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\section*{Notes}}
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\section{Introduction}
Game theorists represent games in two standard ways: in \emph{normal form},
where
each agent simply
chooses a strategy, and in \emph{extensive form},
using game trees, where
the agents make
choices over time.  An extensive-form representation has the
advantage that it describes the
dynamic
structure of the
game---it  explicitly represents the sequence of decision problems
encountered by agents.  However, the extensive-form representation
purports
to do more than just describe the structure of the game; it also
attempts
to represent the information
that players have
in the game, by the use of
\emph{information sets}.  Intuitively, an information set consists of a
set of nodes in the game tree where a player has the same information.
However,
as Halpern \citeyear{Hal15} has pointed out,
information sets may not adequately represent a player's information.

Halpern makes this point by considering the following single-agent game
of imperfect recall, originally
presented
by Piccione and Rubinstein
\citeyear{PR97}:
\begin{figure}[htb]
\centerline{\psfig{figure=game2.ps,height=2.2in}}
\caption{A game of imperfect recall.}
\label{fig2}
\end{figure}
The game starts with nature moving either left or right, each with
probability $1/2$.  The
agent
can then either stop the game
(playing move $S$) and get a payoff of 2, or continue, by playing move
$B$.  If he continues, he gets a high
payoff if he matches nature's move, and a low payoff otherwise.
Although he originally knows nature's move, the information set that
includes the nodes labeled $x_3$ and $x_4$ is intended to indicate that
the player forgets whether nature moved left or right after moving $B$.
Intuitively, when he is at the information
set~$X$,
the agent is not
supposed to know whether he is at $x_3$ or at $x_4$.

It is not hard to show that the strategy that maximizes expected utility
chooses action $S$ at node $x_1$, action $B$ at node
$x_2$, and action $R$ at the information set $X$ consisting of $x_3$ and
$x_4$.  Call this strategy $f$.  Let $f'$ be the strategy of choosing
action $B$ at $x_1$, action $S$ at $x_2$, and $L$ at $X$.
Piccione and Rubinstein argue that if node $x_1$ is reached, the
player
should reconsider, and decide to switch from $f$ to $f'$.
As Halpern points out, this is indeed true, provided that the
player
knows
at each stage of the game
what strategy he is currently using.
However, in that case, if the
player
is using $f$ at the information set,
then he knows that he is at node $x_4$; if he has switched and is
using $f'$, then he knows that he is at $x_3$.  So, in this setting, it
is no longer the case that the
player
does not know whether he is at
$x_3$ or $x_4$ in the information set; he can infer which
state
he is at from the strategy he is using.

In game theory, a \emph{strategy} is taken to be a function from
information sets to actions.
The intuition behind this is
that, since an agent cannot tell the nodes in an information set apart,
he must do the same thing at
all these nodes.  But this example shows that if the agent has imperfect
recall but can switch strategies, then he can arrange to do different
things at different nodes in the same information set.
As Halpern \citeyear{Hal15} observes,
`\,``situations that [an agent] cannot
distinguish'' and ``nodes in the same information set'' may be two quite
different notions.'  He suggests using the game tree to describe
the structure of the game, and using the runs and systems framework
\ijcai{
introduced by
Moses \citeyear{Mosthesis} and by
Halpern and Fagin \citeyear{HFfull}, and further developed by
Fagin et al.~\citeyear{FHMV}}
\ijcaionly{\cite{FHMV}}
to describe the agent's information.
The idea is that an agent has an internal \emph{local state} that
describes all the information that he has.  A strategy (or
\emph{protocol} in the language of \cite{FHMV}) is a function from local
states to actions.  Protocols capture the intuition that what an agent
does can depend only what he knows.  But now an agent's knowledge is
represented by its local state, not
by
an information set.  Different
assumptions about what agents know (for example,
whether
they know their current strategies) are
captured by running the same protocol in different \emph{contexts}.
If the information sets appropriately represent an agent's knowledge in
a game, then we can identify local states
with
information sets.  But, as
the example above shows, we cannot do this in general.

A number of \emph{solution concepts} have been considered
in the game-theory literature, ranging from Nash equilibrium
and \emph{correlated equilibrium} to refinements of Nash equilibrium
such as \emph{sequential
equilibrium} and weaker notions such as \emph{rationalizability} (see
\cite{OR94} for an overview).
The fact that game trees represent both the game and the players'
information has proved critical in defining solution concepts in
extensive-form games.  Can we still represent solution concepts in a
useful way using  runs and systems to represent a player's information?
As we show here,  not only can we do this, but we can do it in a way
that gives deeper insight into solution concepts.
Indeed, all the standard solution concepts
in the literature
can be understood as instances of a single
\emph{knowledge-based (kb) program} \cite{FHMV,FHMV94}, which captures
the underlying intuition that a player should make a best
response, given her beliefs.
The differences between solution concepts arise from
running the kb program in different contexts.

In a kb program,
a player's actions depend explicitly on the player's knowledge.
For example, a kb program could have a test that
says ``If you don't know that Ann received the information, then
send her a message'',
which can be written
$$\quad \mathbf{if}\ \neg B_i(\mathrm{Ann\  received\ info}) \
\mathbf{then}\  \mathrm{send\ Ann\ a\  message}.$$
This kb program has the form of a standard {\bf{if}} \ldots {\bf{then}}
statement, except that the test in the {\bf{if}} clause is a test on $i$'s
knowledge (expressed using the modal operator $B_i$ for belief; see
Section~\ref{sec:background} for a discussion of the use of knowledge
vs.~belief).

Using such tests for knowledge
allows
us to
abstract away from low-level details of how the knowledge is obtained.
Kb programs have been applied to a number of problems in the computer
science literature (see \cite{FHMV} and the references therein).  To see
how they can be applied to understand equilibrium,
given a game $\Gamma$ in normal form, let
${\cal S}_i(\Gamma)$ consist of all the
pure
strategies for player
$i$ in $\Gamma$.
Roughly speaking, we want a kb  program that says that if player $i$
believes that she is about to perform strategy $S$ (which we express
with the formula $\doact_i(S$)), and she believes that she would not do
any better with another strategy, then she should indeed go ahead and run
$S$.
This test can be viewed as embodying rationality.
There is a subtlety in expressing the statement ``she would not
do any better with another strategy''.  We express this by saying ``if
her expected utility, given that she will use strategy $S$, is $x$, then
her expected utility if she were to use strategy $S'$ is at most $x$.''
The ``if she were to use $S'$'' is a \emph{counterfactual} statement.
She is planning to use strategy $S$, but is contemplating what would
happen if she were to do something counter to fact, namely, to use $S'$.
Counterfactuals have been the subject
of intense study in the philosophy literature (see, for example,
\cite{Lewis73,Stalnaker68}) and, more recently, in the game theory
literature (see, for example, \cite{Aumann95,Hal28,Samet96}).
We write the counterfactual ``If $A$ were the case then $B$ would be
true'' as ``$A \RCond B$''.  Although this statement involves an ``if
\ldots then'', the semantics of the counterfactual implication $A \RCond
B$ is quite different from the material implication $A \Rightarrow B$.  In
particular, while $A \Rightarrow B$ is true if $A$ is false, $A \RCond B$
might not be.

With this background, consider the following kb program for player $i$:
$$\begin{array}{ll}
\mathbf{for\ each}\  \mathrm{strategy}~S \in {\cal S}_i(\Gamma) \ \mathbf{do}\\
\quad \mathbf{if}\ B_i(\doact_i(S) \land \forall x(\EU_i = x \Rightarrow\\
\qquad \bigwedge_{S' \in {\cal S}_i(\Gamma)}
(\doact_i(S') \RCond (\EU_i \le x))))\  \mathbf{then}\  S.
\end{array}$$
This kb program is meant to capture the intuition above.
Intuitively, it says that if player $i$ believes that she is
about to perform strategy $S$ and, if her
expected utility is $x$, then
if she were to
perform another strategy $S'$,
then
her expected utility would be
no
greater than $x$,
then she
should perform strategy $S$.
Call this kb program $\EQNF^\Gamma$ (with the individual instance for
player~$i$ denoted by $\EQNF^\Gamma_i$).
As we show, if all players
follow $\EQNF^\Gamma$,
then they end up
playing some type of equilibrium.  Which type of equilibrium they play
depends on the context.
Due to space considerations, we focus on three examples in this
abstract.
If the players have a
common prior on the joint strategies being used, and this common prior
is such that players' beliefs are independent of the
strategies
they use,
then they play a Nash equilibrium.
Without this independence assumption, we get a \emph{correlated equilibrium}.
On the other hand, if players have possibly different priors on the
space of strategies, then this kb program defines \emph{rationalizable}
strategies \cite{Ber84,Pearce84}.

To deal with extensive-form games, we need a slightly different kb
program, since agents choose moves, not strategies.  Let
$\EQEF^\Gamma_i$ be the following program, where
$\sfa \in \PM$ denotes that $\sfa$ is a move that is currently possible.
$$\begin{array}{ll}
\mathbf{for\ each}\ \mathrm{move}\ \sfa \in \PM\ \mathbf{do}\\
\quad \mathbf{if}\ B_i(
\doact_i(\sfa) \land \forall x ((
\EU_i = x) \Rightarrow \\
\quad \quad \bigwedge_{\sfa' \in \PM}
(\doact_i(\sfa') \RCond (\EU_i \le x)))) \  \mathbf{then}\  \sfa.
\end{array}
$$
Just as $\EQNF^\Gamma$ characterizes equilibria of a game
$\Gamma$ represented in normal form, $\EQEF^\Gamma$ characterizes
equilibria of a game represented in extensive form.
We give one example here: sequential equilibrium.
To capture sequential equilibrium, we need to
assume that information sets do correctly describe an agent's
knowledge.
If we drop this assumption, however, we can distinguish
between the two equilibria for the game described in
Figure~\ref{fig2}.

