 \documentclass[12pt,epsf]{article}



\usepackage{verbatim}

\newcommand{\comments}[1]{}


\renewcommand\floatpagefraction{.9}
\renewcommand\topfraction{.9}
\renewcommand\bottomfraction{.9}
\renewcommand\textfraction{.1}
\setcounter{totalnumber}{50}
\setcounter{topnumber}{50}
\setcounter{bottomnumber}{50}

\usepackage{hyperref} \hypersetup{breaklinks=true, pagecolor=white,
colorlinks=
true, linkcolor=black, hyperfootnotes= false, urlcolor=blue } \urlstyle{
rm}

\usepackage{graphicx}

\usepackage{amsmath}

 \usepackage{amssymb}


  \reversemarginpar
  \topmargin  -.3in
  \oddsidemargin -.1in
  \textheight  8.5in
  \textwidth 7in
  \baselineskip 16pt



         \begin{document}

\titlepage



 \parindent 24pt

 \parskip 10pt





                   \vspace*{12pt}

\begin{center}

 \begin{Large}

 {\bf   Incomplete Information  in  Repeated Coordination  Games   }
\\

\end{Large}


 \bigskip


January 24, 2017 \\

\bigskip

Eric Rasmusen \\

\bigskip

{\it Abstract}

\end{center}

  \vspace*{-12pt}

   Asymmetric information can  help  achieve an efficient equilibrium in
repeated coordination games.  If there is a small  probability that one
player can play only one of a continuum of moves,  that player can
pretend
to be of the constrained type and other players will coordinate with
him. This hurts efficiency in the repeated battle of the sexes, however, by knocking out the pure-strategy equilibria. 

\bigskip

\begin{small}

\noindent
  \hspace*{20pt} Dan R. and Catherine M. Dalton Professor, Department of
Business
Economics and Public Policy, Kelley School of
Business, Indiana University. Visitor (2007-2008), Nuffield College,
Oxford
University.  Office: 011-44-1865 554-163 or (01865) 554-163. Nuffield
College,
Room C3, New Road, Oxford, England, OX1 1NF.
Erasmuse@indiana.edu.   \url{http://www.rasmusen.org}. This paper:
 \url{Http://www.rasmusen.org/papers/goodtype-rasmusen.pdf}. It is forthcoming in {\it Essays in Honor of Moriki Hosoe}, eds. Woohyung Lee, Tohru Naito, and Yasunori Ouchida, Springer.

\noindent
  I thank  the Indiana Business Economics Brown Bag Lunch and Michael
Baye for
helpful comments.

\end{small}

\newpage



\noindent
  {\bf 1.Introduction}

    It is well
known that coordination games have multiple equilibria, depending on
player expectations, even if one equilibrium is pareto superior and
players can communicate. This multiplicity is present even in the one-
shot game, and just gets worse when the game is repeated.  Few of the
refinements of Nash equilibrium that have been suggested in the context
of other kinds of games help with coordination games, and none has
gained more than minimal acceptance.

 The problem of multiple equilibria in coordination games has attracted
attention from various authors. One way to try to predict which
equilibrium is played out is to use the behavioral idea of ``focal
points'' from Thomas Schelling (1960): that a human's attention is drawn
to
certain equilibria because they look ``different''.  Thus, if a game's
equilibria had payoffs of (1,1), (2,2), and (100,100), the focal point
would be (100,100).  This is a difficult notion to formalize, though: if
the alternatives were (1,1), (99,99), and (100,100) would we predict
that that the players would end up at (1,1) because it is the most
distinctive?

Clearly, the idea of the focal point is important.  Philosopher David
Lewis (1965)   divides the idea of the salience of a choice into two
parts. The choice has ``primary salience'' to a player if he believes it
is salient to he himself; it has ``secondary salience'' if  he believes
it has salience to other players.   Judith  Mehta,  Chris Starmer \&
Robert Sugden (1994, p. 661) add ``Schelling salience'' to primary and
secondary salience, as a choice that ``seems obvious or natural to
people who are looking for ways of solving
coordination problems.''  In their article they report results of
experiments trying to distinguish between primary salience-- the answers
subjects gave to questions when there was no reward for coordination--
and secondary or Schelling salience-- the answers when subjects were
rewarded for successful coordination. They found that subjects indeed
were picking with an eye towards what other subjects would pick; for
example when asked to  write down any day of the year, only 6\% of the
first set of subjects answered December 25, but 44\% did when they were
rewarded for successful cooperation.

