%Paper: ewp-game/9705003
%From: hart@math.huji.ac.il
%Date: Wed, 7 May 97 16:35:19 CDT


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%TCIDATA{LastRevised=Thu Apr 10 16:27:37 1997}
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\begin{document}

\title{Finite Horizon Bargaining and the Consistent Field\thanks{%
This research was started in 1991, as part of the first author's Ph.D.
thesis at Harvard University. We gratefully acknowledge financial support
from: CAPES, Brazil (Gomes); the U.S.-Israel Binational Science Foundation
(Hart and Mas-Colell); the Israeli Academy of Sciences and Humanities
(Hart); the Spanish Ministry of Education (Mas-Colell); and the Generalitat
de Catalunya (Mas-Colell).}}
\author{Armando Gomes\thanks{%
Department of Finance, The Wharton School, University of Pennsylvania,
Philadelphia. \textit{e-mail:} gomes@wharton.upenn.edu.} \and Sergiu Hart%
\thanks{%
Department of Economics; Department of Mathematics; and Center for
Rationality and Interactive Decision Theory, The Hebrew University of
Jerusalem. \textit{e-mail:} hart@math.huji.ac.il.} \and Andreu Mas-Colell%
\thanks{%
Department of Economics, Universitat Pompeu Fabra, Barcelona. \textit{e-mail:%
} mcolell@upf.es.}}
\date{April 1997}
\maketitle

\begin{abstract}
This paper explores the relationships between noncooperative bargaining
games and the consistent value for non-transferable utility (NTU)
cooperative games. A dynamic approach to the consistent value for NTU games
is introduced: the consistent vector field. The main contribution of the
paper is to show that the consistent field is intimately related to the
concept of subgame perfection for finite horizon noncooperative bargaining
games, as the horizon goes to infinity and the cost of delay goes to zero.
The solutions of the dynamic system associated to the consistent field
characterize the subgame perfect equilibrium payoffs of the noncooperative
bargaining games. We show that for transferable utility, hyperplane and pure
bargaining games, the dynamics of the consistent field converge globally to
the unique consistent value. However, in the general NTU case, the dynamics
of the consistent field can be complex. An example is constructed where the
consistent field has cyclic solutions; moreover, the finite horizon subgame
perfect equilibria do not approach the consistent value.

\textit{Journal of Economic Literature} Classification Numbers: C71,
C72.\newpage \ 
\end{abstract}

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\section{Introduction}

This paper belongs to a general research program which studies the
relationships between equilibria of $n$-player noncooperative games and
axiomatically generated solutions for the cooperative game described in
coalitional form. Here we carry out an exploration based on the theory of
differentiable dynamic systems.

For $n$-person situations of pure bargaining (where the cooperation of all
players is needed to achieve an outcome different from the threat values),
the classical solution concept proposed by axiomatic cooperative game theory
is the \emph{Nash }(1950) \emph{bargaining solution.} Interestingly, the
Nash solution has also been arrived at as a limit --- when the cost of delay
in agreement becomes small --- of the subgame perfect equilibria of models
of bargaining in extensive form (in particular, of the St\aa hl (1972) --
Rubinstein (1982) model of alternating offers; see Binmore (1987) and the
book of Osborne and Rubinstein (1990) for these and other models).

Similarly, for $n$-player games with transferable utility (TU), the \emph{%
Shapley }(1953) \emph{value} is a central solution concept derived by
axiomatic cooperative game theory. Again, bargaining models in extensive
form have been proposed, whose solutions coincide with, or converge to, the
Shapley value (Harsanyi (1981), Gul (1989), Hart and Moore (1990), Winter
(1994), Hart and Mas-Colell (1996b)).

The theory is less settled for the general non-transferable utility (NTU)
games in coalitional form. In this paper we focus on the \emph{consistent
(NTU-) value}, an axiomatic solution proposed by Maschler and Owen (1989,
1992), which generalizes both the Nash solution for the pure bargaining case
and the Shapley value for the TU case. The point of departure for our
current research is Hart and Mas-Colell (1996b), which contains an analysis
of an infinite horizon noncooperative bargaining game whose stationary
subgame perfect equilibria are close, when the parameter that measures the
cost of delay in agreement is low, to the consistent values.

The present paper starts by developing a dynamic approach to the consistent
value. It generalizes to NTU games some dynamic processes put forward by
Maschler, Owen and Peleg (1988) for pure bargaining games and by Maschler
and Owen (1989) for hyperplane games (an extension of TU games), and which,
in these cases, globally converge to the unique consistent value. Motivated
by the axiomatic concept of consistency we introduce the concept of \emph{%
consistent (vector) field}. Roughly speaking, the consistent field is
defined, for every payoff configuration at the Pareto frontier, as the
direction to move locally along the frontier in order to reduce the
``inconsistency'' of the payoff. The singularities of the consistent field
are the consistent values and the solutions (or flows) of the dynamic system
associated with the consistent field constitute a natural way by which
players starting from arbitrary payoffs could adjust.

The main contribution of the paper is to show that the consistent field is
intimately related to subgame perfection for finitely horizon noncooperative
bargaining games, providing thus an unexplored link between the cooperative
and noncooperative theoretical formulations. The specific noncooperative
bargaining game we study is the \emph{finite horizon} version of the
bargaining game introduced by Hart and Mas-Colell (1996b). Informally, this
noncooperative game is a sequential game where the players have up to $T$
stages to reach an agreement. At each stage a player is selected at random
to propose a particular way to split the gains from cooperation, and will be
ousted from the game with a probability of $1-\rho $ if an unanimous
agreement is not reached. The subgame perfect Nash equilibrium (SPNE) of
this game is easily obtained by backward induction, and the problem we
address is to develop a characterization of the SPNE payoffs, denoted $%
w(\rho ,T),$ for a low cost of delay factor $1-\rho $ and a large number $T$
of potential rounds of negotiation.

We show that the limit of $w(\rho ,T)$ depends on the relative rates at
which $1-\rho $ converges to $0$ and $T$ converges to infinity. As $\rho $
converges to $1$ and $T$ converges to infinity, in such a manner that the
probability $\rho ^{T}$ of all players remaining at the last stage of the
game converges to $1$ --- thus the convergence of $T$ to infinity is much
slower than the convergence of $\rho $ to $1$ --- the SPNE payoffs $w(\rho
,T)\,$ converge to $r,$ a well defined, efficient point; we call $r$ the 
\emph{Raiffa point}. In the two-player case, this result was obtained by
Sj\"{o}str\"{o}m (1991), the point $r$ being the Raiffa bargaining solution
(see Luce and Raiffa (1957, \S 6.7)).

Next, assume that the rate at which $T$ converges to infinity increases, so
that the probability of all players remaining at the last stage converges
now to some $\mu <1$ (i.e., $\rho ^{T}\rightarrow \mu $ or $T(1-\rho
)\rightarrow -\ln \mu $). We prove that in this case the SPNE payoffs $%
w(\rho ,T)$ converge to the solution, at time $t=-\ln \mu \in [0,\infty ),$
of the dynamic system associated with the consistent field and having the
Raiffa point $r$ as its initial condition at $t=0.$ We also show that if
this solution trajectory starting at $r$ converges, as $t$ goes to infinity,
to a (local) attractor $a$ of the consistent field, then $a$ is the limit of
any sequence of SPNE payoff when $\rho ^{T}$ converges to zero (and, of
course, $\rho \rightarrow 1$ and $T$ $\rightarrow \infty $). Finally, we
show that any point in the limit set of the trajectory of the consistent
field solutions through the Raiffa point can be reached as the limit of SPNE
payoffs of an appropriate sequence of finite horizon games with $\rho
^{T}\rightarrow 0$ (and $\rho \rightarrow 1,$ $T$ $\rightarrow \infty $).

All these results indicate that we can attach significance to the dynamic
properties of the consistent field both on cooperative and noncooperative
theory grounds, and therefore we conclude that it is a vector field well
worth analyzing in more depth. In that vein, we show that the global
dynamics are convergent to the unique consistent value in the pure
bargaining and in the hyperplane games cases. For the general case we
analyze the local dynamics of the consistent field around a consistent
value. We show that this local dynamics is composed of a ``game part'',
which depends only on the particular consistent value, and a ``geometry
part'', which depends only on the curvature of the Pareto frontier at the
consistent value. Exploiting this relationship we can construct examples
with a wide variety of local behaviors: sink, source, saddle point. We can
also, using Hopf bifurcation theory, construct an NTU game where the
consistent field has cyclical solutions, thus indicating that the limit of
SPNE solutions of finite horizon bargaining games could well be a point
which is \emph{not} a consistent value. This is in contrast to the global
convergence of the consistent field in the TU, hyperplane, and pure
bargaining cases. All this confirms once more the intuition that pure
bargaining games and TU (or, more generally, hyperplane) games are the most
well behaved of the NTU games, and thus the easier to analyze. The game
theoretic behavior of general NTU games is however considerably more complex
than what one may be led to suspect from an analysis of these two cases. We
refer to Hart and Mas-Colell (1996a) for further elaboration of this point.

The paper is organized as follows: Section 2 presents the basic model ---
the underlying cooperative game in coalitional form, and the noncooperative
bargaining game --- followed by a preliminary analysis of the subgame
perfect equilibria of the latter. Section 3 recalls the definition of the
consistent NTU-value, and introduces the consistent field and its associated
dynamics. The results connecting the SPNE payoffs with the dynamics of the
consistent field are stated in Section 4. A (local) analysis of the
consistent field is carried out in Section 5; we then provide various
examples for the behavior of the resulting dynamics (and thus, \textit{a
fortiori}, of the SPNE payoffs). Proofs are relegated to the Appendix.

\section{The Model}

Let $\left( N,V\right) $ be a \emph{non-transferable utility (NTU)} $n$\emph{%
-person game in coalitional form}. The set of players is $N=\left\{
1,2,\ldots ,n\right\} $ and $V$ is the coalitional (characteristic)
function. For each coalition $S\subset N$, the set $V(S)$ --- a subset of $%
\Re ^{S}$ --- is the set of all allocations that are feasible for the
members of $S$.

We make the following standard assumptions on $\left( N,V\right) $:

\begin{description}
\item  (A1) For each coalition $S$, the set $V(S)$ is \emph{closed}, \emph{%
convex} and \emph{comprehensive }(i.e., if $x\in V(S)$ and $y\leq x$ then $%
y\in V(S)$). Moreover, $0\in V(S).$

\item  (A2) For each coalition $S$, the boundary (or Pareto frontier) of $%
V(S)$, denoted by $\partial V\left( S\right) $, is \emph{C}$^{2}$\emph{\ }%
(i.e., at each boundary point there is a single outward normal direction,
which varies in a continuously differentiable manner with the point) and 
\emph{nonlevel} (i.e., the outward normal vector at any point of $\partial
V(S)$ is positive in all coordinates).

\item  (A3) \emph{Monotonicity}: If $Z\subset S$ then $V(Z)\times
\{0^{S\backslash Z}\}\subset V(S)$ (i.e., completing a vector in $V(Z)$ with 
$0$'s for the coordinates in $S\backslash Z$ results in a vector in $V(S)$).
\end{description}

The noncooperative game we analyze is the finitely repeated version of the
game introduced by Hart and Mas-Colell (1996b). The \emph{n-person
noncooperative bargaining game} $(N,V,\rho ,T),$ where $\rho \in \left[
0,1\right] $ and $T$ is a positive integer, is described inductively as
follows:

\begin{quote}
The game is a perfect information game consisting of at most $T$ rounds of
negotiation. In each round $t$ there is a set $S\subset N$ of active players
who can reach an agreement, starting in the first round $\left( t=1\right) $
with $S=N$. One player in $S$ is chosen randomly, with all players in $S$
equally likely to be selected. Say player $i$ has been chosen. Then $i$
proposes a feasible payoff vector in $V(S)$ to the other players in $S$.
They can either agree or not (they are asked in some prespecified order).
The game ends with the proposed payoffs if all players in $S$ agree, or with
payoffs equal to $0$ if it is the last round $t=T\,$ and there is no
agreement. Otherwise, the game moves to the next round $t+1,$ where with
probability $\rho $ the set of active players does not change, and with
probability $1-\rho $ it becomes $S\backslash i$. In the latter case, the
payoff of the ``dropped out'' player $i$ is $0$.
\end{quote}

The \emph{subgame perfect Nash equilibrium (SPNE)} of the above finite game
with perfect information can be easily obtained by backward induction.
Suppose that in the last step of negotiation, $T$, the players in the
coalition $S$ have ``survived''. The equilibrium strategies for these
remaining players are as follows. If player $i\in S$ is chosen to be the
proposer (which occurs with probability $1/\left| S\right| $), then $i$'s$\,$
strategy is to propose that vector $e_{S,i}\in \partial V(S)$ such that $%
e_{S,i}^{j}=0$ for all $j\in S\backslash i$ (efficiency then uniquely
determines the payoff of player $i$). The strategy for each $j\in
S\backslash i$ is to accept any $x_{S}\in V(S)$ such that $x_{S}^{j}\geq 0.$
The (expected) equilibrium payoff vector (before the selection of the
proposer) is then $e_{S}:=(1/|S|)\sum_{i\in S}e_{S,i}$ (note that the
convexity of $V(S)$ implies that $e_{S}\in V(S)$). 

