Cautious farsighted stability in network formation games with streams of payoffs

Abstract

We propose a new notion of farsighted pairwise stability for dynamic network formation which includes two notable features: consideration of intermediate payoffs and cautiousness. This differs from existing concepts which typically consider either only immediate or final payoffs, and which often require that players are optimistic in any environment without full communication and commitment. For arbitrary (and possibly heterogeneous) preferences over the process of network formation, a non-empty cautious path stable set of networks always exists. Furthermore, some general relationships exist between cautious path stability and other farsighted concepts.

Appendix

1.1 Appendix A: Brief description of network stability concepts

See Table 2.

Table 2 Network stability concepts in the existing literature. Formal definitions can be found in Supplementary Appendix D

1.2 Appendix B: Proofs

Proof of Lemma 1

Suppose $P=(g_1,..,g_K)$ and $P^{\prime }=(g_K,..,g_{K+N})$, where $g_1=g$, $g_K=g^{\prime }$ and $g_{K+N}=g^{\prime \prime }$. Let $P\in P^{SI}(g_1,G)$ and $P^{\prime }\in P^{SI}(g_K,G^{\prime })$, where $G\cap G^{\prime }\ne \emptyset$ and $g_{K+N}\in G$. Consider $P^{\prime \prime }=P\oplus P_2^{\prime }=(g_1,..,g_K,g_{K+1},..,g_{K+N})$. Below we will show recursively that for any k in the decreasing sequence $K-1, K-2,..,1$, the continuation of path $P^{\prime \prime }$ from step k, $P^{\prime \prime }_k$, is a surely improving path relative to set $G^{\prime \prime }$, where $G^{\prime \prime }$ is any subset of $G\cap G^{\prime }$. Then, as $P^{\prime \prime }_1=P^{\prime \prime }$, the last step of the argument will complete the proof.

Consider $P^{\prime \prime }_{K-1}=(g_{K-1},g_K,g_{K+1},..,g_{K+N})=(g_{K-1})\oplus P^{\prime }$. Suppose that i and j are the players involved in the first-step change on this path, from $g_{K-1}$ to $g_K$, i.e., $g_K=g_{K-1}+ij$ or $g_K=g_{K-1}-ij$. To show that $P^{\prime \prime }_{K-1}\in P^{SI}(g_{K-1}, G^{\prime \prime })$, let us first verify that $P^{\prime \prime }_{K-1}\in P^{I}(g_{K-1})$. This follows from the fact that $P^{\prime }\in P^{I}(g_{K})$ by definition, and players i, j prefer path $P^{\prime }$ to staying in $g_{K-1}$ for $|P^{\prime }|$ steps. The latter is an immediate implication of the fact that P is a surely improving path relative to G, so that by definition, for any ${\widetilde{P}}\in P^{I}(g_K)$ leading to G, including the path $P^{\prime }$, the following inequalities hold: (a) $\pi _i({\widetilde{P}})\ge \pi _i(g_{K-1}^{|{\widetilde{P}}|})$ and $\pi _j({\widetilde{P}})\ge \pi _j(g_{K-1}^{|{\widetilde{P}}|})$, with at least one inequality being strict, if $g_K=g_{K-1}+ij$, or (b) $\pi _i({\widetilde{P}})>\pi _i(g_{K-1}^{|{\widetilde{P}}|})$ if $g_K=g_{K-1}-ij$. Now, given that $P^{\prime }$ is a surely improving path relative to $G^{\prime }$ and hence, also relative to $G^{\prime \prime }\subseteq G^{\prime }$, that is, $P^{\prime }\in P^{SI}(g_K, G^{\prime \prime })$, and inequalities (a), (b) hold for any ${\widetilde{P}}\in P^{I}(g_K)$ that leads to G and hence, also for any improving path that leads to $G^{\prime \prime }\subseteq G$, it follows that conditions (i) and (ii) of the definition of a surely improving path relative to $G^{\prime \prime }$ are satisfied for all steps on the path $P^{\prime \prime }_{K-1}=(g_{K-1})\oplus P^{\prime }$. Thus, $P^{\prime \prime }_{K-1}\in P^{SI}(g_{K-1}, G^{\prime \prime })$.