  A second approach tries to use derive the unique equilibrium from
rationality.    David Gauthier (1975, p. 201) defines the ``Principle of
Coordination'' as  ``in a situation  with one and only one outcome
which is both optimal and a best equilibrium, if each person takes every
other person to be rational and to share a common conception of the
situation, it is rational for each person to perform that action which
has the best equilibrium as one of the possible outcomes.''  Bacharach (
1993)  John Harsanyi \& Reinhard Selten (1988) and  Maarten Janssen (2000, 2001) have pursued
this approach, trying to add axioms for rational behavior  that require
players to avoid dominated equilibria.

Repeating the game does not reduce the number of equilibria, but it does
introduce a new angle: finding the optimal way to play a game starting
without a convention as to the equilibria. What is the optimal strategy
for the two players if they   must first  grope their ways towards
coordination  by  guessing what the other player will do before they end
up at the same action and use it thereafter?  That is the project in
Vincent Crawford \& Hans Haller (1990), who find   a learning procedure that converges
in
finite time.


A third approach is to look at evolution in games.  Glenn Ellison
(1993),
Michihiro Kandori,  George J. Mailath \& Rafael Rob (1993), and Peyton
Young (1993)   take this approach. Start with a population of pairwise-
interacting players with different strategies. They play coordination
games,  and increase or diminish in frequency depending on their
payoffs.  In such settings, ``risk-dominant strategies'' emerge as
equilibria.  In a symmetric two-player setting, this is the strategy a
player would choose if he thought there was a 50\% probability of the
other player choosing each strategy. The risk-dominant strategy is not
necessarily the one with the highest payoff; it balances that against
the loss if discoordination does occur.

Risk-dominant equilibria also arise in the single-repetition ``global
games'' of Stephen Morris \& Hyun Song 
Shin (2003). They ask what happens if players have some small
uncertainty over what game they are playing out. It turns out that
iterated deletion of interim-dominated strategies can then make the
risk-dominant equilibrium the unique equilibrium.


 I will show below that adding incomplete information changes the
repeated game
drastically.   David Kreps, Paul Milgrom, John Roberts \& Robert Wilson (1982) show that adding
a
small amount of carefully chosen incomplete information to the model can
result in cooperation in the finitely  repeated prisoners' dilemma.
Drew Fudenberg \& Eric Maskin (1986) show more generally that adding incomplete
information can  generate any of a wide range of average payoffs  in
finite  repeated games by getting around the backwards induction of the
Chainstore Paradox.   Their theorem does not apply to many  coordination
games,  since it depends on a ``dimensionality condition'' that requires
payoffs to vary enough between players to allow equilibria to be
supported by punishment phases in which one player is able to punish
another without hurting himself. Jean-Pierre Benoit \&  Vijay Krishna (1985), however, show
that if a game has multiple equilibria, as a coordination game does,
then a wide range of equilibria can be obtained if the game is repeated
enough times by using the threat of punishment phase in an inferior
equilibrium to enforce the desired  behavior.




    I will not be be able to   reduce the number of equilibria in the
one-
shot game, but  I will
show  that with a small amount of
incomplete information and enough repetitions any perfect bayesian
equilibrim of  even a finitely repeated  two-player game will achieve
arbitrarily close to the optimal average payoff.

 The results   will   not depend on careful
specification of the incomplete information,  and it is
robust to out-of-equilibrium beliefs.  There will be no assumption that
``Players are either of type $x=0$ or type $x=100$  (with small
probability), but never any other value of $x$.'' Nor will I specify
anything like, ``Out of equilibrium,  the deviating player is believed
to be of type $x \in [0, 34]$.''
  Rather, the  intuition is that in coordination games, no
player  has an incentive to hurt other players, so any attempt to
``fool'' other players by pretending to be of a particular type
will be eagerly accepted by them.  This intuition is  partly present in
the intuition behind the Gang of Four Theorem of
Kreps, Milgrom, Roberts \& Wilson (1982); here, it applies better and so
the result is easier to achieve.



