 % 14 June 1988 . April 9, 2000, contact and cite info.
%   \documentstyle[double]{knart}
 % ERIC: if you need to change back doc style, leave mine as a comment

\documentclass[12pt,epsf ]{article}

   \begin{document}
    \parindent 24pt
\parskip 10pt
\baselineskip 16pt

 \titlepage
 \noindent\today{}\hspace{3.3in} Comments Welcome
 
 \begin{center}
 \begin{Large}
 
COOPERATION IN A REPEATED

PRISONERS' DILEMMA

WITH OSTRACISM

 \end{Large}
 \end{center}

Published: {\it Journal of
Economic Behavior and Organization}    (August 1989) 
12: 87-106 \\

 \begin{quotation}
 The unique Nash equilibrium of the finitely repeated $n$-person
Prisoners' Dilemma calls for defection in all rounds. One way to
enforce cooperation in groups is ostracism: players who defect are
expelled.  If the group's members prefer not to diminish its size,
ostracism hurts the legitimate members of the group as well as the
outcast, putting the credibility of the threat in doubt. Nonetheless,
we show that ostracism can be effective in promoting cooperation with
either finite or infinite rounds of play.  The model can be applied
to games other than the Prisoners' Dilemma, and ostracism can
enforce inefficient as well as efficient outcomes.
 \end{quotation}

 \vspace{.3in}
 \noindent \hspace*{2.75in} David Hirshleifer and Eric Rasmusen
\newline
 \noindent \hspace*{2.75in} Anderson Graduate School of Management
\newline
 \noindent \hspace*{2.75in} UCLA \newline
 \noindent \hspace*{2.75in} Los Angeles, California  90024\newline
 \noindent \hspace*{2.75in} (213) 825-4154 \newline
 Bitnet: IJJ1RAS@UCLAMVS.\\
 File: /papers/ostrac/ostrac.tex.\\
 Draft: 14.13.\\
 UCLA AGSM Business Economics Working Paper \# 86-9.

2003 note: Hirshleifer is now at Ohio State, and Rasmusen is at Indiana University. 

\noindent
 We thank Sushil Bikhchandani, Robert Boyd, Michael Brennan, Ivan
Png, John Maynard Smith, the UCLA Political Science Theory Workshop,
and two anonymous referees for helpful comments, and John Mamer for
stimulating our interest in this topic.

 \begin{small}
               \noindent
\hspace*{20pt} 2000: Eric Rasmusen, 	Professor of Business
Economics and Public Policy and Sanjay Subhedar Faculty Fellow,
Indiana University,
Kelley School of Business, BU 456,
  1309 E 10th Street,
  Bloomington, Indiana, 47405-1701.
  Office: (812) 855-9219.   Fax: 812-855-3354. Erasmuse@indiana.edu.
Php.indiana.edu/$\sim$erasmuse.
 \end{small}


\setcounter{page}{0}
 \newpage
 \noindent {\bf A  Parable.} \nopagebreak

	The Ostracos are a primitive tribe whose members hunt
collectively for large game.  Anyone who does not hunt, fishes by
himself from the tribal lake for a bare subsistence.  Fishing is
subject to constant returns: the catch per capita is independent of
the number of tribesmen.  Hunting is subject to increasing returns:
meat-per-capita increases with the number of hunters.  Since the
inviolable custom of the tribe is that hunters share meat with
non-hunters, fishermen get both fish and meat.  Consequently, nobody
engages in hunting.

	One evening the elders have a pow-wow to discover why the
precious meat is not being brought home to the tepees. In a
divination the spirit Phonos tells them that those who avoid the hunt
are cowards, not tribesmen, and must be driven into the wilderness to
die.  The elders accordingly so ordain.

	The next day, all come to the hunt except Uurguu, a
powerfully built man with a large stomach.  On being chided by his
fellows, he pronounces ``We tribesmen are all equally good hunters
and fishers, and rational men to boot. So we all know that I have
just as much reason to join the hunt tomorrow as any of you.  And we
also know that if I do help with the hunt, there will be more meat
for each of us than if you drive me away.  Let bygones be bygones, to
the benefit of all.''

	To his immense astonishment, Uurguu was immediately expelled.


 \newpage
 \section{Introduction.} \label{intro}

	A topic of continuing interest in social theory is how
cooperation can emerge in the repeated Prisoners' Dilemma and similar
games.  Cooperation in all rounds can be attained as one of many Nash
equilibria in the infinitely repeated Prisoners' Dilemma, a
two-player example of which is presented in Figure 1.  In finitely
repeated games, however, it is well known that this action pair
cannot be part of an equilibrium.  Cooperation is hard to sustain,
because in any proposed equilibrium there ultimately is some round
$t^*$ (if not earlier, then certainly the last round) at which Player
1 foresees no future cooperation from Player 2. In round $t^*$,
whatever Player 2 may do, Player 1 will choose to defect, which
causes the sequence of defections to begin in round $t^*-1$.  The
backwards recursion continues all the way to the very first round, so
both players defect from the very beginning of the game.

\begin{center}
 Insert Figure 1: The Prisoners' Dilemma with Two Players.
\end{center}

	Several solutions have been proposed to achieve cooperation.
Kreps et al. (1982) suggest that incomplete information is important.
If there is the slightest possibility that one's opponent is a type
who, independently of any rational calculation, will cooperate with
you if you cooperated in the past, then cooperation for every round
up to some round close to the end of the game can arise in
equilibrium.  Another source of cooperation is commitment to a
retaliatory strategy, an approach used by Schelling (1960) and
Thompson \& Faith (1981).  If commitment is feasible, the players in
a Prisoners' Dilemma are both better off if they each commit to never
play Defect first and to retaliate heavily if the other player does
defect.  J.  Hirshleifer (1987) argues that emotions of revenge and
gratitude evolve as ways to make retaliation credible. Still another
way to achieve cooperation is to assume that players are altruistic
towards each other.  But none of these solutions address the question
of how cooperation between self-interested individuals can arise
without binding promises, emotional responses, or deception.

   The solution proposed here allows retaliation against
noncooperators by a means other than future noncooperation. The new
form of retaliation is {\it ostracism}: expulsion of the defector
from the group.  Ostracism does not require commitment, and the
equilibrium satisfies the rationality criterion of ``subgame
perfectness'' whether the game has a finite or infinite number of
rounds.$^1$
%\footnote{Roughly speaking, perfectness requires that an
%equilibrium strategy not only be a best response to the other
%players' strategies early in the game, but also remain a best
%response once the game has been partly played out. This rules out
%threats that would not be carried out.}

 The word ``ostracism'' derives from Greek word for the broken shards
on which the citizens of ancient Athens recorded their votes
expelling individuals regarded as threats to the state. More
generally, ostracism is the practice of excluding disapproved
individuals from interaction with a social group.  In one form or
another, this plays an important role in enforcing socially approved
behavior in most groups.  Parliamentary bodies have means by which
they may expell members, and professional societies have means of
decertifying them: disbarring lawyers and taking away the licenses of
doctors and accountants.