At round $T-1$ of the negotiation the SPNE strategies are as follows.
Suppose that $S$ is the set of remaining players at this stage. If player $%
i\in S$ is chosen to be the proposer then $i$'s$\,$ strategy is to propose
that allocation $a_{S,i}\in \partial V(S)$ such that $a_{S,i}^{j}=\rho
e_{S}^{j}+(1-\rho )e_{S\backslash i}^{j}$ for all $j\in S\backslash i.$ The
strategy for each $j\in S\backslash i$ is to accept any $x_{S}\in V(S)$ such
that $x_{S}^{j}\geq \rho e_{S}^{j}+(1-\rho )e_{S\backslash i}^{j}.$

The strategy profile above is the unique subgame perfect equilibrium
strategy for the game starting at round $T-1$. To prove that, observe that
with probability $\rho $ the proposer $i$ will remain for the next and last
stage, and the expected payoff of the remaining $S$ players in the
continuation game is, as seen above, $e_{S}$. With probability $1-\rho $
player $i$ will drop out of the game, and the expected payoff of the
remaining players in the continuation game is $e_{S\backslash i}.$ It
follows that the most player $j$ expects to get by rejecting an offer of the
proposer $i$ is $\rho e_{S}^{j}+(1-\rho )e_{S\backslash i}^{j}$, which
implies that the strategy profile is the unique SPNE of the game.

To formalize this, define a \emph{payoff configuration (p.c.) }$a$ to be a
collection of payoff vectors for all coalitions: $a=(a_{S})_{S\subset N}$
with $a_{S}\in V(S)$ for all $S\subset N.$ The backward induction arguments
are then captured by the following function $F_{\rho }:V\rightarrow V,$
where $V:=\prod_{_{S\subset N}}V(S)$ is the set of all payoff configurations.

\begin{definition}
\label{def-F-rho}The \emph{backward induction function} $F_{\rho
}:V\rightarrow V$ maps each payoff configuration $a=\left( a_{S}\right)
_{S\subset N}\in V$ to a payoff configuration $F_{\rho
}(a)=(b_{S})_{S\subset N}\in V$ given by

\begin{quotation}
(i) $b_{S,i}^{j}=\rho a_{S}^{j}+(1-\rho )a_{S\backslash i}^{j}$ for all $%
S\subset N$ and all $j\in S\backslash i;$

(ii) $b_{S,i}\in \partial V(S)$ for all $i\in S\subset N;$

(iii) $b_{S}=\frac{1}{|S|}\sum_{i\in S}b_{S,i}$ for all $S\subset N.$
\end{quotation}
\end{definition}

The backward induction function provides the expected payoffs $F_{\rho }(a)$
at any stage of the game, given that the payoffs in the continuation game
are specified by the configuration $a=(a_{S})_{S\subset N}$. The function $%
F_{\rho }$ is well-defined because of the assumptions (A1) - (A3) imposed on
the game $\left( N,V\right) $.

The \emph{SPNE payoff configuration} of the noncooperative game $\left(
N,V,\rho ,T\right) $ can be conveniently represented as the payoff
configuration $w\left( \rho ,T\right) =\left( w_{S}\left( \rho ,T\right)
\right) _{S\subset N}$ where $w_{S}\left( \rho ,T\right) $ is the $\left|
S\right| $-dimensional vector representing the unique SPNE payoff vector of
the noncooperative game $\left( S,V_{\mid S},\rho ,T\right) $ restricted to
the coalition $S\subset N$.

We have just proven above that the continuation games of the noncooperative
game $\left( N,V,\rho ,T\right) $ starting at round $T$ and $T-1$ have SPNE
payoff configurations given by $w\left( \rho ,1\right) =e$ and $w\left( \rho
,2\right) =F_{\rho }(e),$ respectively. Now$\footnote{%
We thank Vincent Feltkamp for pointing out that the induction may be
conveniently started at $0.$}$ $e=F_{\rho }(0),$ where $0=(0^{S})_{S\subset
N}$ is the payoff configuration with all coalitional payoff vectors equal to 
$0,$ and $F_{\rho }(e)=F_{\rho }(F_{\rho }(0))=F_{\rho }^{2}(0)$ is the
second iterate of $F_{\rho }$ evaluated at $0.$ Proceeding inductively one
obtains the strategy profile of the SPNE and its corresponding payoff
configuration: $w(\rho ,t)=F_{\rho }(w(\rho ,t-1))$ for all $t$. Therefore

\begin{proposition}
\label{Prop-recursion}The SPNE payoff configuration of the noncooperative
game $\left( N,V,\rho ,T\right) $ is given by $w\left( \rho ,T\right)
=F_{\rho }^{T}(0),$ the $T$-th iterate of the function $F_{\rho }$ evaluated
at the payoff configuration $0$, where $F_{\rho },$ the backward induction
function, is given by Definition \ref{def-F-rho} above.
\end{proposition}

\section{The Consistent Field\label{field}}

We now turn to the study of dynamics associated with the concept of the
consistent NTU-value, which was introduced by Maschler and Owen (1989, 1992)
and analyzed by Hart and Mas-Colell (1996b). In this section we develop the
concept of the \emph{consistent field}. This is a vector field defined over
the Pareto frontier of the game that, informally speaking, gives the
direction that reduces the ``inconsistency'' in the payoff configuration.

We start by recalling the definition of the \emph{consistent value} of an
NTU-game $\left( N,V\right) .$ Similarly to the Shapley value, let $\pi \,$
be a permutation of the $n$ players, and define recursively the vector of
marginal contributions $d_{\pi }$ (with $\pi \left( i\right) $-th coordinate 
$d_{\pi }^{\pi (i)}$) by 
\begin{eqnarray*}
d_{\pi }^{\pi (1)} &=&\max \left\{ a^{\pi (1)}:a\in V\left( \{\pi \left(
1\right) \}\right) \right\} ,\text{and for }i>1\text{ by} \\
d_{\pi }^{\pi (i)} &=&\max \left\{ a^{\pi (i)}:a\in V\left( \{\pi \left(
1\right) ,...,\pi \left( i\right) \}\right) \text{ and }a^{\pi (j)}=d_{\pi
}^{\pi (j)}\text{ for all }j<i\right\} .
\end{eqnarray*}
So, for a given order $\pi $, each player $\pi \left( i\right) $ gets $%
d_{\pi }^{\pi (i)},$ which is the highest possible given that all the
previous players $\pi (j)$ (for $j<i$) got $d_{\pi }^{\pi (j)}$. Consider
now the vector of expected marginal contributions $\Psi \left( N,V\right)
:=(1/n!)\sum_{\pi }d_{\pi }.$ Since $\Psi \left( N,V\right) $ is an average
of vectors on the boundary $\partial V\left( N\right) $ of the convex set $%
V(N)$, it will not in general be efficient. However, for a \emph{hyperplane
game} (where, for each $S\subset N,$ the set $V(S)$ is a half-space, and so
its boundary $\partial V\left( S\right) $ is a hyperplane), the expected
marginal contribution vector $\Psi \left( N,V\right) $ is efficient. It is
the \emph{consistent value} of the hyperplane game\footnote{%
In the special that $(N,V)$ is a TU-game, this is the Shapley value.} $(N,V)$%
. Further, for each coalition $S\subset N$ we have $\Psi \left( S,V\right)
\in \partial V\left( S\right) ;$ the efficient payoff configuration $\left(
\Psi \left( S,V\right) \right) _{S\subset N}$ is called the \emph{consistent
value payoff configuration} of the hyperplane game $(N,V).$

For a general NTU-game, the construction of the consistent value is based on
the hyperplane case. For each efficient payoff configuration $a=\left(
a_{S}\right) _{S\subset N}\in \partial V:=\prod_{S\subset N}\partial V(S)$
with supporting normal vectors $\lambda _{S}\equiv \lambda _{S}\left(
a_{S}\right) \in \Re _{++}^{S}$ to the boundary $\partial V\left( S\right) $
at $a_{S},$ associate the \emph{supporting hyperplane game} $\left(
N,V_{a}^{\prime }\right) $ defined by $V_{a}^{\prime }\left( S\right)
:=\left\{ c\in \Re ^{S}:\lambda _{S}\cdot c\leq \lambda _{S}\cdot
a_{S}\right\} $ for all $S\subset N.$ Let $b\equiv b\left( a\right) :=\left(
\Psi \left( S,V_{a}^{\prime }\right) \right) _{S\subset N}$ be the
consistent p.c. of the supporting hyperplane game $\left( N,V_{a}^{\prime
}\right) $. If $b=a$ then $a$ is a \emph{consistent value payoff
configuration} of the NTU-game $\left( N,V\right) .$

Following Hart and Mas-Colell (1996b, Proposition 4), a consistent value
payoff configuration $a=\left( a_{S}\right) _{S\subset N}$ for the game $%
(N,V)$ can be characterized by

\begin{quote}
(i) $a_{S}\in \partial V(S)$ for all $S\subset N;$

(ii) $\lambda _{S}\cdot a_{S}=\max \left\{ \lambda _{S}\cdot c:c\in
V(S)\right\} $ for all $S\subset N;$ and

(iii) $\sum_{j\in S\backslash i}\lambda _{S}^{i}{}(a_{S}^{i}-a_{S\backslash
j}^{i})=\sum_{j\in S\backslash i}\lambda _{S}^{j}{}(a_{S}^{j}-a_{S\backslash
i}^{j})$ for all $S\subset N$ and each $i\in S.$
\end{quote}

Conditions (i) and (ii) state that the payoff vector $a_{S}$ is on the
Pareto frontier of the coalition $S$ and that $\lambda _{S}$ is an outward
normal vector to the boundary of $V(S)$ there$.$ The last condition (iii)
may be viewed as a ``preservation of average differences'' requirement: the
average contribution to $i$ from the remaining players equals the average
contribution of $i$ to the remaining players. We refer to Hart and
Mas-Colell (1996b) for further details. In particular, under our
assumptions, consistent value p.c.'s exist and are always non-negative.

Maschler, Owen and Peleg (1988) and Maschler and Owen (1989) have proposed
dynamic processes adapted to the consistent value for pure bargaining games%
\footnote{%
In the pure bargaining case, the consistent value coincides with the Nash
bargaining solution.} and for hyperplane games. We proceed to do this here
for the general NTU case.

