Equilibria with vector-valued utilities and preference information. The analysis of a mixed duopoly

Abstract

This paper deals with the equilibria of games when the agents have multiple objectives and, therefore, their utilities cannot be represented by a single value, but by a vector containing the various dimensions of the utility. Our approach allows the incorporation of partial information about the preferences of the agents into the model, and permits the identification of the set of equilibria in accordance with this information. We also propose an additional conservative criterion which can be applied in this framework in order to predict the results of interaction. The potential application of the theoretical results is shown with an analysis of a mixed oligopoly in which the agents value additional objectives other than their own benefit. These objectives are related to social welfare and to the profit of the industry. The flexibility of our approach provides a general theoretical framework for the analysis of a wide range of strategic economic models.

Appendix

Proof of Theorem 2.3

First, note that under the assumptions, all the values $r_j^i(a^{-i})$ are well-defined. Let $a\in \times _{i\in N}A^i$ be a profile of strategies such that for a certain $i\in N, a^i< {\underline{r}}^i(a^{-i})$. Since $u_j^i$ is strictly concave in its own action, then at $a^i, u_j^i$ is strictly increasing for all $j\in J^i$, and by adopting a strategy $a^i+\varepsilon$, with $\varepsilon >0$, agent i will increase all the components of his utility. Therefore, a is not an equilibrium. Analogous reasoning can be applied when $a^i >\bar{r}^i(a^{-i})$. Let $a\in \times _{i\in N}A^i$, such that ${\underline{r}}^i(a^{-i})\le a^i\le {\bar{r}}^i(a^{-i})$ for all $i\in N$. For $i\in N$, if ${\underline{r}}^i(a^{-i})= a^i= \bar{r}^i(a^{-i})$, by moving from his strategy, all the components of the utility of agent i will decrease. If ${\underline{r}}^i(a^{-i})< a^i< {\bar{r}}^i(a^{-i})$, at least one of the inequalities in ${\underline{r}}^i(a^{-i})\le a^i\le \bar{r}^i(a^{-i})$ must be strict. If ${\underline{r}}^i(a^{-i})< a^i$, by reducing $a^i$, the utility corresponding to ${\bar{r}}^i$ will decrease. If $a^i< {\bar{r}}^i(a^{-i})$, by increasing $a^i$, the utility corresponding to ${\underline{r}}^i$ will decrease.

Proof of Theorem 2.5

A first inclusion is proved in Wang (1993). Conversely, let $a^*\in E^w(G)$. For $i\in N$, define the sets $Y^i=\{ x\in \mathbb {R}^{s^i} : u^i(a^i, a^{*-i})\geqq x, \, \text{ for } \text{ some } \, a^i\in A^i\}$, and $X^i=\{ x\in \mathbb {R}^{s^i} : u^i(a^{*})<x\}$. The sets $Y^i$ and $X^i$ are convex and disjoint. Convexity is a consequence of the concavity of $u^i$. To prove that they are disjoint: If $x\in Y^i$ then $a^i\in A^i$ exists such that $x\leqq u^i(a^i, a^{*-i})$. If $x\in X^i$ then $x>u^i(a^*)$. Therefore, if $x\in Y^i\cap X^i$, then $u^i(a^*)<x\leqq u^i(a^i, a^{*-i})$, and $a^i\in A^i$ exists with $u^i(a^*)< u^i(a^i, a^{*-i})$, which contradicts $a^*$ being a weak equilibrium of G. It follows from Minkowski’s separating hyperplane theorem that there exists some non-null vector $\lambda ^i\in \mathbb {R}^{s^i}$ and some constant c such that $\lambda ^i \cdot x\le c$ for all $x\in Y^i$, and $\lambda ^i \cdot x\ge c$ for all $x\in X^i$. It is easy to see that a non-negative $\lambda ^i$ and $c=\lambda ^i\cdot u^i(a^*)$ satisfy this condition. Therefore, for each $i\in N, \lambda ^i\cdot u^i(a^*)\ge \lambda ^i\cdot x$ for all $x\in Y^i$, and since $u^i(a^i, a^{*-i})\in Y^i$, for all $a^i\in A^i, a^{*i}$ maximizes $v_{\lambda }^i(a^i, a^{*-i})$. Since only the direction of the vector matters, $\lambda ^i$ can be taken in $\Delta ^{s^i}$. It follows that $a^*\in E(G_\lambda )$.