\bigskip






\bigskip
\noindent
  {\sc  2. The   Coordination Game  with Complete Information}

 Consider a  ranked coordination game with $n = 2$ players indexed by
$i$  who simultaneously choose actions $x_1,x_2$ from the interval [0,
100]. The per-period payoff to player $i$  is $\pi(x_i, x_{-i})$
with:
\begin{equation}
\begin{array}{l}
(a)  \forall x, \; \frac{ \partial \pi (x,x)}{\partial x} >0\;\;\;\; (b)
  \pi (0,0) >  \pi (x_i, x_{-i} ) \; if\; x_i \neq  x_{-i}
\end{array}
\end{equation}

Assumption (a) says that  a player's payoff rises if he chooses a higher
action and  the other player  chooses the same action as he does.
Assumption (b) says that if the players choose different actions their
payoffs are lower than if they coordinated on (0,0).

 We will normalize to  $\pi (0,0) =0$ and $\pi (100,100) =100$, which is
to say  the per-period payoff is 0  when both   players choose $x=0$ and
100 when they   both  pick $x=100$.  The assumptions then imply that
coordination on  $x>0$ yields positive payoffs and  discoordination
yields negative payoffs.

 If the game is unrepeated and $T=1$  it has a continuum of pure
strategy equilibria  with $x$ on the continuum from 0 to 100, as well as
mixed strategy equilibria. All players prefer the equilibrium in which
$x=100$.

Which equilibrium will be played out    depends on player expectations.
A reasonable prediction is  $x=100$  because it is pareto superior to
all other equilibria, a focal point. An equally special equilibrium,
however,  is $x=0$.  It is easy to imagine how the players could be
caught in any  equilibrium---  if the game were preceded  by a malicious outsider's cheap talk
announcement that he expected them all to choose
$x=1$, for example,   or  if the players 
had a history of playing $x=5$ for many periods.

   Next, let the game be repeated   a possibly infinite number  $T$   times, with
the players observing each other's strategies after each round   and
  no discounting.
The   equilibrium outcomes and strategies both become more
numerous. Let us classify them as follows:

In a  {\bf  time-dependent equilibrium},  some  player's strategy in a
round depends on which round number it is.    If the strategies  are the
same in each round,  the equilibrium is {\bf time-independent}.

In a {\bf history-dependent equilibrium},  some player's strategy in a
round depends on the history of play up to that point.    If the
strategies do not depend on past play, the equilibrium is {\bf history-
independent}.

 Table 1 provides examples. 
 

  \begin{small}
\begin{tabular}{ll|cc}
  &  & \multicolumn{2}{|c}{\bf Time}\\
 & & &  \\
 & &  Independent & Dependent\\
 & & &  \\
\hline
 & & &  \\
 & Independent &(a) Play 10 in each round.  &\begin{tabular}{l} (b) Play 20 in
the first round\\
 and 25 in the second.\\ \\ \\  \end{tabular} \\
{\bf  History} &  \multicolumn{3}{l}{\hfill}\\
 & Dependent & \begin{tabular}{l} (c) Play 30 in each round unless \\someone
deviates, in which case \\
 play 30 in the second round. \end{tabular} &
\begin{tabular}{l} (d) For the first 50 rounds, player 1 \\
 picks 2 and the
other players pick 14.\\
 For   the last 10,000 rounds \\everyone picks
100 unless someone  \\
 deviates. If someone deviates, all pick 0 \\for
the remainder of the game.
\end{tabular}\\
 & & &  \\
\hline
\end{tabular}
\end{small}
\begin{center}
   {\sc Table 1: Four Types of Games}
\end{center}

 Note  that the history-dependent equilibria include equilibria in which the players discoordinate in some periods, receiving flow payoffs of zero.  
 Benoit \& Krishna (1985) show that a wide array of outcomes might be
observed in equilibrium, supported by  punishment strategies similar to
strategy (d) in table 1.   The players choose any  specified pattern of actions   in the first $S$ periods because in  equilibrium they all play
$x=100$ in the last ($T-S$) periods  but if anybody deviates earlier they
all play $x=2$.  The observed actions, for example, might be (10, 2),  (
7, 7),  (8, 3),  and then (100,100) for the last 200 periods.  Thus, mere  
repetition of the game does not solve the problem of multiple
equilibria, and in fact,  even more outcomes become possible. The
average payoff could even be negative, if the equilibrium has many
periods of discoordination, so long as the average payoff is not below
the discoordination payoff.








