Reasoning about reasonable strategy sets for games of conventions
Presentation, ILLC July 2, 2008
Pelle Guldborg Hansen
Section for Philosophy and Science studies, Roskilde University Denmark
“Framing is logically prior to believing.” [Bacharach 2003, 66]
Let me begin by telling you how I got the idea for this paper. Some time ago I wrote a short paper in the
philosophy of law titled ‘Negligent rape and reasonable beliefs’. A basic requirement of criminal law is that
to be found guilty it has to be proven that one performed an action with criminal intent. Consequently you
cannot be found guilty of rape if you believed that the situation was one of mutual consent. However, what
I agued in this paper was that not only is it possible, but we often do as a matter of fact, hold persons liable
for having acted on certain beliefs despite these beliefs comforming to the usual requirements of rationality.
They are in some way unreasonable, so to say. That is, one may be found liable by not being able to give
an acceptable answer to the question “how could you do that?”, but by not being able to give an acceptable
answer to the question “how could you believe that?”.
However, it also struck me that this point squares badly with the account of normative prescriptions
in terms of frustrated expectations; the account usually available in game theoretical accounts of norms in
terms of conventions (a view I happen to embrace). That is, it squares badly with the idea that we can
only reasonably blame someone from failing to act rationally given his preferences and beliefs; the idea that
ultimately rationalizes punishment as a means to induce a change in preferences over actions, if not merely
to take revenge on the kinds of irrationality for which there seem no intelligible cure.
Lewis’theory of convention
Turning to the paper, we may recall that in his Convention: A Philosophical Study (1969) David Lewis
de…ned conventions as a behavioural regularity the conformity to which instantiate one out of multiple strict
coordination equilibria, made salient by precedent and operational by this being common knowledge to the
agents involved. Model 1 illustrates the single most famous example of game theoretical representation of
the structure underlying such conventions, namely the Driving game.
Game 1: A pure coordination game
Game 2: The telephone tag game
player 2
player 1
l
r
l
1; 1
0; 0
player 2
r
0; 0
1; 1
player 1
c
w
c
0; 0
1; 1
w
1; 1
0; 0
Model 2 illustrates the structure thought to underlie another convention discussed by Lewis; that of reconnecting cut o¤ phone calls in his hometown Oberlin, Ohio. The story Lewis recounts of this is that for a
period all phone calls in this city were cut o¤ without warning after three minutes due to some technical
issue. Soon, Lewis reports, “a convention grew up among Oberlin residents that when a call was cut o¤
the original caller would call back while the called party waited” (p.43). Obviously, as Lewis points out
elsewhere (p.11), conformity to some other practice could have solved this problem as well— and perhaps
just as well; for instance, what, given the cost of calling, may be thought of as the more ‘fair’counterpart
practice of having the original caller wait, while the called party would call back.
This game is similar to the driving game. Yet it di¤er from this by requiring the players to coordinate
on performing di¤ erent, though mutually correlated actions— hence, the conventions it gives rise to are
sometimes referred to as asymmetric conventions, cf. [Hansen 2007]. However, according to Lewis, this
di¤erence is spurious.
“There seems to be a di¤erence between equilibrium combinations in which every agent does
the same action and equilibrium combinations in which agents do di¤erent actions. This di¤erence
is spurious, however. We say that the agents do the same action if they do actions of the same
kind, particular actions falling under some common description. But actions can be described in
any number of ways, of which none has any compelling claim to primacy. For any combination of
1
actions, and a fortiori for any equilibrium combination of actions, there is some way of describing
the agents’alternative actions so that exactly those alternative actions in the given combination
fall under a common description... Whether it can be called a combination in which every agent
does the same action depends merely on the naturalness of that classi…cation... But [in Telephone
tag] what makes the …rst pair of action-descriptions more natural than the second? And so what
if it is?” [Lewis 1969, 10-12].