All these solution concepts are based on expected utility.
But we can also consider solution concepts based on other decision
rules.  For example, Boutilier and Hyafil \citeyear{BH04} consider
\emph{minimax-regret} equilibria, where each player uses a strategy that
is a best-response in a minimax-regret sense to the choices of the other
players.  Similarly, we can use
\emph{maximin equilibria}
\cite{AB06}.  As pointed out by Chu and Halpern
\citeyear{CH03a}, all these decision rules can be viewed as instances of
a generalized notion of expected utility, where uncertainty is
represented by a \emph{plausibility measure}, a generalization of a
probability measure, utilities are elements of an arbitrary partially
ordered space, and plausibilities and utilities are combined using
$\oplus$ and $\otimes$, generalizations of $+$ and $\times$.
We show
in the full paper
that, just by interpreting ``$\EU_i=u$'' appropriately, we can capture
these
more exotic
solution concepts as well.
Moreover, we can capture solution concepts in games where the game
itself is not
common knowledge,
or where agents are not aware of all
moves available, as discussed by Halpern and R\^ego \citeyear{HR06}.

Our approach thus provides a
powerful tool for representing
solution concepts, which works even if (a) information sets do not
capture an agent's knowledge, (b) uncertainty is not
represented by probability,
or
(c) the underlying game is not common knowledge.

The rest of this paper is organized as follows.  In
Section~\ref{sec:background}, we review the relevant background on game
theory and knowledge-based programs.  In Section~\ref{sec:mainresults},
we show that $\EQNF^{\Gamma}$ and $\EQEF^{\Gamma}$ characterize Nash
equilibrium, correlated equilibrium, rationalizability, and sequential
equilibrium in a game $\Gamma$ in the appropriate contexts.  We conclude
in Section~\ref{sec:conclusions} with a discussion of how our results
compare to other characterizations of solution concepts.

\section{Background}\label{sec:background}

In this section, we review the relevant background
on games and
knowledge-based programs.  We describe only
what we need for proving
our results.  The reader is encouraged to consult
\cite{OR94} for more on game theory,
\cite{FHMV,FHMV94} for more on knowledge-based programs without
counterfactuals, and \cite{HM98a} for more on adding counterfactuals to
knowledge-based programs.

\commentout{
\subsection{Lexicographic Probability Systems}
An LPS  is just a pair  $(W,\vecmu)$, where $W$ is a sample space and
$\vecmu = (\mu^0, \ldots, \mu^k)$ is a sequence of probability measures
on $W$.  In
our setting, $W$ is finite and the
probability measures $\mu^j$ are defined on all subsets of $W$.
Intuitively, $\mu^0$ is the most significant measure, $\mu^1$ is the
next most significant measure, and so on.
We identify a probability measure $\mu$ with the LPS $(\mu)$ of length
1.
Given a real-valued random
variable $f$ on $W$, we define the expected value of $f$ with respect to
a sequence~%
$\vecmu$, denoted $E_{\vecmu}[f]$, to be the tuple $(E_\mu^0[f]),
\ldots, E_\mu^k[f])$, where the
$j^{\rm th}$
component of the tuple is the
expected value of $f$ with respect to $\mu^j$.

BBD define conditioning
with respect to an LPS
as follows.  Given an LPS $\vecmu =
(\mu^0, \ldots, \mu^k)$ on a
space $W$ and $U \subseteq W$
such that $\mu^j(U) > 0$ for some index $j$, let $\vecmu|U =
(\mu^{m_0}(\cdot \mid U), \ldots, \mu^{m_h}(\cdot \mid U))$,
where $(m_0, m_1, \ldots, m_h)$ is the subsequence of all indices
for which the probability of $U$ is positive.
Note that
$\vecmu|U$ is undefined if $\mu_j(U) = 0$ for all $j$.

}

\subsection{Games and Strategies}
A game in \emph{extensive form} is described by a game tree.
Associated with each non-leaf node or history is either a player---the
player whose move it is at that node---or nature (which can make a
randomized move).  The nodes where a player $i$ moves are further
partitioned into \emph{information sets}.
With each run or maximal history $h$
in the game tree and
player $i$ we can associate $i$'s utility, denoted $u_i(h)$, if that
run is played. A \emph{strategy} for player $i$ is a (possibly
randomized) function from $i$'s
information sets to actions.
Thus a strategy for player $i$ tells player $i$ what to do at each node
in the game tree where $i$ is supposed to move.
Intuitively, at all the nodes that player $i$ cannot tell apart, player
$i$ must do the same thing.
A \emph{joint strategy} $\vec{S} = (S_1,
\ldots, S_n)$ for the players determines a distribution over
paths in the game tree.  A \emph{normal-form game} can be viewed as a
special case of an extensive-form game where
each player makes only one move, and
all players move
simultaneously.

\subsection{Protocols, Systems, and Contexts}

To explain kb programs, we must first describe standard protocols.
We assume that, at any given point in time, a player in a
game is in some \emph{local state}.
The local state could include the
history of the game up to this point, the strategy being used by the
player, and perhaps some other features of the player's type, such as
beliefs about the strategies being used by other players.  A global
state is a tuple consisting of a local state for each player.

A \emph{protocol} for player $i$ is a function from player $i$'s local
states to actions.  For
ease of exposition, we consider only deterministic
protocols, although
it is relatively straightforward to model
randomized protocols---corresponding to mixed strategies---as
functions from local states to distributions over actions.
Although we restrict to deterministic protocols,
we deal with mixed strategies by considering distributions over pure
strategies.

A \emph{run} is a sequence of global states;
formally, a run is a function from times to global states.
Thus, $r(m)$ is the global state in run $r$ at time $m$.
A \emph{point} is a pair $(r,m)$ consisting of a run $r$ and time~$m$.
Let $r_i(m)$ be $i$'s local state at the point $(r,m)$;
that is, if
$r(m) = (s_1, \ldots, s_n)$,
then $r_i(m) = s_i$.
A \emph{joint protocol} is
an assignment of
a protocol for each player; essentially,
a joint protocol is a joint strategy.   At each point, a joint
protocol $\vec{P}$ performs a \emph{joint action}
$(P_1(r_1(m)), \ldots, P_n(r_n(m)))$,
which changes the global state.
Thus, given an initial global state, a joint protocol $\vec{P}$
generates a
(unique)
run, which can be thought of as an execution of $\vec{P}$.
The runs in a normal-form game involve only one round and two time
steps: time 0 (the initial state) and time 1,
after the
joint strategy has been executed.
(We assume that the
payoff is then represented in the player's local state at time 1.)
In an extensive-form game, a run is again characterized by the
strategies used, but now the length of the run depends on the path of
play.
\ijcai{
We remark that in typical extensive-form games at most one player
moves at any given point. This would correspond to at most one player
having a local state that signals that the player should move at a state,
and having $P_j(r_j(m))$ be
the ``{\em null}'' move $\Skip$ for all other players.
A player performing $\Skip$ doesn't affect the state of the system in any way.
}

\ijcai{In order to  study the properties of a joint protocol, and the
knowledge
that agents have when running it, we consider the set of
runs of the protocol,
together with a possibly different
probability measure
on runs for
each player.}
A \emph{probabilistic system} is a tuple
$\PS = ({\cal R},\vecmu)$, where ${\cal R}$ is a set of runs and
$\vecmu=(\mu_1, \ldots, \mu_n)$ associates a probablity $\mu_i$
on the runs of ${\cal R}$ with each player~$i$.
Intuitively, $\mu_i$ represents player $i$'s prior beliefs.
In the special case where $\mu_1 = \cdots = \mu_n = \mu$,
the players have a
\emph{common prior}
$\mu$ on ${\cal R}$.
In this case, we
write just $({\cal R},\mu)$.

We are interested in the system corresponding to a
joint protocol~$\vec{P}$.  To
determine this system,
we need to describe the setting
in which $\vec{P}$ is being executed.  For our purposes, this setting can be
modeled
by a set ${\cal G}$ of global states, a subset $\G_0$ of ${\cal G}$
that describes the possible
\emph{initial}
global states, a set ${\cal A}_s$ of
possible joint actions at each global state $s$, and
$n$ probability measures on $\G_0$, one for each player.  Thus, a
\emph{probabilistic context} is a tuple
$\gamma = ({\cal G}, \G_0, \{{\cal A}_s: s \in {\cal G}\},\vecmu)$.%
\footnote{We are implicitly assuming that the global state that results
from performing a joint action in ${\cal A}_s$ at the global state~$s$ is
unique and
obvious; otherwise, such information would also appear in the context,
as in the general framework of \cite{FHMV}.}
A joint protocol $\vec{P}$ is \emph{appropriate}
for such a context~$\gamma$ if,
for every global state $s$, the joint actions that $\vec{P}$
can generate are in ${\cal A}_s$.
When $\vec{P}$ is appropriate for~$\gamma$, we abuse notation slightly and
refer to~$\gamma$ by specifying only the pair $(\G_0,\vecmu)$.
A protocol $\vec{P}$ and
a context $\gamma$ for which $\vec{P}$ is appropriate
generate
a system; the system depends on the initial states and probability measures
in $\gamma$. Since these are all that matter, we typically
simplify the description of a context by omitting
the set ${\cal G}$ of global states and the sets ${\cal A}_s$ of global actions.
Let $\Rrep(\vec{P},\gamma)$ denote the system generated by joint protocol
$\vec{P}$ in context
$\gamma$.  If $\gamma = (\G_0, \vecmu)$, then
$\Rrep(\vec{P},\gamma) = ({\cal R},\vecmu')$, where ${\cal R}$
consists of
a the run $r_{\vec{s}}$ for each initial state $\vec{s} \in \G_0$,
where
$r_{\vec{s}}$ is the run generated by $\vec{P}$
when
started in state
$\vec{s}$,
and $\mu_i'(r_{\vec{s}}) = \mu_i(\vec{s})$, for $i = 1,
\ldots, n$.