\bigskip

\noindent
  {\sc 3.   Incomplete Information:  The  Single-Action
Player }


Let us  modify the game  in the spirit of Kreps at al. (1982), by adding
a small amount of incomplete information. Players are of two types. With
some arbitrarily small  probability $p>0$,  a player $i$  must 
  play $x_i = z_i$  in every round of the game,
where $z_i$  is chosen  from [0, 100] using  a  atomless density
$f(z_i)$ such that $f(100)>0$. Such a player is ``constrained''; otherwise, 
the player is ``free''. Note that the other players do not observe which players are constrained, their exact types, or even how many there are. 

  What is essential is that there be some possibility a player will
choose $x=100$ and stick with it, which is true of  the specification above.  All that is needed is a possibility, however small, and it can even have probability zero  in the mathematical sense. That is the case in our specification, since any particular value of $z$ has zero probability, despite having positive probability density. What that means is simply that we would predict any particular value of $z$ (e.g. 97.345) with zero probability, even though we would predict a positive probability for any {\it interval} of types (e. g. [97, 98.5]). 

   How do we interpret the incomplete information? It might be  that a constrained player is truly
constrained, or  that he misunderstands the rules of the game, or 
he is irrational and thinks all players will make the same choice as he
does (psychology's  ``magical thinking''; see Brendan Daley \& Philipp Sadowski [2016]).  If we use a different specification, such as that there is a .0001\% probability that a player is constrained to use $z=100$, then we could interpret it as   that the constrained player wishfully thinks that the equilibrium will be the
pareto-optimal one (perhaps having read some of the references above) or
 thinks, for whatever reason, that if he starts with
$x=100$   the other players will join him.

   In the modified game, some equilibria disappear, as Example 1 shows.

\bigskip

\noindent
{\bf Example 1. } Suppose   $T=20$ and the payoff from discoordination
is $-500$.  Is it an equilibrium  for a free
player to follow the strategy $x=5$ in every period and for a
constrained player of type $z$ to play $x=z$?   No.

Consider what happens if player 1  deviates to $x=100$
in the first round. Is it a best response for   player 2  to
play $x=5$ in the second round? That depends on   player 2's
beliefs,which are generated by Bayes's Rule:
\begin{equation} \label{bayes}
\begin{array}{l}
Prob(z_1=100 |x_1=100)
 = \frac{ Prob( x_1=100|z_1=100)* Density(z_1=100)}{  Prob( x_1=100|z_1=
100)* Density(z_1=100) +  Prob( x_1=100|z_1= free)* Prob(z_1= free )} \\
\end{array}
\end{equation}

 The priors tell us that $Prob(player \; 2 \; is \;free)=1-p$ and
$Density(z_1=100)=f(100)p$.
 In the proposed equilibrium, $Prob( x_1=100|z_1=100) =1$ and
$Prob( x_1=100|z_1= free) =0$.  Thus, equation (\ref{bayes}) becomes
\begin{equation} \label{bayes1}
Prob(z_1=100 |x_1=100) = \frac{ (1)* f(100)p}{ (1)* f(100)p+ (0)* (1-p)}
=1.
\end{equation}

   After the first round, Player 2  therefore believes that Player 1's
type  is $z_1=100$, so he concludes that $x_1=100$ for all future
rounds.  Player 2's best response is   not $x=5$, but to imitate Player
1's action, deviating to $x_2=100$. If both players then stick with
$x=100$, their payoffs are $(-500+ 19(100), -500+ 19(100))$ compared to
the $(20(5),  20(5))$ they would have gotten in  the proposed equilibrium. Thus,
Player 1's deviation has been profitable.