That is, Lewis asks, what makes the pair of action-descriptions fcall, waitg more natural than f(call if you’re
the original caller, wait if you’re the original reciever ); (call if you’re the original reciever, wait if you’re
the original caller )g? According to Lewis the answer seems to be ‘nothing’ and the di¤erence without any
consequence.
Whether or not Lewis is right that conventions may be characterized as strict coordination equilibria,
this characterization reveals little about how the systems of concordant mutual expectations settle on these
as well as perpetuate themselves. As to this point Lewis’theory is that conventions work by making salient
by precedent a certain strict coordination equilibrium in the recurrent game played, and this in such a way
that all agents come to expect one another to conform to it because its conspiciousness in this particular
respect is common knowledge to those partaking in the convention.
Lewis’theory of convention stirred a lot of interest almost from the outset. To begin with it was discussed
within analytical philosophy on its merits as conceptual analysis. Soon its in‡uence spread more widely to
shape ideas within the philosophy of law, social philosophy and economics in particular. However, after some
initial di¢ culties in grappling with the relatively novel framework of game theory, some genuine problems
at the foundation of the theory began to emerge. While early interpretations had readily embraced the idea
that salience of a given strategy pro…le followed from common knowledge of a successful precedent along with
suitable assumptions about the players’reasoning capabilities and inductive standards, [Grandy 1977], and
that given such salience of a precedented pro…le coordination would follow, [Heal 1978], it began to dawn
that none of this followed on closer analysis.
First of all, as any successful precedent in the sense of a series of successful combinations of strategies
may be projected in in…nitely many ways into the future, any strategy pro…le in an ensuing stage game will
be salient under some description of the precedented series of pro…les [Gilbert 1989, 333-334]. Second, even
if one particular strategy pro…le in a stage game could be determined as salient by precedent, this would not
give rational players such as those invoked by classical game theory su¢ cient reason to act in accordance
with this, as the instrumental rationality posited by such agents is purely forward looking, [Gilbert 1983].
Though a rational player would surely want to play his part in the salient strategy pro…le if other players
would play their part, he would also know that they would only play their part if he was to play his – but
this is exactly what he himself is still trying to …gure out. Conclusion: the equilibrium selection problem
remains.
Learning in games
During the last two decades evolutionary game theory has spread like …re within the social sciences as
a response to the equilibrium selection problem. When re-imported into the social sciences this framework is brought to serve as an aggregate model of individual learning and social imitation processes, cf.
[Fudenberg & Levine 1998]. Quite naturally, this in turn has spurred a wish to devise more credible learning
theoretic interpretations of equilibrium selection on the individual level; interpretations based on expectations, information and bounded rationality, rather than the mindless reproduction or imitation of successful
strategies.
One of the most widespread approaches to modeling learning in games work on the assumption that
agents adopt actions that optimize their expected payo¤s given what they expect others to do. These socalled best reply models of learning, then encompasses a variety of learning rules that ascribe di¤erent degrees
of rationality and sophistication in the agents’abilities in forming expectations about other agents’behavior,
cf. [Young 1998, 28].
Perhaps the most simple of such rules is …ctitious play. In this, each agent presumes that others are
playing some stationary (possibly mixed) strategy. Then he chooses his best reply at each round of a
recurrent game to the empirical frequency distribution of other agents past observed actions [Brown 1951].
The resulting outcome then enters as past observed actions to help shape expectations in future rounds. One
way of modelling the way expectations are updated in between rounds is by assuming each agent to follow
the Dirichlet rule, cf. [Vanderschraaf 1995].
2
In this model, an agent k updates his probability distribution nk ( ) according to the following formula:
If one of m possible outcomes A1 ; :::; Am can occur in each of n rounds, and nAi is the number of times that
Ai is observed to occur in the n rounds of the game, then
(1)
n
k (Ai )
=
n Ai +
P
n+ j
Ai
;
Aj
P
0, 1
j
m and j Aj > 0. Notice that the value of Ai re‡ects the strength of k’s
where Ai
initial expectation that Ai willP
occur in any given round. Hence before any rounds have occurred, that is,
when n = 0, the quotient Ai = j Aj determines an agents prior probability that Ai occurs.