 If ostracism were a costless way to make threats and promises
credible, the social dilemma would be easily solved.  But ostracism
is usually costly to the group because expelling a member hurts not
just the outcast, but indirectly all the remaining members.  The very
fact that members are in a group indicates some advantage to grouping
which would be reduced by expelling members.  We will call the gains
from joining together {\it aggregation economies}, which may arise
from scale economies in productive technology, gains from trade and
specialization, simple sociability, or network
externalities.$^2$
%\footnote{One need not subscribe to the ``social
%contract'' theory to attempt to explain grouping behavior in terms of
%costs and benefits to individuals from being in a group.  An
%evolutionary outlook, or Aristotle's view of man as a political
%animal, is entirely consistent with grouping behavior being
%influenced by the costs and benefits individuals derive from
%membership.}

  Under conditions that we will specify, ostracism is a credible
threat, and cooperation can be achieved because society can exploit
the end period problem rather than be victimized by it.  In the final
round, defection is a dominant strategy.  But this means that there
are no gains from cooperation in the final round, so (in contrast to
earlier rounds) the outcasts will not be missed.  In the next to last
round, the threat to expel defectors is therefore credible, and the
threat enforces good behavior in preceding rounds.

    Section \ref{game} lays out the model. Section \ref{many}
demonstrates that ostracism can enforce cooperation with either a
finite or infinite number of rounds.  Section \ref{extend} discusses
the assumptions and compares them with other models.  Section
\ref{moral} adds a touch of morality to eliminate multiple
equilibria, and Section \ref{badeq} shows how even Pareto-inferior
outcomes can be supported by ostracism.  Section 7 gives examples of
ostracism and discusses which applications fit our model.


 \newpage
 \section{The Model} \label{game}

  At the start of the game, players form a group to cooperatively
produce a good. If they play Cooperate, more of the good is produced
than if they play Defect. Defection may be thought of as shirking. A
player who is in the group, having not been ostracized in the
immediately preceding round, is called a {\it member}.  We will call
the individual member's vote for ostracizing someone his {\it
blackball}. One blackball suffices for ostracism. The term
``blackball'' refers to the action of a player in voting to expel,
while ``ostracism'' refers to his actual expulsion.  Each player gets
some base level of satisfaction (normalized to zero below) simply by
being in the group, but his satisfaction is greater (a positive
payoff) if the other members cooperate.  The group can exclude a
player from even the base level (i.e., give him a negative payoff) by
ostracizing him, so ostracism is a punishment even when no one
cooperates, and is more painful to the deviator than merely having
other players also Defect.  In the parable, fishing by all members
leads to the base level of welfare; hunting, which requires
cooperation, is the positive level; and expulsion, which prevents a
player from either hunting or fishing, is the negative level.

 We will examine the equilibrium under the following key assumptions:

\begin{enumerate}
 \item {\bf Free Rider Problem}.
  A defector gets a higher payoff than a cooperator in the round in
which he defects.

    \item {\bf Aggregation Economies}.
  Payoffs per member are an increasing function of the number of
members who cooperate.

\item {\bf No Aggregation Economies without Cooperation.}
 The payoffs per member when every member defects do not depend on
the size of the group.

\item {\bf Excludability of  Resources from Non-Members}.
 A player would rather be in the group than ostracized, even if every
member defects.

\item {\bf Costless Enforcement.}
 Blackballing has no direct cost or benefit to those members who
engage in it.


\end{enumerate}

  Assumptions less central to the results include

 \begin{enumerate}
     \item
  Ostracism only lasts one round. A player who is ostracized can
re-enter the next round unless ostracized again.

     \item
  To be ostracized, a player need be blackballed by only one member.

\item
 A player can blackball any number of other players.

\end{enumerate}

  Each round $t$ is divided into two phases: an {\it ostracism
phase}, labelled $t^{os}$, and a {\it dilemma phase}, labelled
$t^{pd}$.  The game starts with $\bar{n}$ players who are all
members, and continues either until round $T$ or forever, depending
on the particular version of the model.  In any round $t$, let $n_t$
denote the number of members who play in the Prisoner's Dilemma at
the end of the round (the dilemma phase) and let $n^{os}_t$ denote
the number at the start of the round (the ostracism phase).  This
will result in $n^{os}_t = n_{t-1}$, as shown in Figure 2.

 

 Figure missing: ostrac1.eps 

  In the blackballing phase of round 1, each of the $\bar{n}$
members may blackball any of the other members.  Any player who is
blackballed at $1^{os}$ is excluded from the next dilemma phase,
$1^{pd}$, and from the next blackballing phase, $2^{os}$. (His
exclusion from $2^{os}$ is not essential for the results.)

  The number of members in $1^{pd}$, denoted $n_1$, may be less
than $\bar{n}$, since some players may be ostracized at
$1^{os}$.  The game is repeated at round 2 with $n^{os}_2$ members
playing in the ostracism phase $2^{os}$, which leaves $n_2$
members to play in the dilemma phase $2^{pd}$.  This continues
through the final dilemma phase $T^{pd}$, or forever if the game is
infinite.

  A player ostracized in $t^{os}$ does not play in the multiperson
Prisoners' Dilemma in $t^{pd}$. His total payoff for the round is
$-Y$, the cost of being a non-member. In addition, he remains a
non-member at $(t+1)^{os}$, which means that he cannot participate in
blackballing in that round.  Unless he is ostracized again at
$(t+1)^{os}$, he is free to rejoin the group at $(t+1)^{pd}$.

 Dropping the $t$ subscript to avoid clutter, let us write the payoff
functions using $n$ for the number of members in the dilemma phase
and $n^c$ for the number of members who cooperate.  All output is
split equally among the members.  The average output per member is
denoted by $f(n^c,n)$, and the cost to a single member of cooperating
is denoted by $X$. For the game to be a Prisoners' Dilemma, defecting
must be a dominant strategy in the one-period game, so we require
that for any number of cooperators $m > 0$ and members $n > 0$,
 \begin{equation} \label{e1}
 f(m,n)  - X < f(m-1,n).
 \end{equation}
 We also require that
 \begin{equation}
 \label{e1a}
 f(m-1, n-1) < f(m,n) \; \; \; ,
 \end{equation}
 which is the mathematical statement that there exist aggregation
economies: the presence of an additional cooperating member raises
per capita output.  Setting $m = n$, inequality (\ref{e1a}) implies
that per capita payoffs are larger in a larger cooperating group.
Let us normalize the output with zero cooperators to $f(0,n) = 0$,
which satisfies ``No aggregation economies without cooperation.'' As
a complement to (\ref{e1a}), we will assume that the presence of a
defecting member does not raise the average output. Therefore,
 \begin{equation} \label{e1b}
  f(m,n-1) \geq f(m,n).
 \end{equation}
  Ordinarily, inequality (\ref{e1b}) would be strict because the
presence of a free-riding member would strictly lower the average
output.


