Socially conscious stability for tiered coalition formation games

Abstract

We investigate Tiered Coalition Formation Games (TCFGs), a cooperative game inspired by the stratification of Pokémon on the fan website, Smogon. It is known that, under match-up oriented preferences, Nash and core stability are equivalent. We previously introduced a notion of socially conscious stability for TCFGs, and introduced a game variant with fixed k-length tier lists. In this work we show that in tier lists under match-up oriented preferences, socially conscious stability is equivalent to Nash stability and to core stability, but in k-tier lists, the three stability notions are distinct. We also give a necessary condition for tier list stability in terms of robustness (the minimum in-tier utility of an agent). We introduce a notion of approximate Nash stability and approximately socially conscious stability, and provide experiments on the empirical run time of our k-tier local search algorithm, and the performance of our algorithms for generating approximately socially consciously stable tier lists.

Appendices

A Results from experiments on k-tier lists

In Tables 8, 9, 10, 11, 12, 13, 14, 15, 16, and 17, we are comparing the results of several algorithmic variants for finding socially consciously stable k-tier lists. We have boldfaced the highest values for utility, friendliness, and robustness over each set of 5 tables (corresponding to unbiased or biased instance generation). Note that, for $k>2$, TriPart almost always produces the best utility and robustness.

Table 1 Results on unbiased TriPart SC stable k-tier lists

Table 2 Results on unbiased Random SC stable k-tier lists

Table 3 Results on unbiased No Siler SC stable k-tier lists

Table 4 Results on unbiased Even SC stable k-tier lists

Table 5 Results on unbiased Uneven SC stable k-tier lists

Table 6 Results on biased TriPart SC stable k-tier lists

Table 7 Results on biased Random SC stable k-tier lists

Table 8 Results on biased No Siler SC stable k-tier lists

Table 9 Results on biased Even SC stable k-tier lists

Table 10 Results on biased Uneven SC stable k-tier lists

B Results from experiments on $\epsilon$-stable tier lists

In Tables 18, 19, 20, 21, 22, and 23. we are primarily considering a single algorithm, namely epsilonSCStableLocalSearch, Algorithm 5.

Table 11 Results of $\epsilon$-stable tests on unbiased 50-agent instances

Table 12 Results of $\epsilon$-stable tests on biased 50-agent instances

Table 13 Results of $\epsilon$-stable tests on unbiased 100-agent instances

Table 14 Results of $\epsilon$-stable tests on biased 100-agent instances

Table 15 Results of $\epsilon$-stable tests on unbiased 200-agent instances

Table 16 Results of $\epsilon$-stable tests on biased 200-agent instances

C Strict core stability for k-TCFGs

In this appendix, we also consider strict core stability stability is more restrictive than core stability. A weakly blocking coalition forms if at least one of its agents can gain utility by blocking, and the remaining agents are unaffected. Because our environment leads to a high number of agent moves in which that agent’s new peers are unaffected by the move, the notion of strict core stability bears some interest.

In this appendix, we present our findings on strict core stability for k-TCFGs. Noting that strict core stability implies core stability by definition, we answer the remaining questions of implication with respect to the three stability concepts considered in the body of this paper.

Definition 1

Let $G=(N, \succeq , k)$ be a k-TCFG and let T be a k-tier list for G. A weakly blocking coalition in a k-TCFG is a nonempty subset of agents C such that:

• 1. There exists exactly one tier $T_A \in T$ such that $T_A \subseteq C$,

• 2. $\forall a \in C: T' \succeq _a T$, and

• 3. $\exists b\in C: T' \succ _b T$,

where there is some $i\le k$ such that $T'$ is the k-tier list formed by moving all agents in C from their current position in T to a new tier at index i.

Definition 2

A strictly core stable k-tier list is a k-tier list for which there exists no weakly blocking coalition.

Observation 1

Strict core stability of a k-tier list with probabilistic matchup-oriented preferences does not imply Nash stability.

This observation follows from Claim 2, which proves that core stability does not imply Nash stability under these conditions. In our proof of Claim 2, we show that for every possible blocking coalition, there is at least one agent that would lose utility, and therefore no blocking coalition exists. As there is no possible blocking coalition without an agent that loses utility, the same k-tier list is also strictly core stable.

Claim 2 depends upon probabilistic preferences, but Claim 5, which fuels the following observation, does not.

Observation 2

Strict core stability of a k-tier list with matchup-oriented preferences does not imply socially conscious stability.