Working on these assumptions the theory of learning in games is able to explain how systems of concordant mutual expectations may settle on one particular out of the multiple available strict equilibria in
a series of recurrent games. For instance, in the driving game above, for any initial con…guration of the
agents’ expectations these settle on one of the strict equilibria corresponding to a convention and when
…rst there, do this forever after. This result holds in general for any 2 2 symmetric two-player game
[Fudenberg & Levine 1998].
However, when the population taken to be playing a recurrent game is larger than two, it is no longer true
that expectations settle on one of the available strict equilibria in asymmetric labelling games like Telephone
tag. Instead, in this game expectations settle (or converges) on the mixed strategy equilibrium if the number
of agents in the population is equal, or as the number of agents in the population is unequal and goes to
in…nity.
The reason for this is straightforward. In these games agents has to perform di¤erent actions in order
to reach one of the strict equilibria available. The reason for this is straightforward. In these games agents
has to perform di¤erent actions in order to reach one of the strict equilibria available. In Telephone tag, for
instance, if on average agents expect others to call back each will wait and hence pull the average towards
wait. On the other hand, if on average agents expect others to wait they will call back and hence pull the
average towards call back. Only when agents on average expect others to do either with equal probability will
no one have reason to prefer one of the strategies to another and only then will the system of expectations
settle down.
This problem may be avoided if it is allowed to argue along the lines of Lewis that no pair of actiondescriptions is more natural than any other. Thus, the strategy sets upon which the learning process operate
may be re-described as Ak = f(call back if orginal caller, wait if orginal reciver);(wait if original caller,
call if original reciever)g. By this argument the game is transformed into one identical to the driving game,
whereby it follows that expectations will settle on one of the strict Nash equilibria of the new game, one
of which characterizes the convention observed in Oberlin. In fact, one need not even work on the ‘strong’
assumption completely transforming the orginal game into the new one. The same result obtains by merely
extending the original game with the new strategies conditionalising on particular features or events.
Conventions as correlated equilibria
One particular elegant way to incorporate explicitly into the model the idea of extending or transforming
the original game is by modeling these features or events as states of the worlds that agents may conditions
on. This then allows for characterizing social conventions as correlated equilibria.1
In general, following [Vanderschraaf 1995, 75] let each agent have a personal information partition Hk of
a probability space , where the elementary events ! 2 are called states of the world. Also, at each !,
every agent k knows that the element Hkj 2 Hk such that ! 2 Hkj has occurred, but does not in general
know which ! has occurred (however, for now it is assumed that they do). Hence, Hkj is taken to represent
k’s private information regarding the states of the world. A function f :
! S de…nes a system of
exogenously correlated strategy n-tuples, that is, for each ! 2 , the agents selects a strategy combination
f (!) = (f1 (!); :::; fn (!)) 2 S correlated with the state of the world !. Then, informally, f is a correlated
equilibrium if at each state of the world ! 2 , each agent k follows a principle of Bayesian rationality by
following f , that is, by playing fk (!), k maximizes his expected payo¤ given his private information and
expectations regarding his opponents.
1 The solution concept of correlated equilibrium was developed by (Aumann xxxx). The idea of modeling conventions as
correlated equilibria was originally suggested by [?] and [?], respectively. Yet, it was not until [Vanderschraaf 1995] that it was
carried out. The approach taken here is that of [?].