  When all members cooperate, each receives a payoff of $f(n,n)-X$.
If some members defect, the payoff to each defector is $f(n^c,n)$ and
the payoff to each cooperator is $f(n^c,n)-X$. When all defect, each
member receives a payoff of zero. A player who is ostracized receives
$-Y$.  The period's payoff to member $i$ is therefore
  \begin{equation} \label{payoffs}
  \pi_{i}  = \left\{ \begin{array}{ll}
  f(n^c,n) -X & {\rm if \;} i {\rm \; \: cooperates}\\
 f(n^c,n)  & {\rm if \;} i { \rm \; \: defects}\\
  f(0,n) =  0      & {\rm if \;all \; players\: defect}\\
  -Y & {\rm if \:} i {\rm \;\; is \;\; ostracized}
  \end{array} \right.
 \end{equation}
 Let $\delta \in [0, 1]$ be a discount factor common to all players.
The total payoff to player $i$ for the entire game is
$\sum_{t=1}^{T}\delta^{t-1} \pi_{it}$.

 In addition to the general assumptions, we must also restrict the
magnitudes of the parameters. Let us assume that
 \begin{equation}
 \label{e2}
 Y >  X,
 \end{equation}
 so that in each round, the penalty from being ostracized is
larger than the benefit from cheating against cooperators.


Finally, let us assume that ``cooperation'' is socially valuable, so
that the per capita payoff net of costs is higher if all cooperate
than if all defect,
  \begin{equation}
 \label{social}
 f(n,n) - X > 0 \; \; \; \; {\rm if}\;\; n > 0 \; \; \; .
 \end{equation}
 As will be shown by the example in Section 6, Assumption
(\ref{social}) is not always necessary for cooperation to be an
equilibrium, but it is used in the proof of the main proposition
below.

  The payoffs in the Prisoners' Dilemma game of Figure 1 satisfy
these assumptions. Let $\bar{n}=2$, $f(2,2) = 30$, $f(1,2) = 5$,
and $X=15$.  According to payoff function (\ref{payoffs}), if both
players cooperate their payoffs are each $f(2,2)-X$, if both defect
the payoffs are each 0, and if only one cooperates, his payoff is
$f(1,2)- X$ while that of the other player is $f(1,2)$.

   %---------------------------------------------------------------

\newpage
 \section{The Equilibrium.} \label{many}

 In this section we will examine a symmetric perfect equilibrium in
which all players adopt a strategy called {\it Banishment}. A player
following {\it Banishment} cooperates along the equilibrium path and
blackballs anyone who deviates from the strategy--- which includes
defectors, players who blackball when unprovoked, players who fail to
blackball defectors, players who fail to blackball those who fail to
blackball defectors, and so forth. On the equilibrium path, everyone
cooperates; if anyone deviates in any way, the others still
cooperate, but they blackball him.  Banishment is forgiving in the
sense that retribution is limited: after a single round of ostracism,
the outcast is permitted to return to the group without prejudice.

	In the final round, defecting is a dominant action because no
punishment can follow. In models without ostracism, this is the fatal
first domino that successively overthrows cooperation back to the
first round.  In using this inductive argument, we are imposing the
requirement, now standard in game theory, that the equilibrium be
subgame perfect: the relevant portions of an equilibrium strategy are
Nash equilibria for every subgame of the original game, whether or
not that subgame is reached in equilibrium.$^3$
%\footnote{One such
%subgame, for example, is the subgame starting after both players have
%cooperated the first two periods and defected in the third. Such
%behavior might never occur in equilibrium, but a player's equilibrium
%strategy must specify what actions he takes if it does occur, and
%those actions must maximize his payoffs for the remainder of the
%game.}
 Both players always defecting is a perfect equilibrium as well as a
Nash equilibrium of the finitely repeated game when ostracism is not
used.

  If players' strategies incorporate ostracism, however, they may
cooperate until the last round even in a subgame perfect equilibrium.
In the last round all players defect, just as in the single round
Prisoners' Dilemma. But in the next-to-last round there is no cost to
expelling defectors, since everyone knows there are no future gains
from having a large group.  The credible threat of expulsion for the
last round enforces cooperation and forces players to blackball
cheaters in earlier rounds.  Even though in early rounds the group
gain from having an additional cooperating teammate is large,
ostracism can still be enforced, because the gain from not
ostracising a deviator is spread among the group, while the
punishment for failing to blackball him falls on individuals.

\pagebreak

\begin{quotation}
\noindent
 {\bf Banishment Strategy}

{\bf Dilemma Phase.} Before round $T$, cooperate unless you have
violated {\it Banishment} in the immediately preceding ostracism
phase, in which case defect. At round $T$, defect.

{\bf Ostracism Phase.} Blackball any player who in the immediately
preceding dilemma or blackballing phases deviated from the strategy
{\it Banishment}, and do not blackball anyone else.

\end{quotation}

 {\it Banishment} specifies cooperation as the player's equilibrium
behavior until round $T$.  The blackballing action rules are
iterative.  In phase $1^{os}$, a player refrains from blackballing.
In phase $2^{os}$, he blackballs any player who either defected in
$1^{pd}$ or blackballed in $1^{os}$.  For $t \geq 3$, in phase
$t^{os}$ he blackballs any player who (i) defected in $(t-1)^{pd}$;
or (ii) blackballed in $(t-1)^{os}$ without provocation; or (iii)
failed to blackball in $(t-1)^{os}$ when he should have in response
to a deviation in round $t-2$.

 We will prove that {\it Banishment} supports an equilibrium with
cooperation in every round but $T$.  Deviators are blackballed,
because failing to blackball properly provokes blackballs against
oneself.$^4$
 %\footnote{An ostracism strategy simpler than {\it Banishment}
%would be to wait and blackball deviators only in phase $T^{os}$, even
%if they deviated much earlier. Such a strategy can support
%cooperation, without the iterative blackballing rule of {\it
%Banishment}.
% But this wait-and-blackball strategy works under a narrower
%parameter range than {\it Banishment}, because the cost of being
%ostracized during phase $T^{pd}$ must outweigh the total benefit from
%defecting in all previous rounds.}
 Since all players will defect at $T^{pd}$ anyway, there are no gains
from having more members at $T^{pd}$, so there is no loss to
ostracizing someone at $T^{os}$ who deviated in round $T-1$.

\noindent
  {\bf Proposition 1:} {\it With sufficiently little discounting,
there exists an equilibrium with cooperation in rounds 1 through
$T-1$ of the $T$-round ostracism game.}

\noindent
 {\bf Proof}: We use backward induction to verify that the strategy
combination in which all players follow {\it Banishment} is a subgame
perfect equilibrium. The outcome is then cooperation by all players
until the last round. To do so, we must show that no player has an
incentive to deviate from this proposed equilibrium in any subgame.
Let us start with the subgame consisting of round $T$ alone.

\pagebreak
 \noindent
  {\bf (1) Phase $T^{pd}$}.\\
  In phase $T^{pd}$, if a member deviated by cooperating he would
receive $-X$ instead of 0.  (In this phase and in any earlier phase,
if no members remain then trivially the equilibrium strategy is not
violated at that point in the game.)