The dynamic approach to the consistent value that we analyze is an explicit
procedure that, starting from an arbitrary efficient p.c., adjusts the
payoffs in the direction indicated by the above characterization of the
consistent value. The adjustment process can be described as follows. Given
an efficient p.c. $a=\left( a_{S}\right) _{S\subset N}\in \partial V,$ the
payoffs for each coalition $S$ are adjusted, assuming that the players in $S$
already agree with the payoffs $a_{Z}$ for the smaller coalitions $%
Z\subsetneqq S$. Considering the supporting hyperplane game $\left(
S,V_{a}^{\prime }\right) $ at $a,$ and fixing the payoffs for the
subcoalitions of $S,$ then, in order to bring about consistency for the
coalition $S$, the payoff $a_{S}$ would need to be changed to a payoff $%
b_{S}\left( a\right) \ $in the hyperplane $V_{a}^{\prime }(S)$ satisfying
the preservation of average differences; i.e., 
\begin{eqnarray}
\lambda _{S}(a_{S})\cdot b_{S}(a) &=&\lambda _{S}(a_{S})\cdot a_{S},\text{
and}  \nonumber \\
\sum_{j\in S\backslash i}\lambda _{S}^{i}(a_{S})(b_{S}^{i}(a)-a_{S\backslash
j}^{i}) &=&\sum_{j\in S\backslash i}\lambda
_{S}^{j}(a_{S})(b_{S}^{j}(a)-a_{S\backslash i}^{j}).  \label{implicit1}
\end{eqnarray}
The change in the payoff $a_{S}$ is equal to $C_{S}(a):=b_{S}(a)-a_{S},$ and
the \emph{consistent field }at $a$ is defined to be $C(a)=(C_{S}(a))_{S%
\subset N}$\emph{.} Thus, the $S$-coordinate $C_{S}$ of the consistent field
vector gives the direction in which to move locally along the efficient
frontier $\partial V\left( S\right) $ so that the consistency of the payoffs
for the players in coalition $S\,$ is reduced, given that the payoffs for
the subcoalitions of $S$ are unchanged. The explicit expression produced by (%
\ref{implicit1}) is given in the following definition.

\begin{definition}
\label{def-C-field}The \emph{consistent field} (or \emph{C-field})
associated with the NTU-game $(N,V)$ is the vector field $C(\cdot )$ over
the boundary $\partial V,$ with $C(a)=(C_{S}(a))_{S\subset N}\,$ for any $%
a\in \partial V$ defined by the expression 
\begin{equation}
C_{S}^{i}(a)=\frac{1}{|S|\lambda _{S}^{i}(a)}\sum_{j\in S\backslash i}\left(
\lambda _{S}^{j}(a)(a_{S}^{j}-a_{S\backslash i}^{j})-\lambda
_{S}^{i}(a)(a_{S}^{i}-a_{S\backslash j}^{i})\right)   \label{cfield}
\end{equation}
for all $S\subset N\,$ and $i\in S$, where $\lambda _{S}(a)$ is the unit
normal vector to the boundary $\partial V(S)$ at $a_{S}.$
\end{definition}

It is a simple computation to verify the equivalence of (\ref{implicit1})
and (\ref{cfield}); it is actually useful to write this expression yet
another way, namely 
\begin{equation}
C_{S}^{i}(a)=\frac{1}{|S|\lambda _{S}^{i}(a)}\left( \lambda _{S}(a)\cdot
a_{S}-\sum_{j\in S\backslash i}\left( \lambda _{S}^{j}(a)a_{S\backslash
i}^{j}-\lambda _{S}^{i}(a)a_{S\backslash j}^{i}\right) \right) -a_{S}^{i}.
\label{cfield2}
\end{equation}
Note that $\lambda _{S}(a)\cdot C_{S}(a)=0,$ thus $C$ is indeed a vector
field over the boundary $\partial V$. The singularities of the consistent
field, i.e., the payoff configurations $a$ such that $C(a)=0,$ are precisely
the consistent value p.c.'s of the game $(N,V)$. Finally, note that whenever 
$a\geq 0$ and $a_{S}^{i}=0$ for some $i$ and $S,$ then $C_{S}^{i}(a)\geq 0$
(see (\ref{cfield2}) and use $\lambda _{S}(a)\cdot a_{S}\geq \lambda
_{S}(a)\cdot (a_{S\backslash i},0),$ which holds since $(a_{S\backslash
i},0)\in V(S)$ by the monotonicity assumption (A3)). Thus the consistent
vector field ``points inward'' at boundary points of $\partial
_{+}V:=\partial V\cap \{a\geq 0\}.$

Associated with the consistent field $C$ over the boundary $\partial V$ is
the dynamic system $da/dt=C\left( a\right) .$ For any $a$ in the
non-negative part of the boundary $\partial _{+}V$, there is a unique
function $\Lambda _{t}\left( a\right) :[0,\infty )\rightarrow \partial _{+}V$
that satisfies 
\begin{eqnarray}
\frac{d\Lambda _{t}\left( a\right) }{dt} &=&C\left( \Lambda _{t}\left(
a\right) \right) ,\text{ and}  \label{def-lambda} \\
\Lambda _{0}\left( a\right) &=&a.  \nonumber
\end{eqnarray}
We will refer to $\Lambda _{t}\left( a\right) $ as the \emph{solution of the
consistent field} starting, at $t=0,$ from the non-negative efficient p.c. $%
a.$ Note that the solutions are defined on the interval $[0,\infty )$
because, by the ``pointing inwards'' property of the field, every solution
that starts in the non-negative part of the boundary will remain there. For $%
t=\infty ,$ we define $\Lambda _{\infty }(a)$ as the $\omega $-limit set of
the solution, i.e., the set of all limit points of $\Lambda _{t}(a)$ as $%
t\rightarrow \infty .$

\bigskip 

\noindent {\large Example 1: Pure Bargaining Games}%
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An $n$-person pure bargaining game satisfies $V\left( S\right) \cap \Re
_{+}^{S}=\{0\}$ for all $S\neq N$. For this particular case, the consistent
field has the same dynamics as the process introduced by Maschler, Owen and
Peleg (1988). For $S\neq N$ we have $\left( \Lambda _{t}\left( a\right)
\right) _{S}\equiv 0;$ so, the only dynamics that matters is for $S=N.$ For
any $a=a_{N}\in \partial _{+}V\left( N\right) $ with supporting normal $%
\lambda (a)=\lambda _{N}(a),$ (\ref{cfield}) becomes 
\[
C^{i}(a)=\frac{1}{n\lambda ^{i}(a)}\sum_{j\in S\backslash i}\left( \lambda
^{j}(a)a^{j}-\lambda ^{i}(a)a^{i}\right) 
\]
(we have dropped the subscript $N$ throughout for ease of reading). The
dynamics of the C-field is simple:$\,\,$ the unique singularity of the
C-field, which is the unique consistent value and coincides with the Nash
bargaining solution, is a global attractor for the field. (This can be
verified by showing that the function $L(a):=\prod_{i\in N}a^{i}$ is a
Lyapunov function for the C-field; see Maschler, Owen and Peleg (1988)).

\bigskip 

\noindent {\large Example 2: Hyperplane Games}%
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Let the hyperplane game $\left( N,V\right) \,$ be given for each coalition $S
$ by the unit normal vector $\lambda _{S}$ and by the number $\nu _{S}$ such
that $V\left( S\right) =\left\{ a_{S}\in \Re ^{S}:\lambda _{S}\cdot
a_{S}\leq \nu _{S}\right\} .$ The expression (\ref{cfield2}) for the C-field
becomes 
\[
C_{S}^{i}(a)=\frac{1}{|S|\lambda _{S}^{i}}\left( \nu _{S}-\sum_{j\in
S\backslash i}(\lambda _{S}^{j}a_{S\backslash i}^{j}-\lambda
_{S}^{i}a_{S\backslash j}^{i})\right) -a_{S}^{i}.
\]

First observe that the C-field is a linear function of $a$ (since $\lambda
_{S}$ does not change with $a_{S}$) and that the unique singularity of the
C-field$\,$ is the unique consistent value. To characterize the dynamics of
the C-field we simply need to determine the sign of the real part of the
eigenvalues of $DC(a),$ the derivative of the field at $a$. The expression
above for $C_{S}^{i}(a)$ immediately implies that the matrix $DC(a)$ is
triangular and all its diagonal entries are $-1$ (indeed: $\partial
C_{S}^{i}(a)/\partial a_{Z}^{j}=0$ if $Z$ is not a subset of $S;$ moreover, $%
\partial C_{S}^{i}(a)/\partial a_{S}^{i}=-1$ and $\partial
C_{S}^{i}(a)/\partial a_{S}^{j}=0$ for $j\neq i).$ Therefore all the
eigenvalues of $DC(a)$ are equal to $-1,$ implying that the solution of the
C-field converges exponentially to the consistent value (e.g., see Palis and
de Melo (1982)). Again, we conclude that the dynamics is very simple: there
is only one consistent value which is a global attractor. A result similar
to this has been obtained by Maschler and Owen (1989) for the ``correction
function'' they propose.

\section{From Subgame Perfect Equilibria to Consistent Field Solutions}

We now address the problem of characterizing the SPNE solutions of the
noncooperative game $\left( N,V,\rho ,T\right) $ as the probability of
breakdown decreases to zero and the number of periods of negotiation
increases$.$ That is, we want to find the limit of the SPNE payoff
configurations $w\left( \rho ,T\right) $ as $\rho \rightarrow 1$ and $%
T\rightarrow \infty .$ Recall the result of Proposition \ref{Prop-recursion}
that $w(\rho ,T)=F_{\rho }^{T}(0).$

This section will show that the solutions $\Lambda _{t}$ of the consistent
field are intimately related to the subgame perfect Nash equilibria of the
finitely repeated noncooperative bargaining game. We start by highlighting a
basic relationship between the consistent field $C$ and the backward
induction function $F_{\rho }.$ All the proofs are in the Appendix.

\begin{proposition}
\label{Prop-relation}The derivative of $F_{\rho }(a)$ with respect to $\rho $
at any point $a\in \partial V$ satisfies 
\[
\frac{dF_{\rho }(a)}{d\rho }_{|\rho =1}=-C(a).
\]
\end{proposition}

This result will be most useful because we are interested in the limit as $%
\rho $ converges to 1 of the iterates of the function $F_{\rho }$. Observing
that for $\rho =1$ we have $F_{1}(a)=a$ for all $a\in \partial V$, the
result roughly states that, for efficient payoffs and $\rho $ close enough
to 1, $F_{\rho }(a)$ can be approximated by $a+\left( 1-\rho \right) C\left(
a\right) .$ This suggests a natural relationship between the limit of $%
F_{\rho }^{T}$ as $T\rightarrow \infty $ and $\rho \rightarrow 1$ and the
solution of the dynamic system associated with the consistent field.

We first consider the case where there is no breakdown, i.e. when $\rho =1.$

\begin{proposition}
\label{delta1}For any payoff configuration $a\in V,$ the limit as $%
T\rightarrow \infty $ of $F_{1}^{T}\left( a\right) $ exists and is an
efficient payoff configuration: $\lim_{T\rightarrow \infty }F_{1}^{T}\left(
a\right) \in \partial V.$
\end{proposition}

In particular, this proposition implies that the limit as $T\rightarrow
\infty $ of the SPNE payoffs, i.e. $\lim_{T\rightarrow \infty
}F_{1}^{T}\left( 0\right) $, exists and is efficient (this was shown by
Sj\"{o}str\"{o}m (1991) in the two-player case). We call this point the
Raiffa point (see Luce and Raiffa (1957, \S 6.7)).

\begin{definition}
\label{def-raiffa}The \emph{Raiffa payoff configuration} of the game $(N,V),$
denoted $r\equiv r(N,V),$ is given by $r:=\lim_{T\rightarrow \infty
}F_{1}^{T}\left( 0\right) .$
\end{definition}

Depending on the rates at which $\rho \rightarrow 1$ and $T\rightarrow
\infty ,$ the SPNE p.c. may converge to different limits. Specifically,
those turn out to depend on the limit of $\rho ^{T},$ the probability that
all players remain up to the last stage of the game$.$ If $\rho \rightarrow
1,$ $T\rightarrow \infty $ and $\rho ^{T}$ $\rightarrow 1,$ meaning that $T$
converges to infinity much slower than $\rho $ converges to $1$, then we
will see that $w(\rho ,T)\,$ converges to the Raiffa p.c. $r$. As the
relative rate at which $T$ converges to infinity increases, so that the
probability $\rho ^{T}$ of all players remaining at the last stage converges
to some $\mu <1,$ then we will see that $w(\rho ,T)$ converges to an
appropriate point on the solution path of the consistent field starting at $%
r $. Formally,

\begin{theorem}
\label{mainlim}Let $r$ be the Raiffa payoff configuration (given by
Definition \ref{def-raiffa}) and let $\Lambda _{t}$ be the solution of the
consistent field (given by (\ref{def-lambda})). Then:%
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\begin{itemize}
\item[(i)]  If $\rho \rightarrow 1,$ $T\rightarrow \infty $ and {$\rho $}${%
^{T}\rightarrow \mu \in (0,1],}$ then $w(\rho ,T)$ converges to $\Lambda
_{-\ln \mu }(r).$

\item[(ii)]  If $\Lambda _{t}(r)$ converges as $t\rightarrow \infty $ to a
local attractor (sink) $a$ of the consistent field, then $w(\rho ,T)$
converges to $a$ as $\rho \rightarrow 1,$ $T\rightarrow \infty $ and {$\rho $%
}${^{T}\rightarrow 0}.$

\item[(iii)]  Any payoff configuration in $\Lambda _{\infty }(r),$ the $%
\omega $-limit set of the solution of the consistent field through $r,$ can
be obtained as the limit of $w(\rho _{k},T_{k})$ for appropriate sequences $%
\rho _{k}\rightarrow 0$ and $T_{k}\rightarrow \infty $ with $\rho
_{k}^{T_{k}}\rightarrow 0$ (as $k\rightarrow \infty ).$
\end{itemize}
\end{theorem}

Note that when $\mu =1$ we get $\Lambda _{-\ln \mu }(r)=\Lambda _{0}(r)=r,$
therefore $w(\rho ,T)$ converges to the Raiffa point $r$ as $\rho
\rightarrow 1,$ $T\rightarrow \infty $ and {$\rho $}${^{T}\rightarrow 1.}$
As for the second part of the theorem, it includes the two cases of pure
bargaining and of hyperplane games, where, as we saw in Examples 1 and 2 of
Section 3, the unique consistent value is a global attractor.