A
probabilistic
system $({\cal R},\vecmu')$ is \emph{compatible} with a
context $\gamma = (\G_0,\vecmu)$
if (a) every initial state in $\G_0$ is the
initial state of some run in ${\cal R}$, (b) every run is the run of some protocol
appropriate for
$\gamma$, and (c) if ${\cal R}({\vec{s}})$ is the set of runs in
${\cal R}$ with initial global state $\vec{s}$, then $\mu_j'({\cal R}(\vec{s})) =
\mu_j(\vec{s})$, for $j = 1, \ldots, n$.
Clearly $\Rrep(\vec{P},\gamma)$ is compatible with $\gamma$.

We can think of the context as describing background
information.  In
distributed-systems applications, the context also
typically includes information about message delivery.
For example,
it may determine whether all messages sent are received in
one
round, or whether they may take up to, say, five rounds.
Moreover,
when this is not obvious,
the context specifies how actions transform the global state;
for example, it describes what happens if in the same joint action two
players attempt to
modify the same memory cell. Since such issues do not
arise in the games we consider, we ignore these facets of contexts here.
For simplicity,
we consider only contexts where
each initial state corresponds to a particular joint strategy of~$\Gamma$.
That is, $\Sigma^\Gamma_i$ is a set of local states for
player $i$ indexed by
(pure)
strategies.
The set $\Sigma^\Gamma_i$ can be viewed as describing
$i$'s types; the state $s_S$ can the thought of as the initial state
where player $i$'s type is such that he plays $S$ (although we stress
that this is only intuition; player $i$ does not
\emph{have}
 to play $S$ at the
state $s_S$).  Let $\G_0^\Gamma =
\Sigma^\Gamma_1 \times \ldots \times \Sigma^\Gamma_n$.
We will be interested in contexts where the set of
initial global states is a
subset $\G_0$ of $\G_0^\Gamma$.
In a normal-form game, the only actions
possible for player $i$ at an initial global state
amount to choosing a pure strategy,
so the joint actions are joint strategies; no actions are
possible at later times.
For an extensive-form game, the possible moves are described by the game
tree.
We say
that
a context for an extensive-form game is \emph{standard} if the
local states have the form $(s,I)$, where $s$ is the initial state
and $I$ is the
current information set.  In a standard context, an agent's knowledge is
indeed described by the information set.  However, we do not require a
context to be standard.  For example, if an agent is allowed to switch
strategies, then the local state could include the history of
strategies used.  In such a context, the agent in the game of
Figure~\ref{fig2} would know more than just what is in the information
set, and would want to switch strategies.

\subsection{Knowledge-Based Programs}
A \emph{knowledge-based program} is a syntactic object.  For our
purposes, we can take a knowledge-based program for player $i$ to have
the form
$$\begin{array}{l}
{\bf if }~\kappa_1 ~{\bf then} ~{\sfa_1}\\
{\bf if }~\kappa_2 ~{\bf then} ~{\sfa_2}\\
\dots,\\
\end{array}
$$
where
each
$\kappa_j$ is a Boolean combination of formulas of the form $B_i \phi$,
in which
the $\phi$'s can have nested occurrences of
$B_\ell$ operators and
counterfactual implications.
We assume that the tests $\kappa_1,
\kappa_2, \ldots$ are mutually exclusive and exhaustive, so that exactly
one  will evaluate to true in any given instance.
The program $\EQNF_i^\Gamma$ can be written in this form by simply replacing
the $\mathbf{for \ \ldots \ do}$ statement by one line for each pure
strategy in ${\cal S}_i(\Gamma)$; similarly for $\EQEF_i^\Gamma$.

We want to associate a protocol with a kb program.  Unfortunately, we
cannot ``execute'' a kb program
as
we can a protocol.
How the kb program executes depends on the outcome of
tests~$\kappa_j$.
Since the
tests involve beliefs and counterfactuals, we need to interpret them with
respect to a system.  The idea is that a kb program
$\Pgkb_i$ for player~$i$
and a probabilistic system $\PS$ together determine a protocol $P$
for player $i$.
Rather than giving the general definitions
(which can be found in \cite{HM98a}),
we just
show how they work in the two kb programs we consider in this paper:
$\EQNF$ and $\EQEF$.

Given a system $\PS = ({\cal R},\vecmu)$, we
associate with each formula $\phi$ a set $\intension{\phi}$ of points in
$\PS$.
Intuitively, $\intension{\phi}$
is the set of points of~$\PS$
where the  formula $\phi$ is true. We need a
little notation:
\begin{itemize}
\item If $E$ is a set of points in $\PS$, let ${\cal R}(E)$
denote the set of runs going through points in $E$; that is ${\cal R}(E) =
\{r: \exists m((r,m) \in E)\}$.
\item Let ${\cal K}_i(r,m)$ denote the set of points that $i$ cannot
distinguish from $(r,m)$:
 ${\cal K}_i(r,m) = \{(r',m'): (r'_i(m') = r_i(m)\}$.
Roughly speaking, ${\cal K}_i(r,m)$ corresponds to $i$'s information set at the
point $(r,m)$.
\item Given a point $(r,m)$ and a player $i$, let $\mu_{(i,r,m)}$ be the
probability measure that results from conditioning $\mu^i$
on ${\cal K}_i(r,m)$, $i$'s information at $(r,m)$.
We cannot condition on ${\cal K}_i(r,m)$ directly: $\mu^i$ is a probability
measure on
runs, and ${\cal K}_i(r,m)$ is a set of points.  So we actually condition, not
on ${\cal K}_i(r,m)$, but on ${\cal R}({\cal K}_i(r,m))$, the set of runs going through
the points in ${\cal K}_i(r,m)$.
Thus,  $\mu_{i,r,m} = \mu^i \mid {\cal R}({\cal K}_i(r,m))$.
(For the purposes of this abstract, we do not specify $\mu_{i,r,m}$ if
$\mu_i({\cal R}({\cal K}_i(r,m))) = 0$.
It turns out not to be relevant to our discussion.)
\end{itemize}

The kb programs we consider in this paper use a limited collection of
formulas.
We now can  define $\intension{\phi}$ for
the formulas we consider that do not involve counterfactuals.
\begin{itemize}
\item In a system $\PS$ corresponding to a normal-form game $\Gamma$, if
 $S \in {\cal S}_i(\Gamma)$, then
$\intension{\doact_i(S)}$ is the set of initial points $(r,0)$ such
that player $i$ uses strategy $S$ in run $r$.
\item Similarly, if $\PS$ corresponds to an extensive-form game,
then $\intension{\doact_i(\sfa)}$ is the set of points $(r,m)$ of $\PS$
at which~$i$ performs action $\sfa$.
\item Player $i$
believes a formula $\phi$ at a point $(r,m)$ if
the event corresponding to formula $\phi$ has
probability 1 according to
$\mu_{i,r,m}$.  That is, $(r,m) \in \intension{B_i \phi}$ if
$\mu_i({\cal R}({\cal K}_i(r,m)) \ne 0$ (so that conditioning on ${\cal K}_i(r,m)$ is
defined) and
$\mu_{i,r,m}(\intension{\phi} \cap {\cal K}_i(r,m)) = 1$.
\item
With every run $r$ in the systems we consider, we can
associate the joint (pure) strategy $\vec{S}$ used in~$r$.%
\footnote{If we allow players to change strategies during a run, then we
will in general have different joint strategies at each point in a run.
For our
theorems in the next section, we restrict to contexts where players do
not change strategies.}
This pure strategy determines the history in the game, and thus
determines player $i$'s utility.  Thus, we can associate with every
point $(r,m)$ player $i$'s \emph{expected} utility at $(r,m)$, where the
expectation is taken with respect to the probability $\mu_{i,r,m}$.
If $u$ is a
real number,
then $\intension{\EU_i = u}$ is the set of points
where player
$i$'s expected utility is $u$;
$\intension{\EU_i \le u}$ is defined similarly.
\item
Assume that $\phi(x)$ has no occurrences of $\forall$.
Then
$\intension{\forall x \phi(x))} = \cap_{a \in
\mbox{$I\!\!R$}}\intension{\phi[x/a]}$, where $\phi[x/a]$ is the result of replacing
all occurrences of $x$ in $\phi$ by $a$.  That is, $\forall x$ is just
universal quantification over $x$, where $x$ ranges over the reals.
This quantification arises for us when $x$ represents a utility, so
that $\forall x \phi(x)$ is saying that $\phi$ holds for all choices of
utility.
\end{itemize}

We now give the semantics of formulas involving counterfactuals.  Here
we consider only a restricted class of such formulas, those
where
the counterfactual only occurs in the form $\doact_i(S) \RCond \phi$, which
should be read as ``if $i$ were to use strategy $S$, then $\phi$ would
be true.
Intuitively, $\doact_i(S) \RCond \phi$ is true at a
point $(r,m)$
if $\phi$ holds in a world that differs from $(r,m)$ only in that $i$ uses the
strategy $S$.
That is, $\doact_i(S) \RCond \phi$ is true at $(r,m)$ if $\phi$ is true at
the point $(r',m)$ where, in run $r'$,
player $i$ uses strategy $S$ and all the other players use the same
strategy that they do at $(r,m)$.
(This can be viewed as an instance of the general semantics for
counterfactuals used in the philosophy literature
\cite{Lewis73,Stalnaker68} where $\psi \RCond \phi$ is taken to be true
at a world $w$ if $\phi$ is true at all the worlds $w'$ closest to $w$
where $\psi$ is true.)
Of course, if $i$ actually uses
strategy $S$ in run $r$, then $r' = r$.
Similarly, in an extensive-form game $\Gamma$,
the closest point to
$(r,m)$ where $\doact_i(\sfa')$ is true (assuming that $\sfa'$ is an
action that $i$ can perform in the local state $r_i(m)$) is the point
$(r',m)$ where all players other than player $i$ use the same
protocol in $r'$ and $r$, and $i$'s protocol in $r'$ agrees with $i$'s
protocol in $r$ except at the local state $r_i(m)$,
where  $i$ performs action $\sfa'$.  Thus,
$r'$ is the run that results
from player $i$ making a single deviation (to~$\sfa'$ at time~$m$)
from the protocol she uses in $r$,
and all other players use the same protocol as in $r$.