\noindent
  {\bf (2) Phase $T^{os}$}.\\
  In phase $T^{os}$, a member is weakly willing to blackball any
player who violated {\it Banishment}'s rules in phases $(T-1)^{os}$
or $(T-1)^{pd}$, because in the final phase, $T^{pd}$, every member
will defect in any case, and the all-defect payoff of zero is
independent of the number of members.

\bigskip
   Let us next consider any round $t < T$, under the inductive
assumption that all players will follow {\it Banishment} in all
subgames starting after round $t$, {\it including} those subgames
which would not arise if the players follow {\it Banishment} through
round $t$.  We first examine the behavior of a given player, whom we
will call player $A$, in the dilemma phase $t^{pd}$.

 \noindent
  {\bf (3) Non-deviation subgame in phase $t^{pd}$}.\\
 Suppose that player $A$ did not deviate in $t^{os}$, though the
other players may have, and either he or other players may have
deviated earlier in the game.  Under {\it Banishment}, those other
players who deviated in $t^{os}$ will defect in $t^{pd}$, and those
who did not will cooperate in $t^{pd}$.  Let $n^{*}_{t}$ be the
number of cooperators in phase $t^{pd}$ if $A$ cooperates.  Then
$A$'s immediate payoff from defecting in $t^{pd}$ is the receipt of
$f(n_t^{*}-1,n_t)$ instead of $f(n_t^{*},n_t) - X$.  But if he
defects, he will be ostracized in phase $(t+1)^{os}$, which yields
him a discounted loss of $-\delta Y$ instead of: (a)
$\delta[f(n_{t+1},n_{t+1})- X] $, if $t < T-1$; or (b) zero, if $t = T
-1$.
If $t = T-1$, the game ends at $t+1$; otherwise, when $A$ returns in
$t+2$ everyone goes back to cooperating through $T-1$, exactly as if
$A$ had not deviated, so $A$'s payoffs in $t+2$ and beyond are the
same regardless of whether he deviates in $t$.  Therefore, the
condition for $A$ to prefer weakly to cooperate is
  \begin{equation} \label{oboe1}
 f(n_t^{*}-1,n_t) - \delta Y \leq \left\{
  \begin{array}{ll}
 f(n_t^{*},n_t) - X + \delta[ f(n_{t+1},n_{t+1})-X]
\; \; \; & {\rm if} \: t<T-1 \\
  f(n_t^{*},n_t) - X & {\rm if} \: t = T-1. \\
 \end{array}
  \right.
 \end{equation}
  Inequalities (\ref{e1a}) and (\ref{e1b}) together imply that $
f(m-1, n) < f(m, n)$, (per capita output is raised by adding a
cooperator), and by (\ref{e2}), $ X < Y$, so for $\delta$
sufficiently close to one, condition (\ref{oboe1}) is satisfied. If
$A$ did not deviate in $t^{os}$, he will not defect in $t^{pd}$.

%---------------------------------------------------------------

\noindent
 {\bf (4) Deviation Subgame in phase $t^{pd}$}.\\
 If $A$ has already deviated in $t^{os}$, then knowing that he will
be ostracized in $(t+1)^{os}$ anyway he bears no additional penalty
to defecting in $t^{pd}$. His payoff if he defects is $f(n_{t}^{*} -
1, n_{t})$, and if does not defect, $f(n^{*},n) - X$.  The net gain
to defecting earned at $t^{pd}$ is positive, by (\ref{e1}), and the
amount $A$ earns in all later rounds is unaffected. This verifies
that if $A$ had deviated from {\it Banishment} in the immediately
preceding ostracism round, he will defect as required by {\it
Banishment}.

We have therefore verified that {\it Banishment} is followed in
$t^{pd}$.

\noindent
 {\bf (5), (6), (7) The decision in phase $t^{os}$}.

 We next calculate $A$'s gain from deviating in $t^{os}$, given that
all the other players will follow {\it Banishment} for the remainder
of the game.  We must now distinguish carefully between the number of
members present in $t^{pd}$ when $A$ deviates in $t^{os}$ versus when
he does
not.  Let $n_{t}(eq)$ be the number of members in $t^{pd}$ if $A$
obeys {\it Banishment} in $t^{os}$, and let $n_{t}( dev)$ be the
number of members in $t^{pd}$ following a deviation called $deviate$
(some pattern of extra or insufficient blackballs that $A$ directs at
different players) by $A$ in $t^{os}$ .  In equilibrium, the number
of members in $(t+1)^{pd}$ is $n_{t+1} = \bar{n}$, the total number
of players, because no player deviates at $t^{os}$ or $t^{pd}$, so no
blackballs are cast in $(t+1)^{os}$.

\noindent
   {\bf (5) $A$'s decision in phase $t^{os}$ given that he will be
blackballed in that phase.}\\
  If $A$ foresees being blackballed in $t^{os}$ (perhaps because he
had deviated in round $t-1$), then he expects to be ostracized during
$t^{pd}$ and his payoff at that phase is unaffected by his decision in
$t^{os}$.  If he  deviated in $t^{os}$, he would obtain a payoff of
$-\delta Y$ from being expelled in $(t+1)^{os}$ instead of the payoff
of $\delta [f(\bar{n},\bar{n})- X]$ from cooperating in $(t+1)^{pd}$.
An exception is if his deviation at $t^{os}$ is a ``clean sweep''
that eliminates all the other members, in which case he earns zero at
$(t+1)^{pd}$, but this payoff is still lower than $\delta
[f(\bar{n},\bar{n})- X]$.  By the inductive hypothesis, his payoffs
are unaffected for the remainder of the game.  So he strictly refers
not to deviate.

 Suppose, on the other hand, that player $A$ does not expect to be
blackballed in $t^{os}$. We will divide the ways in which he might
then deviate in $t^{os}$ into two cases ([6] and [7]).

\noindent
   {\bf (6) $A$'s Decision in phase $t^{os}$ given that he will not be
blackballed in that phase: clean sweep deviation.}\\
 One possible deviation in $t^{os}$ is for $A$ to blackball every
player, when that is not called for by {\it Banishment}.  This
deviation is special because no other members remain to ostracize $A$
in phase $(t+1)^{os}$. By deviating in this way, $A$ earns zero
instead of $ f(n_t(eq),n_t(eq))-X$ in $t^{pd}$.  The deviation is
immediately unprofitable, and by the inductive hypothesis it creates
no change in $A$'s payoffs in $(t+1)^{pd}$ or any later point in the
game.