\section{Local Analysis of the Consistent Field}

The results of Theorem \ref{mainlim} indicate that the dynamic properties of
the consistent field are of importance in describing the solutions of the
noncooperative bargaining games. We now proceed to analyze in more depth the
dynamics of the consistent field.

The dynamics of the consistent field for a general NTU\ game can be
significantly more complicated than the dynamics for pure bargaining games
and for hyperplane games. For these two particular cases, as shown in
Section \ref{field}, there is a unique consistent value which is a global
attractor for the C-field.

We propose to study the dynamic properties of the consistent field in a
neighborhood of a consistent value. As it is well known, the linear system $%
DC(a)\left( x\mathbf{-}a\right) ,$ where $DC$ is the derivative of $C,$ can
be used as an approximation of the consistent field $C(x)$ around the
consistent value $a$. By a standard result in dynamic system (the
Hartman-Grobman Theorem, e.g., Palis and de Melo (1982)), the local dynamics
of the C-field and the dynamics of the linear system are equivalent if the
consistent value is a hyperbolic equilibrium (i.e., if the eigenvalues of $%
DC(a)$ have non-zero real part). Moreover, these dynamics are determined by
the eigenvalues of $DC(a).$

As a first step we develop an expression for the derivative of the C-field
at a consistent value$\mathbf{.}$

\begin{theorem}
\label{local}The derivative of the C-field $DC(a)$ at a consistent value
payoff configuration $a=\left( a_{S}\right) _{S\subset N}$ is a block
triangular matrix. For all $S\subset N,$ the diagonal block matrix
corresponding to $S$ is $DC_{S}(a):T_{a_{S}}\partial V(S)\rightarrow
T_{a_{S}}\partial V(S),$ where $T_{a_{S}}\partial V(S)\,\,$ is the tangent
plane to the boundary $\partial V(S)$ at $a_{S}.$ Moreover, $DC_{S}(a)v_{S}$
for any $v_{S}\in T_{a_{S}}\partial V(S)$ can be expressed as 
\[
DC_{S}(a)v_{S}=\stackunder{\text{game part}}{\underbrace{G_{S}(a)}}\ 
\stackunder{\text{geometry part}}{\underbrace{D\lambda _{S}(a_{S})}}v_{S}\
-\ v_{S}
\]
where $\lambda _{S}(a_{S})$ is the unit length outward normal to $\partial
V(S)$ at $a_{S}$ and $G_{S}(a)=(G_{S}^{ij})_{i,j\in S}$ is the matrix given
by 
\[
G_{S}^{ii}=-\frac{1}{|S|\lambda _{S}^{i}(a_{S})}\sum_{j\in S\backslash
i}(a_{S}^{i}-a_{S\backslash j}^{i})\text{ and}
\]
\[
G_{S}^{ij}=\frac{1}{|S|\lambda _{S}^{i}(a_{S})}\left(
a_{S}^{j}-a_{S\backslash i}^{j}\right) ,\text{ for }i\neq j.
\]
\end{theorem}

The derivative $DC_{S}(a)$ is thus naturally decomposed into a \emph{game
part} $G_{S}(a),$ which depends only on $a$ and $\lambda _{S}(a_{S}),$ and a 
\emph{geometry part} $D\lambda _{S}(a_{S}),$ which is the Gauss curvature
map of the boundary $\partial V(S)$ at $a_{S}\mathbf{.}$ The theorem is
proved in the Appendix.

We now proceed to exhibit a family of NTU games with various dynamics for
the consistent field around a consistent value: repulsor (source), saddle
point, and cycle. (Recall that in the cases of pure bargaining and
hyperplane games there is a unique global attractor.)

\bigskip 

\noindent {\large Example 3: A family of consistent fields with varied local
dynamics}%
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There are $3$ players: $N=\{1,2,3\}$. We begin by fixing the games in all
but the grand coalition: 
\begin{eqnarray*}
V\left( 1\right)  &=&V\left( 2\right) =V\left( 3\right) =\left\{ c\in \Re
^{1}:c\leq 0\right\}  \\
V\left( 12\right)  &=&\left\{ \left( c_{1},c_{2}\right) \in \Re ^{2}:\frac{%
c_{1}}{14}+\frac{c_{2}}{9}\leq 2\right\}  \\
V\left( 23\right)  &=&\left\{ \left( c_{2},c_{3}\right) \in \Re ^{2}:\frac{%
c_{2}}{14}+\frac{c_{3}}{9}\leq 2\right\}  \\
V\left( 13\right)  &=&\left\{ \left( c_{1},c_{3}\right) \in \Re ^{2}:\frac{%
c_{3}}{14}+\frac{c_{1}}{9}\leq 2\right\} .
\end{eqnarray*}

Every subgame is therefore a hyperplane game, and, for all $S\subsetneqq
\left\{ 1,2,3\right\} ,$ the consistent solution $a_{S\subsetneqq }$ of the
subgame $(S,V)$ --- which is unique --- coincides with the Raiffa point (and
the Nash bargaining solution). Specifically: 
\begin{eqnarray}
a_{1} &=&a_{2}=a_{3}=0  \nonumber \\
a_{12} &=&(14,9,\cdot )  \nonumber \\
a_{23} &=&(\cdot ,14,9)  \label{ex3} \\
a_{13} &=&(9,\cdot ,14).  \nonumber
\end{eqnarray}

We now come to the construction of $V_{h}(123),$ which will depend on a
parameter $h=(h_{1},h_{2}),$ where $0\leq h_{1},h_{2}\leq 8.$ Consider first
the family of ellipsoids 
\[
E_{h}:=\left\{ x\in \Re ^{3}:z=M^{T}x\text{ is such that }(z_{1})^{2}+\sqrt{3%
}h_{1}(z_{2})^{2}+\sqrt{3}h_{2}(z_{3})^{2}=3\right\} , 
\]
where $M$ is the orthonormal matrix 
\[
M:=\left( 
\begin{array}{ccc}
\frac{1}{\sqrt{3}} & 0 & \frac{2}{\sqrt{6}} \\ 
\frac{1}{\sqrt{3}} & -\frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{6}} \\ 
\frac{1}{\sqrt{3}} & \frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{6}}
\end{array}
\right) . 
\]
This family is such that $a_{123}:=\left( 10,10,10\right) $ belongs to $%
E_{h} $ and the unit normal to $E_{h}$ at $(10,10,10)$ is $(1/\sqrt{3},1/%
\sqrt{3},1/\sqrt{3}),$ for all $h.$ Moreover, if $V_{h}(123)$ equals $%
E_{h}^{-}:=E_{h}+\Re _{-}^{3},$ the comprehensive hull of $E_{h},$ then it
is immediate to check that the payoff configuration $a$ (given by (\ref{ex3}%
) and $a_{123}=(10,10,10)$) is a consistent value.

However, it is not possible to put $V_{h}(123)=E_{h}^{-},$ because the
monotonicity of the game would be violated. But this can be fixed. Note
first that if we were to take $V\left( 123\right) =W:=\left\{ \left(
c_{1},c_{2},c_{3}\right) \in \Re ^{3}:c_{1}+c_{2}+c_{3}\leq 30\right\} $ ---
which has the same unit normal at $(10,10,10)$ --- then the game would be
monotone. This implies that we can let $V_{h}(123)$ be a set that is close
to $W,$ it coincides with $E_{h}^{-}$ in an appropriately small $\varepsilon 
$-neighborhood of $(10,10,10)$, and is such that all our assumptions on the
game are satisfied.

Next, observe that if $V(123)=W$ then the grand coalition's component of the
Raiffa point is $(10,10,10).$ Suppose we were to alter this set by
modifications that affect only neighborhoods of the intersections of $W$
with the three axes. Then $(10,10,10)$ would remain a consistent value, but
the Raiffa point would move. Moreover, by suitable such small modifications
we can place the Raiffa point anywhere we want in the $\varepsilon $%
-neighborhood of $(10,10,10)$ (for $\varepsilon $ sufficiently small). By
continuity, all this remains true for $V_{h}(123).$ Summarizing: in
constructing $V_{h}(123)$ we can also make sure that the Raiffa point for
the grand coalition is placed at a specific point --- to be determined later
--- in the $\varepsilon $-neighborhood of $(10,10,10).$

We are now ready to exhibit values of the parameter $h$ for which the local
behavior of the consistent field at $a$ is not attracting. A straightforward
computation yields that the two non-zero eigenvalues of the matrix $%
DC_{123}(a)$ are 
\[
\frac{3\sqrt{3}}{40}(h_{1}+h_{2})-1\pm \frac{1}{40}\sqrt{%
27h_{1}^{2}-154h_{1}h_{2}+27h_{2}^{2}}. 
\]

Therefore:

(i) For $h_{1}=h_{2}=5,$ the two eigenvalues have positive real part. Hence
the consistent value $a$ is a repulsor (source).

(ii) For $h_{1}=1$ and $h_{2}=6,$ the two eigenvalues are real, one is
positive and the other is negative. Hence the consistent value $a$ is a
saddle point.

(iii) For $h_{1}=3$ and $h_{2}=40\sqrt{3}/9-3\approx 4.698,$ the eigenvalues
have zero real part and non-zero imaginary part. Moreover, as $h_{1}$ moves
from below $3$ to above $3$ (and $h_{2}$ is unchanged)$,$ the real part
moves from negative to positive. Therefore, by the Hopf Bifurcation Theorem,%
\footnote{\label{hopf}Hopf (1942) Bifurcation Theorem: Let $x^{\prime
}=F\left( x,\alpha \right) \,$ be a one-parameter family of planar systems
with an equilibrium $x\left( \alpha \right) $ and eigenvalues $\lambda
\left( \alpha \right) =\mu \left( \alpha \right) +i\eta \left( \alpha
\right) $. Suppose that, for some value $\alpha ^{0}$ of the parameter, the
equilibrium $x\left( \alpha ^{0}\right) $ is non-hyperbolic with purely
imaginary eigenvalues (i.e., $\mu \left( \alpha ^{0}\right) =0).$ Moreover,
as $\alpha \,$ crosses $\alpha ^{0}$ in some direction, $\mu \left( \alpha
\right) $ changes from negative to positive and $x\left( \alpha \right)
\,\,\,$changes from sink to source. Then, for all $\alpha $ on the side of $%
\alpha ^{0},$ and close enough to it, there is a periodic orbit surrounding
the equilibrium $x\left( \alpha \right) \,$ with radius of magnitude $\mid
\alpha -\alpha ^{0}\mid ^{1/2}.$ Also, if $x\left( \alpha ^{0}\right) $ is
asymptotically stable, then the closed orbit is stable and surrounds the
unstable equilibrium. Otherwise, the closed orbit is unstable and occurs for
parameter values that make the equilibrium a sink.} there must be values of $%
h_{1}$ close to $3$ such that there is a cycle in the $\varepsilon $%
-neighborhood of $a.$ In particular, the Raiffa point for the grand
coalition may be located on the cycle (remember that we can place it
anywhere in the $\varepsilon $-neighborhood, without affecting the local
behavior of the consistent field in this neighborhood). In this case, by
Theorem \ref{mainlim}, we get an example of a sequence of finite horizon
bargaining problems with $\rho \rightarrow 1,T\rightarrow \infty $ and $\rho
^{T}\rightarrow 0$ and such that the subgame perfect Nash equilibrium
payoffs converge to a point (on the cycle) that is \emph{not} a consistent
value.