There is a problem with this approach.
There is no guarantee that, in general, such a
closest point $(r',m)$ exists in the system $\PS$.
To deal with this problem, we restrict
attention
to a class of systems where this
point is guaranteed to exist.  A system $({\cal R}, \vecmu)$ is \emph{complete
with respect to context $\gamma$} if
${\cal R}$
includes every
run generated by a protocol appropriate for context $\gamma$.  In
complete systems, the closest point $(r',m)$ is guaranteed to exist.
For the remainder of the paper, we evaluate formulas only with respect
to complete
systems.
In a complete system $\PS$, we define $\intension{\doact_i(S) \RCond
\phi}$ to consist of all the points $(r,m)$ such that the closest point
$(r',m)$ to $(r,m)$ where $i$ uses strategy $S$ is in
$\intension{\phi}$.  The definition of $\intension{\doact_i(\sfa) \RCond
\phi}$
is similar.
We say that a complete system $({\cal R}',\vecmu')$
\emph{extends}
$({\cal R},\vecmu)$ if $\mu_j$ and $\mu'_j$ agree on
${\cal R}$ (so that
$\mu'_j(A) = \mu_j(A)$) for all $A \subseteq {\cal R}$) for $j = 1, \ldots, n$.

Since each formula $\kappa$ that appears as a test in a kb program
$\Pgkb_i$ for player $i$ is a Boolean combination of formulas of the form
$B_i\phi$, it is easy to check
that if
$(r,m)\in\intension{\kappa}$, then ${\cal K}_i(r,m)\subseteq\intension{\kappa}$.
In other words,
the truth of $\kappa$
depends only on $i$'s local state.  Moreover, since the tests are
mutually exclusive and exhaustive, exactly one of them holds in each
local state.
Given a system $\PS$, we take the protocol $\Pgkb_i^{\PS}$ to be such
that
$\Pgkb_i^{\PS}(\ell) = \sfa_j$ if, for some point $(r,m)$ in $\PS$ with
$r_i(m) = \ell$, we have $(r,m) \in \intension{\kappa_j}$.
Since $\kappa_1, \kappa_2, \ldots$ are mutually exclusive and
exhaustive, there is exactly one action
$\sfa_j$ with this
property.

\commentout{
Joint protocols $\vec{P}$ and $\vec{P}'$ are {\em equivalent in context
$\gamma$}, denoted $\vec{P} \approx_\gamma \vec{P}'$, if
$P_i(\ell) = P'_i(\ell)$ for every local state $\ell=r_i(m)$ with
$r \in \Rrep(\vec{P},\gamma)$.  That is, $\vec{P}$ and $\vec{P}'$ perform the
same joint action in every global state that arises in
$\Rrep(\vec{P}, \gamma)$.
We remark that if $\vec{P} \approx_\gamma \vec{P}'$, then it easily
follows that $\Rrep(\vec{P},\gamma) = \Rrep(\vec{P}',\gamma)$: we simply
show by induction on $m$ that every prefix of a run in
$\Rrep(\vec{P},\gamma)$ is a
prefix of a run in $\Rrep(\vec{P}',\gamma)$, and vice versa.
Finally, we say that}
We are mainly interested in protocols that implement a kb program.
Intuitively, a joint protocol $\vec{P}$ implements a kb program
$\vec{\Pgkb}$ in context $\gamma$ if
$\vec{P}$ performs the same actions as $\vec{\Pgkb}$
in all runs of $\vec{P}$ that have positive probability,
assuming that
the knowledge tests in $\vec{\Pgkb}$ are interpreted with respect to
the complete system $\PS$ extending $\Rrep(\vec{P},\gamma)$.
Formally,
a joint protocol $\vec{P}$ \emph{(de facto) implements} a
joint kb
program $\vec{\Pgkb}$ \cite{HM98a} in a context $\gamma = (\G_0, \vecmu)$ if
$P_i(\ell) =
\Pgkb^{\PS}_i(\ell)$ for every local state $\ell=r_i(m)$
such that $r \in \Rrep(\vec{P},\gamma)$ and $\mu_i(r) \ne 0$,
where $\PS$ is the complete system
extending $\Rrep(\vec{P},\gamma)$.
We remark
that, in general, there may not be any joint protocols that
implement a kb program in a given context, there may be exactly
one, or there may be more than one (see \cite{FHMV} for examples).  This
is somewhat analogous to
the fact that there may not be any equilibrium of a game for some notions
of equilibrium, there may be one, or there may be more than one.

\section{The Main Results}\label{sec:mainresults}

Fix a game $\Gamma$ in normal form.
Let $\Pnf_i$ be the protocol that, in
initial state $s_S \in \Sigma^\Gamma_i$, chooses strategy $S$; let
$\vPnf = (\Pnf_1, \ldots, \Pnf_n)$.
\commentout{
To explain the results, we first need to recall the notion of
\emph{strong independence} \cite{BBD1,BBD2}.  Given a
tuple $\vec{r} =
(r^0, \ldots, r^{k-1})$ of numbers in
$(0,1)^k$
and an LPS
$\vecmu = (\mu^0, \ldots, \mu^k)$, let $\vecmu \, \Box \, \vec{r}$
be the probability measure
$$(1 - r^0) \mu^0 + r^0[(1-r^1) \mu^1 + r^1[(1-r^2)\mu^2 + r^2[\cdots +
r^{k-2}[(1-r^{k-1})\mu^{k-1} + r^{k-1}\mu^k)]\ldots ]]].$$
The random variables $X_1, \ldots, X_h$ are strongly independent with
respect to LPS $\vecmu$ if there exists a sequence
$\vec{r}^j$, $j = 1, 2, \ldots$ of
tuples in $(0,1)^k$
such that
$\vec{r}^j \rightarrow (0,\ldots, 0)$
as $j\rightarrow\infty$,
and $X_1, \ldots, X_h$ are
independent with respect to $\vecmu \, \Box \, \vec{r}^j$ for $j = 1, 2, 3,
\ldots$.}
Let $\STRAT_i$ be the random
variable on initial global states that associates with an initial global
state~$s$ player $i$'s  strategy in $r$.
As we said, Nash equilibrium arises in contexts with a common prior.
Suppose that $\gamma = (\G_0,\mu)$ is a context with a common prior.
We say that $\mu$ is \emph{compatible with}
the mixed joint
strategy $\vec{S}$
if $\mu$ is the probability on pure joint strategies induced by $\vec{S}$
(under the obvious identification of initial global states with joint
strategies).

\begin{theorem}\label{thm:Nash1} The joint strategy $\vec{S}$ is a Nash equilibrium
of the game
$\Gamma$ iff there is a
common prior probability measure $\mu$ on $\G_0^\Gamma$ such that $\STRAT_1,
\ldots, \STRAT_n$ are independent with respect to $\mu$,
$\mu$ is compatible with $\vec{S}$,
and $\vPnf$ implements $\EQNF^\Gamma$ in the context
$(\G_0^\Gamma,\mu)$.
\end{theorem}

\noindent{\bf Proof:}
Suppose that
$\vec{S}$ is a (possibly mixed strategy) Nash equilibrium
of the game $\Gamma$.
Let $\mu_{\vec{S}}$ be the unique probability on $\G_0^\Gamma$ compatible
with $\vec{S}$.
If $\vec{S}$ is played, then the probability of a run where the
pure joint strategy $(T_1, \ldots, T_n)$ is played
is just the product of the probabilities assigned
to $T_i$ by $S_i$, so $\STRAT_1, \ldots, \STRAT_n$ are independent with
respect to $\mu_{\vec{S}}$.
To see that $\vPnf$ implements $\EQNF^\Gamma$ in the context
$\gamma = (\G_0^\Gamma,\mu_{\vec{S}})$,
let $\ell = r_i(0)$ be a local state such that $r = \Rrep(\vPnf,\gamma)$
and $\mu(r) \ne 0$.  If $\ell = s_T$,
then $\Pnf_i(\ell) = T$, so
$T$ must be in the support of $S_i$.  Thus,
$T$ must be a best response to $\vec{S}_{-i}$, the joint strategy
where each player $j \ne i$ plays its component of
$\vec{S}$.
Since $i$ uses strategy $T$ in $r$, the formula
$B_i(\doact_i(T'))$ holds at $(r,0)$ iff $T' = T$.  Moreover,
since $T$ is a best response, if $u$ is $i$'s expected utility with the
joint strategy $\vec{S}$, then for all $T'$, the formula
$\doact_i(T') \RCond (\EU_i \le u)$ holds at $(r,0)$.  Thus,
$(\EQNF^\Gamma_i)^\PS(\ell) = T$, where $\PS$ is the complete system
extending $\Rrep(\vPnf,\gamma)$.  It follows that $\vPnf$ implements
$\EQNF^\Gamma$.