\noindent
   {\bf (7) $A$'s Decision in phase $t^{os}$ given that he will not
be blackballed in that phase: not a clean sweep deviation}.\\
 Consider any deviation by player $A$ in $t^{os}$ that does not
expell all the other players.  Following such a deviation, $A$ will
be blackballed in $t+1$, giving him a payoff of $- \delta Y$ instead
of $\delta f(\bar{n}, \bar{n}) - X$. In addition, we have shown in
Part (4) of this proof that after he has deviated at $t^{os}$, $A$
will defect at $t^{pd}$, for a payoff of $ f(n_t(dev)-1,n_t(dev))$ in
that phase instead of $ f(n_t(eq), n_t(eq))-X$.  $A$'s total gain
from deviating is nonpositive if
 \begin{equation}
 \label{ost-t}
 f(n_t(dev)-1,n_t(dev)) - \delta Y \leq
 \end{equation}
 $$
 \left\{
\begin{array}{ll}
 f(n_t(eq), n_t(eq))-X + \delta [f(\bar{n},
\bar{n})-X] \; \; \; & {\rm if} \: t<T-1 \\
 f(n_t(eq), n_t(eq))-X & {\rm if} \: t = T-1. \\
 \end{array} \right.
  $$
  If part of the deviation is to cast an unwarranted blackball,
reducing $n_t$, then that part of the deviation is unprofitable for
the deviator by (\ref{e1a}), because it diminishes the size of the
group and the number of cooperators equally in the dilemma phase
$t^{pd}$; since we rule out clean sweep deviations here, unwarranted
blackballing never prevents the punishment of ostracism at
$(t+1)^{os}$.  So if deviations that do not include unwarranted
blackballing can be shown to be unprofitable, it will follow that
those that do are also unprofitable.

  We therefore consider the more tempting deviation of failing to
blackball when {\it Banishment} calls for it after another player or
players deviate. We divide this kind of deviation into two cases.

  If at least two members besides $A$ are in the group in phase
$t^{os}$, any player whom {\it Banishment} requires to be ostracized
will still be ostracized even if $A$ were to refrain from
blackballing. Therefore, $n_{t}(dev) = n_{t}(eq)$, and it follows
from (\ref{e2}) that if $\delta$ is near 1, (\ref{ost-t}) is valid
and no deviation involving a failure to blackball is profitable.

 If only one other member is in the group in phase $t^{os}$, then if
$A$ fails to blackball him as required by {\it Banishment}, the
number of members in the group for phase $t^{pd}$ increases from 1 to
2. The strategy {\it Banishment} calls for $A$ to continue by
defecting in the dilemma phase, so his payoff there changes from
$f(1,1) - X $ to $f(1,2)$. We do not know whether this is an increase
or a decrease, since perhaps $f(1,1)>f(1,2)$, but offsetting any
possible increase is the fact that after failing to blackball and
then defecting, $A$ himself will be ostracized in $(t+1)^{os}$ and
his payoff (discounted back to $t$) will fall by at least $ \delta
Y$.  The gain to this deviation is negative if
 $$
 f(1,1) - X >  f(1,2) - \delta Y.
 $$
 By assumption (\ref{e1b}) (adding a non-cooperator does not raise
the per capita payoff), $f(1,2) \leq f(1,1)$, and by assumption
(\ref{e2}), $Y > X$.  Therefore, if $\delta$ is close enough to 1,
player $A$ loses if he fails to blackball appropriately.



Having verified that {\it Banishment} is followed in $t^{os}$, by
induction it is a subgame perfect equilibrium in all rounds.

Q.E.D.


 \bigskip
 \noindent
{\bf Infinite Rounds} \nopagebreak

 It is well known that cooperation can arise in the perfect
equilibrium of the infinitely repeated Prisoners' Dilemma, in
contrast to the game with a large but finite number of repetitions.
This is implied by the ``Folk Theorem'', which says that virtually
any pattern of actions can be generated by the equilibria of
infinitely repeated games, if the discount rate is sufficiently low
(see Fudenberg \& Maskin [1986] and Rasmusen [1987]).  The end round
argument from the introduction that ruled out cooperation in the
finite game does not apply to the infinite game.  Because infinitely
repeated games allow so many patterns of behavior, the fact that
adding ostracism to the game still allows cooperation is
unsurprising, but we will discuss it briefly.

One type of equilibrium strategy for the infinite game is any
non-ostracism strategy that enforces cooperation plus the rule
``Never blackball.''  A slight modification of the strategy {\it
Banishment} also enforces cooperation in the infinite game. Define
{\it Banishment} as before, but eliminate the inapplicable part of
the definition which refers to defecting in period $T$.  By reasoning
similar to Proposition 1's proof, {\it Banishment} enforces
cooperation when discount rates are sufficiently low. If a player
deviates, he will be ostracized for a round and then the game
proceeds as before along the equilibrium path. This is perhaps a more
attractive equilibrium than those which rely on retaliatory
defecting, because the cost of ostracism is heaviest for the
deviator, which fits our sense of what happens in the world.

  Moreover, when the infinitely repeated multiplayer Prisoners'
Dilemma is expanded by introducing ostracism, cooperation is possible
under a wider range of parameters. If there is heavy discounting
($\delta$ near zero), even the infinitely repeated game has
all-defect as its unique equilibrium outcome. But if ostracism is
possible, and the ostracism penalty of $Y$ is large enough relative
to the low discount factor, cooperation can be achieved.


  \newpage
 \noindent
 \section{ Discussion.} \label{extend}


 {\bf Relaxing Assumptions.}

 The arguments explaining how ostracism can enforce cooperation can
be applied not only to the Prisoner's Dilemma, but also to other
social dilemmas such as coordination games (see Rasmusen
[forthcoming] or Sugden [1986] for examples: e.g., the Battle of the
Sexes).  Assumption (\ref{e1}), which characterizes the strong
temptation to defect in the Prisoners' Dilemma, was not crucial in
the proof in the previous section. Its only significance was in
making Defection part of the equilibrium strategy on an irrelevant
off-equilibrium path; even without this assumption, cooperation is
still an equilibrium.

    Perhaps the most important assumption to check in deciding
whether the ostracism model applies is ``No Aggregation Economies
Without Cooperation'': if all players defect, is there no benefit
from a larger group size?  Lack of such a benefit is crucial to the
argument that the members are willing to ostracize a deviator in the
last period.

  The particular blackballing rule is not important.  We assumed that
one blackball sufficed for ostracism, but similar results can be
derived if ostracism requires blackballs by a majority of members, or
even if it requires unanimity aside from the member in question.
Ostracism also works in much the same way if it is irrevocable, i.e.,
if once a player is ostracized he can never rejoin the group.
Executing a player is an example of irrevocable ostracism. In the
last period, execution is just the same as temporary ostracism, at
least from the ostracizer's point of view: the offender disappears
for a round.  Earlier period executions have the same effect on the
group as irrevocably ostracizing the player, and the qualititative
effect on him is also the same: he forever loses the benefit of being
in the group. Both modifications, different blackballing rules and
irrevocable ostracism, require changes to the details of Proposition
1's proof, but do not change the thrust of the argument.

 Execution is not the only punishment that can be modelled as
ostracism.  Imprisonment is another example.  What is required for
the model to apply is that the punishment diminish the deviator's
ability to contribute to the group.  When society imprisons a
criminal, it loses the benefits it would have had from his
cooperation, if that cooperation could have been ensured.  The main
thing that might distinguish ostracism from prison is that in the
ostracism model we assumed that the punishment imposesd no direct
cost on either the group's output or the blackballer.  The tax to pay
for the deviator's stay in jail is a direct cost.  Such a cost would
prevent ostracism at $T^{os}$ and cause the equilibrium to unravel.
On the other hand, we also ruled out possible direct benefits such as
seizing the property of the deviator. Since our equilibrium in the
last round is weak, relaxing these assumptions would lead to either
never ostracizing or ostracizing even without provocation, both of
which would eliminate the equilibrium with cooperation.  In Section
5, we will demonstrate that a small amount of morality (desire to
punish past wrongdoers) converts our weak equilibrium into a strong
one.  If there are direct gains or losses to individuals from
ostracizing, or if in the all-Defect outcome payoffs are not
perfectly independent of the number of players, then morality can
still enforce cooperation if the costs are small.