\medskip \appendix 

\section{Appendix}

This appendix contains the proofs not given in text.

\medskip

\noindent \textbf{Proof of Proposition \ref{Prop-relation}. }Define the
function $b_{S,i}\left( \rho ,a\right) $ by 
\begin{eqnarray*}
b_{S,i}^{j}\left( \rho ,a\right) &=&\rho a_{S}^{j}+(1-\rho )a_{S\backslash
i}^{j},\text{for all }j\in S\backslash i \\
b_{S,i}^{i}\left( \rho ,a\right) &=&h_{S}^{i}\left( \rho a_{S}^{-i}+(1-\rho
)a_{S\backslash i}\right)
\end{eqnarray*}
where\thinspace $a_{S}^{-i}$ denotes the vector $a_{S}$ without the $i$-th
coordinate, and $h_{S}^{i}(c_{S}^{-i})$ is defined as the $i$-th coordinate
of the point on the boundary of $V(S)$ with remaining $S\backslash i$
coordinates equal to $c_{S}^{-i}$. To simplify notation we will omit the
arguments $\left( \rho ,a\right) $ of the function $b_{S,i}.$ By definition, 
$\left( F_{\rho }(a)\right) _{S}=(1/|S|)\sum_{i\in S}b_{S,i}.$

We first obtain the derivative of $b_{S,i}$ with respect to $\rho $: 
\begin{eqnarray*}
\frac{\partial b_{S,i}^{j}}{\partial \rho } &=&a_{S}^{j}-a_{S\backslash
i}^{j}\text{, for all }j\in S\backslash i,\text{ and} \\
\frac{\partial b_{S,i}^{i}}{\partial \rho } &=&\nabla h_{S}^{i}\left( \rho
a_{S}^{-i}+(1-\rho )a_{S\backslash i}\right) \cdot \left(
a_{S}^{-i}-a_{S\backslash i}\right) .
\end{eqnarray*}

Using the fact that $(\partial h_{S}^{i}/\partial a_{j})\left(
a_{S}^{-i}\right) =-\lambda _{S}^{j}\left( a_{S}\right) /\lambda
_{S}^{i}\left( a_{S}\right) $ for a point $a_{S}\in \partial V(S),$ we
obtain that the derivative $\partial b_{S,i}/\partial \rho $ evaluated at $%
\rho =1$ is 
\begin{eqnarray*}
\frac{\partial b_{S,i}^{j}}{\partial \rho }_{|\rho =1}
&=&a_{S}^{j}-a_{S\backslash i}^{j}\text{, for all }j\in S\backslash i,\text{
and} \\
\frac{\partial b_{S,i}^{i}}{\partial \rho }_{|\rho =1} &=&-\frac{1}{\lambda
_{S}^{i}\left( a_{S}\right) }\sum_{j\in S\backslash i}\lambda _{S}^{j}\left(
a_{S}\right) \left( a_{S}^{j}-a_{S\backslash i}^{j}\right) .
\end{eqnarray*}
Substituting into the expression of $\left( F_{\rho }(a)\right) _{S}$
yields: 
\begin{eqnarray*}
\frac{\partial \left( F_{\rho }(a)\right) _{S}^{i}}{\partial \rho }_{|\rho
=1} &=&\frac{1}{|S|}\sum_{j\in S\backslash i}\frac{\partial b_{S,j}^{i}}{%
\partial \rho }_{|\rho =1}+\frac{1}{|S|}\frac{\partial b_{S,i}^{i}}{\partial
\rho }_{|\rho =1} \\
&=&\frac{1}{|S|}\sum_{j\in S\backslash i}\left( a_{S}^{i}-a_{S\backslash
j}^{i}\right) -\frac{1}{|S|\lambda _{S}^{i}\left( a_{S}\right) }\sum_{j\in
S\backslash i}\lambda _{S}^{j}\left( a_{S}\right) \left(
a_{S}^{j}-a_{S\backslash i}^{j}\right) \\
&=&\frac{1}{|S|\lambda _{S}^{i}\left( a_{S}\right) }\sum_{j\in S\backslash
i}\left( \lambda _{S}^{i}\left( a_{S}\right) \left( a_{S}^{i}-a_{S\backslash
j}^{i}\right) -\lambda _{S}^{j}\left( a_{S}\right) \left(
a_{S}^{j}-a_{S\backslash i}^{j}\right) \right) \\
&=&-C_{S}^{i}\left( a\right) ,
\end{eqnarray*}
which proves the result.%
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\medskip

\noindent \textbf{Proof of Proposition \ref{delta1}. }Let $a\in V$ and put $%
b:=F_{1}(a).$ For each $S\subset N$ and each $i\in S,$ let $\alpha
_{S}^{i}\geq 0$ be such that $(a_{S}^{i}+\alpha _{S}^{i},a_{S}^{-i})\in
\partial V(S).$ Then $b_{S}^{i}=a_{S}^{i}+(1/s)\alpha _{S}^{i},$ where $%
s:=|S|.$ Similarly, let $\beta _{S}^{i}\geq 0$ be such that $%
(b_{S}^{i}+\beta _{S}^{i},b_{S}^{-i})\in \partial V(S).$ Because $%
b_{S}^{-i}\geq a_{S}^{-i},$ comprehensiveness implies that $b_{S}^{i}+\beta
_{S}^{i}\leq a_{S}^{i}+\alpha _{S}^{i},$ or $\beta _{S}^{i}\leq $ $\alpha
_{S}^{i}-(1/s)\alpha _{S}^{i}=(1-1/s)\alpha _{S}^{i};$ hence $%
\max\nolimits_{i\in S}\beta _{S}^{i}\leq (1-1/s)\max\nolimits_{i\in S}\alpha
_{S}^{i}.$ Therefore each iteration of $F_{1}$ decreases $%
\max\nolimits_{i\in S}\alpha _{S}^{i}$ by the fixed factor $1-1/s<1,$ so it
converges to $0.$ Together with $F_{1}(a)\geq a,$ this implies that the
sequence $F_{1}^{T}(a)$ converges to a point on $\partial V.$%
%TCIMACRO{\TeXButton{End Proof}{\endproof}}
%BeginExpansion
\endproof%
%EndExpansion

\medskip

We come now to the proof of Theorem \ref{mainlim}\textbf{. }This will be
done by a sequence of lemmas. The idea of the proof is as follows: Assume $%
\rho \rightarrow 1$ and $T\rightarrow \infty $ with $\rho ^{T}\rightarrow
\mu ,$ or $T(1-\rho )\rightarrow -\ln \mu .$ Split $T$ into two parts, $T_{1}
$ and $T_{2}.$ If $T_{1}$ converges to $\infty $ relatively slowly --- in
particular, if $\rho ^{T_{1}}\rightarrow 1$ or $T_{1}(1-\rho )\rightarrow 0$
--- then we prove in Lemma \ref{a-l-eta} that $a:=F_{\rho
}^{T_{1}}(0)\rightarrow \lim_{t\rightarrow \infty }F_{1}^{t}(0),$ which as
seen in Proposition \ref{delta1} exists and is the Raiffa point $r.$
Therefore $F_{\rho }^{T}(0)=F_{\rho }^{T_{2}}\left( F_{\rho
}^{T_{1}}(0)\right) =F_{\rho }^{T_{2}}(a)$ will be close to $F_{\rho
}^{T_{2}}(r).$ Also, once $a$ is sufficiently close to the efficient
boundary $\partial V$ (recall that $r$ lies on $\partial V$), then all its $%
F_{\rho }$ iterates $F_{\rho }^{t}(a)$ will also be close to $\partial V$
--- this is Lemma \ref{a-l-eta2}$.$ Next, Lemma \ref{a-l-f-lambda} shows
that, for $c$ on the boundary, $F_{\rho }(c)$ can be approximated by the
solution $\Lambda _{1-\rho }(c)$ of the C-field $C(\cdot ).$ Iterating this
implies that $F_{\rho }^{T_{2}}(r)$ is close to $\Lambda _{T_{2}(1-\rho
)}(r),$ which in turn converges to $\Lambda _{-\ln \mu }(r)$ since $%
T_{2}(1-\rho )\rightarrow -\ln \mu .$ This completes the proof, once all
approximations are made precise.

All constants appearing below ($\eta ,$ $K,$ and so on) depend only on the
game $(N,V)$. We use the Euclidean norm in each $\Re ^{S},$ and for payoff
configurations we take $\left\| a\right\| :=\max_{S\subset N}\left\|
a_{S}\right\| .$

The first lemma deals with the case when $T$ converges to infinity slowly
relative to the convergence of $\rho $ to $1;$ in this case, starting from $0
$ one gets the Raiffa point $r.$ More generally,

\begin{lemma}
\label{a-l-eta}There exists a constant $\eta >0$ such that, if $\rho
\rightarrow 1,$ $T\rightarrow \infty $ and $T\leq -\eta \ln (1-\rho ),$ then 
$F_{\rho }^{T}(a)\rightarrow \lim_{t\rightarrow \infty }F_{1}^{t}(a)$ for
all $a$ $\in V.$
\end{lemma}

%TCIMACRO{\TeXButton{Proof}{\proof}}
%BeginExpansion
\proof%
%EndExpansion
The function $F(\rho ,a)=F_{\rho }(a)$ is differentiable with continuous
differential over the compact set $\left[ 0,1\right] \times V_{+}.$ Hence 
\[
\left\| F_{\rho }(a)-F_{1}(b)\right\| \leq K\left( (1-\rho )+\left\|
a-b\right\| \right) , 
\]
where $K>\stackunder{(\rho ,a)\in \left[ 0,1\right] \times V_{+}}{\max }%
\left\| DF(\rho ,a)\right\| +1.$ This implies that 
\[
\left\| F_{\rho }(a)-F_{1}(a)\right\| \leq K(1-\rho ), 
\]
and also that 
\[
\left\| F_{\rho }^{2}(a)-F_{1}^{2}(a)\right\| =\left\| F_{\rho }(F_{\rho
}(a))-F_{1}(F_{1}(a))\right\| \leq K\left( (1-\rho )+K(1-\rho )\right) . 
\]
By induction we then get 
\begin{equation}
\left\| F_{\rho }^{T}(a)-F_{1}^{T}(a)\right\| \leq K(1-\rho )(1+K+\cdots
+K^{T-1})\leq \frac{K(1-\rho )K^{T}}{K-1}.  \label{taylor}
\end{equation}
Take $\eta <1/\ln K;$ then $\ln (1-\rho )+T\ln K\leq (1-\eta \ln K)\ln
(1-\rho )\rightarrow -\infty $ (since $\ln (1-\rho )\rightarrow -\infty ).\,$
Therefore $(1-\rho )K^{T}\rightarrow 0,$ implying that the right hand side
of (\ref{taylor}) converges to $0.$ Hence $F_{\rho }^{T}(a)\rightarrow
\lim_{T\rightarrow \infty }F_{1}^{T}(a)$.%
%TCIMACRO{\TeXButton{End Proof}{\endproof}}
%BeginExpansion
\endproof%
%EndExpansion

\medskip

The next three lemmas show that the $F_{\rho }$ iterates of points close to
the boundary $\partial _{+}V$ stay close to it, and moreover that the
distance becomes of the order of $\varepsilon (1-\rho )$ after no more than $%
-(\eta /2)\ln (1-\rho )$ iterations$.$ Recall the notation $b_{S,i}(a,\rho )$
used in the proof of Proposition \ref{Prop-relation} above.