3
Now, applying this to the recurrent game of Telephone Tag, assume that the agents consider the set of
possible worlds = f! 1 ; ! 2 g, where ! 1 corresponds to the world in which agent k is the original caller and
agent i is the original receiver and ! 2 corresponds to the world in which agent k is the original receiver
and agent i is the original caller. Next, let the agents peg their strategies to
according to the function
f:
! fAk1 ; Ak2 g fAi1 ; Ai2 g de…ned by
{(A(A
f (!) =
k 1 ; A i2 )
if ! = ! 1
:
;
A
)
k2
i1 if ! = ! 2
Notice now that f constitute a strict correlated equilibrium, that is, either agent is strictly worse o¤ if he
deviates unilaterally from f . In fact, this equilibrium corresponds to the behavior of Oberlin residents playing
the Telephone Tag game. Further, it is obvious that there exist a function f 0 where the only di¤erence is
that the correlation is inversed with regard to the possible worlds. Hence, f is a convention in so far that it
also satis…es the requirements of common knowledge set up by Lewis.
It is in this way that it has been argued that conventions in general may be regarded as correlated
equilibria, or more precisely as functions of states of the world on which agents expectations has settled and
hence serve to coordinate their actions. Further, this has the advantage over Lewis’de…nition of convention
that it formally incorporates the notion of salience by formalizing the various pieces of information at the
agents’disposal by which they correlate their actions and expectations via f as the elements of a partition
of an event space . Thus, for instance, if the agents follow a pass-on-the-left convention corresponding to
the pure strategy equilibrium (l; l) in the Driving game the salience of this equilibrium is made precise by
constructing the simple event space
= f!g, and de…ning this convention as the correlated equilibrium
f (!) = (l; l), cf. [Vanderschraaf 1995, 78]. In a similar manner it was seen above that in case of the Oberlin
convention the salience of the corresponding equilibrium was made precise by the event space = f! 1 ; ! 2 g.
Presumptuous learning
So, de…ning conventions as correlated equilibria does makes explicit the various pieces of information by
which agents correlate their expectations as the elements of a partition of an event space . Also, given the
possibility of pegging strategies to states of the world, it is now possible to explain how the expectations of
learning agents applying the Dirichlet rule may settle on a convention like f relative to f 0 , or vice versa.
However, it does not explain why as well as how the agents initially come to correlate their expectations
with one particular partition, rather than another.
In the context of the theory of convention, this inadequacy conceals itself in three crucial, but nontrivial assumptions pertaining to how agents represent or frame the situation they are in. The …rst is the
prerequisite of applying a learning rule in the …rst place; namely, that others are playng some stationary
strategy. The second is that learning is taken to occur in small universes. The third is that agents share
the inductive standards. These assumptions make for referring to the resulting learning models as models
of presumptuous learning, which if unaccounted for in the context of convention threatens with explanatory
circularity.
To properly understand the problem with the stationarity assumption one …rst has to notice that a direct
implication of assuming agents to apply the Dirichlet rule is that each agent has to tie his learning process
to a series of coordination problems considered as one and the same recurrent problem by him. Next, this
rule, like …ctitious play, works by each learning agent – possibly incorrectly – presuming that other agents
are playing some stationary (possibly mixed) strategy. In order for this to make sense, however, he must
also be assuming his opponents’ behavior to be at least correlated with the same recurrent problem. If
this was not the case it would be very di¢ cult to understand his presumption of stationarity; what else
should ‘stationary’be taken to mean in this context? But, in turn, if this is to make sense in the context
of the recurrent problem recognized by him, it seems that he must already be taking them to be playing
what amounts to a convention, or at least having a tendency to do so. In other words, the presumption of
stationarity is a presumption of the existence of some convention.
Second, learning accounts of conventions usually approach the question of how agents initially come to
correlate their expectations on a convention on the presumption that only very small and closed strategy sets
are considered by the agents; and rarely this is taken to go beyond what results from a single partition of the
world they are in. Further, it is taken that the agents have common knowledge as to what this event space
is— i.e. what is to be regarded as the unique salient feature of the recurrent problem to peg their strategies
4
on, cf. [Vanderschraaf 1995]. Thus, for instance, the event space constructed in modeling conventions in the
Driving game is the simple event space = f!g, while it in the case of Telephone tag is taken to be that of
= f! 1 ; ! 2 g for all agents. In other words, the agents are assumed to operate in, and assume each other to
be operating in, the smallest possible, yet arbitrary universes necessary for conventions to emerge.
