Auguste Toulmouche (1829-1890) was a French painter.
The topics are
\begin{equation}{\label{a}}\tag{A}\mbox{}\end{equation}
The \(f\)-Core of a Continuum Game.
Let \((N,{\cal B},\mu )\) be a measure space, where \(N\) is a Borel subet of a complete separable metric space, \({\cal B}\) is the \(\sigma\)-field of all Borel subsets of \(N\), and \(\mu\) is a non-atomic measure with \(0<\nu (N)<+\infty\). Each element in \(N\) is called a player and \(N\) is the player set. Let \(n\), a positive integer, be a bound on coalition sizes. Let \({\cal F}\) be the set of all finite subsets of \(N\) containing no more than \(n\) members. Each element \(S\) in \({\cal F}\) is called a coalition. The mixture of a continuum of players and finite coalitions may raise the question of how we should interpret the individual player relative to the total player set. In the present approach the individual player remains the same as in finite models while the total player set is approximated by a continuum. Mathematically, outcomes of cooperation of finite numbers of players are aggregated into outcomes for the total player set by measurement-consistent partitions, to be defined presently. In the traditional approach to a continuum game, where coalitions are of positive measure, the notion of the individual player becomes vague (Kaneko and Wooders \cite{kan96}).
The set of attributes \(A\) is given as a compact metric space with metric \(d\). Let \(A^{*}=\bigcup_{t=1}^{n}A^{t}\), where \(A^{t}\) is the \(t\)-fold Cartesian product of \(A\). An element \({\bf a}\in A^{t}\) is a list of attributes of a \(t\)-member coalition. A characteristic function \(V^{*}\) is a function on \(A^{*}\), which assigns to each \(t\)-vector \({\bf a}=(a_{1},\cdots ,a_{t})\in A^{t}\) a nonempty closed subset \(V^{*}({\bf a})\) of \(\mathbb{R}^{t}\) with the following properties.
- (Comprehensiveness). If \({\bf x}\in V^{*}({\bf a})\) and \({\bf y}\in\mathbb{R}^{t}\) with \({\bf y}\leq {\bf x}\), then \({\bf y}\in V^{*}({\bf a})\).
- (Nontriviality). The following set
\[\left\{{\bf x}\in V^{*}(a_{1},\cdots ,a_{n}):x_{k}\geq\sup V^{*}(\alpha_{k})(???)\mbox{ for }k=1,\cdots ,t\right\}\]
is a nonempty and bounded subset of \(\mathbb{R}^{t}\). - (Anonymity). For all permutations \(\pi\) of \(\{1,2,\cdots ,t\}\), we have
\[V^{*}(a_{\pi (1)},\cdots ,a_{\pi (t)})=\left\{(x_{\pi (1)},\cdots ,x_{\pi (t)}:(x_{1},\cdots ,x_{t})\in V^{*}(a_{1},\cdots ,a_{t})\right\}.\]
The characteristic function \(V^{*}\) assigns a set of attainable payoffs to each list of attributes. The anonymity means that the characteristic function \(V^{*}\) is invariant with respect to permutations of attributes (Kaneko and Wooders \cite{kan96}).
From closedness, comprehensiveness and nontriviality, it follows that there is a real-valued function \(M(a)\) with domain \(A\) such that \(V^{*}(a)\) can be written as \(V^{*}(a)=(-\infty ,M(a)]\) for any \(a\in A\). We define
\[V_{+}^{*}(a_{1},\cdots ,a_{t})=\{{\bf x}\in V^{*}(a_{1},\cdots ,a_{t}):x_{i}\geq M(a_{i})\mbox{ for }i=1,\cdots ,t,\mbox{ for all }
(a_{1},\cdots ,a_{t})\in A^{t},t=1,\cdots ,n\}.\]
The function \(V_{+}^{*}(a_{1},\cdots ,a_{t})\) gives the set of individually rational payoff vectors.