\begin{lemma}
\label{a-l-k1}There is a constant $K$\ such that $\left\| b_{S,i}(a,\rho
)-a_{S}\right\| \leq K(\mathrm{dist}(a_{S},\partial V(S))+(1-\rho ))$\ for
all $i\in S\subset N,$\ all $a_{S}$\ in $V_{+}(S)$\ and all $\rho \in (0,1].$
\end{lemma}

%TCIMACRO{\TeXButton{Proof}{\proof}}
%BeginExpansion
\proof%
%EndExpansion
Let $c_{S}\in \partial V(S)$ be a closest point to $a_{S}$ on the boundary,
so $d:=\mathrm{dist}(a_{S},\partial V(S))=\left\| a_{S}-c_{S}\right\| $.
Denote $b_{S,i}^{\prime }:=\rho a_{S}+(1-\rho )(a_{S\backslash i},0).$ We
have $\left\| a_{S}-b_{S,i}^{\prime }\right\| =(1-\rho )\left\|
(a_{S\backslash i},0)-a_{S}\right\| \leq M(1-\rho ),$ for an appropriate
bound $M$ (both $(a_{S\backslash i},0)$ and $a_{S}$ are in $V_{+}(S)).$ Thus 
$\mathrm{dist}(b_{S,i}^{\prime },\partial V(S))\leq M(1-\rho )+d.$ Let $%
\lambda _{S}$ be the unit supporting normal to $\partial V(S)$ at $c_{S},$
then the distance of $b_{S,i}^{\prime }$ to $\partial V(S)$ along the $i$'th
coordinate is at most $(M(1-\rho )+d)/\lambda _{S}^{i}.$ So $\left\|
b_{S,i}-b_{S,i}^{\prime }\right\| \leq (M(1-\rho )+d)/\lambda _{S}^{i},$ and
altogether $\left\| b_{S,i}-a_{S}\right\| \leq M(1-\rho )+(M(1-\rho
)+d)/\lambda _{S}^{i}.$ The non-levelness assumption implies that the $%
\lambda _{S}^{i}$ are all bounded away from $0,$ and the result follows$.$%
%TCIMACRO{\TeXButton{End Proof}{\endproof}}
%BeginExpansion
\endproof%
%EndExpansion

\begin{lemma}
\label{a-l-k2}There is a constant $K$\ such that $\mathrm{dist}(F_{\rho
}(a),\partial V)\leq K(\mathrm{dist}(a,\partial V)+(1-\rho ))^{2}$\ for all $%
a\in V_{+}$\ and all $\rho \in (0,1].$
\end{lemma}

%TCIMACRO{\TeXButton{Proof}{\proof}}
%BeginExpansion
\proof%
%EndExpansion
The boundary $\partial V(S)$ is $C^{2},$ therefore there is $K_{1}$ such
that, for any $c_{S}$ and $c_{S}^{\prime }$ in $\partial _{+}V(S)$ we have 
\[
0\leq \lambda _{S}\cdot (c_{S}-c_{S}^{\prime })\leq K_{1}\left\|
c_{S}-c_{S}^{\prime }\right\| ^{2}, 
\]
where $\lambda _{S}$ is the unit outward normal vector to $\partial _{+}V(S)$
at $c_{S}.$ Let $c_{S}$ be one of the vectors $b_{S,i}\equiv b_{S,i}(a,\rho
),$ and consider the hyperplane game $V^{\prime }$ with $V^{\prime
}(S):=\{x_{S}\in \Re ^{S}:\lambda _{S}\cdot x_{S}\leq \lambda _{S}\cdot
c_{S}\}\supset V(S).$ Then 
\begin{eqnarray*}
\text{\textrm{dist}}((F_{\rho }(a))_{S},\partial V(S)) &\leq &\mathrm{dist}%
((F_{\rho }(a))_{S},\partial V^{\prime }(S))=\mathrm{dist}(\frac{1}{\left|
S\right| }\sum_{i\in S}b_{S,i},\partial V^{\prime }(S)) \\
&=&\frac{1}{\left| S\right| }\sum_{i\in S}\lambda _{S}\cdot
(c_{S}-b_{S,i})\leq \frac{1}{\left| S\right| }\sum_{i\in S}K_{1}\left\|
c_{S}-b_{S,i}\right\| ^{2} \\
&\leq &K_{1}\left( 2K_{2}\left( \mathrm{dist}(a_{S},\partial V(S))+(1-\rho
)\right) \right) ^{2},
\end{eqnarray*}
where $K_{2}$ is the constant from Lemma \ref{a-l-k1} (recall that $c_{S}$
is one of the vectors $b_{S,i}$, implying the inequality $\left\|
c_{S}-b_{S,i}\right\| \leq 2K_{2}\left( \mathrm{dist}(a_{S},\partial
V(S))+(1-\rho )\right) $ by Lemma \ref{a-l-k1}).%
%TCIMACRO{\TeXButton{End Proof}{\endproof}}
%BeginExpansion
\endproof%
%EndExpansion

\begin{lemma}
\label{a-l-eta2}There exists a constant $d_{0}>0$ such that for every $%
\varepsilon >0$ there is $\rho _{0}(\varepsilon )\in (0,1)$ with the
property that $\mathrm{dist}(F_{\rho }^{t}(a),\partial V)\leq \varepsilon
(1-\rho )$ for all $\rho \in (\rho _{0}(\varepsilon ),1),$ all $t\geq -(\eta
/2)\ln (1-\rho )$ and all $a\in V_{+}$ with $\mathrm{dist}(a,\partial V)\leq
d_{0},$ where $\eta $ is given by Lemma \ref{a-l-eta}.
\end{lemma}

%TCIMACRO{\TeXButton{Proof}{\proof}}
%BeginExpansion
\proof%
%EndExpansion
Let $d_{0}:=1/(4Ke^{4/\eta })$ and $\rho _{0}:=\max \{1-d_{0}/\varepsilon
,1-d_{0},1-2\varepsilon /(K(\varepsilon +2)^{2}),1-\varepsilon /(2d_{0})\},$%
where $K$ is given by Lemma \ref{a-l-k2}. Take $a$ with $\mathrm{dist}%
(a,\partial V)\leq d_{0}.$ For every $t\geq 0,$ denote $x_{t}:=\mathrm{dist}%
(F_{\rho }^{t}(a),\partial V).$ Lemma \ref{a-l-k2} implies that $x_{t}\leq
f(x_{t-1})$\ for all $t\geq 1,$ where $f(x):=K(x+(1-\rho ))^{2}.$ For every $%
\rho \in (\rho _{0},1)$ put $\xi :=\varepsilon (1-\rho );$ we have: (i) $\xi
<d_{0}$ (since $1-\rho <d_{0}/\varepsilon $); (ii) $f(d_{0})<d_{0}$ (indeed: 
$1-\rho <d_{0},$ implying that $f(d_{0})<4Kd_{0}^{2}\leq d_{0});$ and (iii) $%
f(\xi /2)<\xi /2$ (since $1-\rho <2\varepsilon /(K(\varepsilon +2)^{2}))$.
Therefore the equation $f(x)=x$ has two real roots, $x^{*}<x^{**},$ that
satisfy $0<x^{*}<\xi /2<d_{0}<x^{**}.$ Now $x_{t}-x^{*}\leq
f(x_{t-1})-f(x^{*})=f^{\prime }(y_{t})(x_{t-1}-x^{*})$ for an appropriate
intermediate point $y_{t}.$ The sequence $x_{t}$ starts at $x_{0}=\mathrm{%
dist}(a,\partial V)\leq d_{0},$ thus it never leaves the interval $%
[0,d_{0}], $ and so $y_{t}\in [0,d_{0}]$ too. Hence $0<f^{\prime }(0)\leq
f^{\prime }(y_{t})\leq f^{\prime }(d_{0})=2K(d_{0}+1-\rho )<e^{-4/\eta }.$
From this it follows that $x_{t}-x^{*}\leq e^{-4/\eta }(x_{t-1}-x^{*}),$
implying $x_{t}-x^{*}\leq e^{-4t/\eta }d_{0}.$ If $t\geq -(\eta /2)\ln
(1-\rho )$ then $e^{-4t/\eta }d_{0}\leq (1-\rho )^{2}d_{0}<\varepsilon
(1-\rho )/2=\xi /2$ (since $1-\rho <\varepsilon /(2d_{0})),$ therefore $%
x_{t}-x^{*}<\xi -x^{*}$ (recall that $x^{*}<\xi /2),$ which finally yields $%
x_{t}<\xi =\varepsilon (1-\rho ).$%
%TCIMACRO{\TeXButton{End Proof}{\endproof}}
%BeginExpansion
\endproof%
%EndExpansion

\medskip

From Lemmas \ref{a-l-eta} - \ref{a-l-eta2} we obtain the following

\begin{corollary}
\label{a-prop}For every $\rho \in (0,1)$ there is an integer $n(\rho )>0$
with $n(\rho )\rightarrow \infty $ and $n(\rho )(1-\rho )\rightarrow 0$ as $%
\rho \rightarrow 1,$ such that for every $\varepsilon >0$ there is $\rho
_{0}(\varepsilon )\in (0,1)$ satisfying 
\begin{eqnarray*}
\left\| F_{\rho }^{n(\rho )}(0)-r\right\|  &\leq &\varepsilon \text{ and} \\
\mathrm{dist}(F_{\rho }^{n(\rho )+t}(0),\partial V) &\leq &\varepsilon
(1-\rho )
\end{eqnarray*}
for all $\rho \in (\rho _{0}(\varepsilon ),1)$ and all $t\geq 0.$
\end{corollary}

%TCIMACRO{\TeXButton{Proof}{\proof}}
%BeginExpansion
\proof%
%EndExpansion
Let $\eta $ be the constant given by Lemma \ref{a-l-eta}, and let $d_{0}$
and $\rho _{0}$ be given by Lemma \ref{a-l-eta2}$.$ Define $m(\rho ):=-(\eta
/2)\ln (1-\rho )$ (rounded to the nearest integer) and $n(\rho ):=2m(\rho )$%
, then $n(\rho )\rightarrow \infty $ and $n(\rho )(1-\rho )\rightarrow 0$ as 
$\rho \rightarrow 1$ (since $(1-\rho )\ln (1-\rho )\rightarrow 0).$ Put $%
a(\rho ):=F_{\rho }^{m(\rho )}(0)$ and $b(\rho ):=F_{\rho }^{n(\rho
)}(0)=F_{\rho }^{m(\rho )}(a(\rho )).$

Applying Lemma \ref{a-l-eta} with $T=m(\rho )$ and also with $T=n(\rho )$
gives $\lim_{\rho \rightarrow 1}a(\rho )=\lim_{\rho \rightarrow 1}b(\rho
)=\lim_{t\rightarrow \infty }F_{1}^{t}(0)=r\in \partial V,$ so in particular 
$\mathrm{dist}(a(\rho ),\partial V)\leq d_{0}$ and $\left\| b(\rho
)-r\right\| \leq \varepsilon $ for all $\rho $ close enough to $1.$ Using
Lemma \ref{a-l-eta2} (with $a_{0}=a(\rho )$) then yields $\mathrm{dist}%
(F_{\rho }^{n(\rho )+t}(0),\partial V)=\mathrm{dist}(F_{\rho }^{m(\rho
)+t}(a(\rho )),\partial V)\leq \varepsilon (1-\rho )$ for all $t\geq 0$ and
all $\rho $ close enough to $1.$%
%TCIMACRO{\TeXButton{End Proof}{\endproof}}
%BeginExpansion
\endproof%
%EndExpansion

The next lemma shows that, on the boundary $\partial _{+}V,$ applying the $%
F_{\rho }$ function is almost like moving $1-\rho $ along the solution paths 
$\Lambda _{t}$ of the C-field.

\begin{lemma}
\label{a-l-f-lambda}For every $\varepsilon >0$ there is $\rho
_{0}(\varepsilon )\in (0,1)$ such that $\left\| F_{\rho }(a)-\Lambda
_{1-\rho }(a)\right\| \leq \varepsilon (1-\rho )$ for all $\rho \in (\rho
_{0}(\varepsilon ),1)$ and all $a\in \partial _{+}V.$
\end{lemma}