However, realities are much more complex as they o¤er …nite, but huge sets of arbitrary states of the
world to peg their strategies on. Hence, they make available to each agent really big strategy sets. Given
what is stipulated about learning agents in these models there is nothing to prevent agents from considering
alternative strategy sets, or larger and more complex sets. Thus, for instance, besides considering the states
of being the original caller and the orginal reciever in the driving game, an agent could also be pegging his
strategies on states of the world such as whether he was talking when the call was cut o¤, whether he is the
one who would bene…t the most from resuming the call, whether he would like to resume the call, or even
whether he has more or less than ten matches in his pocket, and so on.
In fact, the problem is even more radical than this. Should one come to think that you could use assume
considerations of simplicity in order to have agents only consider the most simple strategies this would not
get you very far, since what the original set is taken to be is arbitrary in exactly the same way and o¤er just
as many alternatives. For instance, in the driving game the agents could just as well consider their problem
in terms of the strategies going towards the sun, going away from the sun.
Finally, it should be noted that practical problems in principle may be the least of concerns for learning
theories of convention. This because any expectation as to the future play of opponents’is underdetermined
by any …nite series of observations. To see this, note that any series of observations made up until round n 1
will put equal weight on an agents’expectations as to whether his opponents’are following the stationary
strategy ‘always play Ai given the state of the world !’ and ‘always play Ai until round n, then play Aj ,
given the state of the world !’. Consequently, as long as such ‘unbehaved’ strategies in principle remains
possible there is no way for an agent usinig the Dirichlet rule to come to a decision as to what is a best
reply in any stage of the recurrent problem he faces. Only by assuming that each agent apply his learning
process to a strategy set formed under the presumption of common inductive standards and that inductive
standards de facto are common are the agents’expectations ever able to settle on a convention.
Tacking stock, then, we see that the learning models of the emerge of conventions developed do really
not explain how conventions in general may emerge from some pre-conventional state, but rather how agents
may come to learn a convention not knowing which convention is in play, but presuming that (1) there is
such a convention, (2) that the possible strategies that he has to consider are those from a simple and closed
universe that is common knowledge, and (3) that he shares his inductive standards with others and that this
is common knowledge.
While such auxiliary assumptions may be considered as perfectly natural by game theorists, they are
considered as deeply suspicious by social scientists. In fact, scrutinizing some of the more abstruse criticisms
made by social scientists of the game theoretic approach in the study of convention these may actually
be read as criticisms struggling with how to formulate and challenge these assumptions. To him the most
interesting and fundamentally signi…cant things seems to be going on at the level of these assumption, not
at the level of the learning process: what are the viable conventions, how do people manage to coordinate
only after a few rounds given the many strategies available, how do conventions connect with the way social
situations are percieved and interpreted, how do they relate to types of norms, ect.
Reasoning about reasonable strategy sets
What I suggest in my paper is that the theory of convention should be enlarged by a theory that explicitly
addresses how agents use and reason about what reasonable strategy sets to ascribe to each other in situations
of coordination. In particular, I suggest that it is possible to justify the small and closed universe assumption
as well as to some extent the assumption of common inductive standards with such a theory. Now, one could
in principle do this within the theory of learning in games. Stipulating huge strategy sets and then wait for
learning agents to slowly converge on certain strategies and let them extrapolate these to other situations.
However, people in the real world seem to be both more intelligent and less patient than would require to
make such an account true. So I think it more reasonable that agents reason about what reasonable strategy
sets to attribute to each other.