Proof of Proposition 3.2

If $a^* \in E_{\Lambda }(G)$, then for each $i \in N$ there exists $\lambda ^i \in \Lambda ^i$ such that $a^*$ is an equilibrium of game $G_{\lambda }$. It follows from Definition 3.1 that $\lambda ^i\cdot u^i(a^{*})\ge \lambda ^i \cdot u^i (a^{i}, a^{*-i})$, for all $a^i \in A^i$. On the other hand, for each $i \in N, \lambda ^i \in \Lambda ^i$ can be written as a convex combination of the extreme points of $\Lambda ^i, \lambda ^i= \sum _{r=1}^{ p^i}\alpha ^i_r {\bar{\lambda }}^i(r)$ with $\alpha ^i_r\ge 0$, and $\sum _{r=1}^{ p^i} \alpha ^i_r=1$. Thus, for each $i \in N, \sum _{r=1}^{ p^i}\alpha ^i_r{\bar{\lambda }}^i(r)\cdot u^i( a^{*})\ge \sum _{r=1}^{ p^i}\alpha ^i_r {\bar{\lambda }}^i(r)\cdot u^i (a^i, a^{*-i})$ for all $a^i\in A^i$. It follows that $a^*$ is a weak equilibrium of game $\{(A^i, v^i_{\Lambda })_{i \in N}\}$.

Proof of Theorem 3.3

A first inclusion is stated in Proposition 3.2. Conversely, first note that, if for each $i, u^i$ is concave in $a^i$, then, $v_{\Lambda }^i$ is also concave. Let $a^*$ be a weak equilibrium of game $\{(A^i, v^i_{\Lambda })_{i \in N}\}$. It follows from Theorem 2.5 that for each $i\in N, \alpha ^i\in \Delta ^{p^i}$ exists such that $\sum _{r=1}^{ p^i}\alpha ^i_r{\bar{\lambda }}^i(r)\cdot u^i( a^{*})\ge \sum _{r=1}^{ p^i}\alpha ^i_r {\bar{\lambda }}^i(r)\cdot u^i (a^i, a^{*-i})$ for all $a^i\in A^i$. Since $\sum _{r=1}^{ p^i}\alpha ^i_r{\bar{\lambda }}^i(r)\in \Lambda ^i$, from Definition 3.1 it follows that $a^* \in E_{\Lambda }(G)$.

Proof of Proposition 3.5

Let $a^*\in E^c_{\Lambda }(G)$ with $\Lambda = \times _{i\in N}\Lambda ^i$. Thus, $\not \exists i\in N$ with $a^i\in A^i$ such that $\min _{r=1,\ldots ,{p^i}} \{ v^i_{{\bar{\lambda }}(r)}(a^i, a^{*-i}) \} > \min _{r=1,\ldots ,{p^i}} \{ v^i_{{\bar{\lambda }}(r)} ( a^{*}) \}$, where ${\bar{\lambda }}^i(r), r=1,\ldots ,p^i$ are the extreme points of $\Lambda ^i$. Suppose on the contrary that $a^*\not \in E_{\Lambda }(G)$. Under convexity assumptions, by Theorem 3.3, $E_{\Lambda }(G)=E^w(G_{\Lambda })$. Hence, $a^*$ is not a weak equilibrium of the game $\{(A^i, v_\Lambda ^i)_{i\in N}\}$. That is, $\exists i\in N$ with $a^i\in A^i$ such that $v^i_{\Lambda }(a^i, a^{*-i}) > v^i_{\Lambda }( a^{*})$. The components of $v_\Lambda ^i$ are $v^i_{{\bar{\lambda }}(r)}$ for $i=1,\ldots , p^i$, and therefore $v^i_{{\bar{\lambda }}(r)}(a^i, a^{*-i}) > v^i_{{\bar{\lambda }}(r)} ( a^{*}), \forall r=1,\ldots ,p^i$. It follows that $\min _{r=1,\ldots ,{p^i}} \{v^i_{{\bar{\lambda }}(r)}(a^i, a^{*-i})\}> \min _{r=1,\ldots ,{p^i}} \{v^i_{{\bar{\lambda }}(r)} ( a^{*})\}$, and this contradicts that $a^*$ is a conservative equilibrium for the game with preference information $\Lambda$.