Next, consider $P^{\prime \prime }_{K-2}=(g_{K-2},g_K,g_{K-1},g_K,..,g_{K+N})=(g_{K-2})\oplus P^{\prime \prime }_{K-1}$. By the same argument as before, $P^{\prime \prime }_{K-2}\in P^{SI}(g_{K-2},G^{\prime \prime })$. Then by analogy, we can construct a sequence of surely improving paths $P^{\prime \prime }_{K-1}$, $P^{\prime \prime }_{K-2}$, $P^{\prime \prime }_{K-3},..,P^{\prime \prime }_{2}$, $P^{\prime \prime }$. Thus, $P^{\prime \prime }\in P^{SI}(g_{1})$, where $g_1=g$. $\square$

Proof of Proposition 2

$(\Rightarrow )$: Suppose that set G is cautious path stable. Then by definition it is a minimal set that satisfies condition (1), and it only remains to verify that it also satisfies condition (2). Suppose that this is not the case, and there exists a pair of networks $g, g^{\prime }\in G$ such that there is a surely improving path relative to G leading from g to $g^{\prime }$. Denote this path by P. Below we show that a smaller set $G^{\prime }=G\setminus \{g\}$ satisfies condition (1). This will contradict the assumption of minimality of set G and thus, complete the proof.

Note that since path P from g to $g^{\prime }$ is surely improving relative to G, it is also surely improving relative to the smaller set $G^{\prime }$. The same is true about surely improving paths from other networks outside G, which by condition (1) have at least one surely improving path leading to G. If for some of these other networks, say, network $g^{\prime \prime }$, a surely improving path to G does not lead to $G^{\prime }$, then it must be that it leads to g. Denote this path by ${\widetilde{P}}$. So, there exist two surely improving paths relative to $G^{\prime }$: ${\widetilde{P}}$ that leads from $g^{\prime \prime }$ to g and P that leads from g to $g^{\prime }$. Then by Lemma 1, path ${\widetilde{P}}\oplus P_2$ is surely improving relative to $G^{\prime }$ and it leads to $G^{\prime }$. Thus, set $G^{\prime }$ satisfies condition (1) and we arrive at the desired contradiction.

$(\Leftarrow )$: Suppose that set G satisfies the conditions of external stability (1), internal stability (2) and it is also a minimal set that satisfies these both conditions (3). We need to verify that set G is, in fact, a minimal set that satisfies condition (1) alone. Suppose, on the contrary, that there exists a proper subset $G^{\prime }\subsetneq G$ which also satisfies (1). Below we argue that such smaller set $G^{\prime }$ either satisfies (2) or contains another proper subset that satisfies both conditions, (1) and (2). In either case, this will contradict the assumed minimality of set G and thus, conclude the proof.

Suppose that $G^{\prime }$ does not satisfy (2), so that there exists a network $g^{\prime }\in G^{\prime }$ and path $P\in P^{SI}(g^{\prime },G^{\prime })$ such that P leads to $G^{\prime }\setminus \{g^{\prime }\}$. The following algorithm constructs a proper subset of $G^{\prime }$ that satisfies both, (1) and (2).

Consider $G_1=G^{\prime }\setminus \{g^{\prime }\}$. $G_1$ satisfies condition (1). Indeed, from $g^{\prime }$ there exists a path P leading to $G_1$ that is surely improving relative to $G_1$. Similarly, from any other network outside $G^{\prime }$, which by condition (1) has at least one surely improving path leading to $G^{\prime }$, this path is also surely improving relative to $G_1$ and it leads either to $G_1$ or to $g^{\prime }$. When the latter is true, so that for some network $g^{\prime \prime }$ outside $G_1$ the surely improving path from $g^{\prime \prime }$ to $G^{\prime }$ ends at $g^{\prime }$, then denote this path by ${\widetilde{P}}$ and consider a longer path ${\widetilde{P}}\oplus P_2$. By Lemma 1, this path is surely improving relative to $G_1$ and it leads to $G_1$. Thus, $G_1$ satisfies condition (1).