\bigskip
\noindent
 {\bf A Comparison with Other Models.}
  \nopagebreak


   The strategy {\it Banishment} is reminiscent of Thompson and
Faith's (1981) model in that cooperation is enforced by sequences of
threats that involve not only threats to punish the defector, but
threats to punish those who fail to punish, and so on.  But there are
important differences.  In Thompson and Faith there is a hierarchical
structure in which players higher in rank punish those lower in rank.
Commitment to punishment is allowed, and the decision hierarchy is a
series of moves in which the different players commit to punishment
strategies in sequence.  The outcome is dictatorial, in the sense
that the most highly preferred outcome of the first mover is
achieved.

   In our model there is no prior asymmetry between players, and
commitment is ruled out by requiring subgame perfectness.  Players
move simultaneously and they are identical, so that a ``democratic''
outcome is achieved, rather than the favored outcome of a special
player (which in context of Thompson and Faith would involve the
dictator playing Defect and all other players Cooperate).  The
sequencing of threats that enforces cooperation is endogenous,
arising strategically from the interaction of identical players.  Our
model is less applicable than Thompson and Faith's to the punishments
of hierarchical organizations, such as excommunication by the Roman
Catholic Church.

  Bendor and Mookherjee (1987) also analyze social outcomes when the
group can threaten punishments, of which expulsion is an example.
Their setting emphasizes the problem of observing whether defection
has occurred, rather than the credibility of the threat of
punishment.

  In their well-known paper on the finitely repeated Prisoners'
Dilemma, Kreps et al. (1982) base cooperation upon incomplete
information. In their game, players defect in the last $k$ rounds,
where $k$ is determined by the parameters but is independent of $T$,
the total number of rounds in game.  If $T$ is large, the fact that
there is defection in the last $k$ rounds is unimportant.  On the
other hand, if $T$ is small, the social inefficiency can be
relatively severe, and if $T$ is less than $k$, the players never
cooperate.  Ostracism works very differently: it does not depend on
incomplete information, and it works as well with a a small number of
rounds as with a large number.

 \newpage
 \section{A Little Morality.} \label{moral}

  Since in many contexts ostracism is not purely selfish behavior, it
seems reasonable to consider the possibility that players are
slightly moralistic, so they gain a little bit of pleasure from
blackballing a deviator. As described above, the equilibrium with
{\it Banishment} is a weak equilibrium: given that the other players
follow {\it Banishment}, a player is also willing to follow it, but
he is indifferent about following certain of its action rules.
Moreover, another weak symmetric equilibrium is {\it Always Defect,
Never Ostracize}. The cooperative equilibrium is Pareto-superior, so
the players may hope for it to be a focal point, but choosing focal
points is always somewhat arbitrary.

 Let us define ``a little morality'' as a small positive payoff from
blackballing a deviator according to the strategy {\it Banishment}.
In the last ostracism phase, the moralistic player is not indifferent
about ostracizing a deviator; by ostracizing, he unambiguously raises
his payoff, if only slightly.  Because of this, {\it Banishment} is a
strong equilibrium for the $T$-period game.  Perhaps even more
importantly, it becomes the unique equilibrium.  The strategy {\it
Always Defect, Never Ostracize}, for example, is no longer an
equilibrium, because players would raise their payoffs by ostracizing
defectors in the next-to-last round.  The proof that the {\it
Banishment} outcome is unique essentially follows that of Proposition
1, except now the inductive hypothesis is that {\it Banishment} is
the {\it only} subgame perfect equilibrium.  {\it Banishment}
behavior at the last round, which is strongly preferred by morality,
deters any kind of action except for following {\it Banishment} at
the next to last round, which enforces a cascade of threats back to
the first round.

 Morality is similar to altruism as an escape from the Prisoners'
Dilemma, but it is not the same.  Altruism achieves cooperation
because some players unconditionally want to improve the welfare of
others.  Morality achieves cooperation because some players want to
reduce the welfare of others, if those others behave wrongfully.  In
fact, altruism in some players would prevent morality from enforcing
cooperation, by making the altruists unwilling to punish evildoers.

If the model were modified to allow for direct costs or benefits of
ostracizing, then as discussed in the previous section, {\it
Banishment} would no longer be an equilibrium strategy.  But if these
costs or benefits are small, morality could still persuade players to
ostracize when appropriate even if it is costly, or to refrain from
ostracizing even if it yields direct gains.  Of course, if morality
were sufficiently strong, cooperation could be supported by an ethic
that called for unconditional cooperation (``the golden rule'').  But
such a scheme places a heavy burden on morality, because it must
overcome the temptation to defect to seize large gains in the dilemma
phases.  With ostracism, the temptation is much smaller: a little
morality goes a long way.



\newpage
 \section{ Bad Equilibria Enforced by Ostracism.} \label{badeq}

  Although ostracism is able to enforce cooperation in the repeated
Prisoners' Dilemma, it is not necessarily a good thing.  We can apply
the notion of ostracism to repeated games that would normally have
desirable outcomes, but in which ostracism causes the players to
engage in undesirable behavior. This is a game theoretic version of
the idea that social custom can result in economic inefficiency
(Akerlof [1976, 1980], Romer [1985]).  Our interpetation of that idea
is that many games have multiple perfect equilibria, and society may
be stuck at an undesirable one for historical reasons.

   We will use a numerical example to illustrate how ostracism can
hurt a group.  Let there be 6 players, who choose $C$ ({\it
Customary} behavior) or $D$ ({\it Desirable} behavior) in a two-round
game with ostracism and no discounting. The actions $C$ and $D$
correspond to ``Cooperate'' and ``Defect'' in the general model, but
here the payoffs are such that $D$ is a better outcome than $C$.  The
reason for this somewhat counterintuitive reversal is that we want to
provide an example in which, absent ostracism, the individual's
temptation is to perform the socially desirable act, and yet with
ostracism he does not.  In the Prisoners' Dilemma without ostracism,
$Defect$ is the action that the individual is tempted to take, and
this feature of the payoff scheme applies to strategy $D$ here. In a
round in which the group has $n$ members who all cooperate, they each
get $5(n -1)$, while if all defect they each get 100.  If only some
players defect, the defectors each get and the cooperators each get
0.  Figure 3 summarizes these payoffs.  If a player is ostracized, he
gets a payoff of $-150$ in the second round.

\begin{center}
 Insert Figure 3.
 \end{center}

  If this game did not have ostracism, the unique equilibrium would
be for every player to play $D$ in both rounds. In the last round $D$
is a dominant action, since by playing $D$ the player gets 100 if all
the others play $D$ and 40 if they do not; following customary
behavior would give him a payoff between 0 and 30 when there are 6
players.  Since everyone will play $D$ in the last round, defecting
is also the Nash strategy for the first round. The equilibrium payoff
is 200 per player.