We require an additional continuity condition. First, let \(d^{*}\) denote the sup metric on each \(A^{t}\), given by \(d^{*}((a_{1},\cdots ,a_{t}),(b_{1},\cdots ,b_{t}))=\max_{k}d(a_{k},b_{k})\) for any \({\bf a},{\bf b}\in A^{t}\). The \(t\)-dimensional Euclidean space \(\mathbb{R}^{t}\) is also endowed with the sup metric, denoted by \(d^{t}\). Let \(d_{H}^{t}\) be the Hausdorff metric for compact subsets of \(\mathbb{R}^{t}\), that is, for compact subsets \(T,W\) of \(\mathbb{R}^{t}\),
\[d_{H}^{t}(T,W)=\max\left\{\max_{{\bf x}\in T}d^{t}({\bf x},W),\max_{{\bf y}\in W}d^{t}(T,{\bf y})\right\},\]
where \(d^{t}({\bf x},W)=\min_{{\bf y}\in W}d^{t}({\bf x},{\bf y})\) and \(d^{t}(T,{\bf y})=\min_{{\bf x}\in T}d^{t}({\bf x},{\bf y})\). We make the assumption
- (Continuity). \(V_{+}^{*}\) is a continuous function on each \(A^{t}\), \(t=1,\cdots ,n\).
Since the attribute space \(A\) is compact and \(V_{+}^{*}(a)=\{M(a)\}\) is continuous on \(A\), \(M(a)\) has a minimum. Thus we have \(\min_{a\in A}M(a)=\min_{a\in A}\max V^{*}(a)>-\infty\) (Kaneko and Wooders \cite{kan96}).
A game with a continuum of players and finite coalitions is determined by an attribute function, which ascribes an attribute (a point on attribute space) to each player in \(N\). The payoff set of a coalition is defined as the value of the characteristic function determined by the attributes of the members of that coalition. Specifically, an attribute function $\gamma$ is a Borel measurable function from \(N\) to \(A\). The game \(V\) determined by the attribute function \(\gamma\) is defined by
\begin{equation}{\label{kan96eq2}}\tag{1}
V(S)=V^{*}(\gamma (i_{1}),\cdots ,\gamma (i_{s}))\mbox{ for all }S=\{i_{1},\cdots ,i_{s}\}\in {\cal F}.
\end{equation}
Note that by anonymity \(V(S)\) does not depend upon the choice of ordering of the members of \(S\).
The continuum player set is consistently connected to finite coalitions through measurement-consistent partitions. A partion \(\wp\) of \(N\) into coalitions is measurement-consistent if and only if for any positive integer \(k\leq n\), \(N_{k}=\bigcup_{S\in\wp ,|S|=k}S\) is a measurable subset of \(N\), and each \(N_{k}\) has a partition, consisting of measurable subsets \(\{N_{k1},\cdots ,N_{kk}\}\), with the following property: there are measure-preserving isomorphisms (A function \(\psi\) from a set \(A\) in \({\cal B}\) to a set \(B\) in \({\cal B}\) is called measure-preserving isomorphism from \(A\) to \(B\) if and only if \(\psi\) is one-to-one, onto, measurable in both directions, and \(\mu (C)=\mu (\psi (C))\) for all \(C\subset A\) with \(C\in {\cal B}\).) \(\psi_{k1},\cdots ,\psi_{kk}\) from \(N_{k1}\) to \(N_{k1},\cdots ,N_{kk}\), respectively, such that \(\psi_{k1}(i)\) is the identity map and \(\{\psi_{k1}(i),\cdots ,\psi_{kk}(i)\}\in\wp\) for all \(i\in N_{k1}\). Let \(\Pi\) denote the set of measurement-consistent partitions. Note that it implies that for any \(S\in\wp\) with \(|S|=k\), we have \(S=\{\psi_{k1}(i),\cdots ,\psi_{kk}(i)\}\) for soem \(i\in N_{k1}\). Thus, for each integer \(k\), the set \(N_{k}\) consists of all the members of \(k\)-player coalitions and \(N_{kt}\) consists of the \(t\)th members of these coalitions. The measure-preserving property of the isomorphisms from \(N_{k1}\) to \(N_{kt}\), \(t=1,\cdots ,t\), expresses the idea that coalitions of size \(k\) as “many” (i.e. the same measure) first members as second members, as many second members as third members, etc.