If $G_1$ also satisfies condition (2), then we obtain the desired contradiction. If condition (2) is not satisfied, then we reduce the set further by constructing $G_2=G_1\setminus \{g_1\}$, where $g_1$ is such a network in $G_1$ from which there exists a surely improving path relative to $G_1$ leading to $G_1\setminus \{g_1\}$. Iterating this reasoning, we can build a decreasing sequence $\{G_k\}_{k\ge 1}$ of proper subsets of $G^{\prime }$, satisfying condition (1). As $G^{\prime }$ has a finite cardinality, and as a set consisting of a single network trivially satisfies condition (2), there exists $K\ge 1$ such that $G_K\ne \emptyset$ and satisfies both conditions, (1) and (2). The existence of such set $G_K$ establishes the desired contradiction. $\square$

Proof of Proposition 6

Throughout this proof we will employ the alternative definition of a CFNS, established by Proposition 2, in terms of three conditions: external stability (1), internal stability (2) and minimality with respect to these first two conditions (3).

$(\Rightarrow )$: Let G be CFNS set. Let us verify that conditions (i), (ii) and (iii) of Proposition 6 hold. In fact, it is enough to verify that conditions (i) and (ii) hold, as then (iii) is satisfied, too. Indeed, if (iii) is not satisfied, then there exists a proper subset of G, $G^{\prime }\subsetneq G$, such that (i) and (ii) hold for $G^{\prime }$. Consider a minimal among such subsets, i.e., $G^{\prime }\subsetneq G$ that satisfies all three conditions, (i), (ii) and (iii).²¹ But then from the proof of sufficiency $(\Leftarrow )$ it follows that $G^{\prime }$ must satisfy conditions (1) and (2) of a CFNS set, which contradicts the minimality of the CFNS set G.

So, let us focus on conditions (i) and (ii). Clearly, condition (ii) follows immediately from the definition of a CFNS set. Also, condition (i) is trivially satisfied when G is a singleton, i.e., $G=\{g\}$. Now, suppose that G contains at least two networks, and condition (i) does not hold. This means that at least one of the two statements, (a) or (b), is true:

• (a) $\exists g\in G$ and $ij\notin g$ such that $g+ij\notin G$, and $\forall g^{\prime }\in F^{I}(g+ij)\bigcap G$ it holds that $\left( Y_i(g^{\prime }), Y_j(g^{\prime })\right) >\left( Y_i(g), Y_j(g)\right)$;²²

• (b) $\exists g\in G$ and $ij\in g$ such that $g-ij\notin G$, and $\forall g^{\prime }\in F^{I}(g-ij)\bigcap G$ it holds that $Y_i(g^{\prime })>Y_i(g)$.

If (a) is true, then the inequality $\left( Y_i(g^{\prime }), Y_j(g^{\prime })\right) >\left( Y_i(g), Y_j(g)\right)$ holds, in particular, for $g^{\prime }={\widetilde{g}}\in F^{SI}(g+ij, G)\bigcap G$. Such network ${\widetilde{g}}$ exists, as $F^{SI}(g+ij,G)\bigcap G\ne \emptyset$ due to condition (1) of the definition of a CFNS set. Then, as $\left( Y_i(g^{\prime }), Y_j(g^{\prime })\right) >\left( Y_i(g), Y_j(g)\right)$ holds for any $g^{\prime }\in F^{I}(g+ij)\bigcap G$, we obtain that a path from g to $g-ij$ (one step) and further—along the surely improving path to ${\widetilde{g}}$ is surely improving altogether, that is, $F^{SI}(g,G)\bigcap G\ne \emptyset$. However, this contradicts the internal stability condition (2) of a CFNS set.

Similarly, if (b) is true, then the inequality $Y_i(g^{\prime })>Y_i(g)$ holds, in particular, for ${\widetilde{g}}\in F^{SI}(g-ij,G)\bigcap G$. As before, such network ${\widetilde{g}}$ exists due to condition (1) of the definition of a CFNS set. This, together with the fact that $Y_i(g^{\prime })>Y_i(g)$ for any $g^{\prime }\in F^{I}(g-ij)\bigcap G$, means that $F^{SI}(g,G)\bigcap G\ne \emptyset$. However, this contradicts the internal stability condition (2) of a CFNS set.