  Ostracism adds another equilibrium, in which all the players
cooperate in the first round and defect in the second:

\noindent
 {\bf Pareto-Inferior Equilibrium Strategy.}\\
Choose $C$ in the first round.\\
 Blackball any player who chose $D$ in the first round.\\
 Choose $D$ in the second round.

  The players defect in the second round because that remains a
dominant action regardless of ostracism. They are willing to
blackball a player who defects in the first round because they
foresee that in the second round the payoffs will be the constant
100, which does not decline if they ostracize some players. Each
player will cooperate in the first round because his total payoff is
then 25 ($ = 5[6-1]$) plus 100, as opposed to the 40 plus $-150$ he
would receive from defecting and being ostracized.  The equilibrium
payoff is 125 per player.

 $C$ can be replaced by one's least-liked custom, as long as the
custom meets the assumptions of the model. Suppose, for example, that
the group is a set of trading parties. $C$ and $D$ could mean
``Customary Wage'' and ``Market-Clearing Wage,'' ``Do Not Lend'' and
``Usury,'' or ``Shun Blacks'' and ``Hire Blacks.'' The members of the
group would obey the bad customs for fear of being excluded from
trade with the other members.  Other examples might be students who
disapprove of cheating on exams, but shun weaseling as even worse, or
societies where hyper-sensitivity to slights and willingness to duel
is enforced by the fear of public contempt. The bad equilibrium with
ostracism is not the only equilibrium of these games--- another
exists in which the players never ostracize and always defect. Many
theorists would predict that the efficient equilibrium would be the
actual one, since it is both simpler and better.  This is based on
the view that simplicity and efficiency are properties of focal
points, and that player who can communicate will settle upon an
efficient, self-enforcing equilibrium. But if historical accident and
psychological factors are important in establishing norms of
behavior, the result can be Pareto-inferior equilibria.


 \newpage
 \section{ Applications.}
 \label{endsection}

  Various practices that groups use for disciplining their members
can replace ostracism in our model, but some practices are very
different.  Unlike the players in the standard Prisoners' Dilemma,
many groups can use punishments such as lump-sum fines that are not
costly to the group, or can precommit to punishments.  Fines and
other forms of expropriation, however, do not fit the technical
requirements of our model, because fining a defector does not harm
the rest of the group; indeed, it benefits them. If fines are
available, it is not at all surprising that the group can enforce
cooperation. But sufficiently severe fines are often infeasible.  For
example, the group may lack the legal authority to expropriate
physical property, and only have the ability to withhold its society.
Or, punishment severe enough to deter transgression might have to be
nonmonetary and might unavoidably impair the wrongdoer's capacity to
contribute to the group.

  Masters (1984) maintains that ``imprisonment, enslavement and death
are particularly important forms of ostracism.''  Already, in Section
4, we have discussed the relation of imprisonment and execution to
ostracism.  Although this expands the coverage of the term
explosively, these forms of punishment fit well within our framework,
because jailing or executing someone sacrifices aggregation economies
by ending his contribution to the group.  (Enslavement is different
because the rest of the group may directly benefit from the services
or sale of the deviator).

  Ostracism can also take forms that are milder than forced exile.
It may have the same incentive effects as exile without requiring a
change of location: the other players might just be impolite or
refuse to converse with the offender.  Voluntary exile to avoid other
punishments is also equivalent to ostracism. Recall that Socrates
could easily have fled Athens to avoid drinking hemlock, and
surprised his friends (and no doubt his enemies) by refusing to do
so.  Embezzlers in Bermuda and U.S. draft evaders in Canada are other
examples.

  Another application is to the problem of monitoring a group's
behavior. Suppose that the manager of a team of workers has available
a costly technology for monitoring and enforcing cooperation (working
rather than shirking, in this case).  Ostracism is a mechanism that
can enforce cooperation more cheaply.  The manager need only state
the Banishment strategy with ``Inform the manager about Mr X''
replacing ``Blackball Mr X.'' The direct cost of informing the
manager is very low, and if it is credible that the manager himself
will carry out the costly punishment, cooperation within the team has
been enforced by inexpensive self-monitoring.$^5$
%\footnote{Thomas
%Schwartz of the UCLA Department of Political Science suggested this
%idea.}

    The exclusion of the member from the benefits of being in the
group is the obvious aspect of ostracism.  This paper stresses the
obverse, that ostracizing a member sacrifices the benefits that he
can provide the group.  Indeed, in ancient Athens ostracism was
applied to some of the most dynamic leaders, and Amsterdam lost an
exceptionally gifted citizen when it banished Spinoza.

  The ostracism model is not intended to apply to all repeated social
dilemmas. The model applies when the group not only faces a repeated
game, but also: (a) Members can be expelled from the group; (b)
Players would prefer membership in the group even if everyone
defects; and (c) If everyone defects, the per capita payoff does not
vary with the number of members.

  These three requirements rule out applying the model to many
situations.  The model does not, for example, fit the application of
the Prisoners' Dilemma that may first come to mind: oligopoly.
Oligopolists generally cannot expel a price cutter from the industry,
except in markets where sellers must be certified by a regulatory
agency controlled by the group.$^6$
 %\footnote{Kessel (1958), however,
%describes suspension of licenses, expulsion from medical societies,
%and denial of hospital staff privileges to doctors associated with
%price-cutting and group health plans.}
  Moreover, when a deviator can be expelled, the elimination of a
competing seller would generally be beneficial to the remaining
sellers.  The problem is not to make ostracism credible, but to
prevent unprovoked ostracism of players who did not defect.

 Many other situations do fit the assumptions of the model.  Trade
sanctions are examples of ostracism, whether by housewives in Lake
Wobegon against a grocery with misdated goods or Common Market
countries against a country such as South Africa for its race
policies. Ostracism from world capital or goods markets can perhaps
provide a clue as to why nations such as Mexico and Argentina are
reluctant to brazenly default on their debts; without ostracism, it
is a puzzle as to why they would be denied new loans just because of
past misbehavior.

 Ostracism is common in social groups, and we will cite only a few
examples of this widespread phenomenon.  Gruter (1985) describes
Meidung (shunning), and excommunication, forms of ostracism practiced
among the Old Order Amish.  These arose as church commandments in
1632 from the Dordrecht Confession of Faith as a means of
disciplining church members.  Boehm (1985) discusses several forms of
ostracism in Balkan tribal society in 19th century highland Albania.
These range from refusal to talk or listen to an individual regarded
as a coward to expulsion from the tribe, and finally execution.
Because clans were obligated to unconditionally avenge wrongs to
members, the society was prone to blood feuds.  However, clan
ostracism was sometimes performed on members who were so reckless as
to be a liability to the group.  Customarily, the clan was not held
liable for the actions of an {\it odlicen} (expelled member).