I think one way of approaching this question is by using parts of the ‘theory of frames’ developed
by Michael Bacharach as part of his Variable Frame Theory, see [Bacharach 1993], [Bacharach 2003], and
5
[Bacharach 2006]. According to this theory conventional game theory, including the learning theories studied
in the last section, confuse the world as seen by the theorist with the world as seen by the decision maker. In
particular, the game theorist is said to be mistaken in imposing particular conceptual schemes on the world
and then proceed on the assumption that the agents of the model reasons or learns within the scheme that
the theorist has imposed.
The elements of framing theory
In framing theory a frame is a set of concepts or predicates an agent uses in thinking about the world. In
a phrase recurrently used by Bacharach and remnisent of Wittgenstein’s Duck Rabbit drawing ‘one just not
just see, but one sees as’ [Bacharach 2003, 63]. Formally, framing theory distinguishes between object of
choice and act-descriptions. To illustrate this distinction consider an experiment where you are asked to
choose one of the symbols in …gure 2.
Figure 2: Three symbols
Objectively, there are three symbols in this …gure. These might have been identi…ed by the designers of
the experiments as x1 ,x2 , and x3 (working from left to right), respectively; assuming that you have never
entertained terms like ‘x1 ’to yourself. Now, in this situation fx1 ; x2 ; x3 g is the set of objects of choice. Yet,
you do not represent your decision problem in terms of this set. Instead it may the case that you describe
these objects to yourself by predicates such as circle, triangle, and cross. Your descision problem is then to
be de…ned by the set of act-descriptions fchoose the circle, choose the triangle, choose the crossg.
In general, let S = fx1 ; :::xn g be a set of objects of choice and P = f 1 ; ::: n g be a set of predicates. A
frame F P , then, is a set of predicates suitable for describing the objects in S, where if i 2 F , then E( i )
is a function that denotes the extension of in S, that is the (possibly empty) subset of S which satisfy .
We say that x and y 2 S are F-equivalent, written x = F y, if
(1)
y 2 E( )
i¤ x 2 E( ) 8 2 F
That is, equation (1) says that if x and y are F-equivalent, then the predicates of a frame do not su¢ ce to
discriminate x from y. In this way a frame F induces a partition, PF , on S, whose cells are F-equivalence
classes [Bacharach 2003, 63].
In Bacharach’s variable frame theory a game is played rationally, but which game gets played is determined
by nonrational (e.g. perceptual) player chracteristics specifying the way agents frame their decision problem.
The term natural framing is used to refer to how people tend to spontaneously frame the situations they face
in absence of manipulation. Bacharach takes it that situations are spontaneously framed by being primed
by contextual factors:
In describing a given set of objects, a given individual may sometimes use one frame, sometimes another, depending on contextual factors which brings to mind, or prime, particular ideas
[Bacharach 2006, 12].
Further, Bacharach holds that spontaneously primed frames are characterized by certain features. One is
that frames are incomplete. That is, as a consequence of the logical priority of framing to believing, the space
of propositions or events on which an agent’s subjective probabilities are de…ned is always incomplete— there
are propositions expressible in English concerning one’s situation about which one has no beliefs one way
or another. Another is that the principle of negative introspection fails for natural framing: I may fail to
know that P, where P is a proposition, and not know that I do not know P. Finally, Bacharach claims that
although people are often unware of relevant aspects of the situation, they are aware of this. That is, they
know that there exist aspects of which they are unaware; they know their frames are incomplete.
Also, certain structures are imposed on frames. In particular, it is assumed that predicates belong to
disjoint sets, called families, where each family is interpreted as a set of mutually exclusive, but not necessarily
6
exhaustive speci…cations of a potential attribute of the objects of choice. For instance, the predicate ‘triangle’
will most likely belong to the family of shapes Fc = fcross; circle; triangle; square; :::g.