Thus, neither (a) or (b) holds, hence, condition (i) is satisfied.

$(\Leftarrow )$: Suppose that set G is such that conditions (i), (ii) and (iii) of Proposition 6 hold. Let us verify that G is a CFNS set, that is, satisfies conditions (1), (2) and (3). In fact, it is enough to verify conditions (1) and (2), as then (3) follows. Indeed, if not, then there must exist a proper subset of G, $G^{\prime }\subsetneq G$, such that $G^{\prime }$ satisfies (1) and (2). But from the proof of necessity $(\Rightarrow )$ we know that conditions (1) and (2) imply (i) and (ii), that is, a proper subset of G, $G^{\prime }$, must satisfy (i) and (ii). This, however, contradicts the minimality of set G established by condition (iii).

Let us focus on conditions (1) and (2). Condition (1) is trivially satisfied, as it is identical to (ii). If condition (2) is also satisfied, then the proof is completed. Note that this is trivially the case when G is a singleton. Suppose now that set G contains at least two networks, i.e., $|G|\ge 2$, and condition (2) is not satisfied. This means that there exists a pair of networks $g,g^{\prime }\in G$ such that there is a surely improving path relative to G that leads from g to $g^{\prime }$. We claim that this violates condition (iii) of minimality in Proposition 6.

Claim: There exists $G^{\prime }\subsetneq G$ that satisfies conditions (i) and (ii).

Let us construct this set $G^{\prime }$. Consider $G_1=G\setminus \{g\}$. Note that $|G_1|\ge 1$ as $|G|\ge 2$. $G_1$ satisfies condition (ii). Indeed, a path from g to $g^{\prime }\in G_1$ that is surely improving relative to G is also surely improving relative to the subset $G_1$. Also, surely improving paths from other networks outside G leading to G are surely improving relative to $G_1$. Note that such surely improving paths from other networks outside G exist since set G satisfies condition (ii). If for some of these other networks, say, network $g^{\prime \prime }$, a surely improving path to G does not lead to $G_1$, then it must be that it leads to g. Thus, we have two surely improving paths relative to $G_1$: one that leads from $g^{\prime \prime }$ to g and another that leads from g to $g^{\prime }$. By Lemma 1, the concatenation of these two paths produces a surely improving path path relative to $G_1$, and it leads to $G_1$. Thus, $G_1$ satisfies (ii).

Now, if $G_1$ also satisfies (i), then the proof is completed. Note that this is trivially the case when $G_1$ is a singleton. So, suppose that $|G_1|\ge 2$, and condition (i) is not satisfied. This means that at least one of the two statements, (a) or (b), is true:

• (a) $\exists g_1\in G_1$ and $ij\notin g_1$ such that $g_1+ij\notin G_1$, and $\forall g_1^{\prime }\in F^{I}(g_1+ij)\bigcap G_1$ it holds that $\left( Y_i(g_1^{\prime }), Y_j(g_1^{\prime })\right) >\left( Y_i(g_1), Y_j(g_1)\right)$;

• (b) $\exists g_1\in G_1$ and $ij\in g_1$ such that $g_1-ij\notin G_1$, and $\forall g_1^{\prime }\in F^{I}(g_1-ij)\bigcap G_1$ it holds that $Y_i(g_1^{\prime })>Y_i(g_1)$.

In particular, the above is true for $g_1^{\prime }={\widetilde{g}}\in F^{SI}(g_1\pm ij,G_1)\bigcap G_1$, where $g_1$ satisfies either (a) or (b). Such network ${\widetilde{g}}$ exists due to the fact that $G_1$ satisfies (ii). This, together with the fact that the payoffs of i and j improve at any $g_1^{\prime }\in F^{I}(g_1\pm ij)\bigcap G_1$ (i.e., the step from $g_1$ to $g_1\pm ij$ is surely improving), means that there is a surely improving path relative to $G_1$ from $g_1$ to ${\widetilde{g}}$: $F^{SI}(g_1,G_1)\bigcap G_1\ne \emptyset$.