As before, the k-tier list examined in Claim 5 is strictly core stable in addition to being core stable, because every possible blocking coalition either contains an agent that would lose utility or contains no agents that would gain utility.

Let Game 5 be a 2-TCFG on the agents $a_{r}, a_{s}, a_{p}$ representing the game pieces of “Rock, Paper, Scissors." Thus, $a_r \triangleright a_s$, $a_s \triangleright a_p$, and $a_p \triangleright a_r$.

Claim 1

The partition $[[a_r, a_s], [a_p]]$ is socially consciously stable and core stable, but not Nash stable or strictly core stable, on Game 5.

Proof

In this partition, $a_s$ currently has utility of -1, and would have utility of 0 if it moved to $T_2$. Thus, $a_s$ can profitably move up, and the partition is not Nash stable. Observe that $a_r$ would lose utility by moving up, and that the only move available to $a_p$ (without changing the number of tiers) is to move to a new tier below all other agents, a move that would keep the utility for $a_p$ equal to 0. Therefore, $a_s$ is the only agent with a profitable move available.

If $a_s$ moves to $T_2$, $a_r$ loses utility, and the utility of $a_p$ does not change. Therefore, $a_s$ does not have permission to move, and the tier list is socially consciously stable.

On the other hand, $a_s$ moving to $T_2$ is equivalent to $\{a_s\} \cup T_2$ forming a new coalition. This coalition, $C = \{a_s, a_p\}$, satisfies $T_2 \subseteq C$, $T_1 \nsubseteq C$, and has ability to form at tier index 2 such that one of its agents improves its utility from the current tier list, and the other of the two agents is indifferent. Therefore, C is a weakly blocking coalition, and the partition is not strictly core stable. However, this is not a blocking coalition under core stability.

The tiers do not have incentive to swap positions under core stability; $a_p$ would neither gain nor lose utility, while $a_r$ would lose utility as a result of this swap and would therefore object. Similarly, the only other possible blocking coalition that contains all agents in exactly one tier, namely $\{a_r, a_p\}$, would not block at either tier, because the utility of $a_r$ would decrease. This k-tier list is therefore core stable.$\square$

This claim shows that neither socially conscious stability nor core stability imply strict core stability on k-TCFGs with matchup-oriented preferences. Further, it offers an example of a k-tier list under deterministic preferences that is not Nash stable, but is core stable and socially consciously stable, supplementing our earlier demonstration that Nash stability is not implied by core stability or SC stability and showing that these two non-implications hold under deterministic preferences.

Let Game 6 be a 3-TCFG with probabilistic matchup-oriented preferences on seven agents $a_1, \ldots , a_7$. The Win relationship between these agents is described in Table 24. Under this game, let $T = [[a_1], [a_2, a_3, a_4], [a_5, a_6, a_7]] = [T_1, T_2, T_3]$.

Table 17 Game 6

Claim 2

The 3-tier list T is Nash stable, but not strictly core stable, on Game 6.

Proof

Table 25 shows the utility each agent would have if it moved to each tier. Note that for $a_1$, moving to $T_2$ means forming a new tier in the middle of the list, and moving the contents of $T_2$ to $T_1$. Likewise, $a_1$ moving to $T_3$ means forming a new tier at the highest index and moving all other tiers one tier lower. For remaining agents, moving to a tier $T_A$ means joining the current $T_A$. Each agent’s current position (and, thus, current utility) is marked with a *.

Table 18 Utility of moves from T

As demonstrated by the utilities seen in Table 25, no agent can improve its utility in T by moving to a different tier. Thus, T is Nash stable.

Now consider the coalition $C = \{a_1, a_2, a_5\}$. Exactly one tier ($T_1$) is a subset of C. Let $T'$ be the 3-tier list that would result from C forming at index 2. That is, $T' = [[a_3, a_4], [a_1, a_2, a_5], [a_6, a_7]]$. In $T'$, the agents in C will see the set $\{a_1, a_2, a_3, a_4, a_5\}$. This gives $a_1$ a utility of 0, $a_2$ a utility of 0.2, and $a_5$ a utility of 1. Thus, $a_1$ and $a_5$ do not change utility as a result of this move, while $a_2$ gains utility, and C constitutes a weakly blocking coalition. $\square$

Observation 2 and Claim 7 depend upon probabilistic preferences. It remains an open question whether the relationship between Nash and strict core stability changes under deterministic preferences.

Furthermore, we note that we do not know whether all instances of k-TCFG with matchup-oriented preferences admit a strictly core stable outcome. This also is an open question, albeit one closely connected with the same question respecting core stability.