In its entirety, variable frame theory is an analysis of rational play in games which takes account of
frames [Bacharach 2006, 14]. It asssumes the existence of an objective game having S = fx1 ; :::; xn g as
objective strategies. It distinguishes this game from the framed game F in which players choose and reason
among act-descriptions 2 F , where F = fF0 ; :::; Fn g is the universal frame having particular frames Fi as
members, and where F0 = fthingg is the generic frame such that E(thing) includes every object of choice.
v(Fi ) is then the probability that Fi is available to agent i. This implies that if a 2 Fi comes to mind, so
does any other in the family. Also, it is assumed that v(F0 ) = 1. That is, some description comes to the
mind of an agent no matter what. Further Bacharach makes the assumption that when an act-description ,
where E( ) is not a singleton, each object xi 2 E( ) has an equal probability of being the object of choice;
when choice is performed in this way, Bacharach refers to it as picking.
Ultimately, the crucial idea in Bacharach’s theory of frames is then that an agent can choose, and can think
about other agents choosing, only those things for which his own frame provides descriptions. In particular,
agents are assumed to be expected utility maximisers choosing according to a principle of coordination given
the probabilities ascribed by them to other agents holding particular frames.
Reasoning about reasonable strategies
The approach I take di¤ers from this by using the theory of frames merely as a framework of thought for
exploring the reasoning strategies used by agents to arrive at what I call reasonable frames in the context
of social convention. In particular, like for presumptuous learning, each agent is taken to reason from the
premise that other agents’are using some stationary strategy to such frames. That is, each agent is reasoning
on the assumption that other agents are following some convention. Thus, instead of treating frames as given
as in variable frame theory the approach taken here develops a theory of how agents may reason about what
frames to adopt given that others are following a convention, or alternatively are trying to reason about
what frames to adopt as well under the same conditions.
Now, take the telephone tag game as an example. We assume the existence of an objective game
having S = fc; wg as the set of objects of choice. We now distinguish this from the framed game F in
which agents choose and reason among act-descriptions
2 F , where F = fF0 ; :::; Fn g is the universal
frame having particular frames Fi as members, and where F0 = fthingg is the generic frame such that
E(thing) includes every object of choice. Now assume that you are playing the telephone tag having the
frame F1 = fcall; waitg 2 F . However, given what I have said above, you know that you will not be able
to coordinate with others by framing the problem in therms of F1 . Unfortunately you also know that if you
begin to consider the kind of alternative strategies for pegging son states of the world making coordination
possible, there will be too many for you to consider. Anyway, we assume that you consider them all.
However, my point now is that using the stationarity assumption you will be able to cut down the set of
strategies that you can reasonably ascribe to others in the situation. For one, if you believe that everyone else
is following some stationary strategy of what amounts to strict equilibrium ensuring recurrent coordination
in the given problem, you will also know that this strategy has to correlate with some state of the world.
Hence, you know that you will not have to consider strategies that do not correlate with such states.
Next, if you believe that everyone else is following some stationary strategy of what amounts to strict
equilibrium ensuring recurrent coordination in the given problem, you will know that they could only achieve
such recurrent success if whatever state of the world that they peg the more basic strategies on correlate their
behaviour almost perfectly. That is, the strategy must be able to be part of a correlated equilibrium, and as
such the state that strategies are pegged on will have to be public to the extent necessary. Consequently, you
throw away all strategies that do not condition on some public state or signal in the situation you experience.
Third, again given the stationarity assumption for the recurrent game, you will know that in order to
ensure recurrent coordination the state or signal that forms a part of the strategies left to consider must not
only be able to correlate expectations in your situations but in all situations in general that you take the
stationary strategy to apply to. Thus, you can throw away all the strategies that are not public in every
such situation.
Fourth, given the stationarity assumption, you will know that such states do not only have to be public
and common to all situations that the strategy apply to, but also that it has to assign each agent to di¤erent
roles in the game for every such situation. That is, you can throw away all strategies that condition on
states of the world
7
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