Minimal farsighted instability

Abstract

We propose the notion of minimal farsighted instability to determine the states that are more likely to emerge in the long run when agents are farsighted. A state is minimally farsighted unstable if there is no other state which is more farsightedly stable. To formulate what it means to be more farsightedly stable, we compare states by comparing (in the set inclusion or cardinal sense) their sets of farsighted defeating states. We next compare states in terms of their absorbtiveness by comparing both their sets of farsighted defeating states (i.e. in terms of their stability) and their sets of farsighted defeated states (i.e. in terms of their reachability). A state is maximally farsighted absorbing if there is no other state which is more farsightedly absorbing. We provide general results for characterizing minimally farsighted unstable states and maximally farsighted absorbing states, and we study their relationships with alternative notions of farsightedness. Finally, we use experimental data to show the relevance of the new solution concepts.

Appendix

1.1 Experimental data

Let $g^{i}$ be a generic network in class $C_{i}$ and $c_{i}\subseteq C_{i}$ a generic proper subset of the corresponding class. We will write $g^{i}\rightarrow g$ with $g\in C_{j}$, and $g^{i}\rightarrow g$ with $g\in c_{j}$, when the generic network $g^{i}$ in class $C_{i}$ reaches with a farsighted improving path all the networks in class $C_{j}$ or only a proper subset $c_{j}$ of $C_{j}$, respectively.

Table 8 Number of defeating networks ($\#\phi (g)$) and number of defeated networks ($\#\phi ^{-1}(g)$) for each network in each class of networks of the experiment

1.1.1 Minimally unstable and maximally absorbing networks in T1

In treatment 1 (T1) the set of networks that can be reached from $g^{i}$ by some farsighted improving path are:

$$\begin{aligned} \phi (g^{\emptyset })= & {} \left\{ g\mid g\in C_{10}\cup C_{11}\right\} \\ \phi (g^{2})= & {} \left\{ g\mid g\in C_{1}\cup C_{10}\cup C_{11}\right\} \\ \phi (g^{3})= & {} \left\{ g\mid g\in C_{1}\cup c_{2}\cup c_{5}\cup C_{10}\cup C_{11}\right\} \\ \phi (g^{4})= & {} \left\{ g\mid g\in C_{1}\cup c_{2}\cup c_{5}\cup C_{10}\cup C_{11}\right\} \\ \phi (g^{5})= & {} \left\{ g\mid g\in C_{1}\cup c_{2}\cup C_{10}\cup C_{11}\right\} \\ \phi (g^{6})= & {} \left\{ g\mid g\in C_{1}\cup c_{2}\cup c_{4}\cup c_{5}\cup C_{10}\cup C_{11}\right\} \\ \phi (g^{7})= & {} \left\{ g\mid g\in C_{1}\cup c_{2}\cup c_{3}\cup c_{4}\cup C_{5}\cup C_{10}\cup C_{11}\right\} \\ \phi (g^{8})= & {} \left\{ g\mid g\in C_{1}\cup c_{2}\cup c_{4}\cup C_{5}\cup c_{7}\cup C_{10}\cup C_{11}\right\} \\ \phi (g^{9})= & {} \left\{ g\mid g\in C_{1}\cup c_{2}\cup c_{4}\cup c_{5}\cup c_{6}\cup c_{7}\cup C_{10}\cup C_{11}\right\} \\ \phi (g^{10})= & {} \left\{ g\mid g\in c_{2}\cup c_{4}\cup c_{5}\cup c_{6}\cup C_{11}\right\} \\ \phi (g^{N})= & {} \emptyset . \end{aligned}$$

Remember that a network $g\in {\mathcal {G}}$ is minimally farsighted unstable if there is no $g^{\prime }\ne g$ such that $g^{\prime }\gnsim ^{\sigma (\subseteq )}g$; i.e., $\phi (g^{\prime })\varsubsetneq \phi (g)$. It follows that $g^{N}$ is the unique minimally farsighted unstable network. Notice that $g^{\emptyset }$ is more farsightedly stable than all networks $g\ne g^{10},g^{N}$.

A network $g\in {\mathcal {G}}$ is maximally farsighted absorbing if there is no $g^{\prime }\ne g$ such that $g^{\prime }\gnsim ^{\alpha (\subseteq )}g$; i.e., $\phi (g^{\prime })\subseteq \phi (g)$ and $\phi ^{-1}(g)\subseteq \phi ^{-1}(g^{\prime })$ with one inclusion holding strictly. Since $\phi (g^{N})=\emptyset$ and $g^{N}\in \phi (g^{i})$ for all $g^{i}\ne g^{N}$, it also holds that $g^{N}$ is the unique maximally farsighted absorbing network.