Let us define $G_2=G_1\setminus \{g_1\}$. $|G_2|\ge 1$ as $|G_1|\ge 2$. Repeating the same argument as before, but with respect to $G_2$ instead of $G_1$, we can show that $G_2$ satisfies condition (ii). If it also satisfies condition (i), then the proof is completed; otherwise, we construct $G_3$, etc. Iterating this reasoning, we can construct a decreasing sequence $\{G_k\}_{k\ge 1}$ of proper subsets of G, each satisfying condition (ii). As G has a finite cardinality, and as a set consisting of a single network trivially satisfies condition (i), there exists $K\ge 1$ such that $G_K\ne \emptyset$ and satisfies both conditions, (i) and (ii). Denoting this set $G_K$ by $G^{\prime }$, we complete the proof of the claim, and of the proposition. $\square$

Proof of Proposition 7

Below we prove each of the two statements in turn.

• 1. To start with, observe that for any vN-MFS set G and any $g\in G$, there are no surely improving paths relative to G that start at g (and lead anywhere in ${\mathbb {G}}$), i.e., $F^{SI}(g,G)=\emptyset$. Clearly, no surely improving path exists from g to any other network in G, and if there existed a surely improving path from g to some network $g^{\prime }$ outside G, then by Lemma 2 we would obtain a contradiction to the internal stability of a vN-MFS set: by definition of vN-MFS set, from any network outside G there exists an improving path to G, and thus, the concatenation of a surely improving path from g to $g^{\prime }$ and an improving path from $g^{\prime }$ to G would give an improving path between two networks in G.

  We now construct a CFNS set to which a given vN-MFS set belongs. Observe that the whole network space ${\mathbb {G}}$ trivially satisfies the external stability condition (1) of Definition 4. If it is also the minimal set that satisfies this condition, then ${\mathbb {G}}$ is CFNS, and the proof is completed. Otherwise, there must exist a network $g_1 \in {\mathbb {G}}$ from which a surely improving path leads to some other network in ${\mathbb {G}}$: $F^{SI}(g_1,{\mathbb {G}})\ne \emptyset$. Note that this network $g_1$ lies outside the vN-MFS set G, as for any $g\in G$, $F^{SI}(g,{\mathbb {G}})\subseteq F^{SI}(g,G)=\emptyset$.

  The smaller set $G_1={\mathbb {G}}\setminus \{g_1\}$ trivially satisfies the external stability condition. If it is also the minimal set that satisfies this condition, then $G_1$ is CFNS, and the proof is completed. Otherwise, we reduce the set further by constructing $G_2=G_1 \setminus \{g_2\}$, where $g_2$ is such a network in $G_1$ from which there exists a surely improving path relative to $G_1$ leading to some other network in $G_1$: $F^{SI}(g_2,G_1)\bigcap G_2\ne \emptyset$. Note again, that $g_2$ does not belong to the vN-MFS set G, as for any $g\in G$, $F^{SI}(g,G_1)\subseteq F^{SI}(g,G)=\emptyset$. The smaller set $G_2$ satisfies external stability: a path from $g_2$ to $G_2$ that is surely improving relative to $G_1$ is also surely improving relative to the subset $G_2$. The same is true about surely improving paths from other networks outside $G_1$, which at this step is just one network $g_1$, that was withdrawn first. Note that a surely improving path from that network to $G_2$ exists: it either leads to $G_2$ directly or via network $g_2$, as in the latter case, the concatenation of two surely improving paths—from $g_1$ to $g_2$ and from $g_2$ to $G_2$—is surely improving (Lemma 1). If set $G_2$ is also the minimal set that satisfies external stability, then $G_2$ is CFNS. Otherwise, we construct a set $G_3=G_2\setminus \{g_3\}$, etc. Iterating this reasoning, we can build a decreasing sequence $\{G_k\}_{k\ge 1}$ of proper subsets of ${\mathbb {G}}$ that (a) satisfy the external stability condition (1) of Definition 4, and (b) contain the vN-MFS set G (as the networks withdrawn at each step lie outside the vN-MFS set G). As ${\mathbb {G}}$ has a finite cardinality, and as the vN-MFS set G cannot be reduced further, there exists $K\ge 1$ such that $G_K$ is CFNS and $G\subseteq G_K$. This proves the first part of the first statement.