 It is not true, however, that  the only equilibrium  in Example 1  is for a  
player to start   with $x=100$   and to choose in the second and
succeeding periods whatever the other player chose in the first period.
If $x=99.9$, it is not worth bearing the initial cost of $-500$ to
deviate. Rather, what we can say is that for large enough $w$ a time-
independent equilibrium strategy must have a player beginning with $x=w$
and then choosing in the second and succeeding periods whatever the
other player chose in the first period. In such an equilibrium, the
equilibrium payoff is $(20w, 20w)$.  The optimal deviation is to
$x=100$, which generates a deviation payoff of
$(-500+ 19(100), -500+ 19(100))$. There is no incentive to deviate from
equilibrium if and only if $ w  \geq 92.5$.

  Example 1 is the essence of this paper. If information is incomplete,
then a player can break out of a bad equilibrium at some cost by
pretending to be of an unusual type. If the game is repeated long
enough, it is worthwhile to bear that cost.
Thus, if  $T$ is large enough, the game  has  a much smaller interval
of equilibria  and the average payoff becomes arbitrarily close to 100.

\bigskip

\noindent
{\bf Proposition 1:} {\it  For any $\epsilon$, there exists $T$ large
enough that  in all  pure-strategy equilibria the average payoff
approaches within
$\epsilon$ of  the
optimum:}
\begin{equation}
 \forall \epsilon>0, \exists T: \frac{ \sum_{t=1}^T \pi_{it} }{T}> 100-
\epsilon.
\end{equation}

\bigskip


\noindent
  {\bf Proof.}  The probability that a player is constrained is an
arbitrarily small $p$, so  the effect that the presence of truly
constrained players have on the average equilibrium payoffs will be less
than $\epsilon$.

    Let the equilibrium with the lowest average payoff call for the
players
to first choose    $(a,b)$   with $a$ or $b$ or both not equal to 100 in
round $t_1$. Without loss of generality, suppose that player 1 chooses
$a \neq 100$.

 The minimum bound on the payoff is set by    player 1  having the
deviation option to choose $x=100$ in that period and convince    player
2 that  player 1 is constrained of type $z=100$.  Both players  would
choose $x=100$ for every succeeding round.  This  would generate  a
payoff of   $  \pi(100, b) + 100(T-1)$, where $\pi(100, b)$ is the
discoordination payoff that arises from that particular deviation, since
there would be one period of discoordination and all other periods will
have per-period payoffs of 100. This strategy will have an average
payoff of
\begin{equation}
  \frac{\pi(100, b)}{T}  + \frac{100(T-1)}{T}= 100  +  \frac{\pi(100,
b)}{T} - \frac{100 }{T}.
\end{equation}
 If $T$ is large enough, the last two terms, which are both negative,
shrink   to less than whatever small amount
$\epsilon$ we might choose.
Q.E.D.



\bigskip

  The equilibria will be in actions with an average payoff in the
interval $[100-\epsilon, 100]$ for some $\epsilon$ that depends on $T$.
This set of equilibria does not depend heavily on the out-of-equilibrium
payoffs--just for one period of discoordination loss--  and therefore it
is not necessarily the same as the set of risk-dominant equilibria. It
could be, for example, that for $x$ in [0,50] the discoordination payoff
if the other player chooses a different $x$ is $-1$, but for $x$ in (50,
100] it is $-5,000$, in which case the risk-dominant equilibrium would be
(50,50), not (100,100).





 








\bigskip

\noindent
  {\bf 4. Three or More Players in the Incomplete Information Game
    }

 Now let us allow for more than two players.
   Consider a  ranked coordination game with $n \geq 2$ players indexed
by $i$  who simultaneously choose actions $x_1, ..., x_n$ from the
interval [0,100].   If $m(x_i)$ players choose the same action $x_i$,
the per-period payoff to player $i$  is $\pi_i(x_i,x_{-i}, m(x_i))$,
with:
\begin{equation}
\begin{array}{l}
(a') \frac{ \partial \pi_i(x_i, x_{-i},m(x_i))}{\partial x_i}  \geq 0\;\;
\;\; \\
 \\
(b')  \frac{ \Delta \pi_i(x_i, x_{-i},m(x_i))}{\Delta m(x_i)} >0,\;\;
\;\;\\
 \\
(c) \frac{ \partial^2 \pi_i(x_i, x_{-i},m(x_i))}{\partial x_i \partial
x_j} =0\\
  \\
 (d)  \pi_i(0, x_{-i},n) >  \pi_i(100, x_{-i},n-1),\;\;\;\; \\
 \\
 (e) \pi_i(w,
x_{-i},l)     > \pi_i(w', x_{-i},l-1) \forall l, w, w'\neq x\\
\end{array}
\end{equation}

  Assumption (a') says that the payoff to player $i$ rises or stays the
same  as the magnitude of the group action $x_i$ rises-- from 88, say,
to 89. Assumption (b') says that the payoff  to choosing action $x_i$
rises from being in a bigger group.