An outcome for the entire continuum game us defined as follows. First we consider a measurement-consistent partition \(\wp\) of the entire player set \(N\). A payoff \(f\) for \(N\) is feasible if and only if each coalition in \(\wp\) can achieve its part \((h_{i})_{i\in S}\) of \(h\). Thus we define the outcome set \(H(\wp )\) relative to \(\wp\) by
\[H(\wp )=\left\{h\in L(N,\mathbb{R}):(h(j))_{j\in S}\in V(S)\mbox{ for all }S\in\wp\right\},\]
where \(L(N,\mathbb{R})\) is the set of measurable functions from \(N\) to \(\mathbb{R}\). Note that \(H(\wp )\neq\emptyset\) for any partition \(\wp\) since, from comprehensiveness, the constant function \(h\), given by \(h(i)=\min_{a}M(a)\) for all \(i\in N\), is in \(H(\wp )\). The entire outcome space is denoted by \(H=\bigcup_{\wp\in\Pi}\). The outcome set \(H\) is not necessarily closed, i.e., limits of sequences in \(H\) are not necessarily in \(H\). We extend the outcome space \(H\) by adding some idealized outcomes to the space \(H\) so that the new space is closed with respect to a suitable concept of convergence. Here we define the extended outcome set \(H^{*}\) by
\[H^{*}=\mbox{$\{h\in L(N,\mathbb{R})$:for some sequence \(\{h_{\nu}\}\) in \(H\), \(\{h_{\nu}\}\) converges in measure to \(h\)\}},\]
where “convergence in measure to \(h\)” means that for any \(\epsilon >0\), \(\mu(\{i\in N:|h_{\nu}(i)-h(i)|>\epsilon\})\rightarrow 0\) as \(\nu\rightarrow \infty\). Note that \(H(\wp )\subset H\subset H^{*}\) for all \(\wp\in\Pi\). We call an element in \(H^{*}\) simply an {\bf outcome}.
Let \(h\) be a function in \(L(N,\mathbb{R})\). We say that a coalition \(S\) in \({\cal F}\) can improve upon \(h\) if for some \(y\in V(S)\), \(y_{i}>h(i)\) for all \(i\in S\). The {\bf \(f\)-core} of the game \(V\) is defined to be the set
\[C_{f}=\{h\in H^{*}:\mbox{no coalition in \(F\) can improve upon \(h\)}\}.\]
An outcome \(h\) in the \(f\)-core \(C_{f}\) is stable in the sense that no coalition can improve upon \(h\), and it is approximately feasible in the sense that \(h\) is almost sustained by feasible outcomes in \(H\). Except for this approximate feasibility, the core notion is the same as in finite games (Kaneko and Wooders \cite{kan96}).
\begin{equation}{\label{b}}\tag{B}\mbox{}\end{equation}
The Nonemptiness of the \(f\)-Core.
To state the main result, one more condition us required. The condition states that if a player receives a larger payoff in a coalition than his
individually rational payoff, then he can transfer some part of his payoff to other players at a given constant rate (which may be very small). Formally, the characteristic function \(V^{*}\) is said to be strongly comprehensive if and only if there is a \(b>0\) such that for any \((a_{1},\cdots ,a_{t})\in A^{t}\), \(1\leq t\leq n\), and any \(k\), \(1\leq k\leq t\), if \({\bf x}\in V_{+}^{*}(a_{1},\cdots ,a_{t})\) with \(x_{k}>\max V^{*}(a_{k})=M(a_{k})\), then, for any \(c\) with \(0<c<x_{k}-\max v^{*}(a_{k})\), \({\bf y}\in\mathbb{R}^{t}\), defined by its components
\[y_{i}=\left\{\begin{array}{ll}
x_{k}-c & \mbox{if \(i=k\)}\\ x_{i}+bc & \mbox{otherwise}
\end{array}\right .\]
belongs to \(V^{*}(a_{1},\cdots ,a_{t})\).
Theorem. (Kaneko and Wooders \cite{kan96}). Assume that the characteristic function \(V^{*}\) satisfies Strong Comprehensivenes. Then any game \((N,V)\) determined from \(V^{*}\) and an attribute function \(\gamma\) has a nonempty \(f\)-core. \(\sharp\)
The following two examples illustrate applications of this theorem, and how we obtain the nonemptiness of the \(f\)-core in a continuum game.
