other words, it is rational of i to choose s i, given is information in type t i, when no other strategy gives him a strictly higher expected value. Definition 1.5 (Rationality - the probabilistic case) Player i is rational at state (σ, t) whenever, for all s i S i, EV ti (s i) EV ti (σ i ) Consider the state (aa, t Ann, u Bob ) in the example from Section Ann is indeed rational at that state: she is convinced that Bob will play A, against which a is the only best response. Her current strategy, a, gives her an expected payoff of 1: EV tann (a) = 1(1/2) + 1(1/2) + 0(0) + 0(0) The expected payoff of choosing b, on the other hand, is 0: EV tann (b) = 0(1/2) + 0(1/2) + 1(0) + 1(0) Bob, however, is not rational at that state. The problem is not that he his choosing a wrong strategy in itself. After all, he successfully coordinates with 14

15 Ann at that state. Bob s irrationality stems from the fact that Bob mistakingly believes that Ann chooses b. Given this belief, Bob s best response would be to choose B, and not A: EV ubob (A) = 1(0) + 0(1) EV ubob (B) = 0(0) + 1(1) One can easily check that, in this example, Ann and Bob are rational at (aa, t Ann, t Bob ). Their respective strategy choices maximize expected value, given their respective types t Ann and t Bob. This last proviso is important. Assessing a player s rationality at a state of a game playing situation means comparing the different actions he could have taken, keeping his information constant. Another way to put this is to say that rationality at a given state is conditional of the information available at that state. The definition of rationality in epistemic plausibility models follows the same Bayesian line, but in qualitative terms. Look again at state (aa, t a, u b ), but this time at (aa) in the model of Figure 2. Here Bob believes that Ann plays b, because the state he considers the most plausible is ba. But in that context he would strictly prefer the outcome bb. Switching his strategy choice from A to B would give him an outcome he strictly prefers, in all eventualities that he considers most plausible. Choosing A is, in other words, a very bad decision for Bob. The following definition of an irrational strategy is intended to capture this idea. Definition 1.6 (Rationality - the qualitative case [25]) Given a state w, we write w[s i /w i ] for the profile σ that is just like σ(w) except that σ i = s i. Player i is irrational at w when there is a s i σ i (w) such that, v i (f(w )) v i (f(w [s i/w i])) for all w max i [w] i. Player i is rational at a state w when he is not irrational at that state. A player is thus irrational at a state w where there is an action of his, different from the one he plays at w, which gives him an outcome he prefers in every situation he considers most plausible. A player is rational when he is not irrational, that is when for each of his alternative action s i there is a state he most plausible where is current strategy choice is at least as good as s i. In the model of Figure 2, Ann and Bob are both rational in that sense at aa. Knowledge of one s own action plays an important role in this definition. It enforces that σ i (w ) = σ i (w), which means both that the information and the action of i at w are kept constant throughout the comparison or, the other way around, that i s rationality is assessed on the basis of the result of his 15

16 current choice according to different combinations of actions and information of the others. Recall that the same idea is at the heart of the definition of rationality in type structure: i s type and strategy choice are kept constant while those of the other players may vary. Type structure and epistemic plausibility models thus give similar answer to the question what is it rational for Ann and Bob to choose? They allows to assess precisely whether a player is choosing rationally, given the information he has. However, they do not fully answer the question of what is, in general, a rational play. Rationality as defined here depends entirely on the player s expectations, and one can legitimately wonder where do these expectations come from. For instance, what possible ground would Ann have for being convinced that Bob will play A? The epistemic program in game theory takes one step towards answering that question, by looking at what are rational choices in games given some more generic beliefs and expectations [9]. For example, instead of looking at what are Ann s rational choices given that she is sure that Bob plays A, one might ask what is rational for her to play in all situations where she believes that Bob is rational or that she believes that he believes that they are both rational. This, of course, requires to define precisely these kinds of general beliefs in epistemic models. References [1] F.J. Anscombe and R.J. Aumann. A definition of subjective probability. Annals Math. Stat., 34: , [2] R.J. Aumann. Interactive epistemology I: Knowledge. International Journal of Game Theory, 28: , [3] R.J. Aumann and J.H. Dreze. When all is said and done, how should you play and what should you expect? Technical report, CORE Discussion Paper, [4] A. Baltag and S. Smets. A qualitative theory of dynamic interactive belief revision. In Giacomo Bonanno, Wiebe van der Hoek, and Michael Wooldridge, editors, Logic and the Foundation of Game and Decision Theory (LOFT7), volume 3 of Texts in Logic and Games, pages Amsterdam University Press, [5] A. Baltag, L.S. Moss, and S. Solecki. The logic of public announcements, common knowledge and private suspicions. In TARK 98,

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