Assumption (c) says that the  payoff
to player $i$ from choices made by players who choose discoordinating
actions does not depend on which actions they choose.  

 Assumption (d) says
that  group size matters more than action size: the payoff to $i$ from
choosing $x_i=0$ in a group of $n$ is bigger than from choosing
$x_i=100$ in a group of size $n-1$.  Assumption (e)  is a more general
version of (d), saying that a larger group always gets a bigger payoff,
no matter what the size of the action.  

 My colleagues Michael Rauh and Michael  Baye suggested the following as 
a payoff function that satisfies assumptions (a') through (e)  
\begin{equation}\begin{array}{l}
 \pi_i(x_i, x_{-i},m(x_i)) = [m(x_i)(1+ \frac{x_i}{1000}) -n] (100/1.1n)
\end{array}
\end{equation}
 or, without our normalization  of $\pi_i(0, x_{-i},n) =0$ and
$\pi_i(100, x_{-i},n) =100$,
\begin{equation}
 \pi_i(x_i, x_{-i},m) =  m(x_i)(1+ \frac{x_i}{1000})
\end{equation}

    The complete information game has the usual continuum of equilibria, just as when there are just two players.  How about the incomplete information game? Consider Examples 2 and 3. 



\noindent
{\bf Example 2.} Let there be incomplete information of the following form:  with 
some arbitrarily small  probability $p>0$,    player $i$   is ``constrained'' and must 
  play $x_i = z_i$  in every round of the game,
where $z_i$  is chosen  from [0, 100] using  a  atomless density
$f(z_i)$ such that $f(100)>0$.  Let there be three players,   and consider  whether it is an equilibrium outcome to play 
  (5, 5, 5) each of $T$ periods. Suppose player 1 deviates to $x_1=10$ in the
first period. A
unilateral switch by one of   the other   two  players from 5 to 10
would be  profitable  in the long run in the incomplete information game.


\noindent
{\bf Example 3.} Let there be incomplete information of the following form:  with 
some arbitrarily small  probability $p>0$,    player $i$   is ``constrained'' and must 
  play $x_i = z_i$  in every round of the game,
where $z_i$  is chosen  from [0, 100] using  a  atomless density
$f(z_i)$ such that $f(100)>0$.  Let there be four players,   and consider  whether it is an equilibrium outcome to play 
  (5, 5, 5, 5) each of $T$ periods. Suppose player 1 deviates to $x_1=100$ in the
first period. Even if player 1 were truly of type $z=100$, a
unilateral switch by one of   the other  three  players from 5 to 100
would be unprofitable  if $\pi_i(x=5, m=3) > \pi_i(x=100, m=2)$. Thus,
 (5, 5, 5, 5)  would be an equilibrium outcome.  Incomplete information does not reduce the number of equilibria. 

The situation changes if we change the form of the incomplete information. What is needed now is a more-than-infinitesimal probability of a given constraint value of $z$.  Recall our alternative specification of incomplete information in which  player $i$ has probability .01  of being constrained to play $x_i=100$. In that specification, the probability is not zero; it is strictly positive.That makes a huge difference, as we see in Example 4. 