For the converse, suppose that $\mu$ is a common prior probability
measure on $\G_0^\Gamma$, $\STRAT_1, \ldots, \STRAT_n$ are independent
with respect to $\mu$, $\mu$ is compatible with
$\vec{S}$, and $\vPnf$ implements $\EQNF^\Gamma$ in the context
$\gamma = (\G_0^\Gamma,\mu)$.
We want to show that $\vec{S}$ is a Nash equilibrium.  It suffices to
show that each pure strategy $T$ in the support of $S_i$ is a best
response to $\vec{S}_{-i}$.   Since $\mu$ is compatible with $\vec{S}$,
there must be a run $r$ such that $\mu(r) > 0$ and $r_i(0) = s_T$
(i.e., player $i$ chooses $T$ in run $r$).  It since $\vPnf$
implements $\EQNF^\Gamma$, and  in the context $\gamma$,
$\EQNF^\Gamma$ ensures that no deviation from $T$ can improve $i$'s
expected utility with respect to $\vec{S}_{-i}$,
it follows that $T$ is indeed a best response.
\vrule height7pt width4pt depth1pt\vspace{0.1in}

\commentout{

\begin{theorem}\label{thm:perfeq} The joint strategy $\vec{S}$ is a perfect
equilibrium of the game
$\Gamma$ iff there is a common prior LPS $\vecnu$ on $\G_0^\Gamma$ such that
$\STRAT_1,
\ldots, \STRAT_n$ are strongly independent with respect to $\vecnu$,
$\vPnf$ is compatible with $\vec{S}$, and $\vPnf$ implements
$\EQNF^\Gamma$ in context $(\G_0^\Gamma,\vecnu)$.
\end{theorem}

\noindent{\bf Proof:}  STILL TO COME \vrule height7pt width4pt depth1pt\vspace{0.1in}

[[SHOULD WE TALK ABOUT PROPER EQUILIBRIUM?  PERHAPS EASIER WOULD BE TO
COMPARE OUR RESULTS TO THOSE OF BBD HERE, AND SAY THAT WE CAN SIMILARLY
CHARACTERIZE PROPER EQUILIBRIUM.  THEN SAY THAT BBD DIDN'T CONSIDER
RATIONALIZABILITY.]]

[Yoram - I support your suggestion here. There is little added
value in redoing for proper without any novel aspect. We could state
a proposition without proof that would claim the same result for proper,
without formal details, or just state it in the running discussion text.]
}

As is well known, players can sometimes achieve better outcomes than a
Nash equilibrium
if they have access to a helpful mediator.
Consider the
simple 2-player game described in Figure~\ref{table1}, where Alice,
the row player, must choose between top and bottom ($T$ and $B$), while
Bob, the column player, must choose between left and right ($L$ and $R$):

\begin{figure}[htb]
\begin{center}
\begin{tabular}{l|c|c|}\multicolumn{1}{l}{}&\multicolumn{1}{c}{$L$}&\multicolumn{1}{c}{$R$}\\
\cline{2-3}
 $T$& $(3,3)$& $(1,4)$ \\ \cline{2-3}
 $B$&$(4,1)$&$(0,0)$\\ \cline{2-3}
\multicolumn{1}{l}{\vspace{-3mm}}
\end{tabular}\\
\caption{A simple 2-player game.}\label{table1}
\end{center}
\end{figure}
It is not hard to check that the best Nash equilibrium for this game has
Alice randomizing between $T$ and $B$, and Bob randomizing
between $L$ and $R$; this gives
each
of them expected utility 2.  They
can do better with a trusted mediator, who makes a
recommendation by choosing at random between $(T,L)$, $(T,R)$, and
$(B,L)$.
This gives each of them expected utility $8/3$.  This is a
\emph{correlated equilibrium} since, for example, if the mediator
chooses $(T,L)$, and thus sends recommendation $T$ to Alice and $L$ to
Bob,
then
Alice considers it equally likely that Bob was told $L$ and $R$,
and thus has no incentive to deviate; similarly, Bob has no incentive to
deviate.  In general, a distribution $\mu$ over pure joint strategies is a
correlated equilibrium if players cannot do better than following a
mediator's recommendation if a mediator makes recommendations according
to $\mu$.  (Note that, as in our example, if a mediator chooses a joint
strategy $(S_1, \ldots, S_n)$ according to $\mu$, the mediator
recommends $S_i$ to player $i$; player $i$ is not told the joint
strategy.)  We omit the formal definition of correlated equilibrium (due
to Aumman \citeyear{Aumann74}) here; however, we stress that a
correlated equilibrium is a distribution over (pure) joint strategies.
We can easily capture correlated equilibrium using $\EQNF$.

\begin{theorem}\label{thm:correlated} The distribution $\mu$ on joint strategies is
a correlated equilibrium of the game
$\Gamma$ iff $\vPnf$ implements $\EQNF^\Gamma$ in the context
$(\G_0^\Gamma,\mu)$.
\end{theorem}

Both Nash equilibrium and correlated equilibrium require a common prior
on runs.
By dropping this assumption, we
get another standard solution concept: \emph{rationalizability}
\cite{Ber84,Pearce84}.  Intuitively, a strategy for player $i$ is
rationalizable if it is a best response to some beliefs that player $i$
may have about
the strategies that other players are following,
assuming that these strategies are themselves best responses to beliefs
that the other players have about strategies that other players are
following, and so on.
To make this precise, we need a little notation.
Let ${\cal S}_{-i} = \Pi_{j \ne i} {\cal S}_j$.   Let $u_i(\vec{S})$ denote player
$i$'s utility if the strategy tuple $\vec{S}$ is played.
We describe player $i$'s beliefs about what strategies the other players
are using by a probability $\mu_i$ on ${\cal S}_{-i}$.
A strategy $S$ for player $i$ is a \emph{best response
to beliefs described by a probability $\mu_i$ on
${\cal S}_{-i}(\Gamma)$}
if $\sum_{\vec{T} \in {\cal S}_{-i}} u_i(S,\vec{T})
\mu_i(\vec{T}) \ge  \sum_{\vec{T} \in {\cal S}_{-i}} u_i(S',\vec{T})
\mu_i(\vec{T})$ for all $S' \in {\cal S}_i$.
Following Osborne and Rubinstein \citeyear{OR94}, we say that a strategy $S$
for player $i$ in game $\Gamma$
is \emph{rationalizable} if, for each player $j$, there is a set $Z_j
\subseteq {\cal S}_j(\Gamma)$ and, for each strategy $T \in Z_j$,  a
probability measure $\mu_{j,T}$ on ${\cal S}_{-j}(\Gamma)$ whose support is
$Z_{-j}$ such that
\begin{itemize}
\item $S \in Z_i$; and
\item for each player $j$ and strategy $T \in Z_j$, $T$ is a best
response to the beliefs $\mu_{j,T}$.
\end{itemize}

For ease of exposition, we consider only
pure
rationalizable strategies.  This is essentially without loss of generality.
It is easy to see that a mixed strategy $S$ for player $i$ is a best
response to some beliefs $\mu_i$ of player $i$ iff each pure
strategy in the support of $S$ is a best response to $\mu_i$.
Moreover, we can assume without loss of generality that
the support of $\mu_i$ consists of only pure joint strategies.

\begin{theorem}\label{thm:rationalizability}
A pure strategy $S$ for player $i$ is
rationalizable iff there
exist probability measures $\mu_1, \ldots, \mu_n$, a set $\G_0
\subseteq \G_0^\Gamma$, and
a state
$\vec{s} \in \G_0$
such that $\Pnf_i(s_i) = S$ and $\vPnf$ implements
$\EQNF^\Gamma$ in the context $(\G_0,\vecmu)$.
\end{theorem}

\noindent{\bf Proof:}
First, suppose that $\vPnf$ implements $\EQNF^\Gamma$ in context
$(\G_0,\vecmu)$.  We show that for each state $\vec{s} \in \G_0$ and
player $i$, the strategy $S_{\vec{s},i} = \vPnf_i(s_i)$ is
rationalizable.
Let $Z_i = \{S_{\vec{s},i}: \vec{s} \in \G_0\}$. For
$S \in Z_i$, let $E(S) = \{\vec{s}  \in \G_0: s_i = s_S\}$; that
is, $E(S)$ consists  consists of all initial global states where player
$i$'s local state is $s_S$; let $\mu_{i,S} =
\mu_i( \cdot \mid E(S))$ (under the obvious identification of global states
in $\G_0$ with joint strategies).
Since $\vPnf$ implements $\EQNF^\Gamma$, it easily follows that $S$
best response to $\mu_{i,S}$.  Hence, all the strategies in $Z_i$
are rationalizable, as desired.

For the converse, let $Z_i$ consist of all the pure rationalizable strategies
for player $i$.  It follows from the definition of rationalizability
that, for each strategy $S \in Z_i$, there exists a probability measure
$\mu_{i,S}$ on $Z_{-i}$ such that $S$ is a best response to $\mu_{i,S}$.
For a set~$Z$ of strategies, we denote by $\tilde{Z}$ the set
$\{s_T: T\in Z\}$.
Set $\G_0 = \tilde{Z_1} \times \ldots \times \tilde{Z_n}$,
and choose some measure
$\mu_i$ on $\G_0$ such that $\mu_i( \cdot \mid E(S)) = \mu_{i,S}$ for
all $S \in Z_i$.  (We can take $\mu_i = \sum_{S \in Z_i} \alpha_S
\mu_{i,S}$, where $\alpha_S \in (0,1)$ and
$\sum_{S \in Z_i} \alpha_S =1$.)
Recall that $\Pnf_i(s_S)=S$ for all states $s_S$.
It immediately follows that, for every rationalizable joint strategy
$\vec{S}= (S_1,\ldots,S_n)$, both
$\vec{s}=(s_{S_1},\ldots,s_{S_n})\in\G_0$, and
$\vec{S}=\vPnf(\vec{s})$.
Since the states in~$\G_0$ all correspond to rationalizable
strategies, and by definition of rationalizability each (individual)
strategy~$S_i$ is a best response to $\mu_{i,S}$,
it is easy to check that $\vPnf$ implements $\EQNF^\Gamma$ in
the context $(\G_0^\Gamma,\vecmu)$, as desired. \vrule height7pt width4pt depth1pt\vspace{0.1in}

We remark that
Osborne and Rubinstein's definition of rationalizability allows
$\mu_{j,T}$ to be such that $j$ believes that other players' strategy
choices are correlated.  In most of the literature, players are assumed
to believe that other players' choices are made independently.
If we add that requirement, then we must impose the same requirement on
the probability measures $\mu_1, \ldots, \mu_n$ in
Theorem~\ref{thm:rationalizability}.