 Often no formal institutions for expulsion exist, but there are ways
in which the group can pressure the deviant into leaving, or deny him
the benefits of society. The latter has been suggested as a problem
in experimental work on the Prisoner's Dilemma. A student subject
deciding whether to defect against students living in the same
dormitory may decide that exclusion from dorm parties may outweigh
the experiment's monetary incentives.  Ostracism of this kind is a
basic part of our culture.  Readers of Dickens' {\it Hard Times}, for
example, will recall that even then, a deviant worker in an English
trade union would be ``sent to Coventry,'' meaning that no other
worker would speak to him.  And the name of Captain Boycott, a 19th
century Irish land agent, entered the common vocabulary when his
neighbors shunned him for cooperating with the English (Churchill
[1958]).


\bigskip \noindent
 {\bf Conclusion and Summary.} \nopagebreak

	This paper has described a possible escape from the repeated
multiplayer Prisoners' Dilemma and related games: ostracizing
defectors, an escape that can enforce cooperation until the final
round.  Ostracism can be effective despite a perfectness problem in
incurring costs to punish defectors after the defection has taken
place. Our model can explain why ostracism, or social norms which
call for censure of wrongdoers can be self-fulfilling in the sense
that it pays for everyone to conform not only to good behavior, but
also to punish wrongdoers, to punish those who fail to punish, and so
on.  Indeed, even socially dysfunctional norms can be supported by
ostracism.

	The key is that since all players defect in the final round,
in that round no cost of ostracizing wrongdoers is incurred.  The
threat of punishment therefore deters both defection and failure to
blackball properly in the next-to-last round, and the argument can be
carried back to the start of the game. There exist equilibrium
strategies involving cooperation at all rounds until the last.  The
equilibrium with a finite number of rounds is weak, since the
decision on whether to blackball in the final round is a razor's-edge
choice.  However, if the slightest amount of morality is added,
cooperation until the last round becomes a unique and strong
equilibrium outcome.


 \newpage

\begin{center}
 \begin{tabular}{lllcc}
   &     &        &\multicolumn{2}{c}{\bf Player 2}\\
  &       &             &    Cooperate  &  Defect  \\
  & & Cooperate & $ 15,15$ & $-10,5$
\\
 & {\bf Player 1:} & & & \\
&  &         Defect     &     $5, -10$   &  $0,0$ \\
 & & & & \\
 \multicolumn{5}{l}{\it Payoffs to: Player 1, Player 2 }\\
 \end{tabular}

   \nopagebreak

\bigskip
  {\bf Figure 1: A Prisoners' Dilemma  with Two Members.} (missing) 
\end{center}

 

\newpage


\begin{tabular}{lllll}
 & &     \multicolumn{3}{c}{ Other Players}\\
            &  & All Customary  & Some Desirable & All Desirable\\
 & Customary&     $5(n-1)$ &       0      & 0\\
Player $i$ & &  & & \\
 & Desirable&       40      & 40     &100\\
 \end{tabular}

({\it Payoffs to Player $i$})

\bigskip

{\bf  Figure 3: A Bad Equilibrium Enforced by Ostracism.} (missing) 

  %---------------------------------------------------------------


\newpage

\begin{center} Notes \end{center}

1. Roughly speaking, perfectness requires that an
equilibrium strategy not only be a best response to the other
players' strategies early in the game, but also remain a best
response once the game has been partly played out. This rules out
threats that would not be carried out.

 2. One need not subscribe to the ``social contract'' theory to
attempt to explain grouping behavior in terms of costs and benefits
to individuals from being in a group.  An evolutionary outlook, or
Aristotle's view of man as a political animal, are both entirely
consistent with grouping behavior being influenced by the costs and
benefits individuals derive from membership.

3. One such subgame, for example, is the subgame starting after both
players have cooperated the first two periods and defected in the
third. Such behavior might never occur in equilibrium, but a player's
equilibrium strategy must specify what actions he takes if it does
occur, and those actions must maximize his payoffs for the remainder
of the game.

4.  An ostracism strategy simpler than {\it Banishment} would be to
wait and blackball deviators only in phase $T^{os}$, even if they had
deviated much earlier. Such a strategy could support cooperation,
without the iterative blackballing rule of {\it Banishment}.  But this
wait-and-blackball strategy works under a narrower parameter range
than {\it Banishment}, because the cost of being ostracized during
phase $T^{pd}$ must outweigh the total benefit from defecting in all
previous rounds.

 5. Thomas Schwartz of the UCLA Department of Political Science
suggested this idea.

6. Kessel (1958), however, describes suspension of licenses,
expulsion from medical societies, and denial of hospital staff
privileges to doctors associated with price-cutting and group health
plans.




\newpage

\begin{center} References \end{center}

Akerlof, George, 1976, The economics of caste and of the rat race and
other woeful tales, Quarterly Journal of Economics, 599-617.

Akerlof, George, 1980, A theory of social custom, of which
unemployment may be one consequence, Quarterly Journal of Economics,
94, 749-775.

Bendor, Jonathan and Dilip Mookherjee, 1987, Institutional Structure
and the Logic of Ongoing Collective Action, American Political
Science Review, 81, 129-154.

Boehm, Christopher, 1985, Execution within the clan as an extreme
form of ostracism, Social Science Information, 24, 309-321.


 Churchill, Winston, 1958, The Great Democracies, New York: Dodd,
Mead and Co.


Fudenberg, Drew and Eric Maskin, 1986, The folk theorem in repeated
games with discounting and with incomplete information, Econometrica,
54, 533-554.

Gruter, Margaret, 1985, Ostracism on trial: the limits of individual
rights,   Social Science Information 24, 101-111.

Hirshleifer, Jack, 1987, On the emotions as guarantors of threats
and promises, in The Latest on the Best: Essays on Evolution and
Optimality, John Dupre, ed.  Cambridge, Mass.: MIT Press.

Kessel, Reuben, 1958, ``Price discrimination in medicine.''  Journal
of Law and Economics 1:20-53 .

Kreps, David, Paul Milgrom, John Roberts, and Robert Wilson, 1982,
Rational co\-op\-er\-ation in the finite\-ly re\-peated pris\-oner's
dilemma, Jour\-nal of Econ\-omic The\-o\-ry, 27, 245-252.

Masters, Roger, 1984, Ostracism, voice and exit: the biology of
social participation, Social Science Information, 23, 877-893.

 Rasmusen, Eric, 1987, A new version of the Folk Theorem, UCLA AGSM
Business Economics Working Paper \# 87-6.

 Rasmusen, Eric, forthcoming, Games and Information, Oxford: Basil
Blackwell Ltd.

Romer, David, 1984, The theory of social custom: A modification and
some extensions, Quarterly Journal of Economics, 99, 717-727.

Schelling, Thomas, 1960, The Strategy of Conflict, London: Oxford
University Press.

Sugden, Robert, 1986, The Economics of Rights, Co-operation and
Welfare, Oxford: Basil Blackwell.


Thompson, Earl and Roger Faith, 1981, A pure theory of strategic
behavior and social institutions, American Economic Review,
71, 366-380, vol. 71 no. 3.


%---------------------------------------------------------------





 \end{document}