\noindent
{\bf Example 4. }Let there be incomplete information of the following form:  with 
some  small   probability $p>0$,    player $i$   is ``constrained'' and must 
  play $x_i = 100$  in every round of the game.  Let there be four players,   and consider  whether it is an equilibrium outcome to play 
  (5, 5, 5, 5) each of $T$ periods. Assume that if the players do not all play the same action, their period payoff is zero.  Suppose player 1 deviates to $x_1=100$ in the
first period.  With probability $ .99^3$,  he is the only player to play 10, and the other players do not imitate him in future periods for the same reason as in Example 3. But with probability  $1-.99^3$,  at least one of the other players is a true constrained player who also plays 10. If that happens, then  in the second and succeeding periods, the remaining two players will play 10. Thus, the expected  payoff from deviating (and returning to playing 5 if no other player player 10 in the first period) will be greater than     $ .99^3*(0 + (T-1)(5))  + (1-.99^3) (0 + (T-1)(100)$ (I say ``greater than'' because it is slightly higher because of the possibility that not just one, but two or even all three of the other players are constrained to play 100). If $T$ is great enough, the deviation payoff is greater than the proposed equilibrium payoff of $5T$. It is worth the high probability of one period with a payoff of 0 in order to have a chance at $(T-1)$ periods with a payoff of 100.  If the deviation is profitable, however, then all four of the players will choose 100 in the first and every period, constrained or not. 

Thus, with four players, or more, incomplete information can still justify a unique efficient equilibrium. The story is a little different, though, because if expectations begin with some action less than 100, the player who deviates puts high probability on his deviation being unprofitable--- it is just that if it does work out successfully  he  gets a very large payoff increase. For this to work, $T$ must be much larger than when there are only two or three players. 
 
 
 

\bigskip

\noindent
  {\bf 5. Mixed Coordination-Conflict Games: The Battle of the Sexes
    }

 Incomplete information can actually  hurt in  mixed coordination-
conflict
games, by destroying the possibility of pure-strategy equilibria. Consider the Battle of the Sexes in Table 2. 
It has two
pure strategy equilibria, $( prizefight,  prizefight)$ and $(ballet, ballet)$, and a
mixed strategy equilibrium.
 in which the man plays $ prizefight$ with
probability $m=A/(A+B)$ and the woman with probability $w=B/(A+B)$.

The total payoff in the two pure-strategy
equilibria are $(A, )$ and $(B, A)$.    The man's one-shot expected
payoff is then     $AB/(A+B)$, which is less than $B$ since $B<A+B$. The
man's expected payoff (and analogously the woman's)  in the  mixed-
strategy equilibrium is lower  even than   in the pure-strategy
equilibrium he likes least.
 

\begin{center}
{\sc Table  2:  The Battle of the Sexes }

 \begin{tabular}{lllccc}
  &       &             &\multicolumn{3}{c}{\bf Woman}\\
   &       &             &   {\it  prizefight}  &   & {\it Ballet}     \\
 &   &  {\it  prizefight}      &    {\bf  A,B} &    &  0,0  \\
 & {\bf Man} &&  & &  \\
  &  &       {\it  Ballet}     &     0,0  &    & {\bf B,A} \\
 &   &&  & &  \\
\multicolumn{6}{l}{\it Payoffs to: (Man, Woman). $A>B$.  }
\end{tabular}
\end{center}



The $T$-repeated game has many subgame perfect equilibria, but let us focus on the     three time-independent and history-independent equilibria  that repeat the single-period equilibria just described. 
 

Now let us add incomplete information, in the form of constrained players. With probability $p_1$, the man is constrained to play $Prizefight$  and with independent  probability $p_2$ the woman is constrained to play $Ballet$. 
This change eliminates $(Ballet, Ballet)$  and $(Prizefight,Prizefight)$ as equilibria. 
 Suppose the woman thought  $(Ballet, Ballet)$ was the equilibrium. The man would begin the game by playing $Prizefight$. The woman would conclude that the man was constrained, and would play $Prizefight$ in all future rounds, so the man would have succeeded in increasing his payoff (if there are enough rounds) to one round of (Ballet, Prizefight) and $T-1$ rounds of $(Prizefight, Prizefight)$. 
 The equilibrium $(prizefight, prizefight)$ would similarly fail. 
  
 The mixed strategy equilibrium survives. If the man deviates to playing $prizefight$ as a pure strategy, the woman will interpret this as a realization of the equilibrium strategy.  This is ironic, however, because the man's ability to knock out the pure-strategy equilibrium of ($Ballet, Ballet$) ends up hurting him: his payoff is higher in that equilibrium than in the mixed-strategy equilibrium that survives. 
 