Example. (A labor market). We concern firms and workers — a labor market. All firms have the same production possibilities and all workers are substitutes for each other. A firm with zero workers can produce nothing. A firm with \(1\) worker can produce \$1.00 worth of output, and a firm with \(2\) workers can produce \$3.00 worth of output, that is, there are increasing returns to scale up to two workers. A firm cannot gain from having more than two workers. An unemployed worker can produce nothing. If there is only a finite number of firms and workers, the core of the game may be empty. For example, consider the case of \(2\) firms. If there are \(3\) workers, the core is empty, while if there are \(4\) or more workers, the core is nonempty. Now suppose that there are \(100\) firms. If there are \(200\) or more workers, the core is nonempty, while if there is an odd number of workers smaller than \(200\), the core is empty. Now suppose that the number of workers is \(181\). In an efficient state, \(90\) firms hire two workers and one firm hires the remaining worker. At least half the workers hired by the first \(90\) firms received no more than \$1.50. The \(91\)st firm and its one worker produces a total of \$1.00 worth of output. If the \(91\)st firm hires an additional worker, then the firm coucld produce \$3.00 worth of output. In this case, even if the firm pays slightly more than \$1.50 to the new worker, the firm and its first worker can be better off. This implies that the core is empty. Note that this argument does not apply to the case of \(100\) firms and \(180\) workers. To illustrate the formalism and the independence of the core of a large game from the exact numbers of players, we describe the example more precisely. The attribute space consists of two points, say \(A=\{f,w\}\). The bound on coalition sizes can be taken as \(n=3\). The characteristic function \(V^{*}\) is defined on lists contained in \(\bigcup_{t=1}^{3}A^{t}\) and is given by
\[V^{*}(f,w,w,)=\{{\bf x}\in\mathbb{R}^{3}:x_{1}+x_{2}+x_{3}\leq 3\}\mbox{ and }
V^{*}(f,w)=\{{\bf x}\in\mathbb{R}^{2}:x_{1}+x_{2}\leq 1\}.\]
All other (essentially different) lists of attributes have a total sum of values equal to zero. Letting the total player set \(N\) be the interval \([0,2.81)\), an attribute function \(\gamma\) is given by
\[\gamma (i)=f\mbox{ for }i\in [0,1)\mbox{ and }\gamma (i)=w\mbox{ for }i\in [1,2.18).\]
The proportional distribution of firms and workers remains the same as that in the above finite example. There is a measurement-consistent partition assigning two workers to each of the firms in \([0,0.905)\) and zero workers to the other firms. Such a partition supports a core payoff giving each of the workes \$1.50 and each of the firms \$0.00. (in the example the approximation of the taking the closure \(H^{*}\) of \(H\) is not necessary.) In the finite case, the emptiness of the core is derived from the behaviour of the firm having one worker. Such a firm disappears in the continuum game, because there is no distinction of odd and even numbers of workers. Even in the finite case, the effect of the last firm becomes less significant as the economy becomes large, suggesting the nonemptiness of approximate cores of large finite games shown in Wooders \cite{wod83} and other players. the \(f\)-core is the limit case of such finite approximate cores. \(\sharp\)
Example. (Assignment Games). An assignment game with a continuum of players is formulated as follows. Let \(\{A_{1},A_{2}\}\) be a partition of \(A\); \(A_{1}\) and \(A_{2}\) are are the sets fo attributes of the buyers and sellers, respectively. Correspondingly, the total player set is divided into the set \(N_{1}\) of players with attributes in \(A_{1}\) and the set \(N_{2}\) of players with attributes in \(A_{2}\), that is, an attribute function \(\gamma\) satisfies \(\gamma (N_{1})\subset A_{1}\) and \(\gamma (N_{2}) \subset A_{2}\). Here the bound on essential coalition size is \(2\). Let \(V^{*}\) be a characteristic funcion on \(A^{*}=A\cup (A\times A)\) with the property that
\[V^{*}(a_{1},a_{2})=V^{*}(a_{1})V^{*}(a_{2})\]
if \(a_{1},a_{2}\in A_{1}\) or \(a_{1},a_{2}\in A_{2}\). This states that a coalition consisting of a pair of players on the same side of the market can do not better than each of the players separately. The game determined by \(V^{*}\) and the attribute function is given by (\ref{kan96eq2}). \(\sharp\)