Up to now we have considered solution concepts for games in normal form.
Perhaps the best-known solution concept for games in extensive form is
\emph{sequential equilibrium} \cite{KW82}.
Roughly speaking, a joint strategy $\vec{S}$ is a sequential equilibrium
if $S_i$ is a best response to $\vec{S}_{-i}$ at \emph{all} information
sets, not just the information sets that are reached with positive
probability when playing $\vec{S}$.  To understand how sequential
equilibrium differs from Nash equilibrium,
consider the game shown in Figure~\ref{fig:game1}.
\begin{figure}[ht]
\centerline{\psfig{figure=game1.eps,height=1.4in}}
\caption{A game with an unreasonable Nash equilibrium.}
\label{fig:game1}
\end{figure}

One Nash equilibrium of this game has $A$ playing down$_A$ and $B$
playing across$_B$.
However, this is not a sequential equilibrium,
since playing across is not a best response for $B$ if $B$ is called on
to play.
This is not a problem in a Nash equilibrium
because the node where $B$ plays is not reached in the equilibrium.
Sequential equilibrium refines Nash equilibrium (in the sense that every
sequential equilibrium is a Nash equilibrium) and does not allow
solutions such as (down$_A$, across$_B$). Intuitively, in a
sequential equilibrium, every player must make a best response at
every information set (even if it is reached with probability 0). In
the game shown in Figure~\ref{fig:game1}, the unique joint strategy
in a sequential equilibrium has $A$ choosing
across$_A$ and $B$ choosing down$_B$.

The main difficulty in defining sequential equilibrium lies in capturing
the intuition of best response in information sets that are reached with
probability 0.  To deal with this, a sequential equilibrium is defined
to be a pair $(\vec{S},\beta)$, consisting of a joint strategy
$\vec{S}$ and a \emph{belief system} $\beta$, which associates with every
information set $I$ a probability $\beta(I)$ on the histories in $I$.
There are a number of somewhat subtle consistency conditions on these pairs
pairs; we omit them here due to lack of space (see
\cite{KW82,OR94} for details).   Our result depends on a
recent characterization of sequential equilibrium \cite{Hal36} that uses
nonstandard probabilities, which can assign infinitesimal probabilities
to histories.  By assuming that every history gets positive (although
possibly infinitesimal)
probability,
we can avoid the problem of
dealing with information sets that are reached with probaility 0.

\ijcai{It is well known that to}
\ijcaionly{To} every nonstandard
real number $r$, there is a closest standard real number denoted
$\stand{r}$,
and read ``the standard part of~$r$'':
$|r-\stand{r}|$ is an infinitesimal.
Given a
nonstandard probability measure $\nu$, we can define the standard
probability measure $\stand{\nu}$ by taking $\stand{\nu}(w) =
\stand{\nu(w)}$.
A nonstandard probability $\nu$ on $\G_0$ is \emph{compatible with} joint
strategy $\vec{S}$ if $\stand{\nu}$ is the probability on pure
strategies induced by $\vec{S}$.   When dealing with nonstandard
probabilities, we generalize the definition of implementation by
requiring only that $\vec{P}$ performs the same actions as $\vec{\Pgkb}$
in runs~$r$ of $\vec{P}$ such that $\stand{\nu}(r) > 0$.
Moreover, the expression ``$EU_i = x$'' in $\EQEF^\Gamma$ is interpreted as
``the standard part of $i$'s expected utility is $x$'' (since $x$ ranges over
the standard real numbers).

\begin{theorem}\label{thm:seqeq}
If $\Gamma$ is a game with perfect recall%
\footnote{These are games where players remember all actions made and
the states they have gone through; we give a formal definition in the
full paper.  See also \cite{OR94}.}
there is a belief system $\beta$ such that $(\vec{S},\beta)$
is a sequential
equilibrium of $\Gamma$
iff there is a common prior nonstandard
probability
measure
$\nu$ on $\G_0^\Gamma$ that gives positive measure to all
states such that $\STRAT_1, \ldots, \STRAT_n$ are
independent with respect
to $\nu$, $\nu$ is compatible with $\vec{S}$, and $\vPef$
implements $\EQEF^\Gamma$ in the standard context $(\G_0^\Gamma,\nu)$.
\end{theorem}

This is very similar in spirit to Theorem~\ref{thm:Nash1}.  The key
difference is the use of a nonstandard probability measure.
Intuitively, this forces $\vec{S}$ to be a best response even at
information sets that are reached with (standard) probability 0.

The effect of interpreting ``$\EU_i = x$'' as ``the standard part of
$i$'s expected utility is $x$'' is that we ignore infinitesimal
differences.  Thus, for example, the strategy $\vPef_i(\vec{s_0})$ might
not be a best response to $\vec{S}_{-i}$; it might just be an
$\epsilon$-best response for some infinitesimal $\epsilon$.
As we show in the full paper, it follows from Halpern's
\citeyear{Hal36} results that
we can also obtain a characterization of \emph{(trembling hand) perfect
equilibrium} \cite{Selten75}, another standard refinement of Nash equilibrium,
if we interpret ``$\EU_i = x$'' as ``the expected
utility for agent $i$ is $x$'' and allow $x$ to range over the
nonstandard reals instead of just the standard reals.

\section{Conclusions}\label{sec:conclusions}
We have shown how a number of different solution concepts
from game theory can be captured by essentially one knowledge-based
program, which comes in two variants: one appropriate for normal-form
games and one for extensive-form games.  The differences between these
solution concepts is captured by changes in
the context in which the games are played: whether players have a
common prior
(for Nash equilibrium, correlated equilibrium, and sequential
equilibrium) or not (for rationalizability), whether strategies are
chosen independently (for Nash equilibrium, sequential equilibrium, and
rationalizability) or not
\ijcai{
(for correlated equilibrium; we can also allow this for rationalizability),}
\ijcaionly{(for correlated equilibrium);}
and whether uncertainty is represented
using a
\ijcai{
standard probability measure (for Nash equilibrium, correlated
equilibrium, and rationalizability) or  a
nonstandard probability measure (for sequential equilibrium).
}
\ijcaionly{standard or nonstandard probability measure.}

Our results can be viewed as showing that each of these solution
concepts \emph{sc} can be characterized in terms of common knowledge of
rationality (since the kb programs $\EQNF^\Gamma$ and
$\EQEF^\Gamma$ embody rationality, and we are interested in systems
``generated'' by these program, so that rationality holds at all
states), and common knowledge of some other features $X_{sc}$
captured by the context appropriate for \emph{sc} (e.g., that strategies
are chosen independently or that the prior).  Roughly speaking, our
results say that
if $X_{sc}$ is common knowledge in a system,
then common knowledge of rationality implies that the
strategies used must satisfy solution concept \emph{sc}; conversely, if
a joint strategy $\vec{S}$
satisfies \emph{sc}, then there is a system where $X_{sc}$ is
common knowledge, rationality is common knowledge, and $\vec{S}$ is
being played at some state.
Results similar in spirit have been proved for
rationalizability \cite{BD87a} and correlated equilibrium
\cite{Aumann87}.
\commentout{
There are also epistemic characterizations of
rationalizability in terms of common knowledge of rationality
\cite{TW88} and a characterization of necessary and sufficient
conditions on knowledge of rationality needed for  Nash equilibrium
\cite{AB95}.
}
Our approach allows us to unify and extend these results and,
as
suggested in the introduction, applies even to settings
where the game is not common knowledge and
in settings
where uncertainty is not represented by probability.
We believe that the approach captures the essence of the intuition that
a solution concept should embody common knowledge of rationality.

redpajama_set_name: RedPajamaArXiv


Tantra Healing Temple is an evening to heal & nourish yourself with the gifts of intimacy, presence and authentic connection. To receive healing touch and to get deeper into your desires & boundaries, to connect to your heart, to reveal yourself, to speak your truth, to be courages, to give and to receive love.
An explorative space for your senses, sensuality and joy for deeper intimacy to unfold.
This evening is a space for you to claim your power to create healing experiences, to feel into yourself and to say YES to yourself, YES to your wishes, desires and tenderness.
Are you longing for meaningful and nourishing encounters, for true and authentic beings, for feeling alive, free of your own limitations and walls that you create around you that prevent living your life fully? Do you have trouble feeling or communicating your boundaries and truly asking for what you want?
Together we will create a safe space to be our selves, to empower ourself, to grow, to allow deep connection and intimacy to happen.
This evening is all about your desires, boundaries, to feel them, express them, live them. Couples, singles, people of any sexual orientation and relationship constellation welcome!
We will activate the body with breath, sound and movement to get more grounded in our actual experience instead of being busy in the mind.
Through simple and powerful group and pair exercises you will get more connected to your sensations, senses, wishes and boundaries. We will use ceremony to connect to our inner wisdom and our heart. We will use conscious consensual touch to expand into feeling and letting go.
I will facilitate an open space for you to explore your desires and boundaries in the ways you would like to be touched or touch or any other way you want to connect with people.
- A cover to put under you if you use oils.
Arrive in time, no entry possible after we started, sorry!
I support you in your transformation with Workshops, Personal Sexual Healing Sessions and Intimacy & Relationship Coaching Sessions , Private Retreats, for single persons or couples.