%TCIMACRO{\TeXButton{Proof}{\proof}}
%BeginExpansion
\proof%
%EndExpansion
The function $F:\left[ 0,1\right] \times \partial _{+}V\rightarrow \partial
_{+}V$ has a compact domain and is of class $C^{1},$ and $\left( dF_{\rho
}(a)/d\rho \right) _{|\rho =1}=-C(a)$ by Proposition \ref{Prop-relation}.
Then we have that, for every $\varepsilon >0$ there is$\ \rho (\varepsilon )$
with$\ 0<\rho (\varepsilon )<1$ such that 
\begin{equation}
\left\| \frac{F_{\rho }(a)-F_{1}(a)}{1-\rho }-C(a)\right\| \leq \varepsilon
\;\text{for all }\rho \in (\rho (\varepsilon ),1)\text{ and all}\ a\in
\partial _{+}V.  \label{exp1}
\end{equation}
By the same reasoning, since $(d\Lambda _{1-\rho }\left( a\right) /d\rho
)\mid _{\rho =1}=-C\left( a\right) ,$ we can take$\ \rho (\varepsilon )$ so
that we also have 
\begin{equation}
\left\| \frac{\Lambda _{1-\rho }(a)-\Lambda _{0}\left( a\right) }{1-\rho }%
-C(a)\right\| \leq \varepsilon \ \text{for all }\rho \in (\rho (\varepsilon
),1)\text{ and all}\ a\in \partial _{+}V.  \label{exp2}
\end{equation}
But $F_{1}(a)=\Lambda _{0}\left( a\right) =a.$ Therefore 
\[
\left\| F_{\rho }(a)-\Lambda _{1-\rho }(a)\right\| \leq 2\varepsilon (1-\rho
)\text{ for all }\rho \in (\rho (\varepsilon ),1)\text{ and all }a\in
\partial _{+}V. 
\]
%TCIMACRO{\TeXButton{End Proof}{\endproof}}
%BeginExpansion
\endproof%
%EndExpansion

\medskip

The next two lemmas are standard.

\begin{lemma}[Gronwall]
\label{a-l-gronwall}There exists a constant $K>0$ such that 
\[
\left\| \Lambda _{t}(b)-\Lambda _{t}(a)\right\| \leq e^{Kt}\left\|
b-a\right\| \ \text{for all }t\geq 0\text{ and all }a,b\in \partial _{+}V.
\]
\end{lemma}

%TCIMACRO{\TeXButton{Proof}{\proof}}
%BeginExpansion
\proof%
%EndExpansion
$C$ is a vector field of class $C^{1}$ defined over the compact set $%
\partial _{+}V$ (for a proof of this well-known result see Palis and de Melo
(1982)).%
%TCIMACRO{\TeXButton{End Proof}{\endproof}}
%BeginExpansion
\endproof%
%EndExpansion

\begin{lemma}
\label{a-l-sink}Assume that $a\in \partial _{+}V$ is a local attractor of
the consistent field. Then there exist constants $\delta >0$ and $K>0,$ and
a norm $\left\| \cdot \right\| ^{\prime }$ on $\prod_{S\subset N}\Re ^{S}$
such that 
\[
\left\| \Lambda _{t}(b)-a\right\| ^{\prime }\leq e^{-Kt}\left\| b-a\right\|
^{\prime }\ \text{for all }t\geq 0\text{ and all }b\in \partial _{+}V\text{
with }\left\| b-a\right\| ^{\prime }<\delta .
\]
\end{lemma}

%TCIMACRO{\TeXButton{Proof}{\proof}}
%BeginExpansion
\proof%
%EndExpansion
See Hirsch and Smale (1974, Theorem in \S 9.1).%
%TCIMACRO{\TeXButton{End Proof}{\endproof}}
%BeginExpansion
\endproof%
%EndExpansion

\medskip

We can now proceed to

\noindent \textbf{Proof of Theorem \ref{mainlim}.}

(i) Let $\mu \in (0,1],$ and consider a sequence $(\rho _{k},T_{k})$ such
that $\rho _{k}\rightarrow 1,T_{k}\rightarrow \infty $ and $\rho
_{k}^{T_{k}}\rightarrow \mu ,$ or, equivalently, $T_{k}(1-\rho
_{k})\rightarrow -\ln \mu .$ We have to prove that $\lim_{k\rightarrow
\infty }F_{\rho _{k}}^{T_{k}}(0)=\Lambda _{-\ln \mu }\left( r\right) .$

Fix $\varepsilon >0;$ let $n_{k}:=n(\rho _{\kappa })$ be given by Corollary 
\ref{a-prop} and put $a_{k}:=F_{\rho _{k}}^{n_{k}}(0).$ Then $\left\|
a_{k}-r\right\| \leq \varepsilon $ and $\mathrm{dist(}F_{\rho
_{k}}^{t}(a_{k}),\partial V)\leq \varepsilon (1-\rho _{k})$ for all $t\geq 0$
and all $k$ large enough (such that $\rho _{k}>\rho _{0}(\varepsilon )$),
say $k\geq k_{0}.$ From now on assume $k\geq k_{0}.$ Also, put $%
T_{k}^{\prime }:=T_{k}-n_{k};$ then $T_{k}^{\prime }(1-\rho _{k})\rightarrow
-\ln \mu $.

Denote $b_{k}^{t}:=F_{\rho _{k}}^{t}(a_{k})\in V_{+}.$ Then $\mathrm{dist}%
(b_{k}^{t},\partial V)\leq \varepsilon (1-\rho _{k})$, and so there is $%
c_{k}^{t}\in \partial _{+}V$ such that $\left\| b_{k}^{t}-c_{k}^{t}\right\|
\leq \varepsilon (1-\rho _{k}).$ Let $K_{1}>0$ be such that $\left\| F_{\rho
}(a)-F_{\rho }(a^{\prime })\right\| \leq K_{1}\left\| a-a^{\prime }\right\| $
for all $\rho \in [0,1]$ and all $a,a^{\prime }\in V_{+},$ and let $K_{2}>0$
be the constant given by the Gronwall Lemma \ref{a-l-gronwall}. We claim
that 
\begin{equation}
\left\| c_{k}^{t}-\Lambda _{t(1-\rho _{k})}(c_{k}^{0})\right\| \leq
(K_{1}+2)\varepsilon (1-\rho _{k})\left( 1+e^{K_{2}(1-\rho
_{k})}+...+e^{(t-1)K_{2}(1-\rho _{k})}\right)  \label{a-eq-induct}
\end{equation}
for all $t\geq 0$ and all $k\geq k_{0}.$ The proof is by induction on $t$
(for each $k$), using the following inequality (recall that $%
b_{k}^{t}=F_{\rho _{k}}(b_{k}^{t-1})$): 
\begin{eqnarray*}
\left\| c_{k}^{t}-\Lambda _{t(1-\rho _{k})}(c_{k}^{0})\right\| &\leq
&\left\| c_{k}^{t}-b_{k}^{t}\right\| +\left\| F_{\rho
_{k}}(b_{k}^{t-1})-F_{\rho _{k}}(c_{k}^{t-1})\right\| \\
&&+\left\| F_{\rho _{k}}(c_{k}^{t-1})-\Lambda _{1-\rho
_{k}}(c_{k}^{t-1})\right\| \\
&&+\left\| \Lambda _{1-\rho _{k}}(c_{k}^{t-1})-\Lambda _{1-\rho
_{k}}(\Lambda _{(t-1)(1-\rho _{k})}(c_{k}^{0}))\right\| .
\end{eqnarray*}
The first term is bounded by $\varepsilon (1-\rho _{k}),$ the second by $%
K_{1}\left\| c_{k}^{t-1}-b_{k}^{t-1}\right\| \leq K_{1}\varepsilon (1-\rho
_{k})$ and the third by $\varepsilon (1-\rho _{k})$ (by Lemma \ref
{a-l-f-lambda}). As for the fourth term, it is at most $e^{K_{2}(1-\rho
_{k})}\left\| c_{k}^{t-1}-\Lambda _{(t-1)(1-\rho _{k})}(c_{k}^{0})\right\| $
(by the Gronwall Lemma \ref{a-l-gronwall}), which is bounded by the
induction hypothesis by 
\[
e^{K_{2}(1-\rho _{k})}(K_{1}+2)\varepsilon (1-\rho _{k})\left(
1+e^{K_{2}(1-\rho _{k})}+...+e^{(t-2)K_{2}(1-\rho _{k})}\right) . 
\]
Adding the four terms yields the right hand side of (\ref{a-eq-induct}).

As $k\rightarrow \infty $ we have 
\[
(1-\rho _{k})\left( 1+e^{K_{2}(1-\rho _{k})}+...+e^{(T_{k}^{\prime
}-1)K_{2}(1-\rho _{k})}\right) \leq \frac{\left( 1-\rho _{k}\right)
e^{T_{k}^{\prime }K_{2}(1-\rho _{k})}}{e^{K_{2}(1-\rho _{k})}-1}\rightarrow 
\frac{e^{-K_{2}\ln \mu }}{K_{2}},
\]
implying by (\ref{a-eq-induct}) that 
\[
\lim \sup_{k\rightarrow \infty }\left\| c_{k}^{T_{k}^{\prime }}-\Lambda
_{T_{k}^{\prime }(1-\rho _{k})}(c_{k}^{0})\right\| \leq \varepsilon
(K_{1}+2)e^{-K_{2}\ln \mu }/K_{2}=:\varepsilon K_{3}.
\]
Now 
\[
F_{\rho _{k}}^{T_{k}}(0)=F_{\rho _{k}}^{T_{k}^{\prime }}\left( F_{\rho
_{k}}^{n_{k}}(0)\right) =F_{\rho _{k}}^{T_{k}^{\prime
}}(a_{k})=b_{k}^{T_{k}^{\prime }},
\]
and we have 
\begin{eqnarray*}
\left\| F_{\rho _{k}}^{T_{k}}(0)-\Lambda _{-\ln \mu }(r)\right\|  &\leq
&\left\| b_{k}^{T_{k}^{\prime }}-c_{k}^{T_{k}^{\prime }}\right\| +\left\|
c_{k}^{T_{k}^{\prime }}-\Lambda _{T_{k}^{\prime }(1-\rho
_{k})}(c_{k}^{0})\right\|  \\
&&+\left\| \Lambda _{T_{k}^{\prime }(1-\rho _{k})}(c_{k}^{0})-\Lambda
_{T_{k}^{\prime }(1-\rho _{k})}(r)\right\|  \\
&&+\left\| \Lambda _{T_{k}^{\prime }(1-\rho _{k})}(r)-\Lambda _{-\ln \mu
}(r)\right\| .
\end{eqnarray*}
As $k\rightarrow \infty ,$ the first term converges to $0$ (it is $\leq
\varepsilon (1-\rho _{k})$). The second is $\leq \varepsilon K_{3}$ in the
limit. For the third one, note that $\left\| a_{k}-c_{k}^{0}\right\| \leq
\varepsilon (1-\rho _{k})$ and $\left\| a_{k}-r\right\| \leq \varepsilon ,$
which implies $\left\| c_{k}^{0}-r\right\| \leq 2\varepsilon $ and thus, by
the Gronwall Lemma \ref{a-l-gronwall}, 
\[
\left\| \Lambda _{T_{k}^{\prime }(1-\rho _{k})}(c_{k}^{0})-\Lambda
_{T_{k}^{\prime }(1-\rho _{k})}(r)\right\| \leq 2\varepsilon
e^{K_{2}T_{k}^{\prime }(1-\rho _{k})}\rightarrow 2\varepsilon e^{-K_{2}\ln
\mu }=:\varepsilon K_{4}
\]
Finally, the fourth term converges to $0$ since $T_{k}^{\prime }(1-\rho
_{k})\rightarrow -\ln \mu .$ Therefore $\lim \sup_{k\rightarrow \infty
}\left\| F_{\rho _{k}}^{T_{k}}(0)-\Lambda _{-\ln \mu }(r)\right\| \leq
\varepsilon (K_{3}+K_{4})$; since $\varepsilon $ is arbitrary, this
completes the proof that $F_{\rho _{k}}^{T_{k}}(0)\rightarrow \Lambda _{-\ln
\mu }(r).$