  The second part follows from the observation that the existence of a CFNS set $G^{**}$ such that $G^{**} \subset G$ would imply that $G^{**}$ is a strict subset of another CFNS set that contains G. This is however a contradiction to the minimality of a CFNS set.

• 2. For the second statement, observe that any CFNS set $G^{*}$ satisfies conditions (i) and (ii) in the definition of the PWFS set, as condition (i) is identical to the one of Proposition 6 and condition (ii) is weaker than the corresponding external stability condition of Proposition 6. If $G^{*}$ also satisfies the minimality condition (iii) of PWFS, then it is PWFS. Otherwise, there exists a proper subset of $G^{*}$ that satisfies all three conditions and is thus PWFS. Indeed, as the cardinality of $G^{*}$ is finite, the sequence of nested subsets of $G^{*}$, each satisfying (i) and (ii), is finite, and the last, “smallest” subset in this sequence is minimal.

  To prove that no PWFS set contains a CFNS set as a strict subset, observe that the existence of such PWFS set, say $G^{\prime }$, would imply that $G\subset G^{\prime }$, where set G is also PWFS. However, this is ruled out by minimality of a PWFS set.

$\square$

Proof of Proposition 8

First, from the external stability of a CFNS set it follows that $\forall$ $g^{\prime }\in {\mathbb {G}}\setminus G$ $F^{I}(g^{\prime })\bigcap G\ne \emptyset$. This, together with the fact that $\forall g\in G$ $F^{I}(g)\bigcap G=\emptyset$, implies that G is a vN-MFS set by definition and a PWFS set by Theorem 3 of Herings et al. (2009).

If in addition $F^{I}(g)=\emptyset$, then the external stability condition in the definition of all concepts (CFNS, PWFS and vN-MFS) implies that G must be a subset of any stable set. Then by minimality, also present in each definition, G is the unique CFNS, PWFS and vN-MFS set. $\square$

Proof of Proposition 9

Suppose that $G^{*}$ is a CFNS set. Below we show that $G^{*}$ satisfies the definition of a pairwise consistent set (see Definition 5 in the Supplementary Appendix), i.e., $\forall g\in G^{*}$, both external and internal pairwise deviations are deterred. The deterrence of external deviations is already established by condition (i) of Proposition 6. Now, we verify that $\forall g\in G^{*}$ an internal deviation to a network $g\pm ij\in G^{*}$ is also deterred as it results in lower or equal payoff(s) either immediately or at the end of some credible improving path starting at $g\pm ij$.

Suppose that this is not the case and there exist $g\in G^{*}$ and an internal deviation to $g\pm ij\in G^{*}$ such that both, immediate payoff(s) and payoff(s) at the end of all credible improving paths from $g\pm ij$ (if any) improve. Formally this means that at least one of the conditions holds:

• (a) $\exists ij\notin g$ such that for $g^{\prime }=g+ij\in G$ and $\forall g^{\prime }\in F^{I}(g+ij)\bigcap G^{*}$ it holds that $\left( Y_i(g^{\prime }), Y_j(g^{\prime })\right) >\left( Y_i(g), Y_j(g)\right)$;

• (b) $\exists ij\in g$ such that for $g^{\prime }=g-ij\in G$ and $\forall g^{\prime }\in F^{I}(g-ij)\bigcap G^{*}$ it holds that $Y_i(g^{\prime })>Y_i(g)$.

We obtain that a one-step path from g to $g+ ij$ in case (a) and from g to $g-ij$ in case (b) is surely improving relative to $G^{*}$. But this contradicts the internal stability of a CFNS set. $\square$