A retired worker of the Nigerian National Petroleum Corporation, Mr. Godwin Elewana, has been sentenced to death by hanging for the murder of one Douglas Ojugbo.
The deceased was killed by Elewana on March 10, 2015 for dating his daughter Mercy.
A Cross River State High Court sitting in Calabar and presided over by Justice Ukpa Ebitam found Elewana guilty after it was decided that the prosecution had proved its case beyond all reasonable doubt.
According to the Judge; "The prosecution team was able to establish ingredients of murder against the accused. On whether it was the act of the accused that caused the death of the victim, the prosecution team placed reliance mostly on the confessional statement of the accused…clearly, the statement linked the murder to the accused.
The prosecuting lawyer Eneji Amajama in reaction to the verdict said; "I thank God that we have industrious and sound judges who still uphold the rule of law. Today, the parents of the deceased would have some respite that the law of the land took its course.
Meanwhile, the victim's mother stated that those who lived by the sword would die by the sword and that she was happy that Justice had been served.


text:
Raundi Moore-Kondo
Raundi Moore-Kondo may have been born in Orange County but her poetry had its second coming in a Long Beach basement. Legend says it got wet after midnight and has never been the same. As the founder of For The Love of Words Creative Writing Workshops and Small Press she pushes poetry on everyone she meets. Raundi as been previously published in Don't Blame the Ugly Mug on Tebot Bach, Aim For The Head on Write Bloody Press and Lummox Vols. I and II.
Raundi has published The Mammary Chronicles: The Hills Are Alive, a chapbook to raise funds for breast cancer charities, the all-ages anthology A Poet is a Poet No Matter How Tall and her first poetry collection, Let The Ends Spill Over Your Lips. Her first full-length novel, The Revival, will be available in spring of 2014. When she isn't pushing poetry on people she plays bass for the bands Hurt and the Heartbeat and Daisy Chain.
Poems of Cadence Collective: A Reminder Of Why You Still Behave That Way, Should I Warn You About The Scars?, Defying Physics, Drowning In Your Blue

redpajama_set_name: RedPajamaCommonCrawl


Archives Photograph Collection - "Lilly Library (rare books and manuscripts) ground breaking"
J. K. Lilly, Jr. was the last Lilly family member to run the family company that is today Eli Lilly & Company (pharmaceutical manufacturers). He was a collector of rare books and manuscripts as well as coins and lead toy soldiers. His rare books and manuscripts were donated to Indiana University along with funds to build a repository in which to house them.
Lilly, Josiah Kirby, Jr. "Joe"
(L to R) Herman B Wells and Josiah K. Lilly.


text:
GM Loses U.S. Sales Crown to Toyota, Ends Run Dating to 1931
David Welch &
Gabrielle Coppola
(Bloomberg) -- Toyota Motor Corp. grabbed the U.S. sales crown from General Motors Co., swiping an honor that the Detroit automaker has held since Herbert Hoover was president.
If GM's explanation is to be believed -- that its 43% fourth-quarter sales decline and 13% tumble for the year stemmed from a semiconductor shortage -- then last year's sales race was really a supply-chain pageant. Whoever could best cajole stretched chip producers for more product came out a winner.
Navigating the squeeze has been a nightmare for the auto industry, and especially for U.S. carmakers. While GM's sales fell for the year, Toyota, Honda Motor Co. and Nissan Motor Co. posted gains.
Toyota may not be No. 1 -- the spot GM had occupied beginning in 1931 -- for long.
"We see it as not sustainable," Jack Hollis, the company's senior vice president of automotive operations, said without elaborating Tuesday at a briefing for reporters.
GM agreed. Steve Carlisle, the carmaker's president for North America, said the company will increase sales this year. Most major automakers reported fourth-quarter U.S. sales on Tuesday, with Stellantis NV, which owns the Jeep and Ram brands, posting full-year sales down 2% and an 18% drop in the last quarter. Ford Motor Co. is expected to release its figures Wednesday.
To retake its leadership position, GM will have to return to something closer to the 2.5 million vehicles the company delivered in 2020. Before the Covid-19 pandemic and semiconductor shortage hit, GM sold about 2.9 million vehicles in 2019.
"Our dealers and our engineering, supply chain, manufacturing and brand teams moved mountains to satisfy as many customers as possible in 2021" Carlisle said in a statement. "In 2022, we plan to take advantage of the strong economy and anticipated improved semiconductor supplies to grow our sales and share."
Market share is vital to GM's long-range target of doubling revenue to $280 billion. That will require keeping buyers of gasoline-burning cars and trucks and stealing new ones who want electric vehicles.
And as production recovers, GM may not want every sale it can get. Domestic automakers typically keep 80 days worth of vehicles on dealer lots, while Japanese automakers keep as few as 50 days worth available. Thinner inventory allows for better pricing and fatter profits.
See also: Chip delivery times are on the rise again, shortages to continue
Consumers today are paying record prices, since automakers don't need rebates and dealers don't need to bargain heavily.
"The domestic producers will run at 50 to 60 days instead of 80 even if it means smaller market share," said Bloomberg Intelligence analyst Kevin Tynan. "It may mean that they're not the top volume automaker, but they will be much healthier from a financial perspective."
For Toyota to stay in the lead it would need a much bigger slice of the U.S. market than it's had before. Toyota's share inched up to 14.3% in 2020 from 14% in 2018. GM was 17.3%, little changed from the 17.1% notched in 2018, according to Bloomberg Intelligence.
Overall, it was a tough year for the industry and ended on a sour note. Carmakers likely sold a seasonally adjusted annual rate of about 12.5 million new vehicles in December, down 23% from a year earlier, according to the average forecast of six market researchers surveyed by Bloomberg.
Stellantis saw shrinking sales for most of its nameplates, but the Ram pickup brand eked out a positive year, with sales up 4% in 2021, while the hybrid Pacifica minivan drove Chrysler brand sales higher. In the fourth quarter, growing deliveries of the all-new Grand Cherokee SUV were offset by steep declines for two other Jeeps: the Wrangler SUV and Gladiator pickup.
For the full year, U.S. auto sales likely came to 14.9 million vehicles, up 2.5% from the coronavirus-stricken days of 2020, according to Cox Automotive. Automakers typically sell more than 16 million vehicles a year.
Chasing Profit
The chip shortage forced GM to allocate supply to its most profitable vehicles. Fourth-quarter sales of the Chevy Silverado fell more than 30% and tumbled 21% for the GMC Sierra, but a spokesman said the company still sold more full-size pickups than anyone.
Sales climbed for the Chevy Tahoe and Suburban, the GMC Yukon and the Cadillac Escalade large sport utility vehicles -- the company's most profitable models.
Toyota's strong 2021 performance was buoyed by sales of sedans such as the Corolla and Camry. The RAV4 remained the automaker's top-selling vehicle, though its sales dropped 5% for the year. Sales of the Corolla and Camry rose 5% and 6.6%, respectively.
Nissan's sales dropped 20% in the fourth quarter but rose 8.7% for the year. The Nissan Rogue and Kicks small SUVs were strong sellers.
Judy Wheeler, vice president of sales for Nissan, said the company has been working to allocate scant semiconductors to the vehicles that it needs most, but that's not always possible. She compared the puzzle to lining up the colors on a Rubik's Cube.
"Each month has gotten stronger," she said. "Our next quarter will be better as far as production. We believe we'll be back to normal production levels. It may not be the ideal mix, but production levels will be normal."
Honda also boosted sales for the year despite a sharp drop at the end. December's tally fell 23% sales rose 8.9% for the year.
The CR-V compact crossover led deliveries, rising 8.3%. The Civic compact and Accord midsize sedans also did well, continuing the dominance of Asian brands in the segment. Among Honda's biggest gainers: its Ridgeline pickup and Passport midsize SUV, both of which were redesigned to showcase more rugged looks.
Hyundai Motor Co.'s namesake brand was one of the big winners last year, logging a 19% increase. But the Korean automaker also lost steam as the year wound down, with a 23% plunge in December sales contributing to a 15% decline in the fourth quarter.
Hyundai had comparable inventory levels as Japanese competitors but maximized tight inventory by shifting its retail strategy.
"You get better at online retailing and get better at pre-selling your pipeline," said Randy Parker, senior vice president of sales at Hyundai Motor America. "That's exactly what we did, and that helped fuel our success in a very difficult year."


Start by making your basic form with the leather- I decided to add a little flair with an iridescent strip of leather.
Now that we have the pieces of the holder all ready, time to stitch it up!
Grab your leather hole puncher to create holes to make stitching easy as 1 2 3!
Start with tying a knot and then grab that needle! I used a slip knot here to keep things secure. I dove my needle right through the slip knot and tightened it down. It worked like a gem.
Instead of a simple running stitch, I doubled back so that I didn't have any empty spaces when you looked at the stitches on the front of the piece.
I decided to make my closure super simple. Using the leather punch, I made a hole slightly larger than my stitching holes.
Fold a piece of cording, leather, or hemp cording in half. Poke the folded end through the hole and then sting the rest of the cording through the loop. Sometimes I feel like I am talking in some sort of secret code. Hopefully, you understand and get it 😉 Now simply fold the flap closure over, wrap your cording around a few times and tuck the ends in casually. I am loving the texture! This one I made a gift to Sam….brown leather is her jam!