A network $g\in {\mathcal {G}}$ is $\#$-minimally farsighted unstable if there is no $g^{\prime }\ne g$ such that $g^{\prime }\gnsim ^{\sigma (\#)}g$; i.e., $\#\phi (g^{\prime }) < \#\phi (g)$. Thus, $g^{N}$ is the $\#$-minimally farsighted unstable network.

A network $g\in {\mathcal {G}}$ is $\#$-maximally farsighted absorbing if there is no $g^{\prime }\ne g$ such that $g^{\prime }\gnsim ^{\alpha (\#)}g$; i.e., $\#\phi (g^{\prime })\le \#\phi (g)$ and $\#\phi ^{-1}(g)\le \#\phi ^{-1}(g^{\prime })$ with one inequality holding strictly. Then, $g^{N}$ is the $\#$-maximally farsighted absorbing network.

1.1.2 Minimally unstable and maximally absorbing networks in T2

Let us denote $B_{g}$ the networks that are adjacent to g,

$$\begin{aligned} B_{g}=\left\{ g^{\prime }\mid g^{\prime }=g+ij\vee g-ij\text {, for some }ij\right\} , \end{aligned}$$

and let ${\overline{B}}_{g}$ be its complement. In treatment 2 (T2) the set of networks that can be reached from $g^{i}$ by some farsighted improving path are:

$$\begin{aligned} \phi (g^{\emptyset })= & {} \left\{ g\mid g\in C_{5}\right\} \\ \phi (g^{2})= & {} \left\{ g\mid g\in C_{1}\cup C_{5}\cup c_{9}\right\} \\ \phi (g^{3})= & {} \left\{ g\mid g\in C_{1}\cup c_{2}\cup C_{5}\cup C_{9}\right\} \\ \phi (g^{4})= & {} \left\{ g\mid g\in C_{1}\cup c_{2}\cup c_{4}\cup C_{5}\cup c_{9}\right\} \\ \phi (g^{5})= & {} \left\{ g\mid g\in C_{9}\cap {\overline{B}} _{g^{5}}\right\} \\ \phi (g^{6})= & {} \left\{ g\mid g\in C_{1}\cup c_{2}\cup c_{4}\cup C_{5}\cup C_{9}\right\} \\ \phi (g^{7})= & {} \left\{ g\mid g\in C_{1}\cup c_{2}\cup c_{3}\cup c_{4}\cup C_{5}\cup C_{9}\right\} \\ \phi (g^{8})= & {} \left\{ g\mid g\in C_{1}\cup c_{2}\cup c_{4}\cup C_{5}\cup c_{7}\cup C_{9}\right\} \\ \phi (g^{9})= & {} \left\{ g\mid g\in c_{4}\cup (C_{5}\cap B_{g^{9}})\cup (C_{9}\setminus g^{9})\right\} \\ \phi (g^{10})= & {} \left\{ g\mid g\in C_{1}\cup c_{2}\cup c_{4}\cup C_{5}\cup c_{7}\cup c_{8}\cup C_{9}\right\} \\ \phi (g^{N})= & {} \left\{ g\mid g\in C_{5}\cup C_{9}\cup C_{10}\right\} . \end{aligned}$$

Note that $g^{\emptyset }$ is more farsightedly stable than all networks $g\ne g^{5},g^{9}$. Moreover, neither $g^{5}$ nor $g^{9}$ are more farsightedly stable than $g^{\emptyset }$. Thus, $g^{\emptyset }$ is minimally farsighted unstable. Note also that it is not always true that $(C_{9}{\setminus } g^{9})\supseteq (C_{9}\cap {\overline{B}}_{g^{5}})$ since $g^{9}$ could belong to $(C_{9}\cap {\overline{B}}_{g^{5}})$. Then, also $g^{5},g^{9}$ are minimally farsighted unstable.