(ii) For the second part, assume that $a$ is a local attractor and that $%
\Lambda _{t}(r)\rightarrow a$ as $t\rightarrow \infty .$ Consider a sequence 
$(\rho _{k},T_{k})$ such that $\rho _{k}\rightarrow 1,$ $T_{k}\rightarrow
\infty $ and $\rho _{k}^{T_{k}}\rightarrow 0,$ or $T_{k}(1-\rho
_{k})\rightarrow \infty ,$ as $k\rightarrow \infty .$ Fix $\varepsilon >0$
and let $\mu >0$ be small enough so that $\left\| \Lambda _{-\ln \mu
}(r)-a\right\| ^{\prime }\leq \varepsilon ,$ where $\left\| \cdot \right\|
^{\prime }$ is the norm given by Lemma \ref{a-l-sink}. By the result of part
(i), if we let $T_{k}^{1}$ be (the integer part of) $\mu /(1-\rho _{k})$ and
put $a_{k}:=F_{\rho _{k}}^{T_{k}^{1}}(0),$ then $\left\| a_{k}-a\right\|
^{\prime }\leq 2\varepsilon $ for all $k$ large enough, say $k\geq k_{0}$.
Assume moreover that $\rho _{k}>\rho _{0}(\varepsilon )$ and $n(\rho
_{k})\leq T_{k}^{1}\leq T_{k}$ for $k\geq k_{0},$ where $\rho
_{0}(\varepsilon )$ and $n(\rho )$ are given by Corollary \ref{a-prop}. From
now on assume $k\geq k_{0}.$

As in the proof of part (i) above, put $b_{k}^{t}:=F_{\rho _{k}}^{t}(a_{k})$
and let $c_{k}^{t}\in \partial _{+}V$ be such that\footnote{%
The $\left\| \cdot \right\| ^{\prime }$ norm is equivalent to the Euclidean
norm $\left\| \cdot \right\| $, so there is $K>0$ such that $\left\|
x\right\| ^{\prime }\leq K\left\| x\right\| $ for all $x$; one needs only to
replace $\varepsilon $ by $\varepsilon /K$ to get the same estimates for $%
\left\| \cdot \right\| ^{\prime }.$} $\left\| b_{k}^{t}-c_{k}^{t}\right\|
^{\prime }\leq \varepsilon (1-\rho _{k}).$ Let $K_{1}>0$ be such that $%
\left\| F_{\rho }(a)-F_{\rho }(a^{\prime })\right\| ^{\prime }\leq
K_{1}\left\| a-a^{\prime }\right\| ^{\prime }$ for all $\rho \in [0,1]$ and
all $a,a^{\prime }\in V_{+},$ and let $K_{2}>0$ be the constant given by
Lemma \ref{a-l-sink}. Then 
\begin{eqnarray}
\left\| c_{k}^{t}-a\right\| ^{\prime } &\leq &\left\|
c_{k}^{t}-b_{k}^{t}\right\| ^{\prime }+\left\| F_{\rho
_{k}}(b_{k}^{t-1})-F_{\rho _{k}}(c_{k}^{t-1})\right\| ^{\prime }  \nonumber
\\
&&+\left\| F_{\rho _{k}}(c_{k}^{t-1})-\Lambda _{1-\rho
_{k}}(c_{k}^{t-1})\right\| ^{\prime }+\left\| \Lambda _{1-\rho
_{k}}(c_{k}^{t-1})-a\right\| ^{\prime }  \label{a-eq-induct-2} \\
&=&(K_{1}+2)\varepsilon (1-\rho _{k})+\left\| \Lambda _{1-\rho
_{k}}(c_{k}^{t-1})-a\right\| ^{\prime },  \nonumber
\end{eqnarray}
for all $t\geq 0$ and all $k\geq k_{0}.$ We now use induction on $t$ (for
each $k).$ Assume that $\left\| c_{k}^{t-1}-a\right\| ^{\prime }$ $\leq
\delta /2,$ where $\delta $ is given by Lemma \ref{a-l-sink}; this holds for 
$t=0$ if $\varepsilon $ is chosen less than $\delta /4,$ since $\left\|
c_{k}^{0}-a\right\| ^{\prime }=\left\| a_{k}-a\right\| ^{\prime }\leq
2\varepsilon .$ Then $\left\| \Lambda _{1-\rho _{k}}(c_{k}^{t-1})-a\right\|
^{\prime }\leq e^{-K_{2}(1-\rho _{k})}\left\| c_{k}^{t-1}-a\right\| ^{\prime
},$ so (\ref{a-eq-induct-2}) applied inductively yields 
\begin{eqnarray*}
\left\| c_{k}^{t}-a\right\| ^{\prime } &\leq &(K_{1}+2)\varepsilon (1-\rho
_{k})\left( 1+e^{-K_{2}(1-\rho _{k})}+...+e^{-(t-1)K_{2}(1-\rho _{k})}\right)
\\
&&+e^{-tK_{2}(1-\rho _{k})}\left\| c_{k}^{0}-a\right\| ^{\prime } \\
&\leq &(K_{1}+2)\varepsilon (1-\rho _{k})\frac{1}{1-e^{-K_{2}(1-\rho _{k})}}%
+2\varepsilon e^{-tK_{2}(1-\rho _{k})}.
\end{eqnarray*}
Now $K_{2}(1-\rho _{k})/(1-e^{-K_{2}(1-\rho _{k})})\rightarrow 1$ as $%
k\rightarrow \infty ,$ hence the first term above is bounded by $%
2\varepsilon (K_{1}+2)/K_{2}$ for all $k$ large enough. Therefore $\left\|
c_{k}^{t}-a\right\| ^{\prime }\leq \varepsilon \left(
2(K_{1}+2)/K_{2}+2\right) =:\varepsilon K_{3},$ which in particular is $\leq
\delta /2$ if $\varepsilon $ was chosen appropriately small.

To complete the proof, note that 
\begin{eqnarray*}
\left\| F_{\rho _{k}}^{T_{k}}(0)-a\right\| ^{\prime } &=&\left\| F_{\rho
_{k}}^{T_{k}-T_{k}^{1}}(a_{k})-a\right\| ^{\prime }=\left\|
b_{k}^{T_{k}-T_{k}^{1}}-a\right\| ^{\prime } \\
&\leq &\left\| b_{k}^{T_{k}-T_{k}^{1}}-c_{k}^{T_{k}-T_{k}^{1}}\right\|
^{\prime }+\left\| c_{k}^{T_{k}-T_{k}^{1}}-a\right\| ^{\prime }\leq
\varepsilon (1-\rho _{k})+\varepsilon K_{3}.
\end{eqnarray*}

(iii) The third part of the proposition, which states that $\lim_{\stackrel{%
T\rightarrow \infty }{\rho \rightarrow 1}{\ \rho ^{T}\rightarrow 0}}w(\rho
,T)\supset \Lambda _{\infty }\left( r\right) ,$ where $\Lambda _{\infty }(r)$
denotes the $\omega -$limit of the solution of the consistent field through $%
r,$ can be proved using the result of part (i) already established.

Take any $y\in \Lambda _{\infty }\left( r\right) ,$ then there exists a
sequence $\mu _{k}\rightarrow 0$ such that $y_{k}=\Lambda _{-\ln \mu
_{k}}\left( r\right) \rightarrow y.$ For each $k,$ by the result of part (i)
there is $T_{k}$ large enough, say $T_{k}\geq k,$ such that $\left\| F_{\rho
_{k}}^{T_{k}}(0)-y_{k}\right\| \leq 1/k,$ where $\rho _{k}$ is given by $%
T_{k}\left( 1-\rho _{k}\right) =\mu _{k}.$ Hence $F_{\rho
_{k}}^{T_{k}}(0)\rightarrow y$ as $k\rightarrow \infty .$%
%TCIMACRO{\TeXButton{End Proof}{\endproof}}
%BeginExpansion
\endproof%
%EndExpansion
\newline

\noindent \textbf{Proof of Theorem \ref{local}.} We first note that $%
C_{S}(a) $ depends only on $a_{Z}$ for $Z\subset S$ (more precisely: only
for $Z=S$ and $Z=S\backslash i$ for all $i\in S$). Therefore $DC(a)$ is a
block triangular matrix. The diagonals of $DC(a)$ are the matrices $%
DC_{S}:T_{a_{S}}\partial V(S)\rightarrow T_{a_{S}}\partial V(S)$. To
evaluate $DC_{S},$ express $C_{S}\left( a\right) $ as $b_{S}\left( a\right)
-a_{S},$ where 
\[
b_{S}^{i}\left( a\right) =\frac{1}{|S|\lambda _{S}^{i}\left( a\right) }%
\left( \lambda _{S}(a)\cdot a_{S}-\sum_{j\in S\backslash i}\lambda
_{S}^{j}(a)a_{S\backslash i}^{j}\right) +\frac{1}{|S|}\sum_{j\in S\backslash
i}a_{S\backslash j}^{i}. 
\]

We want to evaluate $DC_{S}v_{S}=Db_{S}v_{S}-v_{S}$ for any vector $v_{S}\in
T_{a_{S}}\partial V(S).$ Represent, by an abuse of notation, $b_{S}\left(
a\right) $ as $b_{S}\left( \lambda _{S}(a),\lambda _{S}(a)\cdot a_{S}\right)
,$ where 
\[
b_{S}^{i}\left( \lambda _{S},\zeta _{S}\right) =\frac{1}{|S|\lambda _{S}^{i}}%
\left( \zeta _{S}-\sum_{j\in S\backslash i}\lambda _{S}^{j}a_{S\backslash
i}^{j}\right) +\frac{1}{|S|}\sum_{j\in S\backslash i}a_{S\backslash j}^{i}. 
\]
Using the chain rule we get 
\[
Db_{S}v_{S}=\left( D_{\lambda _{S}}b_{S}+D_{\zeta _{S}}b_{S}a_{S}^{T}\right)
D\lambda _{S}v_{S}, 
\]
since $D\zeta _{S}=D\left( a_{S}^{T}\lambda _{S}(a)\right)
v_{S}=a_{S}^{T}D\lambda _{S}v_{S}+\lambda _{S}^{T}v_{S}=a_{S}^{T}D\lambda
_{S}v_{S}$ (recall that $v_{S}\in T_{a_{S}}\partial V(S),$ and so $%
v_{S}\perp \lambda _{S})$. Evaluating $G=D_{\lambda _{S}}b_{S}+D_{\zeta
_{S}}b_{S}a_{S}^{T}$ yields for $i\neq j$%
\[
G^{ij}=\frac{\partial b_{S}^{i}}{\partial \lambda _{S}^{j}}+\frac{\partial
b_{S}^{i}}{\partial \zeta _{S}}a_{S}^{j}=\frac{1}{|S|\lambda _{S}^{i}}\left(
a_{S}^{j}-a_{S\backslash i}^{j}\right) . 
\]
As for $i=j,$ we get 
\begin{eqnarray*}
G^{ii} &=&\frac{\partial b_{S}^{i}}{\partial \lambda _{S}^{i}}+\frac{%
\partial b_{S}^{i}}{\partial \zeta _{S}}a_{S}^{i} \\
&=&-\frac{1}{|S|\left( \lambda _{S}^{i}\right) ^{2}}\left( \zeta
_{S}-\sum_{j\in S\backslash i}\lambda _{S}^{j}a_{S\backslash i}^{j}\right) +%
\frac{1}{|S|\lambda _{S}^{i}}a_{S}^{i} \\
&=&-\frac{1}{|S|\left( \lambda _{S}^{i}\right) ^{2}}\sum_{j\in S\backslash
i}\lambda _{S}^{j}\left( a_{S}^{j}-a_{S\backslash i}^{j}\right) .
\end{eqnarray*}
But $a$ is a consistent value, hence $\sum_{j\in S\backslash i}\lambda
_{S}^{i}(a_{S}^{i}-a_{S\backslash j}^{i})=\sum_{j\in S\backslash i}\lambda
_{S}^{j}(a_{S}^{j}-a_{S\backslash i}^{j}),$ implying 
\[
G^{ii}=-\frac{1}{|S|\lambda _{S}^{i}}\sum_{j\in S\backslash
i}(a_{S}^{i}-a_{S\backslash j}^{i}), 
\]
and the proof is thus complete.%
%TCIMACRO{\TeXButton{End Proof}{\endproof}}
%BeginExpansion
\endproof%
%EndExpansion

\newpage 

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\end{document}