With the mercury rising and the fizz getting louder, beverage makers are launching new products and campaigns with catchy taglines to grab a share of the $10 billion Indian marked, amid growing consumer base and fierce competition.
PepsiCo, which has started the new year with some product-centric innovations and big brand campaigns, has launched two new flavours of its soft drink Mirinda - Orange Masala and Orange Mango.
"Through this unique local palette-led innovation, we expect to drive consumption frequency and penetration," Deepika Warrier, executive director, marketing, PepsiCo Beverages, India, told IANS.
Venezuela, junto con Haití y tres países centroamericanos, son los más peligrosos para los ejecutivos extranjeros de las empresas multinacionales, según el sexto índice Anual de Seguridad Latinoamericana elaborado por FTI Consulting Ibero America para Latin Business Chronicle.
El mismo informe indica que en Panamá han visto una mejoría. El índice de Seguridad Latinoamericana toma en cuenta cómo anda cada país de la región en cuanto a inseguridad pública, poniendo el foco especialmente en la comunidad de negocios.
To facilitate successful marketing of mangoes using conventional packaging and postharvest handling methods, mangoes destined for import into the USA are harvested at the mature green stage while still firm.
The fruit are then ripened after they arrive in the USA by the wholesaler, retailer, or consumer (Kader and Mitcham, 2008). Kader and Mitcham note that sales of mangoes increase if ready-to-eat fruit are available at retail markets.
One of the challenges to successful marketing of mangoes is their limited shelf-life (typically 14 to 28 days at the mature-green stage and up to a week at the ripe stage). Postharvest technology that would extend the shelf-life of mangoes without adversely affecting their quality at consumption would be of considerable value to the industry.
Whether in an effort to flood the domestic market and bring pricing of mangoes in Mexico to a more favorable price or an orchestrated attempt to raise the price of mangoes in the USA market, Drug Cartels in Mexico have ordered a shutdown of mango export packing operations for one week in the State of Michoacan, Mexico.
This is not EMEX or the Exporters that want to stop by any means.
The order comes at the local level from organized crime organizations in Michoacan who are limiting the harvest of fruit in the local Orchards in Michoacan.
The question of food origin breaks down into several parts. Do we care where it originated? Or, do we care where it was processed? Does where the food comes from impact its quality? Does a food's origin impact its safety? Can we ever know as much about the sources of our food as we would like?
The answers to these questions will vary, but the bottom line is that it is a complex issue.
For some people, knowing where their food comes from is very important. For others, it may not matter at all.
Mexico's Knights Templar, or the Caballeros Templarios, allegedly claims to be the authors of a series of 'narcomantas' reportedly spotted in cities throughout the state of Michoacan.
The banners claim that the well known alleged criminal organization encouraged without blackmail or violence the decrease in food prices. Banners were found in Morelia, Zitacuaro, Lazaro Cardenas, Uruapan, Apatzingan, Ocampo, as well as Tuxpan.
The banners, hung in the early morning hours of Monday April 2nd, arrived after a recent truce called by the organization in preparation for the upcoming Papal visit to Guanajuato, a neighboring state of Michoacan.


text:
Pedro Campos Costa Lisboa (10 de maio de 1972) é um arquiteto, curador, autor e professor português premiado, fundador do atelier de arquitetura Campos Costa Arquitetos.

Biografia
Nascido em Lisboa em 1972, licencia-se em arquitetura pela Faculdade de Arquitetura da Universidade do Porto em 1997. Trabalha, durante 4 anos, na Holanda, passando por diferentes ateliers. Quando regressa a Portugal, ingressa no Promontório Arquitetos, em Lisboa, durante 7 anos até abrir o seu próprio atelier.

Desde então desenvolve uma prática multidisciplinar R&D que se afirma como "anti-catalogo". Participando em diferentes projetos com as mais variadas temáticas e escalas, realiza edifícios públicos, residenciais, comerciais, escritórios, espaços públicos e master plan, assim como participa em numerosos concursos nacionais e internacionais.

Foi Professor convidado no Politecnico di Milano, na Faculdade de Arquitetura de Cagliari,  no departamento de arquitetura da Universidade Autónoma de Lisboa, na Faculdade de Arquitetura da Universidade de Lisboa e no departamento de arquitetura da Universidade de Évora e participou, em qualidade de docente, em numerosos seminários de projeto e workshops internacionais. É regularmente convidado em conferências e em participar de júris de concursos de arquitetura e design em Portugal e no estrangeiro.

Foi ainda editor em diferentes revistas de arquitetura em Portugal e Itália, exercendo funções como redator da revista milanesa D´Ars, de 2000 e 2009 e como membro do Conselho Editorial e da Redação da equipa Editorial do JA, de 2012 e 2014.

Obras
Entre os mais conhecidos trabalhos de Campos Costa Arquitetos: a Extensão do Oceanário de Lisboa, a reabilitação do Hotel Ozadi em Tavira, o Centro de Empresas In Castro, em Castro Verde, o Parque Urbano de Arruda dos Vinhos, o Consulado Geral de Portugal no Rio de Janeiro e Mercado Municipal de Quarteira

Prémios
Recebeu numerosos prémios de prestígio nacional e internacional. Entre eles:
 Premio Anual do IAB - RJ 2017 Rio de Janeiro;
 Prémio FAD 2015 (Best Iberian Architecture Project);
 Prémio Valmor 2011 (Menção Honrosa);
 APRIZE 2012, Exposynergy, Triennale de Milano, Milan;
 EUROPE 40 UNDER 40 2012, The European Centre for Architecture Art Design and Urban Studies, Europe;
 NEXT GENERATION 2006 AWARD Metropolis (architecture magazine), New York.

Publicações
Publicou ensaios e artigos e é autor e editor dos livros Duas Linhas e Sete Círculos (Campos Costa, P., Louro, N. (2009). Duas linhas/Two lines. Edição de autor; Campos Costa, P., Byrne, G., Costa Pinto, E., Tavares, M.G., Domingues, A. (2016). Sete Círculos | Seven Circles. Edição: Circo de Ideias).

Exposições
Desenhou  a exposição Museu de Outro Mundo - José de Guimarães, no Museu do Oriente de Lisboa (16 Março-3 Junho de 2018) e Volta ao mundo. Obra gráfica de José de Guimarães , na Biblioteca Nacional de Portugal (out. 2019- 31 mar. 2020) e foi o curador do Pavilhão Português na 14ª edição da Bienal de Veneza (2014) com o projeto Homeland, news from Portugal. Desenvolve ainda projetos de cenografia e instalações (Interni Mutant Architecture & Design - Salone del Mobile, Milano 2011 com a instalação "13 Chaise Longue"; Illuminação de Natal 2011, Praça de Luís de Camões, Lisboa, entre outras).

Ligações externas
 Site oficial

Ver também
 Arquitectos de Portugal
 Arquitetura de Portugal

redpajama_set_name: RedPajamaWikipedia


There's also an art walk in Little Havana on Friday night, a free back-to-school event at Sunny Isles Beach and a splash bash at Mercy Hospital.
Take advantage of free entrance into the Deering Estate at Cutler on Saturday during the free "Play Day" from 10 a.m. to 4 p.m. You can take a self-guided tour of the historic home and meander around the grounds while admiring the waterfront view. Learn about tropical fruits at the display by the experts from Fruit & Spice Park. Kids can enjoy a visit by the Zoo Squad with fun characters like Gigi the giraffe and Kaz the rhino.
The free author readings at the Coral Gables location of Books & Books make the place a regular hot spot. To add to the rich cultural vibe, the store features free live jazz performances on Friday and Saturday nights from 7:30-11:30 p.m. in the courtyard. Playing this weekend are guitarist Maria Alejandra Rodriguez and the duo of guitarist Jorge Garcia and violinist Federico Britos.
Tucked away off Bird Road is an art scene that's been gaining followers for several years now, and the district hosts a monthly art walk. The free event includes more than 16 galleries showcasing paintings, sculptures, multimedia pieces and installations. They've even got a trolley that takes you to different stops in case your feet get sore. Come out on Saturday from 7-10 p.m. and see what the buzz is about.
This month's free GoGo event at the Museum of Art + Design in the Freedom Tower on Saturday explores the abstract landscape art world of Cuban artist Emilio Sanchez. Kids of all ages can create their own Sanchez-inspired artwork using painter's tape and paint and then create a Sanchez-inspired pop-up mural landscape. And speaking of pop-up, make sure to check out the museum's pop-up shop for some unique gifts. The event runs from noon to 4 p.m.
Cycling has become a recent phenomenon with bike rental programs popping around South Florida. There is a free Bike Expo in Fort Lauderdale on Saturday from 9 a.m. to 7 p.m. and Sunday from 10 a.m. to 6 p.m. Whether you're shopping for a new set of wheels, accessories, products or biking apparel, chances are they've got the gear you need.


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Home Sports Inzamam to step down as chief selector after July 30
Inzamam to step down as chief selector after July 30
LAHORE: Chief selector Inzamamul Haq Wednesday announced that after completing his term on July 30, he would say goodbye to Pakistan Cricket Board (PCB).
Addressing a press conference here, he said that he would step down after completing his term as chief selector. After returning from the UK, he said, "I informed the PCB that I do not want to continue and the new people with new ideas should come." He thanked the PCB and the team for their cooperation and said that he had spent good time since 2016.
To a question about future, he said that being cricketer, he could consider any role, except selection, the PCB offers him. However, h added, there no offer from PCB as yet.
He expressed satisfaction over the performance of the national team in World Cup, saying that Pakistan had defeated the two finalists, won four matches.
Previous articleSanjrani meets opposition leader Raja Zafarul Haq
Next articleNAB summons Shahid Khaqan Abbasi tomorrow
Winter Sports Festival kicks off in Swat
Bangladesh announces 15-member squad for Pakistan T20I series
Mohammad Hafeez announces retirement after T20 World Cup


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