The maximally farsighted absorbing networks are $g^{\emptyset },g^{5}$ and $g^{9}$. Indeed:

$$\begin{aligned} \phi ^{-1}(g^{\emptyset })= & {} \left\{ g\mid g\in C_{2}\cup C_{3}\cup C_{4}\cup C_{6}\cup C_{7}\cup C_{8}\cup C_{10}\right\} ;\\ \phi ^{-1}(g^{5})= & {} \left\{ g\mid g\in C_{1}\cup C_{2}\cup C_{3}\cup C_{4}\cup C_{6}\cup C_{7}\cup C_{8}\cup c_{9}\cup C_{10}\cup C_{11}\right\} \end{aligned}$$

and

$$\begin{aligned} \phi ^{-1}(g^{9})=\left\{ g\mid g\in C_{2}\cup C_{3}\cup C_{4}\cup C_{5}\cup C_{6}\cup C_{7}\cup C_{8}\cup c_{9}\cup C_{10}\cup C_{11}\right\} . \end{aligned}$$

From Table 8, we have that $g^{\emptyset }$ is the $\#$-minimally farsighted unstable network. Notice that the network $g^{\emptyset }$ is $\#$-more farsightedly absorbing than the network $g^{2}$, but not more than $g^{5}$ and $g^{9}$.

1.1.3 Minimally unstable and maximally absorbing networks in T3

In treatment 3 the set of networks that can be reached from $g^{i}$ by some farsighted improving path are:

$$\begin{aligned} \phi (g^{\emptyset })= & {} \left\{ g\mid g\in C_{7}\right\} \\ \phi (g^{2})= & {} \left\{ g\mid g\in C_{1}\cup C_{7}\cup c_{10}\right\} \\ \phi (g^{3})= & {} \left\{ g\mid g\in C_{1}\cup c_{2}\cup C_{7}\cup C_{10}\right\} \\ \phi (g^{4})= & {} \left\{ g\mid g\in C_{1}\cup c_{2}\cup c_{5}\cup C_{7}\cup c_{10}\right\} \\ \phi (g^{5})= & {} \left\{ g\mid g\in C_{1}\cup c_{2}\cup C_{7}\cup c_{10}\right\} \\ \phi (g^{6})= & {} \left\{ g\mid g\in C_{1}\cup c_{2}\cup c_{4}\cup c_{5}\cup C_{7}\cup C_{10}\right\} \\ \phi (g^{7})= & {} \left\{ g\mid g\in \left( C_{7}\setminus g\text { s.t. }d_{i}(g)=d_{i}(g^{7})\text { for all }i\in N\right) \cup C_{10}\right\} \\ \phi (g^{8})= & {} \left\{ g\mid g\in C_{1}\cup c_{4}\cup C_{7}\cup C_{10}\right\} \\ \phi (g^{9})= & {} \left\{ g\mid g\in C_{1}\cup c_{2}\cup c_{4}\cup c_{5}\cup c_{6}\cup C_{7}\cup C_{10}\right\} \\ \phi (g^{10})= & {} \left\{ g\mid g\in C_{1}\cup c_{2}\cup c_{4}\cup c_{5}\cup c_{6}\cup C_{7}\cup c_{8}\cup C_{10}\setminus g^{10}\right\} \\ \phi (g^{11})= & {} \left\{ g\mid g\in C_{7}\cup C_{10}\right\} . \end{aligned}$$

Note that $g^{\emptyset }$ is more farsightedly stable than all networks $g\ne g^{7}$. Moreover, $g^{7}$ is not more farsightedly stable than $g^{\emptyset }$. Thus, $g^{\emptyset }$ and each $g^{7}$ are minimally farsighted unstable networks. The maximally farsighted absorbing networks are also $g^{\emptyset },g^{7}$ and $g^{10}$. Indeed:

$$\begin{aligned} \phi ^{-1}(g^{\emptyset })= & {} \left\{ g\mid g\in C_{2}\cup C_{3}\cup C_{4}\cup C_{5}\cup C_{6}\cup C_{8}\cup C_{9}\cup C_{10}\right\} ;\\ \phi ^{-1}(g^{7})= & {} \left\{ g\mid g\in C_{1}\cup C_{2}\cup C_{3}\cup C_{4}\cup C_{5}\cup C_{6}\cup c_{7}\cup C_{8}\cup c_{9}\cup C_{10}\cup C_{11}\right\} \end{aligned}$$

and

$$\begin{aligned} \phi ^{-1}(g^{10})=\left\{ g\mid g\in c_{2}\cup C_{3}\cup c_{4}\cup c_{5}\cup C_{6}\cup C_{7}\cup C_{8}\cup C_{9}\cup C_{10}\setminus g^{10}\cup C_{11}\right\} . \end{aligned}$$

From Table 8, we have that $g^{\emptyset }$ is the $\#$-minimally farsighted unstable network. Remark that the network $g^{7}$ is $\#$-more farsightedly absorbing than all other networks except $g^{\emptyset }$.