The Chi-compromise value for non-transferable utility games
The paper extends two solution concepts formulated earlier for bargaining games, which are of the 'compromise' type in that it is a compromise between players' maximum and minimum aspirations. One of the solutions is the well known Kalai-Smorodonsky solution to pure bargaining games where solutions have to be unanimous, while the other was formulated earlier by the authors for bargaining games where partial agreements are possible and utility is transferable. The solutions are extended to general bargaining problems where the players may reach partial agreements and the utility is not necessarily transferable. The proposed solution is the efficient vector lying on the segment between the vectors of maximal and minimal aspirations, where the maximal aspiration for a player is his maximal (among all coalitions) marginal contribution and the minimum aspiration is the maximum remainder a player can get by going with a coalition of players and offering them their maximal aspirations. Some details follow. Let \(N=\{1,\dots,n\}\), where \(n\geq 2\), denote the finite set of players and \(V \) denote the characteristic function which assigns to each non-empty subset \( S\) of \(N\) a non empty subset of \(\mathbb R^{S}\). A cooperative game with non-transferable utility (NTU) is an ordered pair \((N,V)\). Let \(V_{N}\) denote the class of such games. Some standard properties of these are assumed. For each player \(i\), there is an individually rational payoff \(w_{i}\in\mathbb R\) such that \(V(i)=\{x\in\mathbb R\mid x\leq w_{i}\}\). Also, for each coalition \(S\), the set \(V(S)_{+}=\{x\in V(S)\mid x\geq w_{s}\}\) is bounded. A game is non-level if for each coalition \(S\) and for any \(x,y\in V(S)_{+}\) satisfying \(y\geq x\geq w_{S}\) and \(x\neq y\) there exists \(z\in V(S)\) such that \(z>x\). A game is totally essential if \(w_{S}\in V(S)\) for all \(S\subseteq N\). Denote by \(C_{N}\) the subclass of non-level and totally essential games with NTU. A solution on a subclass of games \(G_{N}\subseteq V_{N}\) is a function \(\varphi :G_{N}\rightarrow \mathbb R^{N}\) which assigns to each \((N,V)\in G_{N}\) a vector \( \varphi (N,V)\in V( N) .\) A player \(i\)'s maximum aspiration in a game \((N,V)\in V_{N}\) is defined by: \[ M_{i}(N,V) = \max_{S\subseteq N,i\in S} \{t\in R\mid (t,x) \in V(S) _{+},\;x\in P(V(S\backslash i))\}. \] A player \(i\)'s minimal aspiration in a game \((N,V)\in V_{N}\) is defined by: \[ m_{i}(N,V)= \max_{S\subseteq N,i\in S} \{t\in R\mid (t,M_{S\backslash i}(N,V))\in V(S)\}. \] Denote by \(M\) and \(m\) the, respective, aspiration vectors for a given game \( ( N,V) \). \ The Chi-compromise value of the game \((N,V) \) is \(\chi ( N,V) \equiv \gamma M+\left( 1-\gamma \right) m\) where \(\gamma \) is the largest number in \([0,1]\) satisfying \(\chi(N,V) \in P(V(N)) .\) It is shown that for all \(( N,V) \in C_{N}\) there exists \(\chi ( N,V) .\) Different characterizations of the Chi-compromise value in terms of properties such as Pareto Optimality, symmetry, covariance, monotonicity etc is obtained. A non-cooperative game is formulated whose subgame perfect Nash equilibrium payoff implements the Chi-compromise value.
scientific article; zbMATH DE number 1949083