 


 


 

\bigskip
\noindent
 {\bf Closing Remarks}

 Thus, we see that in repeated  Ranked Coordination, the  efficient equilibria are robust to incomplete information but the inefficient equilibria are not, whereas in the Battle of the Sexes the opposite is true. This model has used a particular specification of incomplete information, to be sure, but if we added other incomplete information without removing the possibility of constraint in this model, the results would often stay the same. The key to the result is that if there is some chance that a deviation beneficial to the deviator will be interpreted as predicting that he will choose the same action in the future, the player will have incentive to deviate. 


\newpage

\noindent
{\bf References}

 Bacharach, Michael (1993)  ``Variable Universe Games,''
  {\it  Frontiers of Game Theory}, 1993.

Benoit, Jean-Pierre  \& Vijay Krishna (1985) ``Finitely Repeated
Games,'' {\it Econometrica,}   53: 905-922.


 Binmore, Ken \&  Larry Samuelson (2006) ``The Evolution of Focal
Points,''  {\it Games and Economic Behavior},  55:  21-42.

Daley, 
Brendan  \& Philipp Sadowski  (2016) ``Magical Thinking: A Representation Result,'' forthcoming, {\it Theoretical Economics}. 

  Carlsson, Hans \& Eric van Damme (1994) ``Global Games and Equilibrium
Selection Hans Carlsson,'' {\it Econometrica},  61:   989-1018.

  Crawford,   Vincent P.  \&  Hans Haller (1990)  ``Learning How to
Cooperate: Optimal Play in Repeated Coordination Games,'' {\it
Econometrica,}    58: 571-595.



    Glenn Ellison  (1993) ``Learning, Local Interaction, and
Coordination,''  {\it 
    Econometrica,} 61: 1047-1071.


Fudenberg, Drew \& Eric Maskin  (1986)  ``The Folk Theorem in Repeated
Games with Discounting or with Incomplete Information,'' {\it
Econometrica}, 54: 533-554.

Gautier, David (1975) {\it Coordination}, Cambridge University Press. 

Goyal, Sanjeev \& Maarten C. W. Janssen (1996)  ``Can We Rationally
Learn to Coordinate?'' {\it  Theory and Decision,}  40: 29-49.

Goyal, Sanjeev \& Maarten C. W. Janssen (1997) ``Non-Exclusive
Conventions and Social Coordination,'' {\it  Journal of Economic
Theory}, 77: 34--57 (1997).

  Janssen, Maarten C. W. (2001)	``On the Principle of Coordination,''
 {\it Economics and Philosophy}   17:  221-234.

 Janssen, Maarten C. W. (2001) ``Rationalizing Focal Points,'' {\it
Theory and Decision,}  50: 119-148.

 Kandori, Michihiro, George J. Mailath \& Rafael Rob (1993) 
``Learning, Mutation, and Long Run Equilibria in Games,'' {\it 
Econometrica}, 61:  29-56.



Kreps, David, Paul Milgrom, John Roberts, \& Robert Wilson (1982)
``Rational Cooperation in the Finitely Repeated Prisoners' Dilemma,'' {
\it Journal of Economic Theory},  27:  245-252.


Lewis, David (1969)  {\it Convention,}  Harvard University Press,
Cambridge, Mass.

  Mehta, Judith,  Chris Starmer \& Robert Sugden (1994) 
``Focal Points in Pure Coordination Games: An Experimental Investigation,'' {\it 
 Theory and Decision}, 36: 163-185. 

 Morris, Stephen \& Hyun Song Shin (2003) ``Global Games: Theory and
Applications,''
  in {\it Advances in Economics and Econometrics (Proceedings of the
Eighth World Congress of the Econometric Society)}, edited by M.
Dewatripont, L. Hansen and S. Turnovsky.  Cambridge, England: Cambridge
University Press (2003).

Schelling, Thomas (1960) {\it The Strategy of Conflict}. 

   Sugden,   Robert (1995) ``A Theory of Focal Points,'' {\it The
Economic Journal,}    105:  533-550.

 Young, H. Peyton (1993) ``The Evolution of Conventions,'' {\it 
Econometrica},  61: 57-84.

  